Children's Mathematical Understandings of Tessellations

Copyright

by

Robert Scott Eberle

2011

The Dissertation Committee for Robert Scott Eberle Certifies that this is the

approved version of the following dissertation:

Children’s Mathematical Understandings of Tessellations:

A Cognitive and Aesthetic Synthesis

Committee:

Guadalupe Carmona-Dominguez, Supervisor

Leema Berland

Susan Empson

Nathalie Sinclair

Michael Starbird

Walter Stroup



by

Robert Scott Eberle, B.A.; M.A.; M.Ed.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

December, 2011

Dedication

To Lucia, for her enduring support and her inspiring love of teaching

And to Ana, Christina, and Sara, for their love of geometry

v

Acknowledgements

First of all, I want to thank my family, and especially Lucia, for their support,

encouragement, and patience during the pursuit of this doctorate. I thank my father for

introducing me to the beauty of mathematics via the writings of Martin Gardner.

Secondly I want to thank my supervisor for her incredible support and detailed feedback

during all phases of my dissertation. Her input was beyond all expectations.

My thanks also go to everyone else who contributed to the success of this

research: the members of my committee for their great feedback—many of the ideas in

this dissertation were inspired by my discussions with them; my colleagues here at UT

Austin who gave me encouragement and help; and friends all over the world who sent my

wife and me a steady stream of encouraging messages to persevere to the end. My special

thanks go to all those in SIM and many churches who encouraged me to follow this road.

Without their support, this doctorate would not have been possible.

Most of all, my thanks goes to God for the strength, ability, and desire he gave me

to complete this work. To him be all the glory.

vi



Robert Scott Eberle, Ph.D.

The University of Texas at Austin, 2011

Supervisor: Guadalupe Carmona-Dominguez

Tessellations have a rich mathematical structure and are especially appropriate as

a context for teaching geometry in the middle grades. Few studies have researched how

children conceptualize and learn tessellations in spite of their international use in

educational contexts. This exploratory study looks at how fourth grade students

conceptualize tessellations before instruction. The analysis is done from a Piagetian,

cognitive viewpoint and from an aesthetic viewpoint. It is argued that the aesthetic

viewpoint is crucial and foundational to children‘s mathematical understanding, just as it

is for mathematicians. A series of clinical interviews was conducted with six fourth grade

children. The results identified common themes of children‘s understanding, strategies,

reasoning, and aesthetic criteria for tessellations. Children‘s ontology varied between

object and process conceptions of tessellations. Children struggled especially with the

infinite space of mathematical tessellations. Children‘s aesthetics, including symmetry,

influenced their choices in creating tessellations and are shown to have played a cognitive

role in children‘s mathematical exploration of tessellation structures. Mathematics

influences students‘ aesthetic appreciation of tessellations and, more importantly,

aesthetics drives the study of the mathematical structure of tessellations. Children‘s

aesthetic criteria were the same as mathematicians‘, but with much different emphases.

vii

Other results are discussed, including the mathematical content elicited by the tasks, the

influence of the tools used to create tessellations, the children‘s epistemology of their

tessellations, and the role symmetry played in giving children confidence.

Recommendations for future research and possible implications for curriculum and

instruction are noted.

viii

Table of Contents

List of Tables ........................................................................................................ xii

List of Figures ...................................................................................................... xiii

Chapter 1: Introduction ............................................................................................1

Problem ...........................................................................................................1

Approach .........................................................................................................2

Research Question ..........................................................................................2

Chapter 2: Literature Review ...................................................................................4

Theoretical Framework ...................................................................................4

Cognitive Perspective ............................................................................4

Piaget.............................................................................................5

Vygotsky .....................................................................................10

Process and Object Conceptions .................................................14

Aesthetic Perspective ...........................................................................19

Nature of Mathematics ................................................................20

Mathematical Aesthetics .............................................................27

Mathematical Aesthetics for Education ......................................46

Connection to the Cognitive Perspective ....................................55

Summary of the Aesthetic Perspective .......................................57

Research Related to This Study ....................................................................59

Mathematical Theory of Tessellations .................................................59

Definitions...................................................................................60

Symmetry and Transformations ..................................................62

Polygonal Tilings ........................................................................67

Regular Polygons ........................................................................69

Periodicity and Symmetry...........................................................72

Lack of Symmetry.......................................................................75

Non-Regular Polygons ................................................................80

ix

Polyforms ....................................................................................82

Color ...........................................................................................84

Vocabulary Summary .................................................................85

Children‘s Understandings of Tessellations ........................................87

Geometric Properties in Tessellations

(van Hiele-Geldof and Fuys et al.) .....................................87

Rectangular Arrays of Squares (Outhred & Mitchelmore) .........89

Combining Tessellations With Art

(Upitis, Phillips, & Higginson) ..........................................91

Unitizing (Wheatley & Reynolds) ..............................................93

Covering Shapes With Tiles (Owens & Outhred) ......................93

The Logic of Mixing Tiles (Vitale & Zinder) .............................94

Other Geometry-Related Research ......................................................95

Children‘s Understanding of Infinite Space ...............................95

Children‘s Understanding of Symmetry .....................................97

Summary .....................................................................................................101

Chapter 3: Methodology ......................................................................................104

Clinical Interviews ......................................................................................104

Theory of Clinical Interviews ............................................................105

Methodology of Clinical Interviews ..................................................105

Validity and Reliability ......................................................................108

Method ........................................................................................................114

Pilot Study ..........................................................................................114

Population ..........................................................................................118

Pre-test ...............................................................................................121

Interviews ...........................................................................................122

Criteria ......................................................................................122

Overview of the Interviews .......................................................124

Structure of the Interviews ........................................................126

Computer software ....................................................................132

Analysis..............................................................................................133

x

Aesthetic Analysis ....................................................................136

Symmetry Analysis ...................................................................137

Chapter 4: Results ................................................................................................139

Cognitive Understanding ............................................................................139

Piagetian stage ...................................................................................154

Tiling Criteria.....................................................................................156

How Shapes Fit (Gaps and Overlaps) .......................................157

Infinite Space ............................................................................159

Ontology of Tilings (Process versus Object) .....................................164

Epistemology of Tilings .....................................................................168

Children‘s Strategies ..........................................................................173

Children‘s Reasoning .........................................................................177

Other General Results ........................................................................180

Aesthetic Influences ....................................................................................181

Children‘s Evaluation Criteria ...........................................................187

Mathematicians‘ Evaluation Criteria .................................................197

Children‘s Creation Criteria ...............................................................199

Interaction of the Two Perspectives ............................................................205

Tasks and Tools .................................................................................206

Role of Symmetry ..............................................................................210

The Cognitive Role of Other Aesthetic Criteria ................................214

Differences Between Schools or Across Time ...........................................217

Differences Between Schools ............................................................217

Differences Across Time ...................................................................217

Chapter 5: Conclusions ........................................................................................222

Discussion of the Results ............................................................................222

Children‘s Cognitive Understanding of Tessellations .......................222

Children‘s Aesthetic Understanding of Tessellations ........................228

Interaction of the Two Perspectives ............................................................234

Implications for Future Research ................................................................236

xi

Implications for Curriculum and Instruction ..............................................238

What Students Learn With Tessellations ...........................................239

Mathematical Concepts Elicited by the Tiling Tasks ...............239

The Teaching of Tessellations ...........................................................244

Importance of the Study of Tessellations ...........................................246

Appendix A: Pre-test............................................................................................248

Appendix B: Protocols .........................................................................................251

Appendix C: Software..........................................................................................267

Appendix D: Analysis Codes ...............................................................................270

Appendix E: Results by Task ...............................................................................285

Appendix F: Analysis of Change .........................................................................303

Appendix G: Analyzing Greater Symmetry ........................................................307

References ............................................................................................................309

Vita .....................................................................................................................318

xii

List of Tables

Table 1: Important tessellation terms ....................................................................86

Table 2: Summary of Clement‘s viability criteria ...............................................112

Table 3: Student participants in this study ...........................................................120

Table 4: The task sets ..........................................................................................126

Table 5: Interview structure .................................................................................132

Table 6: Aesthetic themes ....................................................................................188

Table 7: Some unanimous preferences at both schools during Comparisons

in Pairs ............................................................................................196

Table 8: Escher drawings used in the sixth interview .........................................265

Table 9: Codes used in the analysis of the data ...................................................270

Table 10: Groups formed by student sorting .......................................................301

Table 11: Analysis of code changes across first five interviews .........................303

xiii

List of Figures

Figure 1: Bhaskara II‘s visual proof of the Pythagorean Theorem.

Color and labels have been added to show a2, b

2, and c

2..................22

Figure 2: Analysis of a near-Penrose pattern from the Alhambra

(Lu & Steinhardt, 2007) ....................................................................23

Figure 3: A small portion of the Mandelbrot set near ..................40

Figure 4: Some symmetries of a tessellation .........................................................63

Figure 5: Rhombus tiling showing image of rhombus after dilating by factors

of 3 and 5 ..........................................................................................65

Figure 6: An example of a tiling with dilation symmetry .....................................65

Figure 7: A monohedral tiling that is neither isohedral nor edge-to-edge ............67

Figure 8: The ―brick wall‖ tiling of rectangles is not edge-to-edge. .....................68

Figure 9: Six edges of a brick ................................................................................68

Figure 10: Another non-edge-to-edge tiling..........................................................68

Figure 11: Portions of the three regular tessellations ............................................69

Figure 12: Two tessellations with squares and triangles .......................................70

Figure 13: The 3.3.4.3.4 semi-regular tessellation ................................................71

Figure 14: The six other semi-regular tessellations ..............................................71

Figure 15: Two symmetric semi-regular tessellations ..........................................74

Figure 16: A random tessellation of polygons all different from each other ........76

Figure 17: A non-periodic tiling of squares and triangles occurring naturally in

crystal growth (Krumeich, Conrad, Nissen, & Harbrecht, 1998) .....77

Figure 18: The 36/3

2.4.3.4 tiling with three dodecagons highlighted ....................78

xiv

Figure 19: Non-periodic, asymmetric, tiling created by rotating three

dodecagons 30° .................................................................................78

Figure 20: A 4-fold symmetric tiling without translation symmetry ....................79

Figure 21: Isosceles triangle tiling with 8-fold symmetry transformed into a

spiral tiling with rotation symmetry but no reflection symmetry .....80

Figure 22: Tiling of a non-convex quadrilateral ...................................................81

Figure 23: Four of the several possible isohedral tessellations with a scalene

triangle. The two on the right are not edge-to-edge. .........................81

Figure 24: Order-3 and order-4 polyominoes and their common letter names .....83

Figure 25: Polyiamonds up to order 5 ...................................................................84

Figure 26: A triabolo that can be made in two different ways ..............................84

Figure 27: A tiling whose symmetry depends on whether color is taken

into consideration ..............................................................................85

Figure 28: Rhombus tiling with ―saw‖ (red) and ―ladder‖ (orange) highlighted ..88

Figure 29: Regular triangle tiling with oppositely oriented triangles

colored differently .............................................................................94

Figure 30: The six finite tiling problems .............................................................130

Figure 31: Screenshot of ―Polygon Tiler‖ software ............................................133

Figure 32: Some of Rachel‘s tilings during the first interview ...........................140

Figure 33: A student‘s idea of how four equilateral triangles fit around a

common vertex................................................................................142

Figure 34: Kelsey‘s parallelogram tiling drawing...............................................143

Figure 35: The beginning of Michelle‘s regular pentagon tiling ........................144

Figure 36: Moses points out the inevitable gap left by regular pentagons. .........145

Figure 37: The 36/3

2.4.3.4 tiling made by Kobe ..................................................148

xv

Figure 38: A step in Kelsey‘s attempt to squeeze five triangles around a

square corner ...................................................................................148

Figure 39: Kelsey‘s responses to the first pre-test problem and the first

problem of Set B .............................................................................149

Figure 40: An attempt by Michelle to tile a right scalene triangle by using

rectangular units ..............................................................................150

Figure 41: Kelsey‘s L-tromino tiling ..................................................................151

Figure 42: Beginning of an L-tromino tiling by Michelle drawn on

blank paper ......................................................................................152

Figure 43: Kelsey‘s random triangle and hexagon tiling ....................................154

Figure 44: An I-tromino drawn by Michelle in a diagonal orientation ...............155

Figure 45: S-tetromino tiling by Kobe ................................................................156

Figure 46: Tilings by Rachel (left), Marie (middle), and Moses (right) where

equilateral triangles have been stretched to fit gaps large enough

for two triangles ..............................................................................158

Figure 47: Kelsey‘s first two triangle tessellations .............................................161

Figure 48: Rhombus tiling border by Moses .......................................................164

Figure 49: Failed attempt envisioned by Kobe for fitting

T-tetrominoes together ....................................................................172

Figure 50: Semi-random rhombus tiling by Kelsey ............................................173

Figure 51: Two stages of Marie‘s growing tower in the

software environment......................................................................175

Figure 52: Two regular polygon tilings by Michelle ..........................................178

Figure 53: Tiling of a parallelogram on Michelle‘s pre-test ...............................180

Figure 54: Two of the tilings from the Comparisons in Pairs task .....................183

xvi

Figure 55: Both schools thought the mathematician would prefer this tiling. ....183

Figure 56: The two least favorite Alhambra tilings at both schools ...................184

Figure 57: Escher‘s ―Regular Division of the Plane #58,‖ the second most

popular Escher tiling at the private school ......................................185

Figure 58: Rachel‘s favorite from her created triangle tilings ............................191

Figure 59: 4.82 tiling chosen by both schools for the kitchen .............................191

Figure 60: Tiling J in the Comparisons in Pairs task ..........................................192

Figure 61: L-tromino tiling perceived to have pleasing complexity ...................193

Figure 62: The favorite Alhambra tiling at both schools ....................................195

Figure 63: Escher‘s ―Regular Division of the Plane #66,‖ a favorite tiling at

both schools ....................................................................................195

Figure 64: Three I-tromino tilings of increasing complexity by Kobe ...............202

Figure 65: Kelsey‘s rhombus tessellation drawing .............................................202

Figure 66: Kelsey‘s tilings of the acute isosceles, right scalene, and obtuse

scalene triangles ..............................................................................203

Figure 67: Kelsey‘s attempt to draw a tiling with an obtuse scalene triangle .....207

Figure 68: Kelsey‘s tiling of an obtuse scalene triangle in the

software environment......................................................................207

Figure 69: 2-isogonal tiling creation by Rachel in the software environment ....208

Figure 70: How Michelle extracted a symmetric tiling from a

semi-random one .............................................................................212

Figure 71: Comparison of mathematicians‘ and children‘s aesthetic themes .....220

Figure 72: Two tilings with unusual symmetry in the aesthetic

evaluation tasks ...............................................................................231

Figure 73: Mutual influence of aesthetics and mathematics ...............................233

xvii

Figure 74: First page of the pre-test, shown half size .........................................248

Figure 75: Second page of the pre-test, shown half size .....................................249

Figure 76: Third page of the pre-test, shown half size ........................................250

Figure 77: Pre-test Item 1 ....................................................................................256

Figure 78: Set B, Problems 1 and 2 .....................................................................257



Figure 81: Students‘ tilings for the first aesthetic task ........................................262

Figure 82: Most of the tilings for three of the aesthetic tasks .............................263

Figure 83: The seven Alhambra tilings ...............................................................264

Figure 84: Mo and I discussing square and hexagon tilings with Polydrons ......288

Figure 85: Kelsey‘s drawing of a pentagon tiling with Polydrons ......................289

Figure 86: One of Michelle‘s attempts at making a tiling with squares

and triangles ....................................................................................290

Figure 87: Michelle‘s attempt to solve problem B1 ............................................293

Figure 88: Marie‘s attempt to solve Problem B6 ................................................295

Figure 89: Michelle‘s attempt to solve Problem B6 ...........................................295

Figure 90: Rachel‘s attempt to solve Problem B6...............................................296

Figure 91: Moses‘ ―mail with wings‖ .................................................................297

Figure 92: Tessellation with orbifold notation *632 ...........................................307

1

Chapter 1: Introduction

The study of tessellations is a rich topic that connects many geometric concepts.

Tessellations are appropriate for study in some form for all ages, from kindergarten

through college. Between third grade and eighth grade, tessellations connect with much

or even most of the geometry curriculum, as well as with other areas of mathematics.

Many states, provinces, and countries that rank high on international tests of mathematics

require students to study tessellations in these grades. A few teaching experiments have

demonstrated that tessellations provide a powerful learning environment in geometry for

this age group.

PROBLEM

Despite their importance, there has been very little study done on children‘s

understanding of tessellations. There have been almost no studies to understand

children‘s initial conceptualizations of tessellations. This study was designed to explore

fourth grade children‘s understandings before they received any classroom instruction on

tessellations.

Tessellations (or tilings) are rich in mathematical structure. The simplest

tessellation, a square grid, is a basic concept in mathematics, essential for elementary

studies of area and graphing. Studies (Battista, Clements, Arnoff, Battista, & Borrow,

1998; Outhred & Mitchelmore, 1992, 2000) have shown that children do not achieve a

thorough understanding of finite square arrays before third or fourth grade. Before age 8,

most children do not conceptualize square arrays as structured in rows and columns. No

one has yet thoroughly investigated how children conceptualize the structure of other

types of tessellations.

2

APPROACH

I study children‘s understandings of tessellations from two complementary

perspectives: the cognitive viewpoint frequently used for such studies; and the aesthetic

viewpoint, using a framework suggested by Nathalie Sinclair (2006). The cognitive

viewpoint seeks to understand how children perceive tessellations and how they

understand and find the mathematical patterns they are making. The aesthetic viewpoint

looks at the motivating factors behind children‘s creation and evaluation of tessellations.

Both the aesthetic and cognitive viewpoints seek to understand what children are

focusing on as they tile, how children choose the patterns they make, and how children

understand the mathematical structure of their tilings.

The aesthetic viewpoint is usually neglected in studies of this sort. However,

recent studies have begun to show the importance of mathematical aesthetics in

education. Mathematics is itself an aesthetic study and, contrary to popular opinion,

mathematicians use aesthetics in deep ways in their research. Children are capable of

using their aesthetic abilities to pursue mathematics just as mathematicians do, though of

course in different ways. Indeed, it is not even possible to do more than rote mathematics

without using aesthetics, whether children are aware of it or not. By investigating how

children understand tessellations aesthetically as well as cognitively, we can obtain a

more complete understanding of children‘s conceptualizations of tessellations and deduce

how to develop children‘s mathematical aesthetics in classroom settings.

RESEARCH QUESTION

My research question can be stated this way: As understood from both cognitive

and aesthetic viewpoints, how do fourth grade children from various schools

3

conceptualize geometric tessellations in the context of creating and evaluating

tessellations? In particular:

1. What are children‘s cognitive understandings of tessellations? What strategies and

types of reasoning do they use to create tessellations? What mathematical

concepts are elicited?

2. What aesthetic considerations guide students‘ thinking about tessellations? How

do they evaluate tessellations made by themselves and by others? What

mathematical value is there in their aesthetics of tessellations?

3. How do the cognitive and aesthetic viewpoints inform each other for a more

complete understanding of children‘s conceptualizations of tessellations?

4

Chapter 2: Literature Review

In this study I draw from a number of different areas. This literature review is

divided into two parts. The first part (Theoretical Framework) informs the methodology

of this study and the two viewpoints from which the data are analyzed. First I introduce

the background needed for understanding the cognitive viewpoint, and then I present the

theory of the aesthetic viewpoint in mathematics.

The second part of this literature review (Research Related to This Study) focuses

on studies directly related to my research question on tessellations. I begin with a brief

presentation of the mathematical theory of tessellations, followed by a summary of the

research related to children‘s understanding of tessellations.

THEORETICAL FRAMEWORK

This study seeks to understand how children conceptualize tessellations. The

approach is from two viewpoints which are complementary: a cognitive perspective and

an aesthetic perspective (page 19).

Cognitive Perspective

This research is supported by the psychological constructivist perspective where

the individual is understood to construct his or her own knowledge. This perspective is

rooted in the work of Jean Piaget, and I spend some time in this section discussing the

part of his work that informs my study, especially his explanation of how children‘s

understanding of geometry develops. From this foundation, I continue by discussing the

contributions of Lev Vygotsky as well as process and object theories based on Piaget‘s

work. While there are differences between these theoretical viewpoints, they are

compatible. Each focuses on different aspects of cognition from a constructivist

5

perspective. As some researchers (e.g. Bergsten, 2008; Cobb, 2007) have pointed out,

mathematics education researchers must often turn to several theories in order to cover all

the perspectives needed for addressing their research questions. Each theory brings its

own contribution by studying human cognition from a slightly different viewpoint: Piaget

focused on genetic epistemology; Vygotsky sought to understand cognition in its social

context; and process and object theorists explain students‘ ontological perceptions. All of

these viewpoints can be understood as part of a larger constructivist framework which

attempts to understand how students construct their understanding in their local context.

It is within this larger framework that I analyze how fourth grade children initially

understand tessellations before they receive any formal instruction in this subject.

Piaget

Jean Piaget was a Swiss biologist who spent most of his career in the area of

developmental psychology. Much of Piaget‘s research focused on genetic epistemology,

the study of the origins of knowledge. According to Piaget, knowledge is organized in

schemata, mental representations of the world that have been abstracted from repeated

actions. When a subject encounters a novel situation that seems to fit an existing schema,

that new knowledge is assimilated into the schema. If perception of the situation

contradicts existing schemata, or reveals previously unnoticed contradictions between

schemata, cognitive dissonance occurs and the subject must accommodate this new

knowledge by changing or abandoning certain schemata. Assimilation is easier than

accommodation, so if data can be interpreted to fit existing schemata, they will be, even if

unwarranted assumptions or changes have to be made in order for the information to fit.

We tend to hear what we expect, even if this distorts what was actually said (Duckworth,

1996). The important point to Piaget‘s theories is that we never start with a blank slate.

6

All knowledge is filtered through existing schemata. Knowledge is constructed by the

subject as schemata are created, organized, modified, and abandoned.

An important aspect of Piaget‘s theory is that all conceptual knowledge is rooted

in sensorimotor experiences in the first few years of life. Piaget referred to this as the

sensorimotor stage of development. A baby begins by making movements at random, and

then discovers that certain types of movements have consistent consequences. By

noticing these patterns, the child develops schemata that abstract understanding of the

world. From there, the child continues to develop increasingly sophisticated concepts of

space, quantity, logic, and physics as various events lead the child to assimilate new data

into existing schemata or to accommodate data into new schemata. These concepts are

developed jointly and are not always distinguished for young children. Consider, for

example, Piaget‘s well-known experiment showing that preoperational children (before

age 6 or 7) do not understand that quantity is conserved; they believe that when tokens

are spread out they become more numerous. This is explained by the fact that for young

children, space, quantity, and logic are not fully distinguished. The child believes that

collections that take more space are necessarily more numerous, which is a correct

conclusion in some other situations. It is only when children learn that operations (such

as ―spreading out‖) can be reversed that they can understand how number is conserved

and is therefore a property of the objects independent of space.

Note that a proper adaptation to the world requires a balance of assimilation and

accommodation. If children were only to assimilate, they would never create new

concepts. All sensorimotor data would be forced into pre-existing schemata and learning

would never take place. If children were only to accommodate, then a new schema would

be created for every new experience. Children would never generalize.

7

Students‘ schemata can be studied through clinical interviews, a technique

developed by Piaget, which I describe in Chapter 3. Piaget noted (1971) that many people

felt confident inventing various psychological theories by simply depending on

reflection. Piaget moved away from such speculation by depending on experimentation to

analyze how children think and perceive the world. His clinical interview method (or

méthode critique, ―critical method,‖ as he preferred calling it in the original French) was

used to put children in situations that revealed how they thought about the world.

Stages. Piaget is best known for his four developmental stages of children‘s

understanding: sensorimotor, preoperational, concrete operational, and formal

operational. For my purposes, I focus just on the stages of geometric understanding as

Piaget described them in Psychogenèse et histoire des sciences (Piaget & Garcia, 1983).

Though Piaget does not discuss tessellations or symmetry in this work, his analysis of

children‘s understanding of geometric ideas is applicable to those concepts.

According to Piaget, children pass through three developmental stages of

understanding geometry. The first stage is the intrafigural stage, which fits into Piaget‘s

preoperational stage of development. At the intrafigural stage a child considers geometric

objects holistically without appreciating the relationships between them. At this stage, the

child only focuses on what is inherent in a figure and not the relationship between two

geometric objects. For example, young children cannot copy a vertical chimney on a

sloping roof. Instead of drawing the chimney vertically, they draw it perpendicular to the

roof. A right angle can be drawn as a single figure, but young children cannot coordinate

the chimney and the roof in order to form the proper acute angle between them. Young

children can see that the locus of points equidistant from themselves is a circle because

this involves a single figure. However, children at the intrafigural stage cannot conceive

8

the locus of points equidistant from two people because this requires understanding the

relationship between two different measurements.

The second geometric stage is the interfigural stage, which corresponds to

Piaget‘s concrete operational stage. In this stage, children are able to coordinate multiple

figures as well as distances. They begin to understand rigid transformations such as

translations, reflections, and rotations and can visualize how figures are changed under

such transformations. As research by Richard Kidder (1976) shows, children at this stage

can only understand these transformations at the simplest, most basic level and do not

understand principles (such as distance preservation) associated with these

transformations. Children at this stage have interiorized objects and are able to

manipulate a geometric figure mentally in simple ways to see its relationship with

another figure.

Piaget believed there are several factors that allow children to move into this

stage. The first is the homogenization and filling of space. At the intrafigural level,

children do not understand space to be homogenous. If a barrier is placed between two

objects, young children will say the distance between the objects is now shorter. At the

interfigural stage, children realize that distance in space is constant and does not depend

on the objects it contains. Another factor that allows children to begin to reason at the

interfigural stage is the ability to coordinate distance and direction. Children at this stage

realize that it takes two coordinated measurements to locate a point on a sheet of paper.

Both of these factors—homogenization of space and coordination of distance and

direction—are necessary conditions for the most basic understanding of tessellations and

the transformations that assure that tessellation patterns can continue indefinitely in all

directions. Without the ability to relate different figures, it is not possible for children to

9

understand how a single tile is transformed to cover the entire plane, nor is it possible to

understand basic ideas of symmetry beyond a simple appreciation of reflective vertical

symmetry, which is present even in infants (Muir, Humphrey, & Humphrey, 1994). It

therefore would seem that children younger than 6 or 7 cannot be expected to understand

tessellations to any significant depth. And in fact researchers (e.g. Vitale & Zinder, 1991)

have found that young children seem unable to appreciate that tessellations are patterns

that can be understood to continue indefinitely.

Piaget‘s third geometric stage is transfigural and is reached when a person moves

into Piaget‘s formal operational stage. At this stage, people can not only coordinate

individual figures, but also two simultaneous systems of figures, even if such

coordination requires a calculation. For example, if a snail moves along a board a certain

distance and then stops, after which the board itself moves a certain distance carrying the

snail, children at the interfigural stage can visualize and deduce the total movement of the

snail. But if the snail and the board are moving at the same time, a person must be able to

reason at the transfigural stage in order to visualize and deduce the resulting motion, even

though the calculation is precisely the same. Children at the interfigural stage cannot

conceptualize such complex motions to be a single transformation because it requires

coordinating two different systems of motion. I conjecture that children at the interfigural

stage can also not conceptualize a glide reflection as a single transformation, making such

transformations beyond the reach of children at that stage.

Piaget did not believe that children should wait until the transfigural stage to

begin studying geometry, though this was the predominant curriculum worldwide until

recent decades. It was partly because Piaget believed there were appropriate geometric

10

concepts that children can and should learn at each stage that geometry is now taught at

every primary school level in most countries, including the United States.

I chose to study children in fourth grade (age 9) because they should all be firmly

into the interfigural stage. This is the earliest stage at which children can meaningfully

study tessellations. Some second grade students might still be in the intrafigural stage

where they cannot understand that tessellations continue indefinitely in all directions, do

not tend to perceive patterns in more than one direction, and cannot relate an individual

shape to the overall pattern. The National Council of Teachers of Mathematics (NCTM,

2006) recommends studying tessellations in fourth grade because children at this age can

begin to study transformations. At this age, children can coordinate shapes with global

patterns, appreciate symmetry, and create patterns in a variety of relatively sophisticated

ways.

For my study, I used the clinical interview method developed by Piaget for his

studies. Piaget recommends that researchers have a theoretical expectation for what they

will find in a clinical interview, though of course this expectation may not always be met.

According to Piaget, children do not move from the interfigural to the transfigural stage

until about 11 or 12 years of age. Therefore I expected the fourth grade children I worked

with to be still at the interfigural stage, and in fact they showed evidence they were. They

had little difficulty handling very simple translations, reflections, and rotations, but

showed no evidence of noticing glide reflections.

Vygotsky

Lev Vygotsky was a Soviet contemporary of Piaget who unfortunately died after

only 10 years of contributions to psychological research. Vygotsky both conducted his

own research and summarized the research of others as he investigated the psychological

11

origins of thought and speech. Vygotsky‘s distinctive contribution was his insistence that

these psychological phenomena had social and cultural origins. Knowledge is a cultural

product which we develop in the context of social interactions with others. Vygotsky

believed that knowledge started at a social level, and then was later internalized, perhaps

after an extended period of learning (Vygotsky, 1978).

Both Piaget and Vygotsky were interested in analyzing the origins of thought.

Like Piaget, whom he admired, Vygotsky distanced himself from quantitative

experiments of the sort that measured how many correct answers a child could get on a

psychological test, and instead performed qualitative experiments that looked at what

children were doing and how they were doing it. Also like Piaget, Vygotsky did not

believe that people could learn by direct instruction alone. Forming simple association

bonds, as behaviorists suggested, was insufficient for the complex learning needed to

develop higher order thinking skills. On the other hand, Vygotsky also thought that

formal knowledge was distinctly different from the spontaneous knowledge we develop

from everyday interactions. Such ―scientific‖ knowledge could only be learned in formal

classroom settings (Vygotsky, 1978).

Zone of proximal development. Though Vygotsky held that formal knowledge

could only be learned in school, the classroom setting Vygotsky had in mind was not of

the lecture and rote learning type. Higher order mental processes could only come about

through social influences. Vygotsky realized that having children drill knowledge they

had already mastered was useless. Equally useless was trying to teach them knowledge

they were not ready to understand. Between these two extremes, there lies a zone of

proximal development, where the abilities children are ready to begin to learn are found.

For Vygotsky, two children capable of solving tasks for eight-year-olds are not

12

necessarily at the same level. One might be ready to learn new tasks all the way up to the

twelve-year-old level, while the other might only be ready to begin learning tasks

appropriate for nine-year-olds. This difference between the children‘s actual level and

their potential level was what Vygotsky called the zone of proximal development.

Vygotsky criticized Piaget for focusing only on the actual developmental level of the

child and neglecting the potential level.

Classroom instruction therefore should involve the student working with teachers

or peers who help him accomplish a new task slightly beyond what he is currently

capable of handling. As the child learns the new material, help is slowly withdrawn and

the child internalizes this new knowledge, which is actually an understanding he shares

with others.

The zone of proximal development was the main inspiration behind the modern

practice of scaffolding, where a teacher provides various support and guidance as a

student learns a new topic and then gradually withdraws the support as the student

succeeds in constructing the new concepts for himself (Bigge & Shermis, 1999).

The zone of proximal development does not have an impact on the analysis of my

data because I did not conduct a teaching experiment. However, this concept is important

for understanding certain aspects of the role of mathematical aesthetics.

Tools. Vygotsky took interest in the fact that human action is mediated by tools

and signs. Signs are inwardly oriented and tools are outwardly oriented. Both tools and

signs allow humans to work and reason in ways that are not predetermined by our genetic

development.

Tools could be physical or psychological. Physical tools are concrete devices that

enable us to accomplish a task, such as hammers, pencils, or pattern blocks. Later

13

researchers extended this research to conclude that tools have a profound influence on

mathematical thinking and the way students develop mathematical understanding (Cobb

& Bowers, 1999). It is clear that students may not understand a concept with one tool, but

may succeed with another. (See Ward, 2003, for an example with tessellations.) The

effect of the tool on the student‘s mathematical thinking has focused in recent years on

the case of computer technology (Hollebrands, 2003), but the tool shapes thinking in

important ways no matter what the environment. In this research study, I look at the

impact of using different tools to create tessellations.

Tools are not only physical; they can also be psychological, such as sign and

language systems and mnemonic devices. Vygotsky was primarily interested in

psychological tools and believed that speech was the most important tool at our disposal.

Most of Vygotsky‘s work centered on the relationship between thought and speech.

Thought and speech. Vygotsky took issue with Piaget‘s portrayal of private

speech1 as a mere indication that young children were not able to understand others‘

viewpoints. Piaget had portrayed private speech as an intermediate form bridging the

autistic thinking2 of babies and the phenomenon of social speech. Instead, Vygotsky

showed that private speech was a tool that young children used to reason with. Social

speech actually precedes private speech, and therefore private speech cannot be an

intermediate form leading to social speech. Young children take the social phenomenon

of speech and use it as a reasoning tool to enable themselves to solve problems they are

not capable of solving with non-verbal reasoning. One experiment showed that young

1 The older term ―egocentric speech‖ was used in translations of Vygotsky‘s work. ―Private‖ or

―egocentric‖ speech refers to the phenomenon of young children talking out loud to themselves. 2 ―Autistic thinking‖ in Piaget‘s and Vygotsky‘s writings does not refer to the pathological condition

known as ―autism‖ today, but rather the extreme egocentrism of babies, who do not yet comprehend the

distinction between self and other. Autistic thinking is disconnected from the reality of the external world.

14

children who were prevented from using private speech became incapable of solving

problems that they otherwise would have been able to solve. Vygotsky determined that,

rather than fading away as children lose their egocentrism, private speech is internalized

and becomes inner speech, the tool we use as adults to reason logically with (Vygotsky,

1934/1986). Piaget later came to agree with this view. Piaget agreed with Vygotsky that

logic had deeper roots than language (Piaget, 1971), though Piaget still believed an

important phenomenon of egocentrism had been overlooked by Vygotsky. Piaget also

agreed that learning and reasoning had a social nature. ―All logical thought is socialized

because it implies the possibility of communication between individuals‖ (Piaget, Vakar,

& Hanfmann, 1962, p. 10). In the section on Mathematical Aesthetics, I will connect

these ideas of thought and speech to the role of mathematical aesthetics in cognition.

Process and Object Conceptions

Piaget described how actions are encapsulated as thematized objects through one

of three possible mental operations. Empirical abstraction operates on physical objects to

abstract their properties. Pseudo-empirical abstraction is reasoning on operations on an

object (such as counting or ordering) to form concepts. Reflective abstraction operates on

these mental constructs to create more abstract representations (Piaget, 1977).

Educational researchers have used this idea of reflective abstraction to investigate the

construction of mathematical concepts. Piaget identified two types of thinking—

figurative, which is thinking about a static concept, and operative, which deals with

transformations and actions (Piaget & Inhelder, 1971). Many different terms (with

slightly different meanings) have been used by subsequent researchers to describe

figurative and operative thinking with respect to mathematics. The most popular terms

have been object and process.

15

Anna Sfard (1991) noticed that mathematicians tend to have visual, concrete

images of abstract mathematical ideas—they ―see‖ these ideas and manipulate their

conceptualizations of them. She compared this to Piaget‘s figurative thinking. She

referred to these abstract conceptualizations as structural objects, as opposed to

operational processes, such as algorithms, which she compared to Piaget‘s operational

thinking. Sfard described a series of three transitions that transform processes into

objects, the last of which she called reification.

Sfard extended these psychological ideas by analyzing the ontological differences

between processes and objects. It is this ontological difference that is of particular interest

for my research. Sfard noted that reification—the transition from an operational, dynamic

process to a static object—may happen rather suddenly and represents an ontological

shift in the student‘s thinking about the concept. In order to work with mathematical

concepts at a higher, more abstract level in connection with other concepts, it is usually

necessary to conceptualize them first as objects. Sfard also notes that objects may

sometimes be ―debased‖ when an inferior conceptualization is reified in place of the

preferred mathematical one, for example when students adopt a prototype in place of the

full concept. Objects and processes are not mutually exclusive conceptualizations; in fact,

they are ―different sides of the same coin.‖ Those who have an object conception can still

operate with the process conception of a mathematical idea.

Sfard notes that one important exception to the sequence of moving from

processes to objects is in the field of geometry, which is a highly visual subject by its

very nature. Many geometric ideas, such as shapes and symmetry, are grasped first as

objects and then are analyzed until they are also understood as processes. According to

Sfard, symmetry is first understood as a visual property of a shape, which is a static

16

concept—an object. Later it is understood as a transformation, which is a dynamic

operation—a process. Sfard does not discuss the case of geometry beyond this.

Ed Dubinsky. Piaget‘s theories focused on the development of knowledge in

children, though many of his ideas are applicable to all people. Ed Dubinsky and other

researchers have developed a practically-oriented extension of Piaget‘s theories which

helps to explain how mathematical thinking develops in older students. This theory is

known by the acronym APOS, which stands for the four main elements of the theory:

action, process, object, and schema (Dubinsky & McDonald, 2001). Researchers use this

framework to describe how students use reflective abstraction on procedural knowledge

to construct mathematical objects.

According to APOS theory, mathematical understanding usually starts as an

external algorithm, called an action. The steps for solving a mathematical problem are

not yet understood, but are in the form of a list of instructions to follow. If the steps are

memorized at all, it is only at the level of rote memory. When the action becomes

internalized to the point that the student can think about the sequence of steps, including

how to reverse them or compose them with other actions, the concept is called a process.

However, the concept is still in the form of an algorithm. At some point, the student may

understand the process as a totality, knowing what transformations can be carried out on

it as a whole. The student no longer thinks about the steps at all, unless needed to carry

out a calculation. The concept has then become an object, which enables flexible

thinking. A collection of actions, processes, and objects that are connected in a common

framework is a schema, as in Piaget‘s theory. The sequence from action to process to

object is not a simple linear process. When a student is internalizing a process, he may

17

return to the action level in order to further develop the process. However, in general

actions and processes must precede objects and schemata.

Dubinsky agrees that the case of geometry is different. Like Sfard, he considers

geometry to be object-based. Instead of processes that are encapsulated to form objects,

processes are used to study the geometric objects they act on. Geometry concepts are

developed largely through empirical abstraction. Actions are used mainly to study a

geometric object, rather than to form the basis of a geometric concept. Geometric

concepts also develop through reflective abstraction as students develop linguistic

categories to describe the concepts.

David Tall and Eddie Gray support APOS theory, but question whether it is

capable of analyzing all student mathematical thinking. In particular Tall (1999)

questions the primacy of action for all types of mathematical thinking. He has shown, for

example, that it is possible to teach certain calculus concepts in such a way that the object

conceptualization develops in parallel with procedural understanding. Like Sfard and

Dubinsky, Tall sees that for the domain of geometry certain types of object thinking can

precede process thinking. He also sees formal, axiomatic mathematics as requiring yet

another kind of thinking not well analyzed by APOS theory. According to Tall, analysis

of process and object thinking applies principally to symbolic mathematics such as

algebra and calculus.

To supplement APOS theory, Gray and Tall (1991; 1994) propose a slightly

different understanding of the relationship between processes and objects. They noticed

that symbols are ambiguously related to either processes or objects. This flexibility is

ideally utilized by students when they understand both conceptualizations. Tall and Gray

introduced the idea of procept to describe this flexible thinking. They first defined an

18

elementary procept as ―the amalgam of three components: a process that produces a

mathematical object, and a symbol that represents either the process or the object‖ (1994,

p. 121). They then noted that symbols are flexible and several symbols may point to the

same object, from which they drew their definition of procept as ―a collection of

elementary procepts that have the same object‖ (1994, p. 121). These procepts are the

result of pseudo-empirical and reflective abstraction so that the student focuses on

conceptual rather than perceptual understanding. Children who focus on the appearance

of a prototypical square with horizontal and vertical sides are not likely to recognize one

presented at a non-standard orientation. Gray and Tall noted that procepts are not fixed.

Students can revise their encapsulated procedures as needed. Like APOS theory, they saw

procept theory as useful mainly for symbolic mathematics. They understood geometry as

reflective abstraction on perceived, geometric objects, rather than on processes, as is the

case for symbolic mathematics.

Hartwig Meissner explored the procept idea for geometry and found that it could

be profitably applied to most concepts in this domain as well. He showed that geometric

representations such as nets can behave as symbols in procept theory (2001). A geometric

concept, such as triangle, is both a percept and a concept—it is an object (2006). This

object conceptualization undergoes change as properties are associated with the triangle

and these properties are actively put to use in problem solving. The triangle now requires

proceptual thinking—understanding as both process and object. Meissner shows how

other geometric concepts, such as basic theorems, must be understood as both object and

process in order to be used to solve problems.

What all of these theories have in common is that many mathematical ideas must

be understood as both object and process in order to be used flexibly in problem solving.

19

Geometry is more object-oriented than other domains of mathematics and therefore

students can understand many geometric ideas first as objects before analyzing them as

processes.

In the case of tessellations, we are dealing with a geometric construct that cannot

be fully represented in finite space and therefore cannot be fully grasped by empirical

abstraction alone. Some form of proceptual thinking will therefore be required for a full

understanding of tessellations. Tessellations need to be understood both as mathematical

process and as mathematical object. In this dissertation I analyze the students‘ ontological

conceptions of tessellations and the relationship of those conceptions to students‘

understanding of how tessellations extend into infinite space.

Aesthetic Perspective

La beauté apparaît souvent aux festins où l’on n’avait invité que l’utilité ou la vérité.

(―Beauty often shows up at banquets where only usefulness or truth was invited,‖ Le

Lionnais, 1948, p. 437)

In addition to the traditional cognitive perspective of understanding children‘s

mathematical thinking, this research considers the aesthetic perspective of understanding

what children find appealing in tessellation tasks. This perspective is much newer than

the cognitive perspective for modern mathematics education (though it has roots in

antiquity) and deserves more careful explanation. I first explore the issue of aesthetics in

mathematics generally and then in children‘s mathematics education specifically. I focus

primarily on carefully developing the reasons why aesthetics is foundational to a

complete understanding of mathematics and children‘s mathematical thinking.

In order to explore mathematical aesthetics, I first lay some groundwork by

discussing certain philosophical issues concerning the nature of mathematics.

20

Nature of Mathematics

In order to properly understand the role of aesthetics in mathematics, we must

first be clear on the nature of mathematics. I believe the foundational role of aesthetics

can be seen by understanding the nature of mathematics. However, this understanding is

not simple, all the more so because the understanding of what exactly mathematics is has

changed over the course of history and there may not be an agreed, universal view of the

nature of mathematics.

Historical development. In antiquity, mathematics began as a purely practical

matter. Arithmetic developed over commercial needs and geometry developed to solve

problems in surveying and architecture. The Greeks were the first to take these practical

matters and begin to explore the reasons for the formulas that had been discovered.

Possibly because of logical concerns, such as the paradoxes of Zeno, the Greeks refused

to depend directly on visual evidence and instead began to derive theorems by deductive

methods. They found reasoning by deductive methods to be more beautiful than mere

induction because it led to order and consistency (Kline, 1964).

Pythagoras (6th century BCE), one of the earliest Greek mathematicians, founded

a mystic society that considered mathematics to be an expression of divine reality.

Number and form were not considered to be abstract concepts, but were rather the only

true reality. Mathematical beauty was the highest possible aesthetic, as seen in the

harmonies of number and form. ―The Pythagoreans were overwhelmed by the aesthetic

appeal of the theorems they discovered and were perennially preoccupied with the

interconnectedness of the mathematical and the aesthetic‖ (Sinclair & Pimm, 2006, p. 4).

The Greeks pursued mathematics for mathematics‘ sake in just the same way as

people pursue art for art‘s sake—for purely aesthetic reasons (Sinclair & Pimm, 2006).

Aristotle confirmed the supreme place of mathematical aesthetics in his Metaphysics, ―Of

21

what is fair, however, the most important species are order and symmetry, and that which

is definite, which the mathematical sciences make manifest in a most eminent degree‖

(Aristotle, trans. 1896, p. 358). For the next few centuries, the Greeks considered

mathematics more of an art than a science (Le Lionnais, 1948).

Greek mathematics reached its highest point with Euclid‘s Elements (c. 300

BCE). This work, the foundation of Western mathematical thought for two millennia, laid

out geometry and number theory as a purely deductive system based on five common

sense axioms and five geometric postulates, which were considered visually obvious.

Though the geometric theorems were accompanied by diagrams, the reasoning was

supposedly entirely based on deductive logic, not on any visual inferences from the

drawings.3

Other cultures developed mathematics differently. Indians, for example,

prioritized arithmetic, and later algebra, inventing a symbol for zero and our current

Hindu-Arabic numeral system. Geometry was the basis and inspiration for their

mathematics, but geometry was never developed to the same extent as arithmetic, nor did

Indians take much interest in deductive proofs in geometry. When Indians did begin to

include geometric proofs, these proofs tended to be diagrams that were meant to be

understood visually in a single step (Arnheim, 1969), such as Bhaskara II‘s well-known

12th

century proof of the Pythagorean theorem (Figure 1 below), which was actually

discovered by the Chinese many centuries previously (Schattschneider, 2006). The only

text that accompanied Bhaskara II‘s visual proofs was the single word, ―Behold!‖ For

3 There were no obvious visual inferences from the diagrams. As we will see, when mathematicians began

to develop more rigorous methods in the 19th

century, it was discovered that there were many unstated

assumptions in Euclid‘s Elements and that the proofs sometimes did, in fact, depend upon inferences from

the drawings.

22

whatever reason, Indian mathematicians understood the need for proofs in arithmetic and

algebra, but were satisfied with visual reasoning in geometry.

Figure 1: Bhaskara II‘s visual proof of the Pythagorean Theorem. Color and labels have

been added to show a2, b

2, and c

2.

During the European Middle Ages, Arabic mathematics began to soar. Muslim

mathematicians took the best from Greek and Indian mathematics, adopting the Hindu

numeral system and inventing algebra to solve ever more important problems. The

development of Islamic geometry was slow and complex. The most important event was

the translation of Euclid‘s Elements, which ―caused a cultural revolution in the Islamic

world in the ninth century similar to the one it created in the West after the Latin

translation by [Adelard] of Bath in 1142‖ (Powell, 1998). Islamic writers were hesitant to

call mathematics beautiful (though some did), preferring to appreciate it for being

constant and certain (Sinclair & Pimm, 2006). Muslims did not hesitate, however, to

apply geometry to works of beauty. Their work with geometric mosaics is well known.

Mathematics was a tool for art, rather than an art itself. Muslims also believed that

geometry provided a door to the spiritual world allowing us to glimpse perfection (Abas,

2001).

23

Islamic artisans and mathematicians worked together to develop increasingly

interesting mosaic designs (Özdural, 2000). The Alhambra in Spain (14th

century)

contains a rich variety of tessellations, including examples of most of the 17

mathematically possible periodic patterns (Grünbaum, 2006). I used postcards of some of

these tilings in the final clinical interview in this study. By 1500, Islamic artisans were

even creating patterns which are now known to be based on nearly perfect Penrose

tilings, an especially interesting and mathematically complex aperiodic tessellation not

even known to exist in the West until the 1970s (Lu & Steinhardt, 2007). (See Figure 2

below.)

Figure 2: Analysis of a near-Penrose pattern from the Alhambra (Lu & Steinhardt, 2007)

In the Western tradition after the Greeks, mathematics became more closely

linked to science. Nevertheless, until the 18th

century, Western philosophers continued to

claim that mathematics was aesthetic because mathematics was the height of reason, a

pure expression of eternal truth (Chernyak & Kazhdan, 1996), though this claim of

beauty was rarely made explicit by mathematicians. In practice, mathematics was often

seen as a mere tool for analyzing the real world. Mathematicians and philosophers in the

24

late 18th

century, such as Euler and Kant, defined mathematics simply as the study of

quantity (J. Gray, 2008).

In the 19th century Western mathematics began to face a crisis as mathematicians

began trying to bring greater rigor to their subject. Attempts to provide a more rigorous

basis for calculus proved very difficult. In geometry, the need for greater rigor seemed

especially urgent. Euclid‘s Elements was discovered to have unstated assumptions and

had depended at times on visual reasoning, rather than strict deduction. Consequences of

these flaws were easily seen as mathematicians created deliberately fallacious proofs (e.g.

all triangles are isosceles) within Euclid‘s system. Most importantly, dissatisfaction with

Euclid‘s fifth postulate, the parallel postulate, eventually led to the discovery of non-

Euclidean geometries—complete and consistent axiomatic systems which seemed to

describe geometries that did not exist in the real world.

The development of projective geometry and other geometries during the 19th

century drove mathematicians to the realization that mathematical definitions are

arbitrary. In certain geometries, the terms ―line‖ and ―point‖ could be interchanged, along

with ―concurrent‖ and ―collinear,‖ resulting in entirely new theorems, equally as valid as

the originals. This and other developments led to the realization that such words should

be understood uniquely in terms of their axioms without any connection to their

traditional representations. Definitions, axioms, theorems, and proofs became nothing

more than a logical, abstract manipulation of otherwise meaningless symbols.

Similar changes in other branches of mathematics during the late 19th

century led

to a decisive break between pure mathematics on the one hand and applied mathematics

and the natural sciences on the other. Jeremy Gray (2008) credits the sense of uncertainty

created by 19th

century questions of error and rigor with the desire to separate pure

25

mathematics from the natural sciences in order to avoid ―contamination.‖ Universities set

up separate departments for mathematics and physics towards the end of the 19th

century.

The modernist transformation of mathematics, which began at the end of the

19th

century, saw a definitive shift in the ontology and epistemology of mathematics (J.

Gray, 2008). The ontology of mathematics shifted from an objective description of the

real world to an aesthetic creation of the human mind, which could not be both ―true‖ and

provable. Its epistemology was intensely debated in the context of at least three important

movements: formalism, logicism, and intuitionism.

Formalism evolved from the previously mentioned view that pure mathematics is

a completely abstract study with no necessary connection with the real world. Axioms

and definitions are entirely arbitrary; they need no connection with any real

representation. Logicists attempted to set mathematics on a symbolic logical foundation.

One of the logicists‘ hopes was that mathematics would be reduced to pure symbolic

logic and that their system would lead to an algorithmic method by which all

mathematical theorems could be proved—a calculus ratiocinator. Mathematics would

then have a perfect foundation that would bring absolute certainty to every theorem.

Intuitionists pointed out some of the limits of logicism and formalism. They felt that,

even if axioms were in some sense arbitrary, they could still be rooted in some common

sense notions. For example, Henri Poincaré, who was the leading and most read figure in

public debates in the early 20th

century, noted that all attempts to define the number zero

inevitably contained within the definition the common sense notion of ―zero.‖ The

number ―zero‖ was sometimes defined as the number of elements in the null set; but the

null set was conceived as the set containing no (zero) elements. If one needed to have a

previous understanding of this and other primitive notions in order to create rigorous

26

definitions, why not simply depend on these common sense notions to begin with (J.

Gray, 2008)? Poincaré believed that only aesthetics could drive the identification of the

most fruitful axiomatic systems. Poincaré believed that the human mind, guided by an

interior aesthetic, was the only possible mechanism for discovering rich and deep truths

in mathematics (Poincaré, 1908/2000).

In 1931 Gödel showed the limits of logicism when he proved that it was not

possible to set up an axiomatic system that could prove all true arithmetic propositions.

There must by necessity be theorems which cannot be proved from within the system.

Logic alone is insufficient for establishing mathematics.

Today, mathematicians continue to hold a variety of philosophical positions

about the nature of mathematics. However, it is generally agreed that axioms are

basically arbitrary and not necessarily connected with the real world (though mathematics

proves to be remarkably useful for describing it). As such, pure mathematics is an

aesthetic subject because it is pure invention. Pure mathematicians study mathematics for

mathematics‘ sake, much as artists study art for art‘s sake. It is also recognized, as a

result of Gödel‘s theorem, that no calculus ratiocinator can be devised to find and verify

all theorems in mathematics. Without appeal to human abilities other than logic,

mathematics is deadlocked and can neither get started nor progress further.

Definition of mathematics. Mathematics is an aesthetic study. But we are still

left with the question of what exactly mathematics is. It is surprisingly difficult to pin

down a precise definition of mathematics. Many dictionaries and encyclopedias define

mathematics in terms of its primary domains: arithmetic, geometry, analysis, and so forth.

But this does not explain what unifies these domains. The Encyclopedia Britannica

defines mathematics as ―the science of structure, order, and relation‖ (Mathematics,

27

2006), which is a more satisfying definition. G. H. Hardy saw mathematics as the study

of patterns, a definition many others have agreed with. ―A mathematician, like a painter

or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because

they are made with ideas‖ (Hardy, 1940, p. 84). Warwick Sawyer agreed, defining

mathematics as the ―classification and study of all possible patterns‖ believing that

―where there is pattern there is significance‖ (1955/1982, pp. 12 and 36, author‘s italics).

Many mathematicians today agree with these definitions. It is this idea that pattern and

structure are the objects of study in mathematics that will lead us to understand more

fully the fundamental role of aesthetics in mathematics.

Mathematical Aesthetics

―It‘s a thing that nonmathematicians don‘t realize.

Mathematics is actually an aesthetic subject almost entirely‖

(John Horton Conway, quoted in Spencer, 2001, p. 165).

Before turning to the field of education, I define mathematical aesthetics more

precisely. I look at the foundational role given to aesthetics by the modernist

transformation of mathematics, pause to look at the theory of aesthetics in general and its

connection to mathematics, propose precise criteria for identifying mathematical

aesthetics, consider the problem of subjectivity in mathematical aesthetics, and conclude

with a summary of the importance of aesthetics to the field of mathematics.

If mathematics is the study of patterns, it is natural that mathematics should be

aesthetic because patterns are naturally appealing. This could even suggest that

mathematics begins with beauty as we are drawn to study its patterns. Indeed, some

mathematicians admit that they are driven by the initial beauty of certain patterns to

investigate the mathematics of these patterns.

28

As I discussed above, the modernist transformation of mathematics at the end of

the 19th

century brought about a radically different understanding of mathematics.

Whereas mathematics had been previously understood to be a rigorous study rooted in

―obvious‖ axioms, common sense definitions, and logical deductions, by the end of the

19th

century it was understood that axioms and definitions are basically arbitrary. If the

foundations are arbitrary and not based on either empirical evidence or self-evidence,

how are they selected? Being ―obvious‖ is no longer considered a valid criterion. Instead

the criteria for choosing axioms include aesthetic criteria such as simplicity,

connectedness, and elegance, as well as certain logical criteria such as consistency,

mutual independence, and relative completeness.

Not only were the foundations of mathematics partially based in aesthetics, but

the methods were also seen to depend on aesthetics. Gödel proved that a calculus

ratiocinator could never exist to logically crank out all mathematical proofs. Aesthetics is

needed to determine the most fruitful paths for discovering important theorems and

proofs. Poincaré took special interest in writing about this creative process of finding

patterns and relationships, which requires an aesthetic sense to find generative

combinations of logic. Aesthetics is important for filtering what enters the conscious

(Poincaré, 1908/2000). As the Greeks found aesthetics in the ontology of mathematics,

Poincaré found aesthetics in the epistemology of mathematics.

I conclude from the above discussion that mathematics today depends

fundamentally on aesthetics in at least two respects. First, axiomatic systems are now

considered arbitrary rather than ―obvious.‖ In theory, any set of consistent axioms could

establish a mathematical theory; however in practice mathematicians use aesthetics to

derive systems that are useful, elegant, and simple. Second, once an axiomatic system is

29

chosen, no purely logical, mechanical procedure can exist that will deduce the truth of all

possible theorems.4 Aesthetics is needed to pursue a useful and important system of

theorems that can be proved from the axioms. Aesthetics is also needed in the search for

proofs of these theorems. In actual practice, mathematicians usually explore new areas of

mathematics first, and then establish an axiomatic foundation to support it last.5 But in

any case, the entire mathematical enterprise is based on and guided by aesthetics.

Aesthetics. At this point I should say something about aesthetics in general. So I

pause here briefly to discuss the theory of aesthetics in general and show how it connects

to the question of aesthetics for mathematics.

For the Greeks, beauty was an objective quality, a permanent property of the thing

itself. Aristotle considered order and symmetry to be the most important aesthetic criteria

(Aristotle, trans. 1896).

For the modern Western tradition, many of our ideas about aesthetics can be

traced back to Kant‘s Critique of Judgement (Kant, 1790/2008). According to Kant, there

are four kinds of judgment: the agreeable, the good, the sublime, and the beautiful. The

agreeable is what appeals to my senses and is purely subjective. The good, on the other

hand, is objective. It refers to that which is morally or ethically right. Both the agreeable

and the good address our interests—our interests in pleasure or moral right. The sublime

4 Indeed, from a purely formalist point of view, no theorem in mathematics is actually ―true‖ because

everything is based on arbitrary assumptions without real world connections. Theorems can only be

―proved‖ with respect to a set of axioms. This is a distinction mathematicians recognize in theory but

usually ignore in practice, except in certain cases where the provability of a theorem depends on the set of

axioms chosen. For example, the Banach-Tarski paradox gives the counter-intuitive result that a solid ball

can be cut into a finite number of rigid pieces which can be rearranged to make two solid balls the same

size as the original. This theorem is only ―true‖ if one accepts the ―truth‖ of the Axiom of Choice, an

―obvious‖ axiom which mathematicians found controversial in the early 20th

century because of similar

counter-intuitive results, but which is usually accepted today. 5 In fact, Keith Devlin (2008) notes that even when axiomatic systems change, the body of mathematics

supposedly built on those axioms generally is not affected and remains the same.

30

and the beautiful do not address interests. The sublime is a subjective concept describing

that which is awe-inspiring, perhaps to a point bordering on fear. The sublime appeals to

us because it shows reason to be superior to our imagination. For this reason mathematics

may sometimes be sublime when we study, for example, infinity. We delight in the fact

that our reason can understand a concept that we cannot fully imagine.

Unlike the agreeable or the sublime, Kant considered the beautiful to be an

objective category. Like the sublime, the beautiful describes an object which evokes

delight in a disinterested way. By ―disinterested‖ he means that it pleases us not because

of any further benefit we might obtain, but because of the sheer pleasure of

contemplating the object.

For Kant, beauty is its own end, not the means to satisfying some other desire.

Neither the sublime nor the beautiful have any ulterior purpose, and yet beauty has a

certain ―purposiveness‖ or ―finality‖ (Zweckmässigkeit) to it, as if it had an inner

purpose. Geometric figures have a fruitfulness in their use that may be related to their

beauty. Unlike the agreeable, which is judged agreeable only because it gives us pleasure

first, the beautiful is first of all beautiful, and as a result of this judgment of beauty we

take pleasure in it. I know that something agreeable to me may not be agreeable to others.

But there is a sense that what we find beautiful is really objectively beautiful. Even

though we know that all aesthetics is inherently subjective and that taste may need to be

acquired to appreciate certain beauty, we feel that others ought to be able to perceive the

same beauty that we perceive. Kant also noted that beauty requires variety. The overly

familiar does not appeal to us.

Kant anticipated some future developments, seeing for example that mathematics,

like poetry, was ―pure invention‖ (1938/1993), nearly a century before the modernist

31

transformation. Many of Kant‘s ideas have been rejected in the last century, but his ideas

still run deep in Western philosophical thought. Some of his ideas, such as the objectivity

of beauty, are commonly accepted by popular opinion (though rejected by many

philosophers). Other ideas have been accepted by one or several branches of modern

philosophy.

One philosopher who extended and contradicted some of Kant‘s ideas was John

Dewey, who in his book Art as Experience (1934) argued that art was an intensive form

of common experience. Each experience is unified by a pervasive quality. This quality

does not arise by analysis of the parts of the experience, but from the overall pattern or

structure of the experience. This quality that rounds out the experience is the aesthetic.

The aesthetic emerges therefore from the complete experience, not from any of its

elements. Dewey rejected Kant‘s distinctions of the beautiful, sublime, and so forth,

because such labels were too restrictive. Just as Poincaré showed that logic alone could

not create mathematics, Dewey argued that reason alone, without aesthetics, was

insufficient for attaining truth in any domain.

This idea of the fundamental role of aesthetic knowing has been taken up by many

other philosophers. Robert Root-Bernstein (2002) argued strongly for the recognition of

―aesthetic cognition,‖ the ability that all scientists depend on for the source of their

insights. He argues against the commonly held idea that only ideas that can be verbally

expressed can be actually thought. Aesthetic cognition precedes the verbal logic in which

it is later expressed. Just as we depend on a sense of judgment, and not mathematical

formulas, for physical activities such as throwing a ball, so scientists depend on an

aesthetic feeling to direct them. We use this aesthetic cognition to reason in ways that are

not linguistic. Linguistic and aesthetic cognition are quite different because linguistic

32

reasoning must be linear, whereas aesthetic cognition is not—it is a different way of

reasoning. Root-Bernstein quotes Kant, ―The intellect can intuit nothing. The sense can

think nothing. Only through their union can knowledge arise‖ (Root-Bernstein, 2002, p.

69).

Many of Root-Bernstein‘s ideas come from Poincaré, such as the insistence that

discoveries are made through our intuition (which Root-Bernstein equates to aesthetic

cognition), and then afterward confirmed by verbal logic. Logic‘s role is only to prove; it

is useless for identifying which paths will be fruitful. Poincaré wrote, ―Without

[intuition], the geometrician would be like a writer well up in grammar but destitute of

ideas‖ (Poincaré, 1914/2010).

Root-Bernstein accused philosophers of an obsession to understand the logic of

science and of neglecting the role of aesthetics, resulting in a failure to understand

creativity in science. Root-Bernstein believes that philosophers may also avoid the study

of aesthetics in connection to mathematics because of aesthetics‘ deep connection to the

emotions. It is sometimes supposed that emotions impede logical thinking, but in reality

emotions are necessary for all rational decisions. Experiments have shown that people

whose emotions are impaired cannot make decisions, even if their logical abilities are

intact (Root-Bernstein, 2002). Israel Scheffler (1991) has pointed out that scientists

depend on a wide range of cognitive emotions for their research, including love of truth,

concern for accuracy, abhorrence of error, admiration of theoretical achievement, respect

for good arguments, shame at failing, self-respect, joy of verification, and surprise at new

discoveries. Root-Bernstein explains the apparent paradox of the mutual dependence of

feelings and thinking by noting that thinking is ultimately grounded in our senses, and

33

therefore in aesthetics. ―The best science, like the best art, is that which appeals to the

widest range of emotion and intellect‖ (Root-Bernstein, 2002, p. 65).

Root-Bernstein‘s idea of aesthetic cognition is similar to ideas by other writers.

The psychologist Rudolf Arnheim (1969) argued that perception was not ontologically

different from cognition. Dewey (1934) argues that we first intuit an aesthetic quality

from a global situation before analyzing it into discrete logical parts. Note, however, that

after we analyze and understand the parts, we may come back to a greater and different

appreciation of the aesthetic whole. Our understanding enhances aesthetic appreciation.

Language is a tool which we adopt in order to reason logically about situations,

but language is not the totality of our reasoning ability. We all have an innate reasoning

ability which is not verbal. This primary capacity, variously termed aesthetic cognition,

intuition, visual reasoning, meta-logic, super-logic, etc., is where our reasoning begins.

This reasoning is non-verbal, and therefore does not depend on symbolic logic. For

example, as I look at a rectangle, I may notice a pleasing symmetry that leads me to ―see‖

that the diagonals must be congruent and mutually bisecting. This realization takes place

before I have had a chance to carry out a symbolic proof of this fact, or even verbalize it.

Such knowledge is not innate—it does not occur spontaneously to children—but it is

built up through geometric experience with rectangles and their properties. This non-

verbal knowledge is then translated through the tool of inner speech into a form that can

be communicated and checked for logical consistency. According to Root-Bernstein, the

mathematical expression that is communicated is not the actual content of our

understanding, but is merely a way of expressing it. Rather than being some superfluous,

illogical feeling, aesthetic cognition is seen to be a foundational step in any cognitive

task.

34

Nor is aesthetic cognition vague. Mark Johnson (2007) points out that writers will

often struggle to find just the right word to express their idea. The idea is not yet verbal—

they have not found the word for it—and yet it is quite precise, as evidenced by the effort

to find the perfect word to express it.

Aesthetics is therefore foundational not only for establishing the axiomatic basis

of mathematics and for guiding mathematicians towards generative mathematical ideas,

but aesthetics is also foundational for the reasoning process itself. Root-Bernstein writes,

―Only when we feel that we know and know what we feel do we truly understand‖ (Root-

Bernstein, 2002, p. 70). Mathematical aesthetics is a ―way of knowing‖ (Sinclair, 2006)

mathematics prior to verbal reasoning.

One consequence for the field of education is that both aesthetic and logical

cognition should be taught, as well as the connection between the two. Students must

develop their sense of mathematical aesthetics to guide them in problem solving. They

must also learn how to translate their non-verbal ideas into communicable, logical

descriptions, just as they must learn to interpret others‘ verbal descriptions in a way that

makes sense to them. As mathematician Michael Starbird would say, the task of

education is to teach students to say what they think and think what they say (personal

communication, September 24, 2010). Helping children form this deep connection

between the verbal and the non-verbal—the cognitive and the aesthetic—is fundamental

to a clear understanding of mathematics.

So we have now returned to the question of aesthetics in mathematics. Next we

need to know how to recognize mathematical aesthetics. What form does the aesthetic

take in mathematics?

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Criteria for mathematical aesthetics. The idea that pure mathematics is an

aesthetic study is generally accepted by most mathematicians today (Burton, 1999). As

we saw above, since the modernist transformation aesthetics is at the very foundation of

mathematics. Aesthetics is essential for establishing axioms, for providing direction, and

even for the reasoning process itself. I mention these roles first because they are

frequently overlooked.

The role of mathematical aesthetics most commonly recognized today is in the

evaluation of results, particularly theorems and proofs. Morris Kline noted that

mathematicians are rarely satisfied with an ugly theorem or proof. Instead, they are

always seeking to refine theorems and proofs to be as elegant as possible. Kline believed

that this desire to refine proofs showed that mathematics had an artistic nature. ―Much

research for new proofs of theorems already correctly established is undertaken simply

because the existing proofs have no aesthetic appeal‖ (1964, p. 470). Wolfgang Krull

noted that some mathematicians have even refused to publish correct theorems for years

until they found an aesthetically satisfying proof. He stated in his inaugural address at

Erlangen, ―Mathematicians are not concerned merely with finding and proving theorems;

they also want to arrange and assemble the theorems so that they appear not only correct

but evident and compelling. Such a goal, I feel, is aesthetic rather than epistemological‖

(1930/1987, p. 49).6 Krull went on to assert that even a flawed work by Klein (whose

work was usually highly visual and aesthetic) ranked far above many similar flawless

(but less aesthetic) mathematical works by other mathematicians. The goal of

aesthetically pleasing proofs is to demonstrate a theorem in a convincing way which

leads the reader to see that a particular truth could not be any other way. Such aesthetics

6 Of course the goal is also epistemological, but it is primarily aesthetic.

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may even lead mathematicians to prove the same theorem in a variety of ways. ―I believe

this state of affairs shows beyond doubt that aesthetic viewpoints play a large role in

mathematics‖ (Krull, 1930/1987, p. 51).

There have been various attempts to define modern mathematical aesthetics.

English-speaking mathematicians often refer to the work of G. H. Hardy (1940) in

defining what makes a mathematical result aesthetic. Hardy tied his understanding of

mathematics as the study of patterns to the aesthetic appeal of mathematics. Hardy

identified significance, unexpectedness, inevitability, and economy as factors that

contribute to a mathematical aesthetic. Doris Schattschneider (2006) proposes aesthetic

criteria of elegance, ingenuity, insight (the quality that makes you go ―Aha!‖—think of

Bhaskara‘s one-step visual proofs), connections, and ―paradigm of technique‖ (novel,

useful methods that evoke admiration, such as the pigeon-hole principle). Nathalie

Sinclair (2006) suggests including connectedness, visual appeal, apparent simplicity, and

surprise as further criteria for what mathematicians find beautiful.

French mathematicians often turn to the writer and mathematician François Le

Lionnais (1948), who identified two general types of mathematical aesthetics (which I

look at in more detail later). ―Classical‖ aesthetics is characterized by elegance,

simplicity, and a sense of control. ―Romantic‖ aesthetics is that which shocks and

overturns established ideas—the non-conforming and bizarre.

Root-Bernstein claims that elegance is a key criterion. ―The ability to concentrate

meaning and connections maximizes understanding and its emotional impact, whereas

simply following a logical path to a conclusion often yields neither insight, connections,

surprises, nor joy‖ (Root-Bernstein, 2002, p. 70). He also notes that Einstein identified

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the aesthetic criteria for science as simplicity, connections, and significance, claiming

that these were needed for convincing, rather than mere conviction.

I propose summarizing all these criteria in five categories: significance, surprise,

simplicity, connectedness, and visual appeal.

1. Significance – Though significance alone is rarely sufficient to make a

mathematical theorem aesthetic, a sense of significance adds to the perceived

beauty. The Fundamental Theorem of Calculus is considered beautiful partly

because of its extreme importance in mathematics and partly because it connects

two branches of mathematics (differential and integral calculus) in a very elegant

way. A method meeting Schattschneider‘s criterion of ―paradigm of technique‖ is

significant because it has many uses. ―Depth‖ could refer to significance in the

sense of not being trivial, or to the many connections the result makes.

2. Surprise, unexpectedness, shock, new ideas, and insight – This criterion tends to

be mentioned by mathematicians more than any other (Wells, 1990). Sinclair

(2006) notes an example where students were casually told that angles inscribed

in a semicircle are always right angles. Students were so shocked they excitedly

interrupted the lesson to explore this unexpected fact. Le Lionnais mentions that

we are delighted by the appearance of order where we expected only chaos.

However, surprising disorder where we expect order can also create beauty, such

as in Gödel‘s Incompleteness Theorem or Penrose tilings. Israel Scheffler (1991)

identifies surprise as a crucial cognitive emotion for all of science because it leads

us to change our currently held theories. Only by being open to such challenges to

our beliefs can we grow in our understanding.

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3. Simplicity, elegance, economy, control, and ingenuity – Mathematicians find

beauty when a theorem or proof is delightfully shorter or more powerful than one

would expect. A brief, clever solution creates a feeling of joy. Euler‘s formula

( ) is often considered the most beautiful equation in mathematics

because of its extreme simplicity in relating the five most important mathematical

constants (Wells, 1990). Many people who discuss mathematical aesthetics cite

Euclid‘s proof of an infinite number of primes as a supreme example of beauty,

probably because of its ingenious nature, being far simpler than one would

anticipate. Hardy associated ―inevitability‖ with economy: Simple proofs leave no

room for doubt; they compel the reader towards an inevitable conclusion.

Simplicity is often associated with surprise. There are many trivial results in

mathematics which are not considered beautiful because they are not expected to

be complex. Wells notes that complexity is paradoxically also an aesthetic

criterion. For example, Penrose noted that his aperiodic tilings are more beautiful

than simple square tilings (Wells, 1990). In this case, I believe we are dealing

with depth. We find results beautiful when they have complex, significant order,

not random complexity. However, results that are overly complex fail to be

beautiful. Beauty arises from simple expressions of complex depth.

4. Connectedness – A sense of connectedness brings a feeling of unity and

inevitability to our appreciation of mathematics. Euler‘s formula defines a

surprising connection between two numbers, e and π, which otherwise have no

apparent relationship. Non-trivial results have many connections and are often

said to be ―deep.‖ Le Lionnais waxes eloquent over Descartes‘s connection of

geometry with algebra or Klein‘s connection of group theory with geometry. I

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believe tessellations to be beautiful not only because of their visual appeal, but

also because they connect so many different ideas in geometry.

5. Visual appeal – Bhaskara‘s proof of the Pythagorean Theorem is often

considered beautiful because it relies entirely on visual reasoning. Le Lionnais

(1948) points out that the beauty of Klein‘s work is often tied to its visual nature.

Even non-visual results are considered more appealing if they can be explained by

graphs, diagrams, or other visual means. Presence of visual symmetry, unusual

shapes, a moderate amount of visual complexity, or patterns that are clear to the

eye may make the result more beautiful.

Beautiful ideas in mathematics almost always satisfy more than one criterion. The

Mandelbrot Set (Figure 3) is considered by many to be one of the most beautiful objects

in mathematics and seems to satisfy all five of the above criteria: It is significant for

helping to open the field of fractal geometry; its infinite complexity is very surprising

given its extremely simple generating function;7 it has surprising properties such as self-

similarity and topological connectedness; it connects several fields of mathematics,

including complex numbers, geometry, topology, analysis, and complexity theory; and it

has enormous visual appeal as evidenced by the popularity of images and videos in books

and on the Internet.

7 The recursive complex function that defines the Mandelbrot Set is

; . If zn does not

grow arbitrarily large as n goes to infinity, c is in the Mandelbrot Set. That‘s it. The result is surprisingly

complex and beautiful. Colors are usually added according to how long it takes zn to get large. This

increases both comprehension and visual appeal.

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Figure 3: A small portion of the Mandelbrot set near

Trying to be more specific than a list of criteria in trying to define exactly what

makes a mathematical theorem or proof beautiful continues to be difficult, probably

because most mathematicians have not given it much thought (Burton, 2001).

Buckminster Fuller claimed that he never thought about beauty unless a solution was not

beautiful. In that case the lack of aesthetics told him that something was wrong (Fadiman,

1985). In a similar way, mathematicians show concern over ugly theorems, but rarely

give explicit thought to the philosophy of mathematical aesthetics. Questions of

mathematical beauty have not preoccupied modern mathematicians as they did the

ancient Greeks. It is perhaps for this reason that the general public seems unaware of the

role of aesthetics in mathematics.

It is important to note how aesthetics is tied to importance in mathematics.

Criteria such as significance and connectedness are not only criteria which are appealing

to mathematicians, but also important for identifying those elements of mathematics that

will move the field forward. The criterion of surprise is similar to the concept of

41

cognitive dissonance; it implies delight in novel situations that advance our ideas. The

criterion of visual appeal is perhaps related to that of inevitability (simplicity). When we

can perceive a mathematical truth with our eyes, we become more convinced of its

correctness because we more readily understand the result at an intuitive level. The visual

perception may even trigger an unexpected insight (surprise). Sinclair (2006) notes that

simple equations and round numbers (mathematical objects having simplicity) are more

generative. This intersection of aesthetic criteria with what is important and generative

for doing mathematics suggests a close relationship between the aesthetic and the

cognitive in mathematics.

Subjectivity in mathematical aesthetics. Another reason that a mathematical

aesthetic has been difficult to define is that aesthetics is partly contextual, varying

between cultures, across time, and even between individuals of the same time and culture.

If we claim that a mathematical aesthetic exists, then it is important to explain these

differences. There is a certain subjectivity in any aesthetic. But is mathematical aesthetics

so hopelessly subjective that it is futile to think of considering it in a classroom setting?

Or is there a certain objectivity about mathematical beauty that all students should learn

to appreciate?

The fact that aesthetics varies from culture to culture is well known in the arts. In

the West, for example, there is a tradition that values symmetry and balance. In Eastern

tradition, artists make deliberate attempts to break symmetry, considering perfect

symmetry to be too artificial and simplistic (Sinclair, 2009). And yet, people from both

cultures can appreciate the works of art from the other culture. In fact, appreciation of

both perspectives requires an understanding of symmetry. Both types of aesthetics yield

artistic creations which are considered beautiful by everyone, though perhaps for

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different reasons. In a similar way, some mathematicians may value certain aspects of

aesthetics differently from other mathematicians. And yet, there seems to be a general

agreement that, for whatever reason, certain mathematical ideas are consistently

considered more beautiful than others. Certain aesthetic qualities, in mathematics as in

any domain, are universally recognized as aesthetic even though individuals‘ personal

taste may prefer one quality over another.

Perhaps more bothersome to the idea of the importance of mathematical aesthetics

is the fact that aesthetics seems to change over time. What may be beautiful to one

generation of mathematicians may seem trivial to another. Schattschneider (2006) notes

that the Pythagorean theorem was probably a wondrous thing when it was first

discovered, but of course today it is just a basic fact to be learned. Nevertheless, many

mathematical truths remain beautiful from one generation to the next, and even when

theorems begin to become too commonplace to be considered truly beautiful,

mathematicians can still often appreciate the beauty that previous generations of

mathematicians must have found in these now familiar theorems. Even though the

Pythagorean Theorem is now too common to be considered beautiful by most

mathematicians, certain surprising visual proofs of the theorem can still show us the

original beauty today (Schattschneider, 2006). Even today, geometry students often find

Bhaskara‘s proof of the Pythagorean Theorem beautiful when they first see it. (See

Figure 1 on page 22.) The writer and mathematician Lewis Carroll (1890) believed the

Pythagorean Theorem was still just as beautiful today as it was to the ancient Greeks.

The change in aesthetics that the mathematical community experiences across

generations is similar to that of individuals as their understanding of mathematics grows.

Kant found it natural that we grow tired of frequent exposures to the same beautiful

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objects. Beauty requires a certain novelty. We are more likely to find new and surprising

theorems beautiful than old and familiar ones. Le Lionnais (1948) noted that ideas that

once seemed wonderful and mysterious to him became banal once their inner workings

were well understood. This will be important when we consider mathematical aesthetics

for education because it means students will be attracted to new and important ideas

rather than old, familiar ones.

I have explained how aesthetics may vary with time and culture and yet retain a

certain consistent objectivity. But we are still left with the argument that mathematical

aesthetics may be hopelessly subjective because it varies from person to person in a way

making it entirely inappropriate for the classroom. Just as aesthetics may differ from

person to person in other domains, different mathematicians may have somewhat

different criteria for mathematical beauty. Some aesthetic criteria, such as surprising

simplicity, are agreed upon by nearly all mathematicians. But differences do exist, and

mathematicians frequently disagree whether certain theorems are beautiful or not.

These differences are not arbitrary, however. They may, for example, depend on

the mathematician‘s field of interest (Wells, 1990). They may also indicate important

distinctions in what exactly a mathematician is looking for or considers significant. Le

Lionnais (1948) distinguished two types of mathematics, the classical and the romantic.

The former is the mathematics of equilibrium, harmony, and order in mathematics.

Classical mathematics tries to identify general rules that are true globally. Its beauty lies

in surprising order where we expected to find only chaos. What could be more beautiful

than the surprising unity expressed by Euler‘s simple identity ( ) which brings

together the five most important constants in mathematics without any superfluous

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numbers?8 Romantic mathematics, on the other hand, is the mathematics of irregularities,

paradoxes, anomalies, and even ―monstrosities.‖ It is emotional and non-conformist.

Such beauty includes the bizarre and the unusual, such as one often finds with initial

developments like set theory9 or the upsetting discovery by Gödel that a logical

foundation for mathematics could not be both complete and consistent. Romantic

mathematics, such as the discovery of non-Euclidean geometries, upsets the established

order. Classical mathematics, such as Klein‘s Erlangen Program, restores order. Both are

beautiful. The fact that two such different aesthetics should exist is no more surprising

than the fact that such differences exist in all the arts. And just as different artists have

different styles, so might some mathematicians be more drawn to certain types of

mathematical aesthetics rather than others.

What is important to realize is that both types of mathematics—the classical and

the romantic—are needed, so both of these very different aesthetics are useful for

different purposes. The classical aesthetic seeks to find the general rules that bring order

and harmony to mathematics, such as various general theorems about continuous

functions. The romantic aesthetic leads mathematics forward by blazing trails into new

territory and by finding the limits of classical mathematics, such as Riemann and

Weierstrass‘s discovery that there exist continuous functions which are nowhere

differentiable, a fact that greatly troubled and repulsed 19th

century mathematicians.

8 On the one hand, Le Lionnais (1948) noted that Euler‘s formula had lost most of its original beauty

among mathematicians because it is now so well understood. The overly familiar no longer attracts us as it

once did. On the other hand, a more recent poll of mathematicians (Wells, 1990) found that mathematicians

still rank Euler‘s formula as the most beautiful result in mathematics, even when expressed in the less

aesthetic form . 9 According to Le Lionnais (1948), set theory was initially accused of being ―un insolent défi au bons

sens,‖ an insolent challenge to common sense.

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Romantic ideas sometimes take time, but are often eventually accepted by the

expert community. Most mathematicians today find continuous, non-differentiable

functions fascinating and beautiful. We have no trouble today appreciating the aesthetic

behind them. There are many examples in mathematics of the romantic aesthetic, often

labeled according to their initially repellant nature: the ―monster‖ group; ―pathological‖

functions; ―irrational,‖ ―complex,‖ and ―transcendental‖ numbers; and so forth. Le

Lionnais (1948) notes that it is significant that mathematicians named these objects

according to their emotional aesthetic, rather than some term more clearly linked to their

definitions. All of these fascinating discoveries represent important advances in

mathematics.

This leads us to the conclusion that, just as in other aesthetic domains, in

mathematics there exist different types of aesthetics according to the goals that one is

pursuing. When aesthetic criteria change, they often do so in order to serve different

purposes. Different aesthetics reveal different truths and guide in different ways. In some

ways, conflicting aesthetics is like cognitive conflict—it may reveal important ideas that

have not yet been assimilated by the mathematics community. It is therefore wrong to

suppose that conflicting aesthetics means that aesthetics is completely subjective and not

useful. Instead, we conclude that different aesthetics have different purposes. Most

importantly, we see that aesthetics changes according to what we are capable of

appreciating. Aesthetics is tied to our cognitive comprehension.

Importance of mathematical aesthetics. I conclude that aesthetics is essential in

mathematics and varies according to purpose. We have already seen that aesthetics is

essential for laying the foundations of mathematics, as well as for guiding

mathematicians in order to find interesting theorems and promising paths towards their

46

proofs. Aesthetics is even foundational to the reasoning process itself. Aesthetics also

seems to be fundamental in mathematicians‘ evaluations of results. Such aesthetic

evaluation is important because it leads to mathematics that is more compelling and

therefore more likely to be free of error.

Some aesthetic criteria seem to be universal, while other aesthetic criteria may

vary from person to person. But different aesthetics serve different needs and all lead to

important results. Certain mathematical criteria are nearly universally valued by

mathematicians, though individual mathematicians may rank these criteria differently. I

maintain that this sense of aesthetics is needed not only by mathematicians, but by

anyone doing true mathematics, including children. Le Lionnais did not hesitate to speak

of his fascination with mathematical beauty as a young student. It is especially important

to note that the things he found beautiful as a student were precisely those things that he

was beginning to learn and had not yet mastered. When a mathematical idea becomes

well known and commonplace, it is no longer as beautiful. Mathematical aesthetics draws

us towards those ideas that are fresh and new and still a bit mysterious. Therefore I claim

that mathematical aesthetics are actually of importance in guiding and driving children to

those aspects of mathematics that are most important for them.

Mathematical Aesthetics for Education

I have made the argument that mathematical aesthetics attracts mathematicians

and children alike towards ideas that are important. Aesthetic criteria such as significance

and connectedness apply not only to what is beautiful but also to the identification of

important and generative results. Furthermore, when a person has sufficient knowledge to

grasp a concept, but not enough to fully understand its working, the concept may appear

beautiful—surprising and delightful. As such, aesthetics often serves as a natural

47

motivation to work with others in our zone of proximal development, enticing us to

contemplate facts and methods that we only half understand. These are the reasons I have

argued mathematical aesthetics is important and should have a place in education. In this

section I now concentrate on the reasons promoted by other researchers as I look at the

role of mathematical aesthetics in education.

In spite of its clear importance to mathematics, most mathematicians have not

given much thought to aesthetics, and educators have given even less thought to the place

of aesthetics in mathematics education. The National Council of Teachers of

Mathematics (NCTM, 2000) has called for a greater appreciation of aesthetics in

mathematics instruction, but few curricula have given aesthetics more than a cursory

glance. There is almost no place for aesthetics in most school mathematics curricula, nor

is aesthetics taken into consideration by teachers10 except in rare artistic activities.

Seymour Papert wrote, ―If mathematical aesthetics gets any attention in the schools, it is

as an epiphenomenon, an icing on the mathematical cake, rather than as the driving force

which makes mathematical thinking function‖ (1980, p. 192). Nevertheless, some

educators, going back to John Dewey, have seen the importance of aesthetics for

education in general and have argued for its inclusion in the curriculum, though their

arguments have largely gone unheard by the general public. Other researchers who have

written recently of the importance of mathematical aesthetics in education include

Reuben Hersh, Vera John-Steiner, André Mack, and Rena Upitis (Hersh & John-Steiner,

2011; Mack, 2006, 2007; Upitis, Phillips, & Higginson, 1997).

In recent years, Nathalie Sinclair has researched mathematical aesthetics for

education extensively. She has argued that children have a natural attraction to

10 At least, they do not consider aesthetics consciously. Sinclair (2008) has shown that aesthetics is often

implicit in good teaching.

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mathematical beauty. Many of her arguments are laid out in Mathematics and Beauty:

Aesthetic Approaches to Teaching Children (2006). As Dewey argued for all academic

domains, Sinclair contends that Western society has introduced a false dichotomy

between mathematics, which is seen as the domain of logic, and art, which is seen as the

domain of beauty. (This false conception is perhaps at the root of many students‘ dislike

of mathematics.) All people have a sense of aesthetics, probably because of an innate

desire to impose order on their perception of the world. As such, it is only natural to

suppose that aesthetics, this innate ability to see order in our experience, would guide all

people who do mathematics, which we have seen to be the study of pattern and structure.

Seen in this way, aesthetics gives us the ability to do mathematics. Without aesthetics,

our ability to do mathematics is limited to that of a mindless machine programmed to

calculate without appreciating what is important. As mathematicians such as Poincaré

argued a century ago, true mathematics cannot be done in this way.

Most curricula include justification and reasoning as important mathematical

skills to be learned. In this area we seek to teach students to find logic that is compelling.

As Krull (1930/1987) noted, such compelling logic will naturally appeal to our aesthetic

sense. As we have seen, most mathematicians would agree that it is rarely sufficient to

find a proof that is correct. We need mathematics to make us say, ―Oh, of course that

must be right.‖ A good sense of aesthetics is needed to appreciate a good argument in

mathematics just as much as it is needed to craft a good argument in, say, literature or

philosophy. Studies have found that many American students do not learn to reason in

proof-oriented geometry courses, probably at least partly because they are taught to do

two-column proofs by rote rather than learning to appreciate a good argument (Mayberry,

1981; Usiskin, 1982).

49

Sinclair (2006) shows through several examples that it is not just mathematically

inclined people that are attracted to the beauty of mathematics, but all children. Children

have a natural desire to put things in order and to create patterns and this should be

encouraged. In order for students to make the proper connections, they must have an

expectation of regularity. Otherwise learning dissolves into a mess of unrelated facts. The

student should always desire to integrate everything into a pleasing whole.

Sinclair proposed three primary roles of mathematical aesthetics: the motivational

role, the generative role, and the evaluative role. These roles are true for everyone who

does mathematics, whether mathematician or schoolchild.

Motivational role. Aesthetics motivates us to do mathematics. Indeed, we have

seen that the aesthetic appeal of mathematical patterns is often what drives the initial

investigation of mathematics. We choose problems because they are aesthetically

appealing. One powerful aesthetic motivator, especially for children, is symmetry.

Sinclair (2006) gives the example of Casey, a student who was struggling with a problem.

Instead of asking the child to consider the type of heuristics questions that Pólya

recommends we use, Sinclair helped the child focus on the symmetry of the problem.

Once Casey grasped the aesthetics of the problem, she would not let go again until she

had a satisfactory solution. Likewise, I found that children in this study tried to introduce

symmetry into the tasks I gave them. Such aesthetic aspects of a mathematical task give

strong intrinsic motivation.

Aesthetics gives us a sense of enjoyment when we do mathematics. This element

of enjoyment is important. Mann (2006) has noticed that most future elementary school

teachers‘ most memorable childhood mathematics experience was unpleasant. This

removes any intrinsic motivation to do mathematics. Clearly children will progress much

50

further if they have a sense of enjoyment when doing mathematics. Children generally

like math in the early grades. Hersh and John-Steiner (2011) noted that attitudes polarized

after fourth grade with about 40% saying they hate math, perhaps because of the way

mathematics is taught. The harm we do children by training children to detest

mathematics is incalculable. It is all the more inexcusable when we realize that aesthetics

is not an artificial gimmick added to make math more attractive, but a genuine,

overlooked aspect of the foundations of mathematics.

Aesthetics dictates how mathematicians choose the problems they work on and

this choice is what motivates them to pursue a solution. Unfortunately, students usually

cannot choose the problems they work on. Problems are usually imposed by the teacher,

or else the curriculum may not be covered. Clearly this removes a large motivational

factor. In order to overcome this problem, educators often try to make exercises more

appealing, either by turning them into ―real world‖ problems, or else by adding fun

activities that are exterior to the problem. The clear message that children get is that

mathematics is unappealing and needs sugar coating. Instead of such artificial motivators,

it may be preferable to realize that children have a natural attraction to the exploration of

pure mathematics.

Generative role. One of the most important roles of mathematical aesthetics is

the generative role. This is the role that Poincaré explored. Directions of mathematical

thinking that are likely to be productive have aesthetic appeal. When mathematicians

look for a solution to a problem, their belief that the solution should be beautiful leads

them to explore directions with aesthetic appeal. This generative role is just as important

for children as for mathematicians. When children are given an open-ended task, they

tend to examine possibilities that they find appealing. I believe that, more often than not,

51

these turn out to be the same possibilities which lead to mathematically correct solutions.

This study supports this belief. I show, for example, that when children incorporated

aesthetic symmetry into their solutions, they were more likely to find a mathematically

valid solution and to be confident of their answer.

There is very little empirical research in the area of mathematical aesthetics for

children, though sometimes other research touches on this area. Lehrer et al. (1998)

describe a rich five-week second grade geometry activity based on a study of quilt

patterns. Children frequently found the quilts to be ―cool.‖ The authors note that at the

beginning of the unit, ―cool‖ was associated with colors. Later in the unit, talk of ―cool‖

quilts included aspects of symmetry, shape complexity, transformations, and the aesthetic

constraints of certain pattern choices. In other words, as children grew familiar with the

mathematical aspects of their unit on quilts, their aesthetics focused more on

mathematical structure. Again we see a connection between the aesthetic and the

cognitive in mathematics.

Sinclair notes that the generative role is usually harder to observe than the other

two roles because it takes time to manifest. Aesthetics does not typically lead to

immediate flashes of insight. The process is slower and may not be as easy to observe as

the motivational or evaluative role.

Evaluative role. We have noted that mathematicians are frequently unsatisfied

with perfectly correct proofs for aesthetic reasons alone. A sense of aesthetics pushes

mathematicians to seek what they consider to be ―cleaner,‖ more elegant results. A sense

of aesthetics can also help students to evaluate the quality of their solutions. Sinclair

(2006) notes an example of an open-ended sorting exercise where one child, Zoe,

defended her choice as ―better‖ by pointing out that the choice of another group of

52

students was too ―simple.‖ Zoe‘s choice revealed something deeper and less obvious

(symmetry again) about the shapes that were being sorted, compared to the simple side

counting that the other group was engaged in. Zoe‘s aesthetic sense led her to evaluate

her sorting as better and more mathematically interesting.

Unfortunately, many people believe that school mathematics is a finished product.

Students learn what they need to learn as efficiently as possible so that they can give what

is considered to be the single correct answer to each problem. Such mathematics has no

need to be evaluated. The answer is either right or wrong. Without a better understanding

of the place for open-ended inquiry in mathematics, and of the need to evaluate solutions

to such problems, there is little hope that aesthetics will find a place in the current

American curriculum.

In fact, much of this discussion of mathematical aesthetics is of little use for

classrooms where students cannot choose either the problems they will solve or the

methods they will use to solve them. If students are given the methods they must use, and

a system of external rewards and punishments to motivate them to do the problems, then

the aesthetic will play little or no role in their work. This type of mathematics instruction

bears as much resemblance to real mathematics as multiple choice grammar exercises

resemble creative writing. However, it is a reality that some classrooms are like this, and

for those classrooms mathematical aesthetics will be of limited use. In this study I am

assuming that students have at least some opportunity to do authentic mathematics.

Resistance to incorporating mathematical aesthetics in education. Sinclair

(2009) has pointed out that many see the consideration of aesthetics for mathematics

education as either frivolous or elitist. Those who see it as frivolous either wrongly

53

associate aesthetics with style and fashion or else believe aesthetics to be peripheral to

mathematics rather than the foundation on which mathematics is built.

The view that mathematical aesthetics for education is elitist comes from a

Platonic view that aesthetics is an inherent property of mathematical objects, independent

of culture (Sinclair, 2009). As such, it is believed that aesthetics can be properly

evaluated only by mathematics experts.11 Students do not always share the same aesthetic

appreciation for solutions as mathematicians; such evaluation is therefore believed to be

beyond children‘s abilities. Many see the purpose of mathematics education to be the

learning of skills needed for life and other coursework—the mere application of what

mathematicians have developed. Aesthetics, it is argued, is at best a concern for

mathematicians alone.

Children‘s mathematical aesthetics is indeed somewhat different from that of

experts, but appears to serve their needs better than expert aesthetics would because it

highlights the aspects of mathematical learning that are still new to the child and not

completely understood. For example, whereas mathematicians seek elegant solutions,

many students actually prefer messy solutions that encapsulate the history of their

struggle to find a solution. They seem to be more interested in telling the story of how

they solved the problem than in erasing those traces in order to present a ―clean‖ solution.

Sinclair (2006) recommends allowing such students to pursue the kind of mathematics

they find aesthetically pleasing, rather than imposing expert aesthetics that they may not

be ready for. The process of doing mathematics is an important aspect of mathematics

11 Certain mathematicians, such as Hardy and Poincaré, are unfortunately also partly to blame, as Sinclair

also points out. They tended to believe that mathematicians had special ability and that others could not

fully understand and do significant mathematics.

54

that students are still learning. It is therefore natural, instructive, and important that their

aesthetics should include telling the story of their mathematical discoveries.

Aesthetics is therefore just as important for children, at least when they are

involved in genuine mathematical inquiry. Many researchers (e.g. Hiebert et al., 1996)

have emphasized the importance of incorporating inquiry in mathematics education.

Brown, Collins, and Duguid (1989) have argued that learning should be situated in an

authentic context. But if students are to experience such an authentic cognitive

apprenticeship, their learning must include genuine inquiry guided, not by heuristics, but

by aesthetic criteria. An authentic cognitive apprenticeship in the domain of pure

mathematics should involve the exploration of abstract mathematical problems.

Researchers from diverse perspectives have emphasized that it is an error to attempt to

situate all learning in ―real-world‖ contexts (Cobb & Bowers, 1999). It is wrong to

believe that pure mathematics is somehow ―context-free,‖ in need of an artificial

situation. Pure mathematics is as valid a context as applied mathematics and both pure

and applied mathematics should be used appropriately in instruction. In the case of pure

mathematics, it must be recognized that aesthetics is a powerful and important driving

force. One of the goals of this research is to investigate the role of aesthetics in children‘s

mathematical thinking when they are engaged in open-ended tessellation tasks.

Sinclair (2008) notes that enculturation into mathematical aesthetics is in fact

already taking place implicitly in many classrooms. When teachers point out a ―secret

weapon,‖ a surprising fact, a discussion of mathematical style, or a humorous numerical

example, they are really exploiting their students‘ sense of mathematical aesthetics.

Children‘s aesthetics could be more finely developed if teachers gave more explicit

guidance in the aesthetics of mathematics. By becoming more aware of the ways

55

aesthetics guides thinking, we can become better able to give explicit guidance to

students on the aesthetic aspects of mathematics.

Connection to the Cognitive Perspective

In this discussion of mathematical aesthetics we have seen the tight relationship

between the aesthetic and the cognitive in several ways.

All cognition is rooted in the aesthetic. Thinking begins with aesthetic cognition

and is reasoned about and communicated through verbal logic.

Aesthetic appreciation is developed through increased cognitive understanding.

The criteria that mathematicians have identified for describing mathematical

aesthetics are seen to guide towards cognitive understanding.

Aesthetics guides mathematicians and children alike towards the most generative

cognitive ideas. The aesthetic guides our cognitive perception and the cognitive

informs our aesthetic perception.

Aesthetics adapts to our cognitive level. We tend to find most beautiful those

things that we do not yet fully understand. Mathematical aesthetics draws us into

our zone of proximal development as we work with others to understand these

new ideas.

We can analyze any open-ended mathematical task from either viewpoint. The

cognitive viewpoint looks at how students understand a concept from a logical point of

view—their strategies, reasoning, and conceptual understanding. The aesthetic viewpoint

looks at what students find appealing—their choices, preferences, and aesthetic criteria.

These two viewpoints are closely related and may at times be hard to distinguish,

especially when dealing with non-verbal thinking. Doris Schattschneider (1978a) notes

that in the case of tessellations, artistic (aesthetic) symmetry and mathematical

56

(cognitive) symmetry are the same. Does someone create a symmetric pattern because it

is understood how it will help make the tiling easier (a cognitive reason) or simply

because the symmetry is more pleasing (an aesthetic reason)? And to what extent does

such a distinction matter? This research is at the early exploratory stage of trying to

understand children‘s thinking about tessellations, so I am not concerned about overly

fine distinctions. However, I analyze both students‘ cognitive reasoning and their

aesthetic values for tessellations in order to gain as complete an understanding as possible

of children‘s thinking.

Research design. I note at this point what the close relationship of the cognitive

and the aesthetic means for the design of this research. It is natural that I, the researcher,

should have a mathematical aesthetic that is different from the aesthetic of the student

participants because of my different cognitive understanding. My aesthetic is drawn from

my understanding of the mathematics of tessellations, which is naturally much closer to

the canonical understanding of the mathematics research community than fourth graders‘

understanding would be. This aesthetic unavoidably influenced the design of this

exploratory study. Because I am interested in exploring children‘s understanding of

tessellations in relationship to the standard mathematical theory of tessellations and what

I believe children should study in schools, the tasks were designed to explore those

elements that are valued by the mathematical community. This does not mean that

children‘s aesthetic values were overlooked in the design of the tasks. All tasks were

designed so that children‘s aesthetic values could be analyzed. Some tasks were designed

to explore specifically the aesthetic values children had expressed during the pilot study,

as well as aesthetic values typically valued by mathematicians. The type of tessellation

that the task allowed necessarily imposed certain aesthetic constraints. However, this did

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not prevent the children from expressing their aesthetic values. For example, none of the

tasks allowed children to choose the color of the tiles. Nevertheless, the analysis revealed

that color was one of the most commonly discussed aesthetic values.

Summary of the Aesthetic Perspective

Pure mathematics, which is best defined as the study of pattern and structure, is

entirely an aesthetic subject and, like the arts, it has been pursued for its own sake in

Western civilization from the Ancient Greeks to modern mathematicians. The aesthetic

appeal of patterns drives the study of mathematics. The foundations of mathematics are

partly determined by aesthetic criteria; the structure of mathematics is explored and built

up guided by aesthetic forces; and mathematical results are evaluated by aesthetic

criteria. This aesthetic epistemology became clear beginning in the late 19th

century with

the modernist transformation and the arguments of mathematicians, such as Poincaré,

who pointed out that aesthetics is part of all mathematical work. Rigor and logic alone are

insufficient for establishing and developing mathematics.

Many researchers believe that cognition takes place primarily at an aesthetic level.

Such understanding is translated into verbal expression for communication and logical

reasoning, but this verbal expression is not the understanding itself. Authentic

mathematics requires reasoning at both the verbal-symbolic and aesthetic levels.

Although a clear theory of mathematical aesthetics has not been developed in

modern times, mathematicians have described it using criteria of significance, surprise,

simplicity, connectedness, and visual appeal. The fact that these criteria also point to

what is important and generative in mathematics suggests a deep connection between the

aesthetic and the cognitive. It seems likely that an innate desire to impose order on our

perception of the world is at least partly responsible for the aesthetics that drives our

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understanding of mathematical patterns and structure. Mathematical aesthetics drives

cognitive understanding and cognitive understanding develops our mathematical

aesthetics.

For mathematics education, Sinclair (2006) has proposed viewing aesthetics

through its three primary roles of motivation, generativity, and evaluation.

Mathematical aesthetics, like all beauty, is in some sense objective. There is some

agreement among mathematicians as to what is beautiful in mathematics and what some

of the criteria for this beauty are. However, mathematical aesthetics also has subjective

aspects and mathematicians may disagree on details. Mathematicians generally agree on

the criteria for a mathematical aesthetic, though they may value and apply these criteria

differently. Two important ways that mathematical aesthetics varies are in how aesthetics

adapts to our purpose and to our understanding. For example, Le Lionnais (1948) noted

that when our purpose is to confirm and build up existing mathematical structure, our

aesthetic tends in that direction (the ―classical‖); but when our purpose is to explore the

edges and push the limits of our knowledge, our aesthetic (the ―romantic‖) may value the

seemingly bizarre. Likewise, children may have different purposes from mathematicians

in their pursuit of mathematics, and their aesthetic adapts accordingly.

More importantly, aesthetics adapts to our understanding. For an idea to be

beautiful, we must understand it enough to appreciate it without understanding it too

deeply. Ideas that are common and well understood no longer seem as beautiful as they

once did, either to individuals or to the mathematical community. The fact that the most

beautiful ideas are those whose inner secrets we do not yet completely understand makes

mathematical aesthetics an ideal driving force, guiding us to study what is most

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important. This is true for anyone doing mathematics, whether mathematician or

kindergartner.

Criticism that mathematical aesthetics is frivolous can be answered by pointing

out its foundational role in modern mathematics. Criticism that it is elitist can be

answered by noting how it serves children as well as mathematicians.

Because aesthetics is integral to mathematics at its very foundations, it should be

valued from a child‘s earliest years. Mathematics that relies on heuristics alone, with

artificial contexts for motivation, lacks the genuineness that an authentic apprenticeship

demands. Aesthetics is an essential foundation for any true mathematics education.

The cognitive and the aesthetic are deeply related in many ways. Both the

cognitive and the aesthetic viewpoints are needed for a complete understanding of

children‘s mathematical thinking.

RESEARCH RELATED TO THIS STUDY

In this second section of the literature review I summarize some of the research

that has been done directly relating to children‘s understanding of tessellations. I focus on

research conducted in the field of mathematics education, but first I start by explaining

certain aspects of the mathematical theory of tessellations, as well as introducing some

vocabulary and notation. After this, I summarize the scant research that has already been

done on children‘s understanding of tessellations. Finally, I look at the research that has

been done on children‘s understanding of related concepts, namely infinite space,

symmetry, and transformations.

Mathematical Theory of Tessellations

Tessellations have been explored as an art form since prehistoric times in all

cultures and are also found in nature (Grünbaum & Shephard, 1986). They have been

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used extensively in art and architecture in every major civilization since antiquity. And

yet no mathematical study was made of them until Kepler‘s Harmonices Mundi

(1619/1997), which was ignored by mathematicians until the early 20th

century. After

Kepler‘s 17th

century work, no mathematical studies on tessellations were published until

the late 19th

century (Grünbaum & Shephard, 1986). Many problems in tessellations

remain unsolved. Their study has proven especially important in modern times for the

field of crystallography, but also for engineering, biology, geology, physics, computer

science, metallurgy, communication theory, and other scientific domains.

In order to be clear about the subject I am exploring, I briefly describe the theory

of tessellations as it is currently understood in the mathematical literature. I focus on

those elements that are important for mathematical interpretations of the tilings students

could make and evaluate in the tasks they were given in this study. It is important to be

clear about the terminology, especially because there is some variation in the literature. I

give a summary of vocabulary terms12 associated with tessellations at the end of this

section on page 85. The mathematical theory will also form the basis for the design of the

tasks.

Definitions

In the most general definition, a tessellation, or tiling, is a complete covering of a

space by a collection of non-overlapping, closed sets called tiles. By ―complete,‖ we

mean there are no gaps—the union of the tiles is the entire space. (I.e. the tiling is a

mathematical covering.) By ―non-overlapping‖ we mean that the interior of the each tile

is not shared with the interior of any other tile (the interiors are pairwise disjoint), though

12 This terminology is solely for the purposes of analysis and clear communication in this dissertation.

Simple everyday terms were used with the children and most of the concepts here were not shared with

them at all.

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of course points on the boundaries of the tiles will belong to two or more tiles. (I.e. the

tiling is a mathematical packing.) In general tiling theory, the tiling can be embedded in

any geometric space and the tiles can be any shape at all, including unusual shapes that

are not connected or bounded. But for the purposes of this study, I limit my discussion to

tessellations created by a countable collection of closed topological disks13 in the

Euclidean plane. Each tile will usually be a filled polygon (a polygon together with its

interior), though in the case of artwork, such as Escher tilings, the tiles may have curved

edges.

The edges of the tessellation are made up of the pairwise intersections of the tiles.

A point where three or more tiles come together is called a vertex.

We will almost always require the shapes of the tessellation tiles T1, T2, … to be

chosen from a finite set P of possible shapes (prototiles) P1, P2, …, Pn called the

generating set. In other words, a generating set of n prototiles P = {P1, P2, …, Pn} will be

chosen and the tessellation will be constructed in such a way that each tile Ti is congruent

to one of the prototiles Pj from the generating set. If all the tiles in a tessellation are

congruent to a single shape, i.e. n = 1, then we say that the shape P1 can tessellate or tile

the plane, and the tessellation itself is said to be monohedral. Children understand

tessellations easily in reference to floor tilings, which are usually monohedral, or at most

dihedral (n = 2). They understand quickly that tilings should not have gaps or overlaps

between the tiles. There is no general method for deciding if a given generating set can

actually make a tessellation. In general, you cannot look at a prototile or set of prototiles

and know if they can tessellate without actually trying. Even if a shape can tile an

13 A topological disk is basically a closed and bounded set whose boundary is a simple closed curve. We

therefore exclude unbounded tiles, tiles with holes, tiles with several parts connected only at points, tiles

with empty interior, and other complex shapes. The edges of such simple tiles always lie on the boundary

of the tile.

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arbitrarily large area, there is no general test to know if it can continue to tile the whole

plane. There are therefore many open problems in tiling theory (Schattschneider &

Senechal, 2004).

Symmetry and Transformations

Most interesting tessellations are symmetric. Symmetry of a tessellation is

described in terms of isometries. An isometry is a rigid transformation of the plane that

preserves shape and distance. If the isometry maps the tessellation back onto itself in

such a way that each tile is mapped onto a tile, then the isometry is called a symmetry of

the tessellation. In other words, after applying the isometry the tessellation looks exactly

as if it had not been transformed at all, though actually each tile may be in a new location.

The tiles have all ―switched places,‖ so to speak, while still preserving their relationships.

There are only four types of isometries in the Euclidean plane: translations,

rotations, reflections, and glide reflections.14 Every isometry is one of these four types.

All four of these can be seen in the example in Figure 4 below. (Of course, in all figures

the tiling must be imagined to continue indefinitely in all directions throughout the entire

plane with the same pattern. These figures are only finite portions of tessellations.) For

example, from the point O, the tessellation can be translated up one tile, or to the right

one tile. There are several possible reflection symmetries, all horizontal or vertical. The

14 Glide reflections can be decomposed into a ―glide‖ (translation) and a reflection across a line parallel to

the direction of the glide. Psychologically, glide reflections seem more complex than the other three kinds

of isometries, but mathematically, no isometry is more complex than another. A glide reflection is

considered a single transformation. Any isometry, not just a glide reflection, can be decomposed into two

other types of isometries and the composition of any two isometries is always one of these four isometries.

However, glide reflections are hard to visualize as a single movement and some textbooks omit them. They

are rarely included in elementary curricula. Glide reflections are psychologically more complex than

translations, reflections, and rotations. At the very least, they probably require Piaget‘s transfigural stage,

unless they are conceived as the separate applications of a translation followed by a reflection. It is not

expected that fourth graders be able to conceptualize glide reflections. However, patterns with glide

reflection symmetry are visually appealing and symmetric, and children may still notice a symmetry that

they cannot analyze or describe.

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tessellation could be reflected across line l or line m, for example. There are 180°

rotations about hexagon centers (such as point A) or the centers of any two touching

triangles (such as point B or C). A glide reflection can be created in Figure 4 by

combining a horizontal or vertical translation with a horizontal or vertical reflection

respectively. There are, of course, infinitely many symmetries in this tessellation.

Figure 4: Some symmetries of a tessellation

In each of these symmetries, one must imagine the entire plane being mapped to

itself in such a way that every hexagon moves to a hexagon and every triangle moves to a

triangle. When we say that the tiling is translated up, we mean that every tile is moved up

the same distance in the same direction. This idea is sometimes difficult for students to

grasp because they focus on the movement of a single tile rather than the entire plane.

Some mathematicians (e.g. Grünbaum & Shephard, 1986) have found it helpful to

imagine the tessellation copied onto an infinite transparency.15 The transparency is then

15 Laurie Edwards (2003) points out that a transformation is not really a motion and therefore it is wrong

for students to conceive them with the transparency model, especially because it leads to errors. She sees

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moved, rotated, flipped, or glide reflected to a new position that matches the original

tessellation.

We sometimes speak of one other type of symmetry that is not an isometry, but

which is nevertheless visually powerful and useful for creating and analyzing some

tessellations. Dilation symmetry (or scale symmetry) exists when a tessellation can be

mapped onto itself with a dilation (possibly in combination with an isometry). The

tessellation is shrunk or expanded in such a way that single tiles are not mapped onto

single tiles, but rather a single tile is mapped onto a finite collection of tiles (the tiling

expands), or vice versa (the tiling shrinks). This creates self-similarity such as one finds

in fractals or in certain complex tessellations like the Pinwheel Pattern (see Burger &

Starbird, 2005, p. 254 ff). Dilation symmetry also exists in simple tessellations such as

the rhombus tessellation in Figure 5 below. One of the rhombi has been highlighted, as

have its images after dilating (expanding) by a factor of 3 and a factor of 5. The dilated

images are mapped onto sets of 32 = 9 and 5

2 = 25 rhombi. Some form of dilation

symmetry was used by children in this study as a strategy for creating some of their

tessellations. A more complex example of dilation symmetry is in Figure 6 below.

the perceived motion as a remnant of our embodied understanding, in contradiction to a rigorous definition

of transformation as a function. However, mathematicians such as Grünbaum & Shephard (1986)

encourage such visualizations, as long as we keep in mind that this is only an aid and that we really only

care about the final position, and not the motion that got us there. The errors Edwards identifies have to do

with visualizing the transformation of only the object, rather than the entire plane as Grünbaum &

Shephard‘s model encourages us to do.

65

Figure 5: Rhombus tiling showing image of rhombus after dilating by factors of 3 and 5

Figure 6: An example of a tiling with dilation symmetry

Some tessellations are isogonal (vertex-transitive), isotoxal (edge-transitive), or

isohedral (tile-transitive). By this we mean that we can map any vertex, edge, or tile to

any other vertex, edge, or tile by some isometry which preserves the entire tessellation. In

less formal terms, all the vertices of an isogonal tiling are the ―same.‖ An ant standing on

any vertex of an isogonal tiling could not know which vertex it is standing on because

they all look exactly alike—the surrounding tessellation looks the same no matter which

vertex the ant stands on. (The same is true for any edge in an isotoxal tiling or any tile in

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an isohedral tiling.) This is not the case for the non-isogonal tessellation in Figure 4 on

page 63 where there are two types of vertices: one where hexagons and triangles alternate

around the vertex (e.g. vertex B) and one where they do not (e.g. vertex D). An ant

standing on B would know by the pattern of surrounding edges and tiles that it is not

standing on D. There is no isometry that could map this tessellation onto itself in such a

way that vertex B would be mapped to vertex D. The tiling in Figure 4 is not isogonal;

however we can say that it is 2-isogonal, meaning it has precisely two types of vertices—

the kind at B and the kind at D. In general, tessellations with k types of vertices are said

to be k-isogonal. If a k-isogonal tessellation is made of regular polygons, it is called k-

uniform. In a similar way, we can speak of k-isohedral and k-isotoxal tilings. The tiling in

Figure 4 is 2-isogonal, 2-uniform, 3-isotoxal, and 2-isohedral.

A tiling that is isohedral is necessarily monohedral, but not vice versa. These

terms are easily confused. ―Monohedral‖ simply means that only one type of tile is used

to create the tessellation; all the tiles are congruent. ―Isohedral‖ means that you can map

any tile to any other tile using an isometry which preserves the tessellation. The regular

square tessellation is isohedral; any square can be mapped by a symmetry to any other

square. The tiling is Figure 4 is not monohedral (it is dihedral), so it cannot be isohedral.

The tiling in Figure 7 below is monohedral, but not isohedral. All the tiles are congruent,

but there does not exist an isometry of the tessellation that maps, say, one of the central

rectangles to one of the horizontally oriented tiles. Similarly, the tessellations in Figure

20 and Figure 21 on pages 79 and 80 are also monohedral but not isohedral.

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Figure 7: A monohedral tiling that is neither isohedral nor edge-to-edge

Polygonal Tilings

All of the tasks where students created tilings in this study involved tilings of

polygons where the sides are straight line segments. For the rest of this section, I focus on

tilings of polygons. We call the edges of polygons sides and the vertices of polygons

corners. This is in order to avoid confusion with the terms ―edge‖ and ―vertex‖ as applied

to tilings. Usually the polygon sides will be the same as the tiling edges and the polygon

corners will be the same as the tiling vertices, but this is not always the case. When they

are the same, we say that the polygonal tessellation is edge-to-edge. It is often assumed

that tilings are to be created in an edge-to-edge manner, but unless specifically stated, this

is not a requirement. Figure 7 above and Figure 8 below are examples of tilings that are

not edge-to-edge. Note that the ―brick wall‖ tessellation in Figure 8 could be considered

edge-to-edge if the ―bricks‖ were considered hexagons with two straight-angled corners.

(See Figure 9 below for the six edges of the orange brick.) I do not consider the bricks as

hexagons because such interpretations are very unlikely among children, but one

mathematician interviewed in this study did make such observations. Some tilings, such

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as Figure 10 below cannot be construed as edge-to-edge no matter how you consider the

sides of the polygons; the ―zigzag‖ edges between pairs of tiles are made of at least three

sides each.

Figure 8: The ―brick wall‖ tiling of rectangles is not edge-to-edge.

Figure 9: Six edges of a brick

Figure 10: Another non-edge-to-edge tiling

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Regular Polygons

Certain types of tilings have special interest. Mathematicians, including Kepler,

have frequently focused on tilings created from regular polygons. For monohedral tilings

(where all the tiles are congruent), the only regular polygons that can be used to tessellate

the plane are the equilateral triangle, the square, and the regular hexagon. When the tiles

are placed edge-to-edge, there are just three possibilities, called the three regular

tessellations, which are illustrated in Figure 11 below.16 (As always, the patterns must be

imagined to continue indefinitely in all directions.) The regular tessellations are isogonal,

isotoxal, and isohedral.

Figure 11: Portions of the three regular tessellations

If we create a tessellation with more than one type of regular polygon, then there

are infinitely many possibilities. For example, we could make the hexagon and triangle

tessellation in Figure 4 on page 63. Or we could make the tiling on the left in Figure 12

below with one row of triangles between each row of squares. Or we could make the 2-

isogonal tiling on the right in Figure 12 with two rows of triangles between each row of

16 Two tilings are considered the same if there exists a similarity mapping from one to the other. In other

words, we do not say that rotating, flipping, or shrinking a tiling creates a different tiling. In this sense,

there is only one possible way of creating an edge-to-edge tiling with an equilateral triangle, a square, or a

regular hexagon. This is the accepted view in mathematics (Grünbaum & Shephard, 1986). Children and

non-mathematicians, on the other hand, may or may not consider various orientations or different sizes of a

tiling as being the same tiling.

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squares. Or we could make a similar tiling with any numbers of rows of triangles and

squares in any pattern we wanted, or no pattern at all. Tessellations such as the one on the

right in Figure 12 are neither isogonal, isotoxal, nor isohedral. The tessellation on the left

is isogonal (vertex-transitive), but not isotoxal or isohedral. (It is not isohedral because

there are two types of tiles: squares and triangles. It is not isotoxal because there are three

types of edges: square-square, square-triangle, and triangle-triangle).

Figure 12: Two tessellations with squares and triangles

Of special interest among tessellations using more than one type of regular

polygon are the eight semi-regular (or Archimedean) tessellations. These are isogonal

tessellations made of two or more regular polygons, such as the tiling at left in Figure 12.

The only other semi-regular tessellation made only of triangles and squares is in Figure

13 below. This tessellation is visually much more complex. However, one child in my

pilot research attempted to create it as her own idea, and nearly succeeded. The other six

possible semi-regular tessellations are in Figure 14 below.

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Figure 13: The 3.3.4.3.4 semi-regular tessellation

Figure 14: The six other semi-regular tessellations

At this point it is helpful to introduce some notation. When tilings are isogonal,

we can define them by describing the pattern of polygons around each vertex. If the tiles

are regular polygons, we can simply list the number of sides of each polygon in order

around the vertex. By convention, we start with the smallest polygons. The notation for

the regular triangle tessellation (Figure 11 on page 69) would be 3.3.3.3.3.3 or 36 because

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there are six triangles around each vertex. The regular square tessellation is 44 and the

regular hexagon tessellation is 63. The semi-regular tessellation on the left in Figure 12

above would be 33.4

2. The tessellation in Figure 13 has the notation 3.3.4.3.4 (or

32.4.3.4). For 2-isogonal tessellations, we list the polygons around each type of vertex.

The tessellation on the right in Figure 12 would be described as 36/3

3.4

2 because some

vertices are surrounded by six triangles and others by three triangles and two squares.

Periodicity and Symmetry

Up to this point we have mainly considered tessellations that are symmetric and

periodic. A periodic tessellation consists of a finite region which repeats indefinitely in

the same orientation throughout the plane. It can be shown that all periodic tessellations

have two linearly independent (non-parallel) translation symmetries. For example, the

regular 44 unit square tessellation has horizontal and vertical translation symmetry. We

can translate the unit square tessellation up one unit, or to the right one unit, in order to

map it onto itself. All other translation symmetries of the unit square tessellation can be

expressed as a linear combination of these two base translations. Every square in the

tiling is the image of any other square under one of these translation symmetries. It can

also be shown that if an edge-to-edge tiling has translation symmetry in one direction, the

tiles can be used to make a tiling with translation symmetry in two directions (Grünbaum

& Shephard, 1986). However, it is also quite easy to make tessellations which have

translation symmetry in only one direction, such as the example in Figure 7 on page 67,

which only has horizontal translation symmetry.

Periodic tessellations can be further analyzed by the various types of symmetry

they possess other than translation symmetry. We can classify the symmetry of a

tessellation by the number and order of distinct, independent rotations, reflections, and

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glide reflections that map the tessellation onto itself. The set of all possible

transformations form a group with respect to composition of the transformations. There

are a finite number of possible symmetry groups. It can be proved that there are exactly

seven possible symmetry groups with translation symmetry in only one direction. These

are known as frieze groups. There are also 17 possible periodic groups (known as

crystallographic groups or wallpaper groups) of planar patterns possessing two

independent translation symmetries. All repeating tessellations have the same basic

symmetries as one of these 24 possible patterns.

In this study, I worked with children who had only a basic notion of reflection and

translation symmetry, and perhaps a visual sense of other symmetries. (See the summary

of research on Children’s Understanding of Symmetry on page 97.) Instead of using a

complete symmetry analysis, it was sufficient merely to note that certain patterns have (1)

(simple) reflection symmetry, (2) rotation symmetry of order n, or (3) n-fold symmetry.17

This analysis can be applied equally to tessellations or to finite patterns. Further analysis

was not really needed for this study.18

(1) Simple reflection symmetry means that there is one line of reflection

symmetry and no rotation symmetry.

(2) Rotation symmetry of order n means that we must rotate the pattern at least

360°/n about a point of rotation symmetry to map the pattern onto itself.19 The semi-

regular tessellation on the left in Figure 15 below has rotation symmetry of order 6 about

17 Note that the simple fact that a tile is symmetric does not mean that the tiling will inherit the same

symmetry. For example, the X-pentomino has 2-fold reflection symmetry, but can only form a tiling with

simple rotation symmetry (Schattschneider, 2010). 18 Appendix G describes the one case where deeper mathematical analysis was perhaps needed. However

the results from that analysis were inconclusive. 19 Many mathematicians (e.g. Grünbaum & Shephard, 1986) use the term ―n-fold rotational symmetry,‖

but I will reserve the term ―n-fold‖ to indicate presence of reflections in order to make a greater distinction.

Many other ways of describing symmetry are also in common use.

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the center of any hexagon. A rotation of 60° about such a center will map the tessellation

onto itself. It also has rotation symmetries of order 3 and order 2 about certain other

points on the triangles. I generally only note the highest order of rotation symmetry when

discussing a tiling.

Figure 15: Two symmetric semi-regular tessellations

(3) A pattern has n-fold symmetry if there are n axes of reflection symmetry that

intersect through a common point. There are six axes of reflection that intersect each

vertex in the regular triangle tessellation, so we say it has 6-fold symmetry. (Other points

happen to have other orders of reflection symmetry, but six is the maximum.) A pattern

with n-fold (reflection) symmetry (n > 1) necessarily has rotation symmetry of order n as

well. This is because the combination of two non-parallel reflections makes a rotation.

Therefore I invent the convention of saying such patterns have n-fold symmetry without

further mention of reflections or rotations. The semi-regular tiling on the right in Figure

15 above has both horizontal and vertical20 reflection symmetry as well as two oblique

axes of reflection symmetry about the centers of each tile, so it also has order-4 rotation

symmetry about those centers. I simply say these tilings have 4-fold symmetry. Note,

20 Reflection symmetry is said to be horizontal if the axis of reflection is horizontal. Vertical reflection

symmetry has a vertical line of reflection.

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however, that rotation symmetry does not imply reflection symmetry. For example, the

tiling on the left in Figure 15 above has order-6 rotation symmetry, but no reflection

symmetry.

I use these simplified analyses for finite patterns as well as tessellations. When a

pattern is finite, the symmetry must map the border of the pattern onto itself. Under this

restriction, the finite pattern on the right in Figure 15 above has horizontal reflection

symmetry, but no other kind of symmetry.

Lack of Symmetry

Note that a tessellation does not have to have any symmetry at all.21 A tessellation

can even be non-periodic (lacking translation symmetry). A random pattern of polygons

all different from each other, as in Figure 16 below would be one such example, as long

as it is imagined to continue randomly forever in all directions. However, tessellations

can still have great structure and be built from a small set of prototiles without having any

symmetry. Such tessellations gained special interest in the 1960s and 70s when aperiodic

prototiles were discovered—any tessellation made with such prototiles was necessarily

non-periodic. The most famous of such tilings are Penrose tilings, discovered by Roger

Penrose in 1974. These tilings are created from a generating set of only two prototiles

which force mathematically and aesthetically interesting non-periodic tilings (Gardner,

1977).

21 We are, of course, not considering the identity mapping as a symmetry, though technically, from a

mathematical point of view, it is. If we did consider the identity mapping, we could say that a tiling must

have a symmetry other than the identity mapping to be considered symmetric.

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Figure 16: A random tessellation of polygons all different from each other

It is quite easy to make simple non-periodic tessellations even with regular

polygons. One of the simplest examples would be to lay down dominoes at random, half

horizontal and half vertical, aligned without gaps or overlaps, making a tessellation

without any pattern. As long as reasonable care is made not to leave square-shaped holes,

the tiling will lack symmetry. Another type of random tiling is Voronoi (or Dirichlet)

tilings, which have been extensively studied by researchers in recent years. There are

many non-periodic tessellations with triangles and squares.22 In fact, children in this

study sometimes attempted to make such tessellations and these tessellations also

frequently occur in nature (see Figure 17 below). Making such tessellations by placing

the tiles at random, as the children did, is tricky because it is possible to get ―caught‖ in a

position where the tiles no longer fit. This sometimes occurs in nature as well—such

natural tessellations usually contain gaps in the pattern. But if the squares and triangles

22 In fact, there are uncountably many tessellations such that every vertex is surrounded by two squares and

three triangles. A complete description of all such tessellations has not yet been made (Grünbaum &

Shephard, 1986).

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are placed with reasonable care, convex patterns can always be extended to the whole

plane and the children in the pilot study did not leave holes in their patterns.23

Figure 17: A non-periodic tiling of squares and triangles occurring naturally in crystal

growth (Krumeich, Conrad, Nissen, & Harbrecht, 1998)

It is easy enough to show that non-periodic tessellations exist by simply creating

one. Start with the periodic 36/3

2.4.3.4 tessellation in Figure 18 below. (One of the

children in my study created this tessellation. See Figure 37 on page 148.) Next, rotate at

least three dodecagonal groupings, as in Figure 19. The flaw introduced into the pattern

destroys all symmetry, including translation symmetry (periodicity).

23 To be precise, it can be proved that any convex region of squares and triangles in which all vertices are

surrounded by two squares and three triangles can be extended to a tessellation of the entire plane with all

vertices surrounded by two squares and three triangles (Grünbaum & Shephard, 1986, p. 99).

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Figure 18: The 36/3

2.4.3.4 tiling with three dodecagons highlighted

Figure 19: Non-periodic, asymmetric, tiling created by rotating three dodecagons 30°

Several children in this study tried to create random tilings similar to those

occurring in crystals. However it is impossible to know if such tilings will eventually

contain a gap and this sometimes created uncertainty in the minds of the children. There

is no such uncertainty when there is translation symmetry in two independent directions.

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Note also that it is easy to create tessellations which have rotation or reflection

symmetry without translation symmetry. For example if we had rotated only two

dodecagons in Figure 19, the pattern would have lost translation symmetry, but would

still have retained rotation (and possibly reflection) symmetry. Another example is in

Figure 20 below which is a monohedral tiling made of L-trominoes (a type of non-convex

hexagon). It has 4-fold symmetry, with a unique center of symmetry from which the rest

of the pattern appears to radiate visually. Because it has only one center of symmetry, it

has no translation symmetry. One of the children in my pilot study attempted to make a

pattern similar to this one, but with 2-fold symmetry. I say that patterns with rotation and

reflection symmetry about a point without translation symmetry are radial patterns.

Figure 20: A 4-fold symmetric tiling without translation symmetry

A tessellation without translation symmetry can also be created from isosceles

triangles as on the left in Figure 21 below, which is a radial tessellation. If we translate

half the tessellation, we can even make a spiral tessellation which has rotation symmetry,

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but no reflection or translation symmetry, as on the right in Figure 21. (Certain lines have

been highlighted to emphasize the spiral nature of the tiling.)

Figure 21: Isosceles triangle tiling with 8-fold symmetry transformed into a spiral tiling

with rotation symmetry but no reflection symmetry

Non-Regular Polygons

It is interesting to note that, not only equilateral triangles and squares, but in fact

all triangles and quadrilaterals tessellate. That is, any triangle or quadrilateral, even non-

convex, can be used to create tessellations in various ways with various types of

symmetry, for example as in Figure 22 below.24 Triangles and quadrilaterals can actually

tessellate in a wide variety of ways. There are 14 ways to form an isohedral tessellation

with triangles and 56 ways to form an isohedral tessellation with quadrilaterals

(Grünbaum & Shephard, 1986). Figure 23 below shows four possible ways of tiling with

a scalene triangle.

24 It is also interesting to note that this basic fact appears to be not well known among mathematicians.

Rena Upitis (Upitis et al., 1997) recounts a trip to MIT where she casually mentioned her work with

children making tessellations with non-convex quadrilaterals. The mathematicians there did not believe

such tessellations were possible.

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Figure 22: Tiling of a non-convex quadrilateral

Figure 23: Four of the several possible isohedral tessellations with a scalene triangle. The

two on the right are not edge-to-edge.

Pentagon tessellations are much more complex and in fact the precise number of

possible pentagons that tessellate is an unsolved problem in mathematics. Currently there

are 14 known types of pentagons that can make monohedral tessellations. Four of these

were discovered by Marjorie Rice, a San Diego homemaker with a high school education,

who noticed the problem in her son‘s copy of Scientific American and decided to work on

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it. These 14 types of pentagons have many ways of tiling, some of which have been

discovered by high school classes working on the problem (Schattschneider, 1978b).

More recently, it has been discovered that the number of ways of creating monohedral

convex pentagon tilings is actually infinite (Sugimoto & Ogawa, 2000). It is of course

impossible to tessellate with regular pentagons because the angles about each vertex must

add up to 360.

Pentagons represent the only case of polygons which have not been completely

analyzed. There are precisely three types of hexagons which can tessellate, including of

course the regular hexagon. It can be proved that convex polygons with seven or more

sides cannot form a monohedral tessellation (Grünbaum & Shephard, 1986), though there

are non-convex polygons for any number of sides which can tessellate (polyforms, for

example, which I discuss next).

Polyforms

Another type of polygon whose tessellations have been extensively explored are

polyforms, which are polygons made by adjoining copies of some unit shape along their

sides. The most frequently studied polyform is the class of polyominoes—shapes made by

adjoining squares along their sides. A single square is called a monomino. Two squares

are called a domino because of their resemblance to the playing tile. (Other polyominoes

were named as a humorous back formation from ―domino,‖ as if the d- were the Greek

di- or ―two.‖) If we do not count congruent shapes as different, there are two ways to

assemble trominoes (three squares) and five ways to form four squares into a tetromino.

See Figure 24 below for the standard letter names given to these shapes. (The interior

lines are only meant to show the unit squares and have no role in making tessellations,

though some students insisted on adding these lines, perhaps to help them visualize the

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shape.) If we count reflected shapes as different, there are two versions of the L- and S-

tetrominoes, making the seven familiar tetrominoes used in the popular game of Tetris.

All of these shapes can tessellate the plane, as can the 12 pentominoes and the 35

hexominoes, though some shapes are more difficult to tile than others. Figure 10 and

Figure 20 show two ways to tile with the L-tromino. Searching for these tessellations

provides open-ended tasks at a range of difficulties, which I used as one of the sets of

tasks in this research.

Figure 24: Order-3 and order-4 polyominoes and their common letter names

Another common polyform is the class of polyiamonds, which are polyforms

made from equilateral triangles. Figure 25 below shows all the polyiamonds up to order

5. Any polyiamond up to order 6 can tessellate the plane (Grünbaum & Shephard, 1986).

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Figure 25: Polyiamonds up to order 5

A less well-known type of polyform is the polyaboloes, which are made of right

isosceles triangles. The only polyabolo used in this study is the trapezoid triabolo

pictured in Figure 26 below. Notice that this triabolo can be made in two different ways.

Figure 26: A triabolo that can be made in two different ways

Color

Note that it is possible to take color into consideration when analyzing

tessellations. Though normally color is not a consideration in geometry, it may become a

consideration for tessellations when analyzing symmetry and periodicity because we

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expect colored shapes to be mapped to shapes of the same color. The tiling in Figure 27

below would be considered a simple isohedral hexagon tiling with order-2 rotation

symmetry about four different points, as long as the colors are not taken into

consideration. If, however, we take color into consideration and require symmetry to map

tiles to tiles of the same color, the tiling no longer has rotation symmetry, though it is still

periodic. Most people, and especially children, are likely to take color into consideration

when evaluating a tessellation. Many tessellations in works of art, such as those at the

Alhambra or in Escher‘s work, rely on color to restrict the symmetry patterns in

interesting ways, leading the eye to notice symmetry that would not have been obvious

without color.

Figure 27: A tiling whose symmetry depends on whether color is taken into

consideration

Vocabulary Summary

Table 1 below has a summary of important technical terms used to discuss

tessellations.

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Table 1: Important tessellation terms

Term Meaning

Aperiodic Set of prototiles which can only form non-periodic tilings

Corner The term “vertex” has a special meaning for tilings, so “corner” is used to refer to what is usually called the vertex of a polygon.

Dihedral Tiling that uses two prototiles

Dilation symmetry Transformation mapping several tiles onto one tile or vice versa

Edge The intersection of two neighboring tiles; not always the same as a side

Edge-to-edge Polygonal tiling whose edges are the same as the polygon sides

Generating set The set of prototiles for a tiling

Isogonal Vertex-transitive: any vertex can be mapped to any other by a symmetry

Isohedral Tile-transitive: any tile can be mapped to any other by a symmetry

Isotoxal Edge-transitive: any edge can be mapped to any other by a symmetry

Isometry A rigid transformation: translation, reflection, rotation, or glide reflection

k-isogonal Tessellation with k different types of vertices that cannot be mapped to each other by any symmetry

k-uniform k-isogonal tiling made of regular polygons

Monohedral Tiling that uses only one prototile; all tiles are congruent

n-fold symmetry n lines of reflection symmetry through a point; this also implies rotation symmetry of order n (n > 1).

Non-periodic Tiling that lacks translation symmetry

Periodic Tiling with two independent translations; the pattern repeats in all directions

Prototile An initial shape used to tile with; all tiles are congruent to a prototile

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Table 1. cont.

Term Meaning

Radial Patterns with rotation and reflection symmetry about a certain point, but no translation symmetry

Regular An isohedral, edge-to-edge tiling made of regular polygons

Rotation symmetry of order n

A symmetry that is a rotation of 360°/n

Semi-regular Isogonal, edge-to-edge tiling of at least two types of regular polygons

Side The term “edge” has a special meaning for tilings, so “side” is used to refer to what is usually called the edge of a polygon.

Symmetry An isometry of the tiling that maps tiles onto tiles

Vertex The non-empty intersection (endpoint) of three or more edges

Children’s Understandings of Tessellations

Tessellations have been used successfully in many school classes, and are even

required in curricula of many countries and some states. Surprisingly, however, there has

been very little research into children‘s understanding of tessellations. The following

studies comprise the sum of all research that specifically informed the tiling aspect of this

study.

Geometric Properties in Tessellations (van Hiele-Geldof and Fuys et al.)

Perhaps the earliest educational study using tessellations was done by Dina van

Hiele-Geldof, who, with her husband Pierre van Hiele, developed the well known van

Hiele model of geometric reasoning. Dina‘s 1957 doctoral dissertation describes a

teaching experiment with 12-year-olds in a Dutch Montessori secondary school (Fuys et

al., 1984). She used a variety of methods to lead her students to explore many geometric

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ideas. She devoted the second school term to leading her students to construct and

analyze tessellations, including sidewalk paving; polygon tilings; possibilities with

triangles, quadrilaterals, and regular pentagons; parallelograms; and other shapes.

Various geometric properties were discovered in this way. For example, by noticing

parallel lines and congruent angles in certain tessellations, the students found patterns

they called ―saws‖ and ―ladders.‖ (See Figure 28 below.) These patterns led to the finding

of congruent angle theorems with parallel lines and how to reason with such theorems.

Her visual method of placing these theorems in the context of tessellations was much

more successful than the abstract, context-free way these theorems are usually presented

to students. Dina van Hiele reported a significant increase in students‘ ability to reason in

geometry through her teaching experiment. Her method was successfully repeated by the

Brooklyn College Project in the 1980s with similar results (Fuys, Geddes, & Tischler,

1988). These studies demonstrate some of the mathematical power of tessellations in

education.

Figure 28: Rhombus tiling with ―saw‖ (red) and ―ladder‖ (orange) highlighted

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Rectangular Arrays of Squares (Outhred & Mitchelmore)

Most of the studies of children‘s conceptualization of tilings have focused on

finite square arrays in order to understand how children understand the area formula for

rectangles. Outhred and Mitchelmore (1992) noted that children can easily arrange

wooden square tiles in rectangular arrays without necessarily understanding the row and

column structure of the array. This is because the physical tiles cannot overlap and

naturally organize themselves as children place them next to each other. Outhred and

Mitchelmore found that some children learn the area formula for a rectangle without

understanding—indeed without even the possibility of understanding—because they lack

a valid conceptualization of the row and column structure of a rectangular array. Battista,

Clements, Arnoff, Battista, and Borrow (1998) carried this research further, identifying

five levels of understanding that they believe children pass through in arriving at a robust

understanding of row and column structure. Outhred and Mitchelmore (2000) did further

studies, proposing a sequence of principles to be learned in order to understand the array

structure behind the area formula of a rectangle.

The goal of these studies is to lay the foundation for understanding the area

formula for a rectangle. Therefore the authors are only interested in finite examples of

square arrays, rather than infinite tessellations. Nevertheless, the authors report several

findings which are easily extended to all tessellations and therefore have a direct impact

on my study:

Tessellations require a mental model of the shape as well as a mental model of the

array. These models do not exist in young children. Most children do not develop

mental models of square arrays until third grade.25 Even in fourth grade, only

25 The new Common Core State Standards (2010) require students to study square arrays in second grade.

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about half of the students demonstrated a complete awareness of the array

structure in one of the tasks they were given. These mental models come about as

abstractions of one‘s actions in forming tessellations rather than any direct

abstraction of the tiling itself.

Use of concrete manipulatives ―prestructures‖ the tiling so that an accurate

conceptualization is not needed to create a successful tiling. It is easy for young

children to create successful tilings with physical square tiles, even if they have

no understanding of the row and column structure of the array. Other methods,

such as freehand drawing, must be used to analyze the child‘s thinking. (For

investigating student thinking, I found that it was even more helpful to have

children describe a tiling orally first, with one or two physical tiles for support,

before attempting to sketch the tessellation. I discuss this further in Chapter 3.)

Outhred and Mitchelmore also found that freehand drawing promoted

mathematically correct understanding of the structure of a tiling better than

concrete manipulatives.

Freehand drawing alone will not lead children to understand the structure of the

array unless they also realize when their drawings are insufficient. A predict-and-

check method may be best for guiding students to confront mathematically

erroneous schemata. It may be best to have children draw first and then check

with tiles.

Array structure is so well assimilated for adults that teachers may not realize that

array structure is not self-evident for children. This also appears to be the case for

some researchers before 1992, such as Vitale and Zinder (1991, see below), who

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assumed that all fourth graders understand the array structure of a parallelogram

tessellation.

One of the goals of my dissertation is to extend this study of children‘s

understandings of square arrays to children‘s understandings of other types of

tessellations. I show that some results are similar to those found by Outhred and

Mitchelmore for square arrays. I also show how and why Outhred and Mitchelmore‘s

reliance on drawing should be extended to oral discussion in order to get an even better

idea of children‘s initial conceptions of tessellations.

Combining Tessellations With Art (Upitis, Phillips, & Higginson)

Unlike some mathematical topics, tilings have obvious and immediate aesthetic

appeal to most people and many teachers see tessellations as an opportunity to do art

rather than math. One very common activity in schools is to modify the edges of a

tessellation so that the tiles take the shapes of animals or other objects, making an Escher-

type tessellation. These entertaining activities frequently have little or no mathematical

content, though they can.

An interesting study that combined both the mathematical and artistic aspects of

tessellations is recounted in Creative Mathematics: Exploring Children’s Understanding

(Upitis et al., 1997). The three authors—a researcher (Rena Upitis), a teacher (Eileen

Phillips), and a mathematician (Bill Higginson)—describe an aesthetic approach to

tessellations in the second chapter of their book. Rena begins the unit on tessellations

with the third graders five weeks before Eileen starts with the fourth graders, allowing the

third graders to teach certain discoveries to the fourth graders. Rena pursues the unit in a

very artistic manner, but with significant mathematical content, having the children create

patterns with a variety of materials, invent mathematical notations to describe the

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patterns, and make connections with the real world. The children eventually work out that

all triangles and quadrilaterals tessellate, but only some pentagons and hexagons. They

make Escher-type tessellations on dot paper, experiment with pattern blocks, discover

unknown tessellations, and end with a silk card project.

Eileen noticed a significant impact on students‘ ability in other mathematics units,

especially when using geoboards, which Eileen uses to teach certain fraction, geometry,

and multiplication concepts. Three months after the tessellation project, Eileen‘s students

had no trouble with these geoboard activities, though students in her 20 previous years of

teaching had always struggled. Furthermore, the children who studied tessellations

retained all the polygon vocabulary and showed impressive and surprising transfer of

their geometry knowledge in their accurate and complex geoboard results. ―Tessellations

were obviously real—really powerful, really useful, really adaptable‖ (p. 47). Bill notes

how the tessellation unit integrated all five of the bullet points from the NCTM Standards

for K-4 geometry (NCTM, 1989). He remarks that tessellations include a strong aesthetic

appeal. The process of ―fitting‖ is an aesthetic one and is basic to human instinct. This

explains the appeal of games such as Tetris, where the goal is to quickly create a gapless

tiling from tetrominoes. However, Bill notes that it is important for the teacher to have

significant knowledge of the subject. He applauds Rena‘s willingness to allow her

students to explore unknown paths, though she appropriately backed off from pentagons,

which, as we have noted, are still not fully explored by mathematicians.

This is one of the only teaching experiments on fourth grade tessellations in the

literature. It demonstrates the power of tessellations as a context for learning geometry at

this grade level.

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Unitizing (Wheatley & Reynolds)

Wheatley and Reynolds (1996) did a study comparing how children unitized in

arithmetic and tiling problems. The authors found that third grade children who unitized

well in arithmetic also tended to unitize in creating tilings, such as by putting two L-

trominoes together to form a rectangle. For example, the most successful student drew

rectangular units first and then subdivided them into the tiles that made up the unit. For

the student who had the most difficulty, even the tile was not yet perceived as a unit and

she had to draw the polyomino tile slowly one segment at a time. They concluded ―that

the construction and coordination of abstract units is central to mathematical activity in

both numerical and geometric settings‖ (p. 67).

The authors were surprised by the difficulty of the tasks. (Fifth and sixth graders

did much better.) The tiling exercises were done using blank paper or dot paper. The

polyomino tasks in my study are an extension of Wheatley and Reynolds‘s dot paper

exercises. The authors concluded, ―Students are likely to benefit greatly in their

mathematical development from opportunities to construct tilings of geometric shapes‖

(1996, p. 82).

Covering Shapes With Tiles (Owens & Outhred)

Owens and Outhred (1998) studied how children tiled finite areas. These finite,

monohedral tilings were intended to explore concepts necessary for understanding area

principles. Students had much more difficulty tiling with triangles than with rectangles

because triangular tiles had to be placed in two orientations. (Children in my study also

frequently commented on this dual orientation phenomenon. See, for example, Figure 29

below, where triangles oriented differently are also colored differently.) Unfamiliar

shapes to be covered, such as trapezoids, were more difficult than familiar shapes. All

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children had to draw in order to visualize the tessellations and be aware of their structure.

Children who worked on spatial activities improved on the tiling test. The authors

criticize common methods for teaching area concepts such as tiling with potato prints

because these highlight counting rather than area concepts. Children need to draw the

tilings by hand and understand the limitations of their drawing ability.

I based one of the tasks in my study on Owens and Outhred‘s finite tiling tasks in

order to explore students‘ understanding of fit.

Figure 29: Regular triangle tiling with oppositely oriented triangles colored differently

The Logic of Mixing Tiles (Vitale & Zinder)

Vitale and Zinder contributed a chapter on tessellations in Piaget and Garcia‘s

Toward a Logic of Meanings (1991). The subject of the book is on understanding

children‘s intensional logic. In Vitale and Zinder‘s chapter, the focus is on children‘s

intensional logic in deciding which combinations of polygons can form tessellations. This

particular chapter is flawed, in my opinion, because there does not seem to be any

rationale for the research question, either from a mathematical perspective or a child‘s

perspective. (There is no intensional logic in the combination of tiles.) Nevertheless,

there are some interesting points that come from this study. One is the fact that young

children do not understand that rectangular arrays of squares can be continued

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indefinitely. This is very likely due to the fact that young children do not yet

conceptualize tessellations in terms of rows.

Other Geometry-Related Research

The following studies do not explicitly address tiling, but do have results

important for my study concerning children‘s understanding of geometric concepts

important in tiling. In my study I pay special attention to students‘ conception of infinite

space with respect to tessellations and the role of symmetry in students‘ creation and

evaluation of tessellations.

A study by Clements, Wilson, and Sarama (2004) was potentially useful for

analyzing students‘ sense of fit, but the trajectory they proposed was developed for

younger children and did not seem to describe how the fourth grade students in my study

approached composition of units.

Children’s Understanding of Infinite Space

Plato held that space, the place where objects could be situated, was actually

infinite (Monaghan, 2001). Aristotle, however, taught that infinity itself did not actually

exist, but was merely a potentiality—straight lines could be extended as far and as often

as needed, but were not actually infinite. Some researchers have found that these two

ideas continue to be held in our minds, creating contradictory notions of how exactly

infinity is to be understood. Unfortunately, almost all research on infinity has been on

concepts such as limits, continuity, numbers, and the infinitely small. Almost no

researchers have looked at children‘s conceptions of infinite space.

Piaget and Inhelder were the first to look at children‘s conception of infinity, but

they did not cover it extensively. Aside from infinite numbers, they looked at ideas of

continuity, which according to Monaghan (2001) was mostly in the interest of

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establishing topological properties. They only worked with children up to 12 years of

age. Children at the concrete operational stage could imagine dividing a segment into a

large but finite number of pieces. Children at the formal operational stage could conceive

of dividing the segment an infinite number of times (Piaget & Inhelder, 1956).

Fischbein, Tirosh, and Hess (1979) extended Piaget and Inhelder‘s work to older

participants, whom they found to have contradictory ideas. They found that it was

important to distinguish the mathematical (Cantorian) idea of infinity and the self-

contradictory intuitive concepts of infinity. Our schemata are created through finite

contexts, so they are not well adapted to deal with mathematical infinity. They found that

children continued to grow in their understanding of infinity until about age 12, after

which their ideas stabilized, though these ideas were easily open to change. Whether

children gave the ―infinitist‖ answer or not depended on the question. Some questions

elicited 81% ―correct,‖ Cantorian answers and others only 10%, an indication that

students held contradictory notions. High achievers did not give better answers, but did

give wrong answers more consistently. Instruction had little or no effect on the answers,

especially for the non-standard questions.

Monaghan (2001) considered a variety of infinite concepts, including endless

processes, sequences, and operations. He cautioned that it is easy to lose sight of the fact

that infinity is a strange idea for children. Monaghan‘s studies found that students tended

to look at infinity more as a process than as an object. Like Fischbein et al., he found that

their concepts were contradictory, easily changed, and impervious to instruction. He

cautions that Cantorian infinity is not the only mathematically possible approach to

infinity and student answers should not be considered ―wrong‖ if they do not agree with

Cantorian set theory.

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One of the only studies so far of spatial infinity was done by Marchini (2003),

who looked at young children, aged 5–7, as they drew simple pattern sequences. Every

time they came to some sort of end, such as the edge of the paper, they were simply told

to ―continue.‖ Some students simply stopped anyway, whereas others found creative

ways to carry forward, such as making loops or attaching new sheets of paper. The

children made various comments. The data suggested that only a few had consistently

infinite ideas, though a larger number had what he called ―mixed‖ finite and infinite

conceptions. Marchini recognized that his methodology had limited power, but proposed

it as a first step towards understanding young children‘s idea of spatial infinity.

Children’s Understanding of Symmetry

Next I summarize the main findings on children‘s conceptualizations of

transformations and symmetry, which are key concepts for an understanding of

tessellations, as well as key factors in the aesthetics of tessellations. Symmetry connects

the cognitive with the aesthetic. Symmetry is an objective, mathematical concept which

explains much of the structure of a tessellation. As such it is understood from the

cognitive perspective. But symmetry is also universally recognized as an aesthetic

category which contributes to the perceived beauty of a tessellation. Arnheim (1969)

notes that perceptual symmetry can bring together the parts of a figure into a unified

whole, making perception easier. It is therefore important to understand to what extent

children understand symmetry, as well as the closely related topic of transformations,

which give mathematical definition to symmetry.

Most geometric concepts appear to be universal, and therefore innate in all

people. Dehaene, Izard, Pica, and Spelke (2006) confirmed this by testing the geometry

ability of the Mundurukú, an isolated Amazonian indigenous people, in order to explore

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what innate abilities we have without the schooling or other geometric context that

permeates modern society. Mundurukú participants demonstrated basic abilities in

topology, Euclidean geometry, and geometric figures. Both Mundurukú children and

adults scored at the same level as American children. (American adults scored

significantly higher.) The Mundurukú did less well with symmetry and metric properties,

though they still got most problems correct, except those involving chirality at oblique

angles. The Mundurukú did not do well at all with any problems on transformations,

including translations, dilations, reflections, and rotations. ―It is possible that geometrical

transformations are inherently more difficult mathematical concepts. Alternatively, such

transformations may be more difficult to detect in static images‖ (pp. 381–382).

Visualizing transformations may be a skill that requires experience.26

Understanding and working with symmetry, and to a greater extent

transformations, appears therefore to be at least partially a learned skill. Laurie Edwards

(2003) analyzes some of the inherent difficulty with transformations using the theory of

embodied cognition, a viewpoint that understands cognition to be situated in our

sensorimotor processes. Edwards discusses how children and adults struggle with

transformations, particularly rotations. Most people have difficulty conceiving rotations

where the center is not on the object. This difficulty arises because we do not think of the

entire space as being transformed, only the object. Edwards previously thought rotation

errors were just a ―misconception,‖ but now realizes that these errors occur because

students see transformations based on previous experiences. Even after explanation,

students talk about transformations as movements and struggle when centers are not on

26 It is unfortunate that the new Common Core State Standards (2010) postpone students‘ first work with

transformations until eighth grade, despite the heavy importance given to transformations in high school

geometry.

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the object. ―This tension between the mathematician‘s understanding of transformations

as mappings, implemented in the microworld, and the embodied, natural understanding of

motion that the learners brought to the experience is the source of their ‗misconceptions.‘

These ‗misconceptions‘ are in actuality, conceptions that are adaptive and functional

outside of the context of formal mathematics‖ (p. 9, sic for punctuation). Children at the

age studied in my research have a very physical understanding of transformations. It is

normal that they struggle to visualize rotations or rotation symmetry, especially because

the context of tessellations requires visualizing the transformation of the entire plane

rather than a single tile. It was not expected that children be able to verbalize any

recognition of rotation symmetry, and in fact none of them did, though rotation symmetry

clearly added to the aesthetic appeal of some of the tessellations for the children.

Researchers, such as Hoyles and Healy (1997), have found that reflections are

also a difficult transformation to visualize when the line of reflection is oblique. Hoyles

and Healy also found that 12-year-old students did not understand reflection as a

transformation of the entire plane, but rather just transformation of certain points. Some

transformations were difficult for the students because it was not clear how to connect

their visual perception with a formal procedure.

Piaget and Inhelder reported their studies on children‘s ability to understand

transformations in Mental Imagery in the Child (1971). They found that children were

successful with simple translation and reflection tasks from age 6 or 7, though much

younger children could gesture how shapes would be displaced in a translation. Simple

rotation tasks could be answered by age 7, though they sometimes described results with

oval imagery instead of circles. They concluded that children younger than age 7 or 8

have some understanding of motion, but cannot conceptualize transformations, which

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require an understanding of how an object changes due to the transformation. Young

children therefore reason in static configurations only.

Kidder (1976) found that children up to age 13 have difficulty with mental

representations of transformations. Kidder found this surprising because Piaget and

Inhelder believed children could begin to work with transformations at age 9. It should be

noted, however, that Piaget and Inhelder‘s tasks were much simpler than Kidder‘s

activities. The students in Kidder‘s study found translations to be the easiest

transformations. There was no significant difference in ability between rotations and

reflections for Kidder‘s activities. Compositions and inverses of transformations were

difficult for all students. Students were not consciously aware of the implications of

length conservation under any transformation. The numerous errors in students‘ answers

led Kidder to believe that his tasks required Piaget‘s formal operational (transfigural)

stage, which would have just been beginning for the oldest participants in Kidder‘s study.

In more recent studies, researchers have found that rotations are harder for

children to conceptualize than reflections, and that reflections are more difficult than

translations (Clements, 2003). Reflection symmetry is more easily noticed if the axis is

vertical or, to a lesser extent, horizontal (Genkins, 1975).

It was expected that children in my study would easily see vertical and horizontal

reflection symmetry as well as translation symmetry. Oblique reflection symmetry,

rotation symmetry, and glide reflection symmetry may be evident at the aesthetic level,

but it was not expected that students would be able to describe them, nor was it expected

they should be able to analyze any of the symmetry in terms of transformations except in

vague, localized terms.

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SUMMARY

This literature review has been necessarily lengthy because I am studying what is

essentially an unresearched area of mathematics education. There is no single theory that

can provide a sufficient theoretical framework for this study, especially because my

research question is analyzing the question of how children conceptualize tessellations

from two different theoretical viewpoints. These two viewpoints are complementary and

both are needed for a complete understanding of children‘s mathematical thinking, but

they are almost always treated separately in the literature. The second viewpoint

(aesthetic) is not well known, so it has required a more extensive presentation.

For the cognitive viewpoint, I have taken a broad Piagetian constructivist view.

Children construct their knowledge through assimilation and accommodation, passing

through various stages of understanding geometric concepts. Fourth grade children are

expected to be all at Piaget‘s interfigural stage, which is the earliest stage where

significant work with infinite tessellations can be handled. Children at this stage can

understand relations between figures, including basic symmetry and transformations.

They cannot however understand relations between systems of figures, which means they

probably will not understand glide reflections.

This Piagetian base is extended by other theories, all within the broader

constructivist framework. Vygotsky analyzes the social-cultural aspect of learning,

introducing the idea of zone of proximal development, remarking on the important

impact of tools on the child‘s thinking, and providing a theory for the relationship of

thought and speech. Theories by Sfard, Dubinsky, Tall and Gray, and Meissner enable us

to understand the relationship between a process conception and an object conception of

a mathematical idea.

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The aesthetic viewpoint has historically been overlooked in most research, but is

of importance for several reasons. Some researchers believe that cognition takes place

primarily at the level of aesthetic cognition. Aesthetics is also foundational to

establishing axiomatic systems and for finding generative paths of mathematical research.

Mathematicians recognize the important role of aesthetics in evaluating mathematical

results, using criteria of significance, surprise, simplicity, connectedness, and visual

appeal. The fact that these criteria also point to what is important in mathematics suggests

a deep connection between the aesthetic, the cognitive, and the mathematical. Children‘s

mathematical aesthetic is different because they have different purposes from

mathematicians in their pursuit of mathematics, and because children‘s mathematical

aesthetic adapts according to their goals and understanding. Nevertheless, children do

have a mathematical aesthetic, which educators should take into account, especially in the

context of mathematical inquiry. For mathematics education, Sinclair has proposed

viewing aesthetics through its three primary roles of motivation, generativity, and

evaluation.

I have summarized the mathematical theory of tessellations for clarity in analysis

and communication. A summary of technical terms used to discuss tessellations is given

on page 86.

I have also summarized the scant research that has been done directly related to

this study on children‘s understandings of tessellations. This body of research connects

the cognitive viewpoint with the mathematical content of this study. Several studies

demonstrate that tessellation lessons can have a powerful impact on the learning of

geometry by children aged 8–12. Studies on finite square arrays in the 1990s showed

some of the thinking that might be revealed by children in a more general study of

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tessellations, and in particular how the structure of the arrays is not understood by most

children until about age 8. Work by Wheatley and Reynolds showed that geometric

unitizing is an important strategy that is used by students at various levels. Owens and

Outhred investigated children‘s ability to tile a finite area.

In addition to these directly related studies, I also summarized the research on

children‘s understanding of infinite space, symmetry, and transformations. There seems,

in fact, to be no significant research on children‘s conception of infinite space, though

there are related studies on children‘s understanding of infinity in other contexts, showing

that students of all ages have limited ability to conceptualize infinity mathematically. As

Piagetian theory suggests, fourth grade children have only a simple grasp of basic

symmetry and transformations. Children understand translations best, followed by

vertical reflective symmetry, horizontal reflective symmetry, other reflective symmetry,

and then rotations. Glide reflections are probably too complex for fourth grade children to

understand at an analytical level.

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Chapter 3: Methodology

For this study of children‘s understanding of tessellations, I conducted a series of

clinical interviews of six fourth grade children. In this chapter I first describe the

methodology of clinical interviews—theory, practical details, and validity of the

methodology. I then give the details of the method used for this research.

CLINICAL INTERVIEWS

Clinical interviews have their origin in the work of Piaget, who used them

extensively to study children‘s psychological conceptions of their world. It is a method

which is suitable for use by both researchers and teachers, for whom it can give a much

deeper understanding of children‘s thinking than standardized tests.

In this study I am trying to understand children‘s understandings of tessellations

from cognitive and aesthetic viewpoints. What strategies, reasoning, and mathematical

structures do they use and how does aesthetics guide their thinking? Ginsburg (1997)

claims that a constructivist orientation requires the use of clinical interviews or similar

method in order to properly understand children‘s thinking. Indeed, for an exploratory

study such as this one, the clinical interview method is almost the only method that will

work. A more standardized format would limit the possibility of follow-up as new data

reveal children‘s new ways of thinking. The Piagetian constructivist theoretical

framework presented here therefore requires that I use clinical interviews, especially

because this research is exploratory in nature. Other researchers have done teaching

experiments with tessellations, but in this study I am trying to understand children‘s

natural thinking without specifically designing for any type of instruction.

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Theory of Clinical Interviews

A clinical interview is a social interactional event (diSessa, 2007). The goal is to

get children to expose their natural ways of thinking. This is not an easy task and requires

flexibility. In a clinical interview, a task is defined in advance and questions are prepared,

but as thinking is revealed, the researcher invents follow-up questions uniquely tailored

to the situation. These questions allow for deeper exploration of interesting answers.

The researcher is therefore not a passive questioner, but must be constantly

probing and creating hypotheses to explain how the child is thinking. These hypotheses

are tested by follow-up questions created during the dialogue between researcher and

student. The researcher must have background knowledge and an initial theory of what

might be expected in the interview in order to inform what will be looked for (Ginsburg,

1997). Piaget noticed that when new researchers start, ―they either suggest to the child all

they hope to find, or they suggest nothing at all, because they are not on the look-out for

anything, in which case, to be sure, they will never find anything‖ (1929/2007, p. 9). My

literature review informed the initial background of expectations for this study.

Methodology of Clinical Interviews

The interview is prepared in advance with a protocol or rough plan including the

tasks the interviewer will propose to the children. The interviewer must make clear that

the purpose is for the children to explain their thinking and engage in mutual inquiry, not

to display formal school knowledge (Russ, Lee, & Sherin, 2010). During the interview

the researcher can follow up on the children‘s responses with questions and tasks that are

not in the original protocol. Various researchers (e.g. diSessa, 2007; Ginsburg, 1997;

Hunting, 1997) have described techniques for clinical interviews in detail. The goal is to

set interviewees at ease in an informal atmosphere and gain their trust so they can feel

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free to think aloud. Interviewers do not normally teach during the interview. The purpose

of a clinical interview is to discern how the child is reasoning. Robert Hunting found that

35 to 50 minutes is ideal for interviewing 10- to 12-year-old children. The children in this

study are a bit younger (about 9 years old), so I chose to conduct half-hour interviews,

though some interviews lasted a bit longer.

Piaget believed in checking the robustness of children‘s beliefs by challenging

their assertions in order to see the children‘s reaction. This is to check the stability of

children‘s knowledge. Do children change their mind easily when challenged, or do they

maintain their beliefs? Ginsburg suggests that it could also be productive to help, rather

than challenge, the child in order to determine the child‘s fluid knowledge, what is in the

child‘s zone of proximal development. By providing limited help, we can explore not

only what a child already knows, but what the child is now capable of understanding. We

can distinguish concepts that the child is not ready to understand from concepts that are

ready to be grasped.

DiSessa (2007) notes that exposing natural ways of thinking is not easy because

thinking is complex. He tries to look at both robust, stable knowledge and fragile, fluid

knowledge. A schema is considered robust when it is used persistently even while the

child is learning ideas that should modify the schema. By interacting with children‘s

ideas and pushing them to think beyond what they already know, we can discover what

knowledge is stable and what knowledge is easily changed or abandoned. I believe the

exact nature of the interaction between interviewer and child depends on what the

interviewer is trying to investigate, but in any case it is a good idea to distinguish

between robust ideas held by the child and less stable ideas that change under

interrogation. I engaged the students in a variety of activities, as well as discussions of

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the aesthetics of tessellations. One of the tessellation activities was repeated to see if

increased familiarity with tessellations changed their answers. (I show in Chapter 4 that

no change was observed.) In many of the tasks, children were challenged to justify their

answers. This allowed me to check to see if their beliefs were firm, or only weakly-held

hunches.27

Though it could have been revealing to do so, I did not follow Ginsburg‘s

recommendation of helping children because of the length of the interviews. Each child

was interviewed six times. If help had been given during any interview except the last, I

could not have known what children were doing on their own in subsequent interviews

and what they were doing because they had learned from me in an earlier interview. I

believe that looking for consistency across the interviews was sufficient to identify

consistent, robust trends of thinking, especially because of the length of time each child

was interviewed.

After a clinical interview, the data are reviewed and models that explain the

child‘s thinking are created, evaluated, and revised according to the extrinsic criteria I

discuss below. The researcher begins with a broad analysis and gradually narrows the

focus in order to create a model that explains the data.

Clinical interviews have come under some criticism in recent years. In order to

assure the validity of this methodology, it is important to consider the intrinsic and

extrinsic criteria that researchers have established to respond to this criticism.

27 Exploring a child‘s fluid knowledge is in some ways a type of teaching experiment. Some teaching

experiments, in fact, are little more than a series of clinical interviews. This research was not a teaching

experiment because I did not do any explicit teaching. The term ―teaching experiment‖ is used by

researchers with a range of meaning (Cobb, 2000; Cobb & Steffe, 1983; Steffe & Thompson, 2000). The

fact that I looked at how children‘s thinking changed through experience with one of the tasks would lead

some to call what I did a teaching experiment, at least for that task. However, the emphasis of this study

was on discovering children‘s current knowledge and the analysis of possible change was only an isolated

observation.

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Validity and Reliability

Researchers have addressed concerns about the validity and reliability of clinical

interview research by describing the intrinsic and extrinsic criteria that should be used in

such research.

Intrinsic criteria describe how researchers maintain the validity of the clinical

interview process itself. Andrea diSessa (2007) defended clinical interviews against

criticism that the methodology lacked ecological validity, was coercive, and did not

reveal invariant knowledge that could transfer to other situations. DiSessa concedes that

clinical interviews are less developed than some other methodologies. He claims,

however, that the problems clinical interviews are sometimes associated with are often

the fault of the larger program which the clinical interview is a part of.

Some educational researchers have attacked the ecological validity of clinical

interviews, claiming that they are too far removed from the natural environment of actual

classrooms. DiSessa defends the ecological validity of clinical interviews by locating

them in the process of normal human inquiry and sense-making situations. Clinical

interviews are similar to the everyday phenomenon of mutual inquiry where people say,

―I wonder why….‖ Nevertheless, it is true that the clinical interview situation is not quite

the same as ordinary daily experiences, so the researcher must understand the relationship

between the child‘s answers in the context of the clinical interview and the child‘s

general beliefs in the world. DiSessa proposes as the Central Hypothesis that the clinical

interview is ―a form of [familiar] mutual inquiry that is developmentally derivative of

naturally occurring individual and mutual inquiry activities‖ (2007, p. 531). He proposes

as the Primary Constraint on clinical interviews that questions must be sensible in the

sense of being reasonably similar to questions someone might bring up in a normal sense-

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making inquiry. Therefore, questions must not be too difficult and the researcher must

constantly evaluate the participant‘s competence to respond to a given question.

Even if clinical interviews are not as ecologically valid as other forms of research,

they can still address questions that cannot be reasonably answered by other

methodologies. In particular, clinical interviews are ideally suited to studies that seek to

understand a concept that may not occur often in more natural settings. For certain

concepts, it is impractical to follow a student around, waiting for evidence of the

particular kind of thinking under investigation. In a classroom setting, even when the

child displays evidence of the concept of interest, it will likely be in too limited a fashion

to be helpful. The clinical interview provides a means for exploring the concept in depth

by temporarily creating a necessarily artificial situation where follow-up questions and

tasks can be proposed in order to obtain a deeper and more accurate understanding of the

participant‘s thinking.

DiSessa addresses the accusation that interviewers are unduly influencing the

participants and that coercion is producing unnatural responses in the clinical interview.

To address this concern, diSessa proposes four criteria for evaluating whether the intent

of the interview is respected: (1) Is there independent evidence of conviction on the part

of the participant? (2) Is it possible that the viewpoint was prompted? (3) Were the

responses systematic (e.g. always changing one‘s mind after a prompt) or were they

natural? (4) Is there converging evidence? If these criteria show that the participant was

in any way coerced, the interview has failed. During the analysis of my data, the

interviews were reviewed with these criteria in mind in order to be sure that intrinsic

criteria were being respected and that any answers that were possibly coerced were not

considered for the conclusions I reached. For example, when students were asked ―Are

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you sure?‖ and they answered with a quick, non-committal ―Yeah,‖ this was given less

weight than when students expressed long, unsolicited expressions of confidence. Quick

answers may only be an unthinking attempt to please the interviewer. Such answers were

therefore not considered sufficient evidence of student conviction unless supported by

other independent evidence.

Coercion is to be avoided, though diSessa freely admits that he may be

―seductive‖ during a clinical interview in the sense that he deliberately pulls participants

into areas that are most likely to reveal interesting thinking. This is not coercion so long

as there are no leading questions that would bias the participant‘s answer. The fact that

the interviewer retains a certain authority to direct the interview is also not considered

coercion as long as the participant retains the right to judge reasoning. Again, it is

important to make these points clear at the beginning of the interview. Children must feel

free in the clinical interview to express their beliefs without fear of being judged right or

wrong. This is not the usual environment in the classroom, so there may be a more or less

long time of transition at the beginning where this question of authority is negotiated and

the participant accepts that we are engaged in a form of mutual inquiry. This orientation

is maintained during the interview by withholding judgment on responses and

questioning the correctness of an answer only to the extent the participant is vague or is

aware that the answer is problematic. An answer may also be challenged to test

robustness, but the child should understand that this is not a judgment on the correctness

of the answer.

The third criticism to which diSessa responds is that knowledge is too fluid and

the results from the clinical interview do not transfer to the ―real‖ world. DiSessa

responds that researchers must therefore explore the relationship between robust and

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fragile knowledge in their clinical interviews. This can be done, as noted earlier, by

challenging answers in order to see what knowledge is stable. In the case of lengthy

interview time, as in my research, answers can also be checked for consistency across

time.

Extrinsic criteria are those that are used to evaluate the results of a clinical

interview or similar method. How well does the model of thinking proposed by the

researcher explain the data evaluated? John Clement (2000), basing his work on the

criteria offered by Kuhn and Darden for evaluating scientific theories, proposes an

extensive framework of criteria for evaluating the viability of a model derived from the

analysis of clinical interviews. Clement rejects the classical construct of validity,

claiming that it has had too many different interpretations. Instead he proposes the

construct of viability—the strength or believability of a model.

Clement outlines a framework for viability using criteria categories of

plausibility, empirical support, non-empirical support, and external viability. The

plausibility criteria require that the model explain the behavior in a believable way and be

internally consistent, e.g. not mixing conscious and unconscious thoughts. The criteria of

empirical support verify that there are sufficient, strongly supporting data and

insignificant confounding data. There are also non-empirical criteria requiring the model

to be clear and simple, without arbitrary contrived elements (no ―ad hocness‖), as well as

coherent with other known theories. External viability includes generalizability to other

populations (external generalizability) and other contexts (theoretical generalizability).

External viability criteria also include predictive validity, as well as the ability for the

model to be eventually extended to larger contexts and to be fruitful for identifying other

behaviors in line with the model; a model which is only successful in predicting behavior

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in one context would not be very useful. These external viability criteria are especially

important for exploratory studies such as this one where new models are being created,

though they can generally only be tested over time. Clement strongly urges researchers to

be explicit about all these criteria in research based on clinical interviews. The criteria are

summarized in Table 2 below.

Table 2: Summary of Clement‘s viability criteria

Category Criterion Explanation

Plausibility Explanatory adequacy The model is believable.

Internal coherence E.g., not mixing conscious and unconscious thought

Empirical Support Triangulation and number of supporting observations

There are strongly supporting observations.

Strength of connection to each observation

Reliable data directly support the model.

Lack of anomalies There are no significant confounding data.

Nonempirical Criteria Clarity The model is clear.

Simplicity The model is simple.

Lack of “ad hocness” The model is not contrived.

External coherence The model is coherent with other known theories.

External Viability Generalizability The model has external generalizability (to other populations) and internal generalizability (to other contexts).

Predictiveness The model has predictive validity.

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Table 2. cont.

Category Criterion Explanation

Extendibility The model can be extended to larger contexts.

Fruitfulness The model is fruitful for identifying other behaviors in line with the model. It is useful.

Clement also describes a spectrum of four possible approaches for clinical

interview studies, from generative, exploratory studies which investigate an initial model,

to convergent, independent coder studies. The former use larger units of analysis and a

process of abduction, criticism, and revision to interpret observations and propose models

that explain them. Clement proposes a cycle where generative studies lead to well-

defined observations that allow convergent studies which bring new observations to be

explained by more generative studies. There should always be a balance between

theoretical work, which develops models, and empirical work, which provides new data.

He notes that Ptolemy‘s geocentric system was able to predict the motion of the planets

with significant accuracy, but was nevertheless a wrong model. Likewise, a convergent

study with high replicability may still fail to identify a viable model. More in-depth

generative studies are needed alongside the convergent studies. My research is a

generative, exploratory study, developing an initial theory of how children conceptualize

tessellations.

Clement also considers criteria for reliability of the data. He defined viability as

the strength or believability of the model, and he defines reliability as the strength or

believability of the observations. This includes and goes beyond the classical definition

of reliability as consistency. The more convergent approaches have higher reliability than

the divergent, generative ones, but all kinds of studies are needed. Reliability can be

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improved at any level by standardizing procedures, estimating and reporting the relative

strength of observations, and increasing the quality of participants‘ self-reporting, all of

which were used to increase the reliability of this study.

All claims and their supporting data in this study were analyzed for viability and

reliability according to Clement‘s criteria.

Clement also discusses objectivity. Observations are theory laden. This is a

necessary thing, as Piaget pointed out. Researchers have certain expectations of how

children will respond to each situation. However, it is critical for researchers to be open

to new and unusual phenomena. This point was made clear to me during my pilot study

when I realized that I was frequently assuming students understood certain tilings in a

certain way, but further questioning showed that the children had different ideas. For this

reason, I lengthened the time given for each task in the main study, in order to explore

answers in depth and to be more certain that the children really were thinking what they

seemed to be thinking.

METHOD

The method used in this research involved a series of interviews with six children

as they solved various tiling tasks. Diversity was the key to the method design. Because

this was exploratory research, I designed a method where fourth grade students from

diverse backgrounds with diverse ways of reasoning used diverse tools to create tilings

with diverse shapes. In this way, the data were expected to identify a broader spectrum of

children‘s ways of thinking.

Pilot Study

In order to prepare for this study, I conducted a pilot study in which I tried out

several ideas for investigating children‘s conceptualizations of tessellations. The purpose

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of the pilot study was to see how various ideas for tessellation activities would work out

in practice and what kind of insights they might provide into children‘s thinking in order

to plan the dissertation study. I used a variety of tessellation tasks which were prepared in

an open-ended format that allowed tasks to be modified as needed during the interviews.

The 11 participants of the pilot study were in the fourth or fifth grade in two

elementary schools, one a charter school and the other a private school. The charter

school is a demonstration elementary school for a large state university serving students

from low income families. The private school is a small, multilingual elementary school

with accreditation from the French Ministry of Education. Many, but not all, of the

students at the private school come from high income families. The same two schools

were also used for the dissertation study. Three of the children in the pilot study were

French; the other eight were American.

For the pilot study, six of the eleven children were interviewed in pairs, the other

five individually, in whichever language they were most comfortable with. Students were

given a variety of tiling tasks similar to the tasks eventually used for the dissertation

study.

The pilot study showed me various practical ways that my clinical interview

techniques could be improved on for this study. More importantly, it showed that

interesting observations could be made which could throw light on how children

conceptualized tessellations.

From the cognitive viewpoint, the pilot study suggested a number of results,

which were further explored in the dissertation study. There were several strategies that

students used repeatedly in their creations during the pilot study, including making

simple or complex rows, unitizing, making random patterns, making complex patterns,

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and growing patterns. Students also used symmetry and other strategies. Children

justified their answers in a variety of ways, including using empirical evidence, using

previous experience, explicitly using another known tessellation, noting that gaps are

always filled, noting the resemblance with another known tessellation, using symmetry,

and using transformations. These types of strategies and reasoning were used as part of

the initial basis for analyzing cognitive thinking in the dissertation study.

It was also noticed in the pilot study that the children often focused on finite

patterns and did not always show a clear understanding that a tiling could continue

indefinitely. The children sometimes did not see how many polygons would fit around a

vertex. Several observations concerning symmetry and aesthetic influences were also

made. All of these observations formed the basis for the design of the tasks in the

dissertation study and the initial analysis of my dissertation data.

I concluded from my pilot study that an exploratory study of children‘s

conceptualizations of tessellations was feasible and that important conclusions were

possible from both cognitive and aesthetic viewpoints by using clinical interviews of

children as they discussed the possibilities of tiling. The study also resulted in a paper

presented at a PME-NA conference (Eberle, 2010) in which I showed that even though

children‘s mathematical aesthetic is different from that of mathematicians‘, their aesthetic

sense of symmetry was ideally suited for the kinds of explorations that they were engaged

in. Children are not capable of exploring sophisticated non-periodic tessellations to any

significant mathematical depth. And in fact children were dissatisfied with the tilings if

they did not possess symmetry. If symmetry was present, the children were more

confident of their tiling. This confidence was mathematically appropriate—more often

than not, the symmetric tilings were in fact mathematically correct, or at least on the right

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track, whereas the non-symmetric tilings were not. Aesthetics is therefore not only

important for motivational reasons, but it is important because it usually correctly leads to

generative and valid mathematical solutions. This important result from my pilot study

was further confirmed by my dissertation data, as I shall discuss in Chapter 4.

There were three main changes made from the pilot study as I designed the

dissertation study: (1) the dissertation study was considerably longer than the pilot study,

allowing for a greater variety of tasks at greater depth; (2) the dissertation study explicitly

included investigation of the aesthetic elements of the tasks; and (3) there were no

interviews in pairs.

(1) The dissertation study included six half-hour interviews instead of only one

for each student. With the 30-minute interview of the pilot study, I had to cut many of the

tasks short as soon as I thought the children‘s reasoning was sufficiently clear. However,

on later analysis, I realized that I often cut the task short too soon. When I did allow

students to continue, I was often surprised by the result. I clearly needed to allow the

students to continue the tasks further and to probe in greater depth as to how the students

were thinking. With some students in the pilot study, I was not even able to finish all the

tasks. Allowing more time enabled me to probe almost every task at sufficient depth,

letting students have enough time to express their thinking. I was also able to include a

greater variety of tasks in order to explore thinking in more ways.

(2) During the pilot study interviews, I was not yet considering the aesthetic

viewpoint. After analyzing the interviews it became clear that the aesthetics of the tasks

played a very important role in how the students approached the problems. This aesthetic

viewpoint was incorporated explicitly in the design of the dissertation study.

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(3) Interviewing in pairs seemed like a good idea for the pilot study. It

occasionally allowed students to share thinking in a way that made it easier for them to

verbalize thinking. However, as often as not, one student dominated the conversation

making it virtually impossible to know what the other student was thinking. At times,

students would collaborate on a tiling without a clear, common objective, creating a tiling

that was difficult to analyze because the students had different strategies and did not have

a common pattern in mind. I found it was much easier to discern a student‘s mental

model for a tessellation when they worked alone. For this reason, I followed the classic

procedure of interviewing students alone for this study, except for the final interview.

The final interview was done in a group format because it focused entirely on aesthetic

evaluation. There was no danger of confusing two children‘s thinking because no tilings

were created in the sixth interview. Using a group format in the sixth interview allowed

students to verbalize their aesthetic reasoning more easily.

Population

My pilot study, as well as the research discussed in the Literature Review, show

that fourth grade students are capable of beginning to analyze tessellations of varying

complexity, sometimes in rather sophisticated ways. This age group therefore forms a

baseline for early conceptualizations of tessellations at a meaningful level. Most children

in second grade still have not grasped the row and column structure of square arrays and

will therefore certainly not be ready to discern more complex structure in non-square

tessellations. It was therefore decided to analyze children‘s thinking in fourth grade in

order to understand thinking at an age where elementary but sufficiently sophisticated

tessellation ideas could be analyzed.

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For this dissertation study, I aimed to interview fewer students than the pilot

study, but to allow much more time for exploring their thought in depth. I nevertheless

wanted a sufficient number of students in order to gain a reasonable view of the diversity

of fourth grade thinking. It was important that the population be large enough that the

results could be considered generalizable. Individual case studies can provide powerful

insights, but may only be revealing thought that is peculiar to one particular student. I

therefore decided to interview six students in order to have a reasonable overview of how

students likely tend to think. More than six was not needed because this study was only

intended to be exploratory in nature. An exploratory study is intended to identify likely

models, which can then be further explored in a more convergent study, as explained on

page 113.

In order to survey as wide a spectrum of children‘s thinking as possible, the

participants were chosen in such a way as to create as much diversity as possible within

the small population of six children. Children were chosen from the same two schools as

the pilot study in order to sample students from populations of both low and high socio-

economical status using two different curricula, one American and one French. The

French curriculum tends to place a somewhat greater emphasis on geometry and

precision, while the American curriculum frequently emphasizes creativity. There are

therefore somewhat different emphases in the curricula from both cognitive and aesthetic

viewpoints. The goal in continuing to study two schools was not to compare them, but

simply to achieve an additional measure of diversity in the students and their

backgrounds. Most importantly, diversity in the selection of students was achieved

through a pre-test, which I describe below. No student at either school had yet studied

tessellations in their classes at the time of the interviews.

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In the French system, the nearest equivalent to fourth grade is the cours moyen 1

(CM1), though students in CM1 may be a few months younger than American fourth

grade students because of different cutoff birthdates for school admission. Most students

were nine years old at the beginning of the interviews in October, though one in the

French school was eight and one at the charter school was ten. The eight-year-old turned

nine before the final interview in January. The six students selected for the interviews are

described in Table 3 below. The names are not their real names.

Table 3: Student participants in this study

Student Grade Sex Age (1st interview)

Dominant language

Kelsey 4 Female 9 English

Rachel 4 Female 9 English

Moses 4 Male 10 English

Kobe CM1 Male 8 English

Marie CM1 Female 9 French

Michelle CM1 Female 9 English

The tasks for this study were longer and more detailed than the tasks in the pilot

study and some were new. Another girl aged eight years nine months from a public

school was selected to test the tasks before giving them to the six study participants.

Refinements were made to the protocol based on these trial interviews. Results from her

interviews are not reported in this study.

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Pre-test

In order to further diversify the population of students used for this study, students

were not selected at random. Instead, a pre-test was designed, which was administered to

the two classes. The pre-test had three questions assessing different aspects of tiling

possibilities. (See Appendix A.) The first question was a finite tiling question based on

one of the questions from the study done by Battista, Clements, Arnoff, Battista, and

Borrow (1998). A 5-by-7 square array was partially erased and students were asked how

many squares were in the original array. This was intended to evaluate how the students

understood the row and column structure of a square array. Such conceptions were well

studied in the literature and likely to be easier than understanding other tessellation

structures. The second question was to tile a 60° parallelogram, chosen partly to test

students‘ thinking with angles other than 90° and 45°. The third problem was to tile a

right-angled trapezoid with a 45° angle, chosen because it was not symmetric and it

probed understanding of 45° angles. The second problem was in the context of a blank

sheet of paper, whereas the third problem was done on dot paper, exploring tessellations

in the context of two different tools.

The test was created in two versions—in French for the private school (because

math was taught mainly in French in that school) and in English for the charter school.

The French pre-test was discussed with the CM1 teacher and modified according to his

suggestions. The English pre-test was discussed with the mathematics coordinator of the

charter school. The classroom teachers administered the 15-minute pre-test without the

investigator‘s presence.

Pre-tests from each school were analyzed in an attempt to sample a diversity of

thinking while still identifying common ways of thinking. Pre-tests from each school

were analyzed separately; the pre-tests from the two schools were not combined together.

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Each answer was coded for evidence of different types of thinking in comparison with

other answers. Coding was emergent as each test was analyzed. For example, the first

problem was coded according to whether the array was drawn; whether multiplication

was used; evidence of the lack of structural understanding (imperfect rows and columns,

rows but not columns, or neither rows nor columns); and whether students used two

methods with conflicting answers. After all the answers for a school were coded, the pre-

tests were sorted according to codes. I looked for ways to find three students who

represented frequently found solutions and yet who had maximally different ways of

thinking. For each school, I found several different ways of achieving such results,

resulting in the selection of eight children at each school. I then reviewed the sorted

analysis with the teacher of the private school or the coordinator of the charter school and

discussed which students would most likely do well in a clinical interview format

(speaking clearly, offering their own thinking, interacting well with a stranger, speaking

English or French as their first language, etc.). Together, we chose at least three students

at each school. We waited to see which students would return permission slips in order to

make the final selection.

Interviews

In this section, I describe the design of the interview tasks. I begin by noting the

extent of the constraints imposed by the investigator.

Criteria

In order to study students‘ aesthetic thinking, it is important to impose as few

constraints as possible. However, some sort of question or task must be posed using some

sort of tool, and therefore some constraints are inevitable. The goal of the clinical

interviews was to explore students‘ understandings of tessellations, and therefore the

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minimal constraints (no overlaps, no gaps, endless) were imposed to focus the interviews

on this specific mathematical task. Students were also given a specific tool for reasons

that I discuss in the Structure of the Interviews section below.

It was known from the pilot study that only the infinity constraint (tiling an

endless room without walls) was likely to be problematic. The infinity constraint could

have been omitted; the students could have been simply asked to make a ―tiling.‖

However, it was clear from both the pilot study and the pre-test results that many,

perhaps most, students would interpret this to mean ―Create a finite tiling that fits the

paper.‖ Because this is not the kind of mathematical thinking that the study was intended

to explore, it was important to specify this criterion up front. If children had begun with

finite tilings and then been told to switch to infinite tilings, the result might have been

confusing.28 Because this criterion might (and did) take some adjustment time, and

because it was fundamental to the study, it was decided to impose this criterion from the

beginning. The one task that involved tiling a finite figure was not given until the third

interview.

Students were briefly reminded of the criteria at the beginning of most interviews.

Once it was clear that students understood the three criteria, they were given freedom to

create the tilings any way they wanted, including ignoring the criteria. If students seemed

to be ignoring the criteria because they had forgotten, or if they explicitly asked about the

criteria, they were reminded, but then still allowed to create the tilings the way they

wanted. Reminders of the infinity criterion sometimes engendered interesting and

revealing conversations about this constraint.

28 One example of how the initial task can influence all the following tasks is with the special sixth

interview of this study. The sixth interview contained a variety of tasks, the first of which was to tile the

rooms of a house. None of the other tasks involved rooms of a house, and yet some students continued to

interpret all future tasks in the context of which tiling would look good in which room of a house.

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Overview of the Interviews

The interviews were organized as six half-hour sessions spaced at least a week

apart. Interviews were conducted in English or French according to whichever language

the child was most comfortable with. Marie was most comfortable in French, the others

in English. The first five interviews were organized as seven sets of tiling creation tasks,

labeled A through G, plus optional sets H and I, which were not used.

The framework for all the tasks was the problem of tiling a floor. In all but one

task, the floor was said to be infinite and without walls (or that it was so big we did not

need to worry about what would happen at the walls). The finite tiling task was not given

until the third interview. The three criteria were discussed at the beginning of the first

interview in order to check understanding.

I began with a dialog of how to tile a floor with squares in order to discuss the

task and its criteria. The initial interview also contained a discussion of the ground rules

for a clinical interview: why we are here; that there are no right or wrong answers; that

students should think aloud; and so on.

In addition to these largely cognitive tasks, there was a sixth interview which

consistently entirely of aesthetic evaluation tasks, labeled Set J. This final interview was

conducted by school with all three participants together so that students could interact

and share their criteria for evaluating the tessellations with each other. Aesthetic

evaluation tasks included tasks that evaluated tilings the students had made in previous

interviews as well as tilings made by other people.

I describe all tasks briefly below. Detailed protocols of the interviews can be

found in Appendix B.

It was decided to do the aesthetic evaluation after the creation tasks so that

exposure to professional tilings would not influence the children‘s tilings. If the aesthetic

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tasks had occurred earlier, children may have been under the impression that they were to

give artistic responses to the creation tasks, or they may have tried to imitate the

professional tilings. The goal was for the students to work in an environment for the

creation tasks where any aesthetic influences revealed during the analysis would be from

their own initiative.

The aesthetic evaluation tasks allowed me to identify the kinds of tilings the

children found more pleasing, and the reasons for these choices. By having the children

compare tilings I sought to identify the elements that students find beautiful or interesting

(presence of symmetry, type of symmetry, complexity, shapes, mathematical validity,

and so on, as well as non-mathematical criteria such as color and style). By having the

students work in groups and come to joint decisions concerning the choice of tilings, the

protocol focused on getting students to articulate what they found appealing (and

unappealing) about the tessellations and what aesthetic criteria they found most

important.

In addition to this formal aesthetic evaluation interview, Interviews 2–5 also

included evaluation questions about the tilings the children had just created. Did they like

the tiling they created? Which was their favorite tiling for this interview? Why? These

brief questions were generally towards the end of the interview. I did not want the

students to misunderstand the goal of the interviews to be an artistic one. Instead, I

wanted to probe their natural aesthetic reactions to the proposed tiling tasks. Almost no

aesthetic questions were asked during the first interview for the same reason. Analysis of

the interviews revealed no significant change in aesthetic criteria verbalized by the

students between the first five interviews and the final sixth interview. (I discuss this

further in Chapter 4.)

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Structure of the Interviews

The eight task sets and two optional sets are summarized in Table 4 below.

Table 4: The task sets

Set Task Tools

A Basic Tilings (Rhombi, regular hexagons and triangles, parallelograms)

Blank paper, pattern blocks, software

B Finite Tilings (Tiling completion tasks based on Owens & Outhred, 1998)

Prepared sheets

C Pentagons (Regular pentagons)

Polydrons

D Squares and Triangles (Dihedral tilings with squares and equilateral triangles)

Blank paper, pattern blocks, software

E Triangles (Acute isosceles, right scalene, and obtuse scalene triangles)

Blank paper, software

F Polyominoes (Trominoes and tetrominoes)

Dot paper, software

G Regular Polygons (Non-monohedral tilings of regular polygons)

Software

H Polyiamonds (Optional set as time allowed; not used)

Software

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Table 4. cont.

Set Task Tools

I Follow-up (Optional time for following up on any past interview difficulties—not needed)

As needed

J Aesthetic Evaluation Printed tilings, postcards, books

I began with the most basic tilings in Sets A and C in order to probe their basic

understanding of tessellations. Set B explored basic concepts of polygon fit. I then moved

into dihedral tessellations of squares and triangles, which I introduced early in order to

see if their answers to this task changed by the end of the series. (There was no

significant change, as I discuss in Chapter 4.) This was followed by increasingly

sophisticated tasks. Triangle tessellations are very basic, but sometimes difficult for

children to conceptualize (Owens & Outhred, 1998). Polyominoes introduce non-convex

tilings, some of which were seen to be challenging in the pilot study. I expected regular

polygon tilings to be the most complex because students had a large generating set to

draw from. The goal in ordering the tasks in this way was not to make a teaching

experiment, but to ensure that students were not overwhelmed by the more complex tasks

until they had a chance to get used to the tools used for the interviews.

The goal in using a variety of tasks with a variety of tools was to explore students‘

conceptions as thoroughly as possible. As discussed in the theoretical framework, the tool

has a significant impact on thinking and the subsequent task results. It was important to

propose tiling tasks with a variety of tools so that general results would not be due to the

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particular tool that was used. For most of the tiling creation tasks, the format was in three

phases: oral, drawing, and construction.

The oral phase involved asking the student if a shape could tile, and if so how.

Students were allowed to hold or look at one tile, but otherwise had no drawing or

construction tools. Owens and Outhred (1998) had determined that drawing forms

a closer link with the student‘s mental model than physical tiles. However, my

pilot study found that a more fruitful method for investigating initial student

thinking was to have them describe their ideas orally without access to a

construction tool. This was their opportunity to predict how the tiling would work

with the physical tiles. Drawing provided a context in which students could (and

did) make discoveries that changed their initial ideas. Sharing their ideas orally

was less likely to change their ideas, though this method (like any method) also

affected their thinking. Holding a single tile caused them to focus on describing

how neighboring tiles fit around the central tile. It was very rare during this phase

for students to describe the global pattern.

The drawing phase involved giving the students paper and marker (or pencil)

and allowing them to sketch how they thought the shape would tile (or very

rarely, why it would not tile). The oral phase had the constraint of not allowing

the students access to a tool to help their cognition, causing them to focus

thinking on a few neighboring tiles. The construction phase had the constraint of

tiles that would prestructure the tiling in various ways. The drawing phase

allowed students to express their ideas of how they thought the tiling would work

without either constraint.

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The construction phase involved allowing them to explore further than the first

two phases. The students tended to consider this tiling to be the ―real‖ tiling

where they could test their ideas and possibly explore new ideas. The ―real‖ tile

could be a pattern block, Polydron tile, or computer tile. Some tasks allowed

students to create physical tilings with more than one tool. Once students reached

this phase, if they asked to switch to a different tool, they were allowed to do so. I

usually ended the construction phase by asking them to describe how the tiling

would continue beyond what they had constructed.

A few of the tasks had unique formats. The pentagon task included a comparison

to square and hexagon tilings in order to encourage students to explore their reasoning

concerning the impossibility of tiling with a regular pentagon. The polyomino tasks used

dot paper for the drawing phase, which had aspects of a construction phase because of the

constraining nature of the dot grid. For polyominoes, students explored the two trominoes

first, followed by the five tetrominoes in any order they wanted. The regular polygon task

had only a computer phase because the task was too open-ended and complex for paper

and pencil. For the regular polygon task, students were able to choose freely from a

generating set of five regular polygons: triangle, square, pentagon, hexagon, and octagon.

Set B (Finite Tilings) was the only set that did not explore infinite tessellations.

Instead it examined questions of fit in a finite space. The problems were adapted from

Owens and Outhred‘s (1998) study of children‘s ability to cover shapes with tiles. The

six problems are reproduced in Figure 30 below. Children used the colored prototile at

top to tile the shape at bottom. The printed prototiles in the first four problems were the

same size, color, and shape as pattern blocks. These problems went through all three

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phases if needed. The last two had no physical tiles so the problem was limited to oral

and drawing phases. All problems were possible except for the fifth.

Figure 30: The six finite tiling problems

The aesthetic evaluation tasks of Set J were entirely different from the tiling

creation tasks. Rather than creating tilings, students were asked to evaluate previously

created tilings. These tasks took place during the sixth interview with all three students

from the same school together as a group. Tasks were presented in a variety of formats

that encouraged students to discuss with each other their reasons for choosing the tilings

they chose. There were six tasks:

1. The Student Tilings task involved choosing four tilings for four rooms of a house.

These tilings were chosen from a sample of printed tilings students had created in

the computer environment during the first five interviews. Each student chose a

tiling for each of the four rooms, resulting in three tiling proposals for each room.

Then as a group they selected one tiling for each room.

2. In the Other Tilings tasks I presented a sample of tilings ―made by adults,‖ which

were used in three tasks. These tilings had been chosen to represent a wide variety

of possible aesthetic and mathematical criteria.

a. The Student Sorting task asked students to agree as a group how to sort the

tilings and then to decide which set and tilings were their favorites.

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Students were encouraged to discuss their reasons together in order to

reach consensus.

b. For the Comparisons in Pairs task, the interviewer showed students a pair

(or larger set) of tilings and asked the group which was their favorite.

Students had to discuss and come to a group decision. The pairs had been

chosen according to some of the aesthetic criteria revealed in the

theoretical framework or pilot study. The goal in this and the other tasks

was not necessarily to test specific criteria, but to provide a space where as

wide a range of aesthetic values as possible could be expressed.

c. The Professional Preferences task asked which tiling an artist or a

mathematician would choose from among all the adult tilings.

3. The Professional Tilings (Alhambra) task involved showing students postcards of

Alhambra tilings, one at a time. Students worked as a group to put the tilings in

order from best to least favorite and then to justify their choices.

4. In the Professional Tilings (Escher) task I showed students a variety of tilings

from three books on artist M. C. Escher. Students worked as a group to choose the

best tilings and then award and justify first, second, and third prizes to their

favorites.

Table 5 below contains a summary of the structure of the interviews.29

29 Set B had been originally planned to be done after Set A. However, the responses for Set A were so rich

that I decided on the spot to lengthen the time for this set and moved Set B to the third interview. This also

helped assure that students were clear on the infinity criterion before introducing the one task where this

constraint was temporarily set aside. There was also an optional Set H of polyiamond tasks created in the

eventuality that a student would finish one of the tasks early. Sets H and I were not needed.

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Table 5: Interview structure

Interview Format Tasks (30 minutes per interview)

1 Individual Set A: Basic Tilings

2 Individual Set C: Pentagons

Set D: Squares and Triangles

3 Individual Set B: Finite Tilings

Set E: Triangles

4 Individual Set F: Polyominoes

5 Individual Set G: Regular Polygons

Set I: Follow-up

6 Group Set J: Aesthetic Evaluation

After the interviews were completed, I also informally interviewed two

mathematicians in order to obtain experts‘ viewpoints of the aesthetics of tessellations.

Those interviews used the same tessellations as the children‘s sixth interview.

Mathematicians were asked to give their opinion of which tilings they liked best, or

found beautiful, and why.

Computer software

Most of the tools used for the interviews are well known and have been used

extensively by myself and by countless teachers and researchers: pencil, marker, paper,

dot paper, pattern blocks, and Polydrons. The one exception is the computer software

used for certain tasks.

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I considered various possibilities for the software to use for this research. It

quickly became apparent that software was needed that was tailored to the unique

requirements for this research. Therefore I developed my own software environment

(Figure 31 below). Details of the software and the motivating factors behind the design

choices for the ―Polygon Tiler‖ software are in Appendix C.

Figure 31: Screenshot of ―Polygon Tiler‖ software

The purpose of the software was (a) to provide a different tool that may elicit

thinking different from the other tools; (b) to provide a second tool (in addition to the

pattern blocks) where inaccurate drawing does not play a role; and (c) to provide a

greater variety of possible tiles than the pattern block set could provide.

I tested the software informally with several children and also tested the final

version formally with the 8-year-old child with whom I tested all of my interview tasks.

Analysis

The data for this research consisted of the pre-test; video recordings of all

interviews; student drawings made during the interviews, including the finite tiling

worksheets from Set B; screenshots from each step of students‘ software creations; and

notes made during the two mathematicians‘ interviews.

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The interviews were transcribed using F4 transcription software (Lauterbach &

Kunath, 2008). The transcriptions were analyzed in two ways: (1) data were analyzed

using NVivo qualitative analysis software ("NVivo 8," 2009) and an inductive coding

method (Miles & Huberman, 1994); and (2) transcripts were printed and analyzed by

hand, looking for trends, general methods of tiling, and general results. Comparisons

were made across tasks for the same student and across students on the same task. I

constantly referred back to the actual video recordings while reading and interpreting the

transcripts. F4 and NVivo associated the transcripts and the videos via their timestamps,

so it was easy to flip back and forth between the two while coding.

(1) Coded analysis. Initial coding categories suggested by the literature review

and by the pilot study were analyzed using NVivo with the transcript and video data.

Other categories and codes emerged during the analysis. Categories of codes used during

the analysis included:

Aesthetic criteria used by students (and mathematicians);

Level of confidence in their answers;

Polygon used in the task;

Reasons why a tiling is impossible, or must be possible;

Type of reasoning used to justify confidence that the tiling will work;

Sorting criteria used in the sorting task of the sixth interview;

Strategy used to create a tiling;

Symmetry used to create a finite pattern;

Whether students understood how the tiles fit together;

Type of understanding of the tiling (To what extent their understanding was

apparently accurate from a mathematical point of view); and

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Whether students used active or descriptive wording during the first interview.

Other codes included awareness that a freehand drawing is representative and

may not be accurate, changing the center of symmetry, concern for accuracy, making an

explicit reference to symmetry (not necessarily using that word), and tracing a pattern

block to make the drawing. A complete list of codes is included in Appendix D.

Because this is an exploratory study, coding was not carried out at the level of

detailed behaviors (see Clement, 2000, pp. 558–559), but rather each interview was

broken into short episodes and coded according to broad identifications of the types of

thinking each student used. Each episode generally discussed a single idea. Episodes

ranged from about 3 seconds to more than 2 minutes in length, depending on the amount

of dialog in the exchange. Most episodes were less than 15 seconds long. Codes were

used to help find patterns in the responses, which were then analyzed through the process

recommended by Clement: interpretation, criticism, and revision by reading the passages

and reviewing the video recordings to see what interpretation made the most sense in

context.

Notes from the mathematicians‘ interviews were analyzed only according to

aesthetic criteria.

Because the interview process was carried out over an extended period of time,

analysis was also made to see if there was any change between the early interviews and

the later interviews by comparing both aesthetic and cognitive codes from early

interviews with codes from later interviews.

I used NVivo‘s query functions (especially matrix queries) to analyze patterns

relating different codes. In this way, relationships between the codes became apparent.

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(2) General analysis. In addition to coding, I also reviewed the printed transcripts

by hand. I analyzed the data and made notes summarizing what each student was doing

during each task. Unusual statements were noted. Attempts were made to understand the

thinking of each individual student for each task. Responses to each task were also

analyzed across all six students, looking for patterns in the types of answers, strategies,

and reasoning that tended to be used for each task.

I drew from suggested techniques by Clement for analyzing exploratory research

data in order to identify patterns and draw conclusions. Possible models and claims were

critiqued according to Clement‘s criteria of viability. Evidence in support of the claims

was analyzed for reliability.

Aesthetic Analysis

Aesthetics was analyzed according to the framework suggested by Nathalie

Sinclair (2006). Mathematical aesthetics has motivational, generative, and evaluative

roles. For the evaluative role, I coded students‘ aesthetic expressions whenever they

evaluated a completed tessellation, whether their own or someone else‘s. For the

generative role, I analyzed the aesthetic criteria students used in creating their

tessellations. This created two sets of aesthetic criteria: evaluation criteria and creation

criteria. The motivational role was often in the background, keeping the students engaged

on the tasks. (In fact, they were all quite enthusiastic throughout all six interviews.)

Students did not have many choices as to which task they would pursue except in the

fourth and fifth interviews where students were able to choose which prototiles to use.

Students were also able to choose which tool they would use in the later interviews.

Instances where aesthetic motivation played a role in students‘ choices were noted.

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Aesthetic criteria that emerged during the analysis proved to be particularly

diverse, especially for the evaluation criteria. All instances where students gave a reason

why they liked, preferred, or disliked a tiling were coded as an aesthetic criterion. There

were 327 instances in all of verbal aesthetic reasons from students and mathematicians. I

initially created codes for the aesthetic criteria which had been identified in the

theoretical framework or the pilot study. Whenever a participant expressed a positive or

negative aesthetic criterion that was different from the criteria that had been expressed

before, a new code was created. This resulted in the creation of 59 aesthetic criteria, too

many to analyze in a meaningful way, especially because many criteria were only used

once or twice. When all the aesthetic instances had been coded, the criteria were then

grouped into a more manageable number of themes by putting together negative and

positive criteria (such as good color and bad color) and other closely related criteria (such

as multi-colored and insufficient color). This resulted in 18 aesthetic themes which

included all criteria which had been frequently cited. There were 14 other aesthetic

criteria which could not be grouped, but these criteria were not frequently cited; none

were cited more than twice. Most of the aesthetic analysis was carried out on the 18

aesthetic themes.

Symmetry Analysis

Two types of symmetry analysis were considered—finite patterns and infinite

tessellations. Analysis of the finite patterns students made as they developed their

tessellations was straightforward. Patterns were coded according to the amount of n-fold

symmetry or order-n rotation symmetry they possessed. Translation and glide reflection

symmetry is not possible on finite patterns. See Periodicity and Symmetry on page 72 for

an explanation of types of symmetry analyzed in this study.

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Symmetry of infinite tessellations was only appropriate for patterns that clearly

represented infinite tessellations. This was not always evident for children‘s tilings, nor

was it clear how some students envisioned their infinite tessellations (as I discuss in

Chapter 4), so no attempt to analyze the theoretical infinite symmetry of their tilings was

made. We could ask if children preferred infinite tessellations with greater symmetry

during the aesthetic evaluation tasks of the sixth interview when such tessellations were

evaluated. For the most part, this would only apply to the Other Tilings tasks in the sixth

interview. Analysis of the symmetry of Alhambra and Escher tilings was considered too

complex, as evidenced by mathematicians‘ debates over the symmetries in the Alhambra

(Grünbaum, 2006; Jaworski, 2006; Rønning, 1995). Attempts were made to see if

students preferred certain types of symmetry during the Other Tilings tasks. Results were

inconclusive. See Appendix G for details of methodology and results.

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Chapter 4: Results

In this chapter I report on the results from the analysis of my data. I first focus on

the cognitive results, which come from the first five interviews. I describe the general

results from each of the tasks, and then report on certain patterns found in the data across

tasks. These results will help answer the first part of my research question about

children‘s cognitive understandings of tessellations and their strategies and reasoning

about tessellations.

Next I report in a similar fashion on the aesthetic results, most of which come

from the sixth interview, though there are also aesthetic data in the other interviews. This

will address the second part of my research question about how children evaluate

tessellations, how aesthetics guides their thinking, and the mathematical value of their

aesthetics.

Next I look at results which must be looked at from both viewpoints

simultaneously: results concerning the influence of tools, results concerning symmetry,

and the cognitive role of aesthetic criteria. This will address the third part of the research

question about how the aesthetic and cognitive viewpoints inform each other.

Finally, I look briefly at any differences revealed between the two schools. I also

look at differences between the early interviews and later interviews in order to evaluate

any change in thinking as well as differences between creation and evaluation task

results.

COGNITIVE UNDERSTANDING

As described in Chapter 3, there were seven sets of tasks given in the first five

interviews. Each set used a distinct set of tools and shapes, which influenced the kind of

thinking that was revealed. I first briefly summarize the results of each set of tasks. Fuller

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details of the results of each task are given in Appendix E.30 After the summary I discuss

results that cut across all the tasks.

Set A: Basic Tilings. Students were asked about the possibility of tiling with four

basic shapes. The rhombus, hexagon, and triangle tasks used pattern blocks. The

parallelogram task appeared as a paper-and-pencil task on the pre-test and was done in

the software environment during the interviews. Students described tilings through the

three phases described in Chapter 3 (page 128): an oral phase, a drawing phase, and a

construction phase. As an example of the construction phase, Figure 32 below shows

some of Rachel‘s tilings for this set.

Figure 32: Some of Rachel‘s tilings during the first interview

30 In this summary, I sometimes use terms like ―most,‖ ―some,‖ or ―a few students‖ when precise numbers

are not easily given. For example, if four of the six students believe something, and a fifth student hesitates

or gives more than one answer, I simply write that ―most students‖ believed this instead of ―four or five

students.‖ Fuller details are in Appendix E, though even there I sometimes avoid precise numbers if exact

counts are ambiguous. My goal is to indicate general trends, not precise quantitative analysis, which is not

warranted in this qualitative study. Nevertheless, all terms are used with care. ―Most‖ always indicates

more than half. Most of the counts come from the general analysis. (See page 137.) Descriptive statistics

that come from the coding are noted as they occur.

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All six students believed from the beginning that it would be possible to tile with

each of the four shapes, though Marie and Kobe were doubtful about the hexagon until

they tried sketching it. Descriptions tended to be local at first, describing how tiles would

fit around the initial tile. It was rare for students to describe the global structure, though

there were three instances where this happened. Moses and Michelle described how a

rhombus pattern would grow in all directions, and in the pre-test, Michelle had taken a

ruler and drawn a parallelogram grid.

Nearly all tilings in Set A were created edge-to-edge. (Figure 32 above shows one

of the two exceptions.) Aside from orientation, the hexagon and the equilateral triangle

can only tile one way in an edge-to-edge tessellation, but there are infinitely many ways

to create edge-to-edge tessellations with the rhombus and with the parallelogram, as well

as other ways that are not edge-to-edge. Students made a variety of tilings with the

rhombus and with the parallelogram, including tilings with tiles all in the same

orientation, alternate row tilings, non-edge-to-edge tilings, and random tilings. Some

students made several different rhombus and parallelogram tilings, sometimes

discovering new ways to tile through exploration. Two students thought that the standard

tiling was the only possible tiling. Some students considered different orientations to be

the same and others considered them different. Two students compared the process to

doing a puzzle. Students used a variety of strategies for every shape, including dilation

strategies, row strategies, and unitizing strategies. I discuss these strategies for creating

tilings in more depth on page 173.

Students tended to look for finite patterns in their rhombus and triangle

tessellations, identifying hexagons, stars, trapezoids, and rhombi patterns. Even though

there is only one way to make an edge-to-edge tiling with triangle pattern blocks,

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students used a multitude of different strategies to create these tilings. Each student

created several (identical) triangle tessellations, using a different method for constructing

each one. As I discuss on pages 159–168, at least two students did not appreciate that

these triangle tessellations were not actually different.

None of the students saw during their oral descriptions that six triangles could fit

around a vertex. Most students thought that four triangles would be enough, as it was for

the rhombus. During the pilot study, one student even drew how the four triangles would

fit in Figure 33 below. During the drawing phase of this study, Moses was surprised to

discover that it actually took six; he clearly experienced a moment of cognitive

dissonance, resulting in accommodation of the new fact: ―Oh my goodness. So it‘s kinda

like a hexagon.‖ Other students in this study also drew six, though half the students

occasionally drew five.

Figure 33: A student‘s idea of how four equilateral triangles fit around a common vertex

Almost no aesthetic questions were asked during the first set, but students

occasionally made aesthetic comments. Many students saw the standard parallelogram

tiling as three-dimensional, as if looking at a rectangular floor pattern from an angle.

Kelsey thought during the drawing phase that the rows of the parallelogram tiling would

be ―shorter and longer,‖ which would look cool (see Figure 34 below). It is not clear if

she was envisioning some kind of brick wall tiling, or if she thought parallelograms

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would naturally jut out beyond each other. She made a standard edge-to-edge tiling

during the construction phase.

Figure 34: Kelsey‘s parallelogram tiling drawing

At the end of the interview I asked if there were any shapes that were impossible

to tile with. Most students demonstrated that it was impossible to tile with a circle.

Rachel thought all non-polygons would fail to tile and recited polygon criteria she had

learned in third grade. Kobe repeatedly insisted that it was possible to tile with any shape.

Set B was done during the third interview, just before Set E.

Set C: Pentagons. I showed the students tilings made with square Polydrons and

with hexagon Polydrons in order to review these tilings from the first interview and to

introduce the new Polydron material. Then I asked them if it was possible to tile with

pentagon Polydrons. All students believed it was possible, though a few hesitated. Most

seemed to think it would look like the hexagon tiling and drew sketches to that effect,

with three or four pentagons around each vertex. Some even put five pentagons around a

vertex (see Figure 35 below).

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Figure 35: The beginning of Michelle‘s regular pentagon tiling

Students were not sure how to explain the unexpected result that it was, in fact,

impossible to tile with the regular pentagon. Three students simply pointed out the

inevitable gaps when asked why the pentagon cannot tile. (See Figure 36 below.) Kobe

was particularly bothered by the failure, having stated both in the previous interview and

at the beginning of this interview that all shapes can tile. He spent over five minutes

trying to find a solution, apparently trying to resolve his cognitive dissonance. Some

students correctly discerned that other types of pentagons could successfully tile. Kelsey

seemed confused by the tool itself and focused on the tiny tabs that make the Polydrons

snap together. She does not seem to have conceptualized the tiles as pentagons during

any of the phases.

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Figure 36: Moses points out the inevitable gap left by regular pentagons.

One consistent result was that the students perceived the hexagon as having much

more symmetry than the pentagon, possibly explaining the pentagon‘s inability to tile.

The hexagon clearly had six-fold symmetry, but the pentagon was usually perceived to

have only simple reflection symmetry. Some saw the regular pentagon as a square plus a

triangle, like a house. The pentagon was therefore not seen as isogonal and isotoxal. It

had a top and a bottom and the sides needed to play different roles in the structure.

Consequently, students did not see that it would have been sufficient to study how

pentagons fit around a single vertex. The following is Marie‘s description of the

difference between a hexagon and pentagon (with translation). The researcher is Scott.

Original Translation

Scott: Alors pourquoi les hexagones, ça

marche et les pentagones ça

marche pas ?

So why do the hexagons work

and the pentagons don‘t?

Marie: Je crois parce que l‘hexagone

c‘est tout pareil comme ça

[tracing with finger on table].

I think because the hexagon is all

the same like that [tracing with

finger on table].

Scott: Mm-hmm. Mm-hmm.

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Marie: Et celui-là [pentagon] c‘est un

peu différent.

And that one [pentagon] is a bit

different.

Scott: Comment c‘est différent ? How is it different?

Marie: Parce que c‘est comme une

forme de maison.

Because it‘s like a house shape.

Scott: C‘est comme une forme de

maison.

It‘s like a house shape.

Marie: Et pa-, et alors en fait c‘est tout

droit. C‘est comme un carré au

bout d‘un moment ça branche

[referring to “roof” of

pentagon].

And no-, and well in fact it‘s

straight. It‘s like a square and

after a while it branches

[referring to “roof” of

pentagon].

Scott: Uh-huh. Uh-huh.

Marie: Et [unintelligible] en tout cas, un

hexagone c‘est tout le temps,

comme, pareil, …

And [unintelligible] in any case,

a hexagon it‘s always, like, the

same …

Scott: Mm-hmm. Mm-hmm.

Marie: … c‘est, … … it‘s …

Scott: C‘est pareil partout ? It‘s the same everywhere?

Marie: … c‘est toutes les lignes c‘est pas

ça fait pas ça ou quelque chose

comme ça.

… it‘s all the lines it‘s not, it

doesn‘t do that or something like

that.

Set D: Squares and Triangles. Participants were asked if it was possible to make

a (dihedral) tiling with square and triangle pattern blocks. After using pattern blocks, they

were invited to continue exploring on the computer. This activity was repeated at the

beginning of Set G. Most students could not recall during Set G how they had created

their tiling in Set D, but they tended to give very similar answers both times, using

similar reasoning.

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Each student‘s approach was unique. Most thought it was possible from the

beginning, and they all sketched and created a variety of tilings. Several made the semi-

regular 33.4

2 tessellation by using a row strategy. Other common strategies included

putting squares around a central triangle, which often resulted in a radial pattern,

sometimes with 3-fold symmetry, or putting triangles around a core of squares, which

usually proved problematic because they wanted to fit only triangles around the corner of

each square, which cannot work. A few students succeeded in creating complex

mathematically valid tessellations, including one example of the 2-isogonal 36/3

2.4.3.4

tessellation. (See Figure 37 below.) Most tilings incorporated symmetry, though a few

appeared to be random or semi-random.31 Students tried to make the tilings edge-to-edge,

but in some cases which did not seem to be working, they shifted the tiles into non-edge-

to-edge positions in an effort to squeeze in certain tiles. This inevitably left gaps. See, for

example, Figure 38 below, where Kelsey has shifted the two bottom right triangles over a

bit to make more room. During Set D, two students reasoned that these gaps were

insignificant, but during Set G one student decided the tiling was impossible and another

abandoned the attempt and tried a different strategy. This may have indicated some

learning between the two interviews. However, other than this, there does not seem to be

any significant difference in tilings, reasoning, or strategy between the two interviews.

31 By ―random‖ I mean there was no single, clear repeating pattern. I do not mean to imply the students

deliberately incorporated randomness in any statistical sense. ―Semi-random‖ refers to a tiling where some

sections contained patterns, but there was no overall pattern to the tiling. See Figure 70 on page 212 for an

example.

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Figure 37: The 36/3

2.4.3.4 tiling made by Kobe

Figure 38: A step in Kelsey‘s attempt to squeeze five triangles around a square corner

Set B: Finite Tilings. This set was adapted from Owens and Outhred‘s (1998)

study of finite tilings. Students were asked if and how it was possible to tile the ―small

rooms‖ with the given tile. They were also asked how many tiles it would take. As with

the other tasks, the interview went through oral and drawing phases, as well as a

construction phase in the case of some of the pattern block tiles. The six tiling problems

are in Figure 30 on page 130. All the tilings were mathematically possible except for the

fifth problem, which most students saw at once to be impossible.

Students were generally more successful with the problems than I was expecting.

They sometimes misjudged the size of the prototile in both the oral and drawing phases,

particularly in the first few problems. Orientation only caused difficulty in a few cases.

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The most difficult problem proved to be the sixth, in which they needed to tile a right

trapezoid (a triabolo) with a right isosceles triangle. The difficulty seemed to come from

the fact that the prototile needed to be turned in two or three different orientations in

order to tile the trapezoid. Difficulty in judging the size of the prototile caused some

students to believe the tile would fit on the left with the hypotenuse at the bottom. (See

Figure 89 on page 295.)

In both the pre-test and the first finite tiling problem, Kelsey displayed a lack of

understanding of the row and column structure of a square array. (See Figure 39 below.)

In discussing her pre-test, she confirmed that some squares would necessarily ―be bigger

because some of them are shorter.‖

Figure 39: Kelsey‘s responses to the first pre-test problem and the first problem of Set B

As Owens and Outhred (1998) found, students tended to be more successful with

shapes that they were familiar with. It is impossible to compare these results with Owens

and Outhred‘s study because they interviewed both 2nd

and 4th

year students and did not

distinguish the two groups in their results.

Set E: Triangles. For this task, students were shown an acute isosceles triangle, a

right scalene triangle, and an obtuse scalene triangle in the software environment.

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Students initially believed all the triangles could tessellate, though a few changed their

minds if they were unsuccessful during the construction phase.

The acute isosceles triangle was tiled by most students in rows. Two students

attempted to create a radial pattern, one successfully.

The right scalene triangle prototile was oriented at a nonstandard angle. Only one

student realized at first that a rectangular unit could be made from two triangles. Two

other students realized this after exploring with the computer tiles, though one was

unsuccessful at tiling with the rectangular units, perhaps because of the non-standard

orientation. (See Figure 40 below.) Other students tried a variety of strategies, none of

which could make a mathematically valid tiling, though they believed their patterns

would eventually work. It is very likely that students would have seen the rectangular

unit tiling much more easily if the initial prototile had been oriented with horizontal and

vertical sides.

Figure 40: An attempt by Michelle to tile a right scalene triangle by using rectangular

units

Most students pursued strategies for the obtuse scalene triangle that were similar

to their strategies for the right scalene triangle. Upon exploration, many students tried

tilings made with parallelogram or kite units.

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Set F: Polyominoes. For the first part of this set, students were asked about tiling

each of the two trominoes. The drawing and construction phases were replaced by a

phase using dot paper. Three students insisted on dividing the trominoes into their unit

squares, even when I drew the initial tile. See for example Figure 41 below. This made it

very difficult to see the boundaries between the tiles. This may be due to the plastic tile

which served as a model; it had distinct square units. During the pilot study, no such

model was shown and none of the 11 students tried to divide the tromino into unit squares

on the dot paper. This phenomenon could also be an indication that the unit squares were

seen as an important part of the structure of the polyomino and students may have felt

that this needed to be seen clearly in order to have a clearer view of the local structure,

not realizing it would make the global structure nearly impossible to see. They may also

have understood the task as that of making a standard square tiling, in which case the

polyominoes were conceived not as tiles, but merely as units of tiles.

Figure 41: Kelsey‘s L-tromino tiling

Students were successful at creating a variety of tilings, including creative tilings

with the simple I-tromino. (See, for example, Figure 64 on page 202.)

Tilings with the L-trominoes were similar to those found by Wheatley and

Reynolds (1996). Most students made random patterns with the L-tromino. Two students

tiled with rectangular units made of two L-trominoes. Michelle believed there were other

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possibilities if she departed from the grid constraint. She showed, for example, how she

thought a radial pattern could be made with L-trominoes. See Figure 42 below.

Figure 42: Beginning of an L-tromino tiling by Michelle drawn on blank paper

In the second part of the interview, students were shown plastic models of all five

tetrominoes at once. They were asked which could tile and which couldn‘t. They were

then allowed to explore the tilings with any tile they chose using either dot paper or the

computer. Students generally thought most tetrominoes could tile, but in a few cases they

believed the tiling would be impossible. Students created a wide variety of tilings using

various strategies.

The T-tetromino was the most difficult. Three students doubted it was even

possible. Moses presented a demonstration showing why he thought the T-tetromino

could not tile. Three students made random patterns with the T-tetromino, but two got

stuck and decided it was not possible.

Sets F and G were the only sets where students had real choices over what

problems they would tackle first. Most students were motivated to work with the S-

tetromino, claiming it had a ―cool‖ shape. They seemed to be drawn by its rotational

symmetry. They described it as pointing in two directions at once.

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Set G: Regular Polygons. The last tiling creation task was open-ended and did

not have an oral or drawing phase because of its complexity. It was done entirely in the

computer environment. Students were first asked to revisit the squares and triangles task.

This served the purpose of seeing if any thinking had changed, as well as introducing the

next task. Students were slightly more successful with squares and triangles this time, but

for the most part they approached the problem the same way they had before, even if the

approach failed. The only difference was that students were more likely to recognize that

their approach failed during Set G than during Set D. Each student tended to use the same

sort of reasoning in both sets.

Next they were given five regular polygons (triangle, square, pentagon, hexagon,

and octagon) and invited to create tilings by combining any polygons they wished.

Students seemed attracted to trying to use the pentagon. None made reference to the

earlier pentagon task where they had discovered it was impossible to tile with the regular

pentagon. (It is also impossible to tile with the regular pentagon in combination with

other regular polygons, but none of the students realized this.) The numerous attempts to

include a pentagon in the tilings were probably due to the motivational appeal of

uniqueness. They were unfamiliar with any tilings using the pentagon and tried very hard

to find one.

Of the ten tiling attempts that did not use a pentagon, seven were successful.

Three students constructed the 4.82 semi-regular tessellation using squares and octagons.

Students made a wide variety of other constructions, including some complex

symmetrical tilings. All students except Marie were successful in creating at least one

tessellation. Kelsey was the only student to create a random pattern. See Figure 43 below.

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Figure 43: Kelsey‘s random triangle and hexagon tiling

Sets H and I were optional sets, which were not needed. Set H was an extra task

(polyiamonds) in case a student finished or stopped another task early. Set I was a follow-

up time for any unusual answers from the other tasks.

Set J: Aesthetic evaluation. The sixth interview had a completely different

format. Students were involved in evaluating previously created tilings. Data from this

interview were used (in combination with data from the first five interviews) to explore

students‘ aesthetic criteria for tessellations. See the Aesthetic Influences section

beginning on page 181 for a summary of the aesthetic results.

In addition to analyzing how students responded to each task, data were also

analyzed across tasks. The remainder of this section explores the findings from this cross-

task analysis.

Piagetian stage

One interesting incident occurred with Michelle‘s solution of the first problem of

Set B (tiling a 2×3 rectangle with squares). Her attempt to draw the tiling was

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unsuccessful, even though she had an idea how six or seven squares would fit in the

rectangle. The orientation of the square seems to have been the source of the difficulty.

She realized that the square could be turned, and conceptualized the square as rotated for

the rightmost column, but thought the square could be left in its original orientation for

the other two columns. This would seem to be the kind of thinking that occurs in Piaget‘s

intrafigural stage. Another indication that Michelle sometimes reasoned at the intrafigural

stage occurred during the polyomino task when she tried to draw an I-tromino in a

diagonal orientation during Set F. Though three of the sides of the tromino were drawn

correctly, one side and the lines for the unit squares in the tromino were oriented

according to the dot paper grid. See Figure 44 below.

Figure 44: An I-tromino drawn by Michelle in a diagonal orientation

Aside from these two instances, evidence suggests that all students, including

Michelle, were at the interfigural stage throughout all interviews, as expected. Children

had little difficulty handling very simple translations, reflections, and rotations of tiles.

They were able to describe these transformations orally before performing them, as

would be expected at the interfigural stage. They were able to coordinate tiles locally.

Evidence suggests that none had reached the transfigural stage. They created simple

global patterns requiring translations, but could not handle global symmetry requiring

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multiple mental transformations. For example, there was no evidence of noticing glide

reflections, which requires envisioning a translation and a reflection. Kobe did create the

tessellation in Figure 45 below, which could have glide reflection symmetry (depending

on how it is continued). However he gave no indication of noticing the glide reflection

symmetry that the infinite tessellation would have had, nor did he use glide reflections to

construct it. There was no clear evidence that children noticed rotation symmetry in any

tessellation. This may be because such patterns almost always had reflection symmetry as

well, which is easier to see, and children at the interfigural stage should not be able to

perceive two different types of symmetry simultaneously.

Figure 45: S-tetromino tiling by Kobe

Tiling Criteria

As explained in Chapter 3, three constraints were imposed on the tasks: tilings

should be gapless, without overlaps, and endless in all directions. I analyzed the data to

understand the students‘ reactions to these constraints.

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How Shapes Fit (Gaps and Overlaps)

Avoiding gaps and overlaps was natural for the students, even on the pre-test,

where no constraints were given. With some tools, such as pattern blocks and dot paper,

overlapping is basically impossible. Pattern blocks sometimes leave slight gaps, but they

cannot overlap. In the computer environment, which allowed both gaps and small

overlaps, if it was necessary to break a constraint because tiles did not fit perfectly,

students preferred overlapping. It is possible that, because the software rejected large

overlapping, the students depended on the program to protect them from illegal overlaps.

In that case, they may have accepted slight overlaps simply because the computer did.

Another explanation for the preference of overlaps is that gaps are more visible to the eye

than overlaps. On at least one occasion, a student did not notice overlapping of computer

tiles. They always noticed gaps, no matter how small.

Gaps. Two students noted that in real tilings there are small spaces between the

tiles. However all of the students easily accepted the gapless constraint for these

interviews. One student once left deliberate gaps in order to highlight units of tiles, but

even then he clarified that the gaps were only for explanatory purposes. If possible,

students would fiddle with the tiling in an attempt to squeeze out all perceived gaps. The

fit must be precise, or else something must be done about the remaining gap, possibly

moving other pieces around. Students sometimes rotated tiles around a vertex or put them

in a different order around a vertex to try to close up a gap. (This shows that they did not

understand the additivity of angles around a vertex.) If the gap was small, it sometimes

became acceptable to leave it in hopes that the gap was not real, but simply an

imperfection of some sort. It was not always clear if these imperfections were the fault of

imprecise drawing or if tilings might legitimately have tiny gaps. Michelle noted for two

of her tilings that ―there might be about one or two cracks about that big,‖ holding her

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fingers very close. She believed these potential gaps were an artifact of the lack of

symmetry in her tilings. ―It wouldn‘t be an exact pattern, but um, it should work.‖

Slight gaps raised doubts about the validity of the tiling. When drawing, students

would distort shapes in order to avoid leaving gaps. For example, Rachel stretched the

topmost triangle, Marie stretched the rightmost triangle, and Moses stretched the bottom

triangles in their tilings in Figure 46 below to much more than 60° in order to avoid

leaving a gap. In other words, when adding a tile to a drawing, students were often

guided by the need to fit rather than the shape of the tile.

Figure 46: Tilings by Rachel (left), Marie (middle), and Moses (right) where equilateral

triangles have been stretched to fit gaps large enough for two triangles

Overlaps are closely related to gaps. Both are questions of fit. Students generally

tried to avoid overlaps. Marie confirmed the importance of the overlapping constraint by

recounting a story of overlapping house tiles that made her trip. Overlapping was

sometimes cited as the reason circles could not tile a floor. However, the desire to avoid

overlapping seems slightly less strong than the desire to avoid gaps. Marie made a regular

polygon tiling with some slight overlapping. Rather than reject the tiling, she questioned

the overlapping constraint. She thought that with extra work the overlapping could

probably be fixed. Likewise Michelle thought some overlapping might be permissible as

long as it was not much. At one point Rachel asked my opinion whether two computer

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tiles were overlapping. I told her yes, but she decided to leave them anyway until it

became impossible to continue without fixing the problem.

This occasional ambiguity towards overlapping was also found by Owens and

Outhred (1998). In their study they noted that the one task in their study that could not be

tiled without overlapping proved to be the hardest. Students frequently answered it could

be tiled. Upon interview the students indicated that they knew the tiles would overlap, but

had answered that the tiling was possible anyway.

Infinite Space

Students quickly expressed comfort with the idea of a room that goes on forever

and ever without walls. Students repeated this constraint in their own words. For

example, Rachel said, ―Like, um, where it doesn‘t stop or where, it, can keep going and, a

long distance.‖ During the tasks, Moses would occasionally envision infinite tilings as

covering the world: ―Yeah, if it went on forever and ever and ever, around the globe, then

it would stop and connect [on the other side of the world], like a puzzle.‖

No constraints were given on the pre-test, but only one of the six children

interpreted the pre-test tiling problems to mean tiling only the rectangular sheet of paper.

The other students treated the problem in an apparently infinite space. In some cases, this

may be because they simply did not consider the problem of what happens when the tiles

meet the border. Rachel, for example, drew a rectangular space to tile in, but then

answered the problem as if the border did not exist.

During the interviews, the students showed significant signs of struggling with the

infinity constraint, even though they accepted it. This struggle was evidenced in several

ways: forgetting the constraint; finite patterns from the same tessellation representing

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different tessellations; expressing completion with a finite pattern; and, most importantly,

discomfort over tiling in an empty plane.

Forgetting the constraint. Students were briefly reminded at the beginning of

most interviews, and whenever they forgot or asked, that the rooms went on forever and

that we therefore do not worry about the walls—if the walls exist at all they are too far

away to care about. Nevertheless, students frequently forgot about this constraint,

especially during the earlier interviews. They spoke about walls and worried what would

happen when the tiling reached the walls.

Finite patterns from the same tessellation representing different tessellations.

At least two students, and possibly more, did not understand that different finite patterns

can represent the same infinite tiling. This was particularly evident for the equilateral

triangle tilings. Because all students made these tilings edge-to-edge, there was only one

possible tiling (barring orientation differences). Students found several different

strategies for making this tiling, sometimes in rows, sometimes by unitizing. If they

unitized, they might use hexagons, trapezoids, or more complex shapes. Kelsey and

Michelle believed that each strategy produced a unique tiling. This was true even when

both of Michelle‘s patterns were in front of her at the same time. They did not envision

the tiling extended to an infinite pattern, but only focused on the finite units they had

created. Kelsey, for example, insisted that one of her tilings would contain more

hexagons than another, though neither had been created by hexagonal units alone. (See

Figure 47 below.) Michelle, when asked if her favorite triangle tiling was more

interesting because of the way it was made or because of how it would look when it was

finished, stated that it was more interesting because of how it would look when it was

finished. She did not realize that the two tilings would look the same when extended

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indefinitely. Kobe and Rachel were not sure if their different strategies were creating

different tilings or not. Marie and Moses showed a firmer understanding of infinite space.

They explicitly acknowledged that their different finite patterns, if continued indefinitely,

would create the same infinite tiling.

Figure 47: Kelsey‘s first two triangle tessellations

I return to this phenomenon in the section on Ontology of Tilings (Process versus

Object) on page 164.

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Expressing completion with a finite pattern. Students sometimes announced

they were finished when they achieved a finite design, often when it was symmetric and

convex, sometimes without a clear idea of how the pattern would continue. For example,

Rachel talked about a hexagon tiling looking like a flower, ―when you‘re done with it.‖

After laying down a hexagon and six surrounding hexagons like a flower, she paused.

When I asked if she was done, she replied, ―Yes.‖ This would seem to suggest that

Rachel was focusing on how the finite pattern would look, rather than the infinite

tessellation. On the other hand, Rachel later pointed out for triangles that the finite

patterns were only temporary. They are destroyed as soon as you continue, which is what

makes all the tilings really the same. This suggests that Rachel may have been hesitating

between two ways to view tessellations.

During the pilot study, two students created a tower pattern with alternating rows

of squares and triangles. If extended indefinitely, it would become the semi-regular 33.4

2

tiling. However when challenged to extend their tiling further, both students put squares

in the triangle row and triangles in the square row, preserving the tower as a finite

pattern, as well as preserving the vertical reflection symmetry of the tiling. These

students seem to have been focusing on the symmetrical finite structure rather than

conceptualizing an infinite tiling.

Discomfort over tiling in an empty plane. During the pilot study, students

sometimes started a tiling in the corner of the paper, which caused them to create finite

patterns that incorporated the corner as part of the tiling. In order to avoid this problem, I

asked students in this study to begin all of their tilings in the center of the sheet of paper.

The computer software would automatically shrink and center any tiling that approached

the edges of the screen too closely. Marie complained about this. She insisted that she

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understood that there were no walls, but maintained that she needed some walls to help

her think about the tiling. She therefore used corners and walls in many of her tilings in

order to help her structure the tiling. When she did so, she would sometimes apologize,

saying she knew the wall was not really there, but that it was needed to make the task

easier. However, it was not clear that she had a conception of how the tiling continued

beyond her temporary walls.

Instead of working in an empty plane, students sometimes tried to create a tiling

with an implied or created frame. On the pre-test, the sheet of paper was interpreted by

Kobe as a frame within which the tiling had to fit. Marie saw that the computer screen

had an apparent border and tried to fit the tiles along this border.

Moses was more concerned about the borders of his rhombus pattern than the

interior structure, even with reminders not to worry about the walls. He spent a great deal

of time working out how the pattern could be closed off at the edges. He even doubted

that his alternating rows tiling would work because of the difficulty of finding a way to

enclose the pattern. At one point his border became very problematic as he attempted to

define a ―pit‖ in which he could attempt to complete his rhombus tiling. (See Figure 48

below.) In answer to the question whether his rhombus tiling would go on forever, he

talked about finishing it off with side borders.

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Figure 48: Rhombus tiling border by Moses

Some students enjoyed creating radial patterns—non-periodic tessellations with

rotation symmetry. The children in this study may have valued radial patterns

aesthetically. For example, in the Alhambra aesthetic evaluation task, students preferred

the radial patterns over the periodic patterns. (See, for example, Figure 62 on page 195.)

It may also be that students sometimes created radial patterns because they needed an

anchor point for their tessellations. The initial tile can serve as an anchor around which

the rest of the tessellation is constructed. Marie‘s corners may have served as anchor

points for her. Having an infinite, empty plane with no points of reference may be a

daunting idea for children at this age. Further research would be needed to know if this

were true.

Ontology of Tilings (Process versus Object)

There is evidence that participants sometimes conceptualized tiling as an object,

either mathematical or physical, and sometimes as a process or action. It is not likely that

in the context of this study we could distinguish between a mathematical and a physical

ontology, but we can look at evidence that points toward an object or process ontology. If

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the students truly conceptualized the tiling as infinite, we can safely say this is a

mathematical construct. Beyond this, I will not attempt to distinguish between

mathematical and physical tessellations for these children‘s conceptualizations.

One piece of evidence for a process ontology is in the episodes mentioned earlier

during discussion of participants‘ conceptualization of infinite space. Kelsey and

Michelle each created two mathematically identical tilings, and yet they believed the

tilings were different because each was created by different processes. For example, the

tilings in Figure 47 on page 161 were created by different procedures. Though Kelsey

was making the same regular 36 tessellation in each case, she perceived the two

tessellations to be different because of the different actions used to make them. Part of the

discussion of the difference between the two was as follows:

Scott: Ok. Umm, is there another way to tile with triangles?

Kelsey: You could make it like these put these here, just like um the,

[makes a hexagon from six pattern blocks] like this, then make it

do like a pair [adds six triangles to make a star] and then put this

right here, and this [fills in spaces between some star points to

form a more convex figure] and you keep going like that just like

in these. [See the bottom photo in Figure 47 on page 161.]

Scott: Ok. And that will make a different pattern?

Kelsey: Mm-hmm. Yes.

Scott: Ok.

Kelsey: ‗Cause this one has the shapes in the middle like that [points to

shapes near middle].

Scott: I‘m sorry, say it again. How is this different from the other pattern?

Kelsey: Um this one ha- gets these shapes [indicates two hexagonal

patterns on sides] like this and makes ‗em like that and then there‘s

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like one that makes a big triangle [shows a large triangle made of

six triangles at bottom] and one that …

Kelsey‘s perception of each tessellation is tied to how she actually created the

tessellation (in rows or with units) and not to what they currently look like. Even if we set

aside the issue of what the tessellations will look like when extended infinitely, she is

interpreting each tessellation in terms of the process used to create them rather than the

current (finite) result.

On the other hand, Marie and Moses expressed their belief that there was only one

triangle tessellation, no matter what method they used to achieve it. Moses was especially

clear about this. Like Kelsey and Michelle, Moses created triangle tessellations in two

different ways, but said that every method created the same tessellation. ―You could look

at it as a d-, um, in a different, like perspective and like uh if it was diamonds but it

would still be the same thing.‖ This points to an object conceptualization for Marie and

Moses. They distinguished the object tessellation from the process used to create it. Kobe

and Rachel were not sure if each creation was a different tessellation or not. Kobe

hesitated for over three minutes trying to decide whether the different creations were the

same or not.

An object conceptualization does not prevent someone from using a process

conceptualization. In geometrical contexts, an object conception may precede a process

conception, though it is difficult to see how one can completely understand an infinite

tessellation without also having a process conception. It is possible that students would

have access to both conceptions. Proceptual thinking requires flexible use of both.

Further evidence that Kelsey and Michelle used primarily a process conception

and Marie and Moses depended more on an object conception comes from an analysis of

the wording they used to describe their tessellations. Students sometimes used an active

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wording, describing the tiling in terms of their actions, e.g. ―Put this one right here like

that and then do it again.‖ Students also sometimes used descriptive wording, depicting

the tiling as an object they were passively describing, e.g. ―One line‘s gonna be facing

one way, the other, the next line‘s gonna be facing the other way.‖ Instances of such

wording in the first interview were coded by episode. There were 122 instances coded in

all. A ratio was calculated for each student of active wording to descriptive wording. A

high ratio indicated predominantly active wording and possibly a predominant process

conception. A low ratio indicated predominantly descriptive wording and may indicate a

preference for an object conception. Kelsey and Michelle, who interpreted their tilings in

terms of the actions used to create them, had ratios of 4.25 and 2.40 respectively. Marie

and Moses, who recognized the independence of their actions from the final result, had

ratios of 1.25 and 0.41 respectively. (The ratios for Kobe and Rachel, who were not sure,

were in between these two extremes.)

It is natural that participants should use active language to describe their

tessellations because they were in the process of creating them. Even I frequently used

active wording in my questions. However, the fact that Marie, and especially Moses, had

such low ratios, indicates that in spite of the active environment, they had a strong

tendency to view the tessellation as an object. The ratio measures point to the same

students as the observations about distinct tilings. Students with low ratios were precisely

those who were able to recognize the tessellation as an object distinct from their actions

and students with high ratios were not. This correspondence between two independent

sets of observations suggests that certain students predominantly conceptualized the

tessellations as an action and others as an object.

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It may be that all students used both object and process conceptions to some

extent, even if some privileged the process conception. For example, Marie, who seemed

to show an object conception for triangles, created the hexagon tiling in several different

ways and believed each way was different (though in this case the differences may be due

to orientation and focus on finite patterns rather than processes). Another example is

children‘s responses to the pentagon task. It is possible that children‘s reaction to this

task represents various process and object conceptions about pentagon tessellations.

Students believed that it was possible to tile with the pentagon, so when it did not work

they began trying different arrangements of pentagons. They seemed to believe the tiling

was not working because their actions were not yet correct—a process conception.

Eventually students would decide that the pentagon tessellation was impossible. They

usually pointed out a structural problem, the inevitable presence of a gap, to explain the

failure—an object conception.

Epistemology of Tilings

There is evidence that children sometimes had an idea of what the tiling looked

like, and perhaps even some of its structure, before they used any tool to construct it. The

extent to which they had a clear idea of the tiling in advance varied considerably. Usually

the descriptions during the oral phase were vague and local. Global structure was rarely

discussed. Oral descriptions were sometimes hard to understand, even on questioning.

The students would frequently express some doubt about the tiling during the oral phase,

which also contributed to the vagueness. If the tiling was more familiar, the description

was less vague. For example, most students were familiar with the 44 square tiling.

However, none of them could describe in any detail at all what the triangle tiling would

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look like, even though they were certain it existed. A typical partial description of a tiling

is the following oral description of a rhombus tiling by Michelle.

Scott: How would it work?

Michelle: Well if you, you could put one that goes (like) right there [on one

side of the original tile].

Scott: Mm-hmm.

Michelle: And there [a second side].

Scott: Mm-hmm.

Michelle: And there [a third side]. And it just sort of goes around and around

and around. [Spreading hands around table] Out out out.

Notice that this is primarily local in its description. There is no clear description

of the global structure, only how the tiles will fit around the initial tile and that it will

somehow spread ―out out out,‖ presumably by fitting around each tile in the same way.

When students say they could put a tile on each side of the initial tile, they typically do

not describe the orientation of the tiles, even when several orientations are possible. On

the other hand, Moses gave a global description of the rhombus tiling, noting that other

rhombi would line up from the first rhombus in four directions, all having the same

orientation.

Scott: Tell me first [tapping tile] and then we‘ll, we‘ll try it after you

describe what you‘re gonna do.

Moses: Wait. One, [pointing to sides of rhombus] like that like (one) in

there [pointing to spaces at opposite sides of rhombus], and they‘re

all gonna be facing the same way but they‘re gonna be, like it‘s

gonna be kinda like a diagonal line [indicating row coming from

two opposite sides of rhombus].

Scott: Ok.

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Moses: And then this [same gestures showing row from other two sides],

it‘s gonna be just a straight line of them.

Scott: Mm-hmm.

Moses: All facing the same way.

Drawings of tilings were usually more detailed than the initial oral descriptions.

Michelle admitted she really did not know what the hexagon tessellation would look like,

but ―I know that you can, you can connect them.‖ She then traced the hexagon pattern

block with pencil on paper until she discovered what the tiling actually looked like. She

perhaps recollected that she had seen the hexagon tiling before, but had not retained any

knowledge of its structure.

Students sometimes explicitly said they saw new ideas as they created their

tilings, confirming that the final tessellation was a combination of initial ideas and new

discoveries as they tiled. Moses began with his description of simple rhombus rows, but

then suddenly saw that he could make an aesthetically more interesting tiling by

alternating the orientations of the rows. On at least five occasions, Marie said she had just

had an idea while creating her tiling, changing her initial idea for the tiling.

This change from their initial ideas also occurred during the drawing phase.

Moses discovered as he was drawing his first triangle tessellation that six triangles make

a hexagon. At that point he reconceptualized his tiling and planned a radial hexagon

pattern of triangles.

Outhred and Mitchelmore (2000) noted that physical tiles prestructure square

arrays in such a way that students do not have to understand the structure of the tiling in

order to create them. This prestructuring frequently occurred for the tessellations created

by the children in this study as well. When Kobe began to construct his first triangle

tiling with pattern blocks, he placed the first six tiles next to each other and created a

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hexagon, at which point he said, ―I don‘t really know how I did that.‖ He later used this

hexagon pattern to help structure his tiling.

Physical tilings were often different from both the drawing and oral phases,

indicating an evolution of ideas, even when students claimed to be using their initial idea.

In one case, Kelsey said that her physical rhombus tiling was the same as her drawing,

but her drawing and oral description seem to have been of the standard tiling with all tiles

in the same orientation, whereas her physical tiling was semi-random. This is not because

Kelsey thought all tilings were the same. She continued to make other tilings that she

claimed were different.

Students sometimes talked about seeing tilings in their head. When asked how he

knew the S-tetromino could tile before trying it, Kobe traced with his finger where the

second tile would go and replied, ―I just pictured how it would work in my head.‖

Michelle explained how she could fix a flaw in her newly created S-tetromino tiling and

said, ―So mainly in my head I‘m looking at a quilt pattern.‖ Moses said he was sure his

triangle pattern would continue forever because ―I can picture, I can picture a lot of

things in my mind.‖

Of course, the structure they imagine in their head may not fit the structure that

appears in drawn explanations. Children often knew not to depend on imprecise

drawings, especially the children from the school with the French curriculum. Michelle

struggled with her explanation that trominoes could be drawn in patterns that did not fit

the dot paper lattice. When she drew Figure 42 on page 152, she saw that she had not

placed the tiles in the outer layer quite the way she wanted:

Scott: Ok. And you‘re sure that would work?

Michelle: Mm. It says so.

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Scott: It says so?

Michelle: From the drawing, no, but in my head yes, so it‘s like

[unintelligible].

Scott: Ah-ha. So in the drawing it doesn‘t look like it would work, but in

your head you‘re sure this will work?

Michelle: Yeah.

She also knew to distrust answers she could see only in her head, ―‗Cause in my

head most of the time I can make anything work.‖

In fact, students frequently had a mental model that could not work physically.

Kobe for example orally described the pattern in Figure 49 below, but when he tried it, he

saw that the tetromino on the left would not fit in the two-space gap left by the two

tetrominoes on the right.

Figure 49: Failed attempt envisioned by Kobe for fitting T-tetrominoes together

Sometimes students had an idea in their heads that they lost while trying to sketch

it. It is possible that their drawing conflicted with the structure they imagined in such a

way that it interfered with their initial idea, causing them to forget it. Or it may be that,

rather than trying to accommodate the new situation into their initial schema, they

discarded the schema altogether. ―Losing‖ an idea is more likely when the idea is vague

to begin with.

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Children’s Strategies

Students used a variety of strategies to create their tessellations. Strategies were

coded and counted. The most popular strategies used, in order of preference, were as

follows:

Simple rows. The most common strategy was to arrange tiles in simple rows with

straight borders between the rows. This was an effective strategy for simple

shapes such as parallelograms and rectangles. This strategy was also used with

squares and triangles to create the 33.4

2 tessellation, or some variation thereof.

Random (or semi-random). Students frequently used the ―Fit one at a time‖

strategy to create random patterns without symmetry. In many cases, this became

a semi-random strategy as students created sections with patterns, but not one

global pattern. The semi-random strategy was particularly popular for the

rhombus. See for example Figure 50 below. Even if they did not deliberately

create local patterns, they usually looked for local patterns and units after the fact.

Figure 50: Semi-random rhombus tiling by Kelsey

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Fit one at a time. When students were not sure what to do, they simply began

placing one tile at a time, as with a puzzle, to see what patterns would emerge.

Sometimes no pattern would emerge, in which case they continued to place tiles

with a random strategy. But often there would be a clear pattern, though not

organized by units or rows. Fitting one at a time was the most common strategy

for hexagons, whose structure was not always evident to the students, but which

was nevertheless not random. This strategy was also frequently used for rhombi,

equilateral triangles, and parallelograms, as well as the Regular Polygons task.

Unitizing. Students often grouped tiles into units that were easier to tile with.

Students did this most frequently when triangles were involved. They also

unitized with the L-tromino, as in Reynolds and Wheatley‘s (1996) study. All

students used the unitizing strategy at least twice, except Marie.

Growing. Students sometimes used some form of growing strategy to make a

small pattern larger and larger. For example, Marie created the tower pattern on

the left in Figure 51 below. When challenged to continue it, she expanded it both

vertically and horizontally, as in the photo on the right. In some cases, students

used dilation symmetry to grow their tiling. For example, Moses created large

triangles with the triangle pattern blocks which he kept adding on to in order to

make still larger triangles. This strategy also describes radial tilings, such as the

tilings in Figure 52 on page 178.

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Figure 51: Two stages of Marie‘s growing tower in the software environment

Alternating rows. For aesthetic reasons, students often preferred creating rows

with alternating orientations. Moses and Kelsey used this strategy frequently, but

every student used it at least once. Edge-to-edge row tilings of equilateral or

isosceles triangles were perceived to be in alternating rows because the triangle

orientations reversed with each row. This alternation may have made triangle

tilings slightly more difficult for the students; they commented on it several times.

Moses thought alternating triangle patterns looked like rows of teeth.

Complex. Sometimes students used a complex strategy that used neither rows nor

unitizing. For example, sophisticated tilings with 6-fold symmetry such as the one

in Figure 37 on page 148 were organized in overlapping units. Such tilings took

time, but students were able to analyze and defend them in a mathematically

correct way. Kobe described how the translation symmetry would continue the

pattern.

Other strategies were used less often. These included making complex rows

without straight borders, creating brick wall type tilings, making a two-dimensional grid,

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and creating spiral tilings. Another uncommon strategy was random units, which put tiles

together in units, but then placed the units together randomly instead of in rows.

Students, tools, and prototiles were coded and compared to strategies used.

Strategies depended more on the prototiles than on the student or the tool. For example,

the ―semi-random‖ strategy was one of the main strategies for the rhombus tilings, the

dihedral square and triangle tilings, and the pentagon tilings, but was hardly ever used for

the other tasks. The ―alternating rows‖ strategy was the main strategy used for the acute

isosceles triangle, and was frequently used for the parallelogram and the equilateral

triangle. But it was hardly ever used for other shapes.

There were few tendencies among the students. The popular strategies were used

by all six children with few exceptions. Rachel tended to avoid any random strategy.

Marie did not use the ―unitizing‖ strategy. Otherwise, everyone tended to use the same

strategies, though not always the same strategy for the same task.

The tool seems to have had little impact on the strategy used. The semi-random

strategy was used mostly with pattern blocks, but was also used with other tools. The

spiral strategy was only used with drawing on blank paper. Otherwise, there were no

clear tendencies associating strategies with students or tools, only with the prototile being

used. This is natural because certain strategies really only make sense with certain

shapes. The L-tromino cannot be used to make simple rows with straight borders unless it

is first put in rectangular units. No simple units can be made with the hexagon, and there

is only one way to fit hexagons together, so the ―fit one at a time‖ strategy makes sense.

However, most shapes can be tiled in more than one way and it is interesting to note that

certain strategies seem to be frequent modes of thinking among all students.

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Children’s Reasoning

Students frequently did not give a clear reason to justify why their tessellations

were valid or to explain how they knew how to construct the tessellation, even when

questioned. When they did give a reason, this was coded and counted. Reasons were

usually one of three kinds: justification by experience, empirical justification, and

justification by known unit.

Justification by experience. Students frequently said they knew a certain shape

would tile because they had seen it before. Kelsey knew that the hexagon could

tile because she had seen walls of hexagon tessellations in the film A Bee Movie.

Other students also mentioned honeycombs either before or after creating the

hexagon tessellation. (This visual, real world connection is actually an aesthetic

quality influencing their thinking.) During the Regular Polygons task, Michelle

made the squares and triangles tessellation on the left in Figure 52 below. Later in

the interview she made the tessellation on the right. When asked to justify how

she knew the second tessellation would continue to work, she suddenly realized

that there was a correspondence between the two tessellations. She had me call up

a copy of the previous tessellation so that she could compare and show me how

they had the same structure.

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Figure 52: Two regular polygon tilings by Michelle

Empirical justification. Students were frequently certain that a tiling was

possible simply because it was working so far. This inductive reasoning was

sufficient to assure them that the tiling would always continue to tile. Students

who used the ―fit one at a time‖ or ―random‖ strategy seemed to be satisfied that

the tiling was valid as long as the tiles kept fitting. Moses justified one of his

tilings by saying, ―I‘m just thinking, I‘m, I‘m not guessing but I‘m just, based on

what has happened already I think it should work.‖ Rachel justified one of her

polyomino tilings by saying, ―Because like, once (you‘ve) like make one, it just,

you have a space for another, and then a little space for another.‖

Justification by known unit. If copies of the prototile can be put together to form

a familiar unit that is known to tile, then the prototile can also tile. For example in

Set A (Basic Tilings), students often knew or discovered that equilateral triangles

can be put together to make a rhombus or hexagon. Students knew from the two

previous tasks that the rhombus and the hexagon could tile, therefore the triangle

could also tile. In Set E (Triangles), students who succeeded in making rectangle

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or parallelogram units usually showed how those units could be used to make a

simple periodic tessellation. Sometimes they were not successful with the new

unit, but they were still confident a tiling with the unit was possible.

Other justifications were used more rarely. Occasionally students would say a

shape was similar to another shape that was known to tile. Rachel said she knew the

rhombus could tile, ―‗Cause it‘s just like a square really, except it‘s kinda more,

smooshed together.‖32 Kobe sometimes used a transformational argument, showing how

his rows or units would repeat periodically.

Several students used an unusual subunit justification. Instead of putting tiles

together to make a larger unit, they broke the tile down into smaller subunits. Kobe

noticed that the parallelogram could be decomposed into four triangles. Rachel saw that

the hexagon could be broken down into six triangles. Both argued that the larger shape

must tile because the smaller shape can tile. From a mathematical viewpoint, this is not

quite correct (not all polyiamonds tile), but such reasoning can help to give an intuitive

understanding of how the shape might tile by imposing it on an imaginary triangular grid,

the same way polyominoes are imposed on a square grid. Moses also used this strategy

by noting that a hexagon could be broken down into two trapezoids, which he apparently

was more familiar with.

Symmetry may have been an unstated reason why students believed some of their

tessellations would work. When a pattern is symmetric, only part of the pattern needs to

be validated in order to be sure that the entire pattern will make a tessellation. Students

who made symmetric tessellations were usually more confident of their answers, as I

discuss later.

32 Steven Greenstein, in his doctoral dissertation (2010), shows how this intuitive qualitative

transformational knowledge is common in children and could be developed in educational settings.

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Other General Results

It is worth noting a few other cross-task results.

Dimensions. It was rare for students to see two dimensions simultaneously. They

usually thought locally or in linear rows. Many patterns were created either in rows,

which allow students to focus on one dimension at a time, or in units, which are local

finite patterns that are then often organized in rows. There were exceptions however.

Michelle showed how the parallelogram could tile on her pre-test by taking a ruler and

drawing a parallelogram grid, as in Figure 53 below. Moses demonstrated two-

dimensional thinking in his oral descriptions of both rhombus and L-tromino tessellations

by saying that the tiles or units would go off in rows in two different directions.

Figure 53: Tiling of a parallelogram on Michelle‘s pre-test

Orientation. Students had differing opinions on whether rotating a tessellation

created a different tessellation. Marie and Michelle both thought that each orientation

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should be counted as a new tessellation. Kelsey and Kobe insisted that orientation did not

matter.

Confidence. Levels of confidence were coded and compared to other codes.

Students displayed a great deal of confidence in their answers, even when those answers

changed. Strong confidence usually correlated with a mathematically accurate

understanding of fit and mathematically valid tilings. Confidence correlated even more

strongly with symmetry, as I discuss later.

AESTHETIC INFLUENCES

―Whoa! Ooh! Wow!‖

(Collective reaction at private school on seeing the first two Alhambra tilings)

In this section we look at the aesthetic values that influenced students‘ strategies

for creating tilings and for judging tilings. Children frequently expressed their aesthetic

criteria explicitly. I identify most of these explicit judgments with the evaluative role of

mathematical aesthetics. In addition to these explicit comments, children seemed to have

implicit preferences for the ways they created their tilings, particularly with respect to

visual symmetry. I identify these preferences with the generative role of mathematical

aesthetics. Occasionally children made choices of what problem they would work on or

what tool they would use. These choices come from the motivational role of

mathematical aesthetics.

In this section I first highlight a few results of the six aesthetic evaluation tasks

and then discuss cross-task observations from all interviews. The cross-task observations

are organized according to whether they are evaluative (Children’s Evaluation Criteria,

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page 187) or generative (Children’s Creation Criteria, page 199). Instances of the

motivational role are noted when they occur.

The aesthetic evaluation tasks were done during the last interview so that they

would not influence students‘ thinking in the other tasks. Further details of the results can

be found in Appendix E.

Student Tilings. Students were given cards with tilings they had previously

created in the other tasks and asked to choose tilings for four rooms of a house. The

charter school chose a finite pattern with slightly overlapping tiles for the bathroom

because it looked fun and resembled a sun. The private school almost picked this pattern

too, but Kobe insisted it was not valid. Both schools chose the 4.82 tiling for the kitchen.

Both schools chose one random pattern and both schools chose one pattern with six-fold

symmetry among their four choices.

Student Sorting. For this and the next two tasks, students were shown cards with

tessellations that others had created. In this task, students were asked to sort the cards into

groups and choose favorite groups and tilings. There was no coordination of criteria in

sorting the tilings to make groups. The private school sorted tilings into six groups with

criteria such as ―blue,‖ ―mazy,‖ and ―crazy.‖ The charter school made seven groups by

using criteria such as ―colorful,‖ ―same colors,‖ and ―maze.‖ Most of the preferred

groups of sorted tilings at both schools were associated with color.

Comparisons in Pairs. I asked students to choose their favorite from among two

(or more) tilings I showed them. Students may have had a slight preference for symmetric

patterns (see Appendix G), though both schools chose the random tiling on the left in

Figure 54 below over the tiling on the right. The explicit reason for choosing the random

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tiling was for its colors. The private school said the rejected tiling had bad patterns

(perhaps because it has no reflection or rotation symmetry).

Figure 54: Two of the tilings from the Comparisons in Pairs task

Professional Preferences. Students were asked which tilings an artist and a

mathematician would choose. Both schools thought the artist would like the colorful,

random pattern on the left in Figure 54 above. Both schools thought the mathematician

would pick the tiling in Figure 55 below because of its illusion of three dimensional

cubes. Other suggestions were also made for both professionals.

Figure 55: Both schools thought the mathematician would prefer this tiling.

Alhambra. Students were asked to put seven postcards of Alhambra tilings in

order from best to worst. Both schools preferred the tiling in Figure 62 on page 195. The

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least favorite tilings were the two in Figure 56 below because they were too simple and

boring. (The mathematicians, on the other hand, chose the tiling on the left in Figure 56

as one of their favorites. It may be that the symmetry was not as evident for the children.)

Figure 56: The two least favorite Alhambra tilings at both schools

Escher. Students were shown many Escher tilings from three books and asked to

award three prizes for best tilings. Both groups chose the tiling in Figure 63 on page 195

among their top two choices. The private school chose the tiling in Figure 57 below for

―2nd

prize‖ because of its perceived Asian style.

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All M.C. Escher works © 2011 The M.C. Escher Company - the Netherlands.

All rights reserved. Used by permission. www.mcescher.com

Figure 57: Escher‘s ―Regular Division of the Plane #58,‖ the second most popular Escher

tiling at the private school

All of the aesthetic evaluation tasks were organized as group activities where

students were told to discuss and make unanimous decisions in selecting the best tilings.

In some ways the revealed aesthetic values were diverse and subjective; in other ways

they were shared among all students. When it came to choosing individual tilings, there

was frequently disagreement between the students. Unanimity was often achieved by

compromise rather than coming to agreement over which tilings were best. In this way,

the students‘ aesthetic values were seen to be subjective. However, analysis of the values

expressed in their discussions revealed many common themes that were shared by all

students. As students discussed the aesthetic criteria, they would usually agree that a

tiling possessed certain criteria. This was objective. However, the decision to value one

criterion over the other was subjective. Everyone might agree that a particular tiling had

vibrant colors and exciting star patterns whereas another tiling had an interesting maze-

like quality and looked futuristic. Where subjectivity entered was in deciding which

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aesthetic criterion was most valuable. One student might prefer the vibrant colors of the

first tiling while another student might choose the maze-like quality of the second. This

was a matter of personal taste. Subjectivity might enter also in preferring certain colors

over others, or in the real world connections they made. A tiling might remind one child

of something at home, which would be outside of the other children‘s experience.

However children usually agreed on the presence of various aesthetic criteria in the

tilings, even if they disagreed over which criteria were most important.

Children sometimes mentioned more than one aesthetic quality for a tiling and I

imagine that the children always saw several aesthetic qualities, just as most people do.

Usually, however, children would express only one aesthetic value, which attracted their

attention above the others, unless the tiling was extremely appealing, in which case they

might express several qualities in its favor.

There were 228 instances of aesthetic expression from the students during all

interviews, and 99 instances from the mathematicians in their interviews. As I described

in Chapter 3, there were 59 aesthetic criteria that emerged from the coding of these 327

instances of aesthetic expression. These verbal aesthetic criteria were almost33 always in

the context of evaluating a tiling. The criteria were grouped into 18 aesthetic themes.

Each theme represents a type of aesthetic value which might be expressed positively or

negatively. The themes represent all codes that could be grouped with other codes, or

codes that could not be grouped but that occurred more than twice. There were 14 codes

that could not be grouped and occurred only once or twice. Most of the following

analysis is done on the 18 aesthetic themes, which represent 94% all aesthetic expressions

in the interviews.

33 The only exception is when Mo explained how he was making his tiling symmetric because he liked the

patterns that symmetry created.

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Analysis of the expressions revealed both diversity and unanimity. Students relied

on a wide range of aesthetic criteria in both their group discussions and in their individual

interviews. However, there were no clear preferences for any one aesthetic theme during

the group interviews. Both schools relied on all of the top 16 aesthetic themes during

their group interviews. There were very few differences between individuals revealed in

the individual interviews. (These are discussed below.)

Children’s Evaluation Criteria

The evaluative role of aesthetics is to judge the mathematical results either of

one‘s own work, or the work of others. This was studied by analyzing children‘s

expressions of aesthetic judgment.

Children could not always express why they liked certain tilings. Some tilings

simply looked ―cool‖ to them and they were not sure why. But when they could express

their reasons, the aesthetic criteria they used were rich and diverse. Table 6 below shows

the 18 aesthetic themes that emerged from the coding analysis of all interviews, in order

of frequency among students. This includes all aesthetic expressions. Most of these

expressions were evaluative, though a few concerned the creation process. The

percentages are calculated relative to each group.

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Table 6: Aesthetic themes

Theme Relative Frequency Description

Students Mathematicians

Real World Connection 21% 5%

Students were excited when the tiling, or a part of the tiling, suggested a real world object or context.

Color 18% 2% Color could be good or bad. Diversity of color was good.

Complexity 14% 16%

Simple tilings were often “boring.” Overly complex tilings were confusing. The best tilings were intriguing.

Uniqueness 6% 4%

Tilings that were different from what students had seen before were attractive. Students often called them “creative.”

Energy 6% 0% Students liked tilings that seemed “crazy,” “energetic,” or suggested exploding patterns.

Interconnectedness 5% 1% Students liked tilings that recalled mazes, puzzles, paths, or knots.

Dimensionality 4% 13%

Tilings which suggested surfaces other than the plane of the tiling were said to be “3D,” which was appealing to all. Mathematicians disliked tilings with only one interesting dimension.

Units 4% 1%

Students were impressed when they found special shapes made by several tiles. If simple units could not be formed, this might be bad.

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Table 6. cont.



Tiles 4% 4% The shape of the prototiles played a role in several aesthetic judgments, e.g. the animals in Escher tilings.

Culture 4% 2% Tilings were attractive if they suggested a foreign style such as “Egyptian,” “African,” or “Asian.”

Alternation 2% 1% Patterns that had alternating orientations were more interesting.

Symmetry 2% 5% Symmetry was a positive factor, but was rarely mentioned explicitly by students.

Surprise 1% 14% Mathematicians particularly enjoyed tilings that surprised them or were thought provoking.

Validity 1% 8% Mathematical validity (e.g. the three criteria) was an important aesthetic criterion to mathematicians.

Connectedness 1% 5%

Tilings that connected two different ideas, or that made a connection to ideas exterior to the tiling, were appealing. Mathematicians liked multiple possibilities in interpretation.

Gender 1% 0%

Students (of both genders) reacted against two “girlish” tilings: one having a pink theme and one having flowers.

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Table 6. cont.



Folds 1% 3% People were intrigued by tilings that seemed to be folded or foldable in surprising ways.

Randomness 0% 6% Mathematicians found randomness sometimes appealing, sometimes not.

Other (not grouped by theme)

5% 9% A wide diversity of qualities were mentioned only once or twice each, such as fun, futuristic, or relaxing.

By far the three most important aesthetic themes were Real World Connection,

Color, and Complexity. Together, these accounted for over half the children‘s aesthetic

expressions.

Real World Connection could be a real world object or symbol that the tiling, or

a part of the tiling, reminded people of. For example, Rachel chose her obtuse scalene

triangle tiling (see Figure 58 below) as her favorite because it reminded her of a vampire

coffin. The Real World Connection theme also refers to occasions when a tiling was

valued because it seemed appropriate for, or was frequently used for, a certain real world

tiling task. For example, the semi-regular 4.82 tiling in Figure 59 below seemed

especially appropriate by both groups of students for use as kitchen flooring, apparently

because they had seen this tiling on kitchen floors before. Some tilings seemed good for

stained glass windows. If the real world connection was repulsive or boring, this aesthetic

could be negative. For example, one tiling reminded Kobe of a ―sewer gate.‖ Real world

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connections were used by most students during the creation tasks, as well as the sixth

interview. If a real world object was perceived in part of their created tiling, this would

attract their attention. Marie especially enjoyed comparing her created tilings to windows,

bookcases, ladders, stairways, farms, houses, robots, judges, cartoon characters, cats, and

flowers.34

Figure 58: Rachel‘s favorite from her created triangle tilings

Figure 59: 4.82 tiling chosen by both schools for the kitchen

Color elicited a variety of comments, which were of two types. (1) Specific

colors could be either good or bad. (2) Having many colors was good; insufficient

34 Many of these comments were not counted in the coding if they served neither to guide Marie in her

tiling creation, nor to evaluate her tilings. If we decided to count them, the Real World Connection

frequency would have a slightly higher percentage.

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coloring was bad. In the following example, students begin to discuss reasons from three

different aesthetic themes, including Color, Uniqueness, and Interconnectedness. The

tilings they discuss are Tiling J in Figure 60 below and Tiling P (a 4.82 tiling with green

squares and octagons in two shades of blue).

Scott: Why do you like J better than P?

Moses: It‘s more colorful and, it‘s more, …

Rachel: It‘s more unique. (I like that.)

Moses: … not co-, yeah it‘s more unique …

Moses: … and it looks like a pathway around.

Scott: How is it unique?

Rachel: If it was just, if it was just maybe uh like a purple background it

w-, it wouldn‘t be as interesting, but it‘s all these different colors

for (that background).

Figure 60: Tiling J in the Comparisons in Pairs task

Complexity. People seemed to think that tilings ranged from too simple to too

complex. The best tilings were somewhere in between. Overly simple tilings were

considered boring. The children sometimes said that anybody could do such tilings.

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Simplicity could also be a good quality, when appropriate. For example, students found

the 4.82 tiling in Figure 59 above to be ―kinda basic.‖ But they also felt that kitchens

should have simpler tilings, so this tiling was appropriate for that purpose. More often,

everyone seemed to prefer tilings that were moderately complex. Such tilings either had a

variety of shapes, or else the shapes were used in a complex way, such as the L-tromino

tiling in Figure 61 below, which had dilation symmetry that the students at the charter

school got excited about.

Figure 61: L-tromino tiling perceived to have pleasing complexity

Differences. Different students had different aesthetic preferences. During the

individual interviews, Kelsey only made four aesthetic comments (three from the

Alternation theme and one from the Uniqueness theme). Rachel made 14 aesthetic

comments from 10 different themes. The other students fell between these two extremes.

No single aesthetic theme was used by all students during the individual interviews. Most

students showed no clear preference for any particular theme, though Kelsey‘s comments

were mostly about alternation and half of Marie‘s comments concerned real world

connections.

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There were no clear preferences when students perceived more than one aesthetic

quality in a tiling. Choices had to be made between tilings with different aesthetic

qualities. There was no hierarchy of themes to guide these choices. Sometimes students

preferred a real world connection over having many colors. Sometimes they preferred a

tiling with good color over one with a real world connection. The children disagreed

among themselves much of the time as to which tilings were aesthetically the best, often

because of these conflicting aesthetic possibilities. But in spite of their disagreements,

they tended to use the same themes of Color, Complexity, and Real World Connection

most of the time to defend their choices.

Students were not always in disagreement. They frequently agreed when a tiling

was ―boring‖ or when it was colorful. Both schools independently chose the tiling in

Figure 62 below as their favorite of the seven Alhambra tilings. Both groups cited criteria

from the themes of Energy and Color to defend this choice. Both groups also

independently chose the Escher tiling in Figure 63 below as one of their three prize-

winners out of the 54 tilings they considered. Both groups cited criteria from themes of

Color, as well as the fantasy/mythological element, to explain this choice. Both schools

were impressed with the way differently colored animals faced in opposite directions in

this tiling.

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Figure 62: The favorite Alhambra tiling at both schools

All M.C. Escher works © 2011 The M.C. Escher Company - the Netherlands.

All rights reserved. Used by permission. www.mcescher.com

Figure 63: Escher‘s ―Regular Division of the Plane #66,‖ a favorite tiling at both schools

Some other common preferences are illustrated in Table 7 below, along with the

aesthetic themes cited to explain the preference. In each case, the tiling on the left was

unanimously preferred at both schools over the tiling or tilings on the right during direct

comparisons, though not for the same reasons. Note that in the first pair the theme of Real

World Connection was used to select the first tiling, not the second; it reminded the

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children of waves and a grandfather‘s bathroom tiling. Another unanimous preference is

illustrated in Figure 54 on page 183.

Table 7: Some unanimous preferences at both schools during Comparisons in Pairs

Preferred Not preferred Aesthetic themes

Charter school: Color, Complexity

Private school: Energy, Real World Connection

Both schools: Color

Private school: Interconnectedness, Energy

Charter school: “It’s fragments …”

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Mathematicians’ Evaluation Criteria

I only interviewed two mathematicians informally, but their responses can still

throw some light on interpreting children‘s mathematical aesthetic. These interviews

were informal because the goal of this research was not to evaluate mathematicians‘

aesthetics. Other studies referenced in my theoretical framework have already

investigated this question. The goal of interviewing two mathematicians was simply to

validate the value of the children‘s evaluation aesthetics. I think it is important to place

children‘s criteria in the context of the aesthetic criteria valued by the mathematics

community. This helps answer my research question ―What mathematical value is there

in children‘s aesthetics of tessellations?‖

For the most part, mathematicians used the same aesthetic themes as the students,

but tended to use the themes very differently. As Table 6 above shows, the only

significant aesthetic theme not used by mathematicians was the Energy theme. Students

frequently were impressed by the energy certain tilings seemed to possess. Students used

terms such as ―crazy,‖ ―lightning,‖ ―weird,‖ ―shocking,‖ ―energetic,‖ and ―exploding,‖ as

well as several invented words like ―Pfoom!‖ to describe tilings that made them excited.

The only significant aesthetic theme not used by the children was Randomness.

According to the mathematicians, randomness was usually appealing, but sometimes not.

This might apply to complete randomness or to semi-randomness where there are

elements of order and chaos combined. Either could be appealing or unappealing to the

mathematicians. Children never explicitly mentioned randomness in their evaluations,

though they did sometimes talk about the appeal of patterns.

Aside from Energy and Randomness, children and mathematicians shared the

same aesthetic themes. The difference was in emphasis. Whereas children turned mainly

to real world connections, color, and complexity for their aesthetic criteria,

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mathematicians turned mostly to complexity, surprise, and dimensionality. The children‘s

themes of real world connections and color were not frequently mentioned by

mathematicians, and the mathematicians‘ theme of surprise was not often brought up by

children. Mathematicians were especially appreciative of tilings that were ―thought

provoking.‖

The mathematicians used the Symmetry, Validity, Folds, and Connectedness

themes much more often than the children. The children used the themes of

Interconnectedness, Units, and Alternation much more often than mathematicians.

Both children and mathematicians shared the importance of complexity and

dimensionality. The only difference in the ways they approached these two themes is that

one of the mathematicians frequently brought up the lack of two interesting dimensions

in the tilings, a quality never considered by the children or by the other mathematician.

Both children and mathematicians shared a fascination for tilings that had the illusion of

having three dimensions.

Recall from the theoretical framework that mathematicians in general usually

identify criteria of significance, simplicity, surprise, connectedness, and visual appeal in

defining a mathematical aesthetic. Mathematicians‘ preference in this study for the

themes of Complexity and Surprise would align with the mathematical aesthetic criteria

of simplicity35 and surprise discussed in the theoretical framework. Connectedness was

noted by mathematicians five times. It was difficult to notice criteria of significance

because the tilings were not mathematical theorems that would contribute significantly to

the field. However, the theme of Validity may have stood for the place of significance in

these evaluations. Mathematicians mentioned the Validity theme eight times—the fourth

35 The Complexity theme contains aesthetic criteria of being simple as well as criteria of being

complicated.

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most common theme for them. All of the criteria could probably be included in the

domain of ―visual appeal,‖ because tessellations are aesthetic primarily because of their

visual qualities. Symmetry is an especially important quality for visual appeal and was

noted by mathematicians much more often than children. The one theme that

mathematicians used with great frequency that does not seem to line up with the five

criteria is Dimensionality. However, Dimensionality was frequently brought up because

tessellations are supposed to be two-dimensional, and one-dimensional patterns or

apparent three-dimensional patterns are breaking this expectation of what a mathematical

tessellation is defined to be. This criterion may therefore be related to significance and

surprise.

Note that it is not so clear how to relate the children‘s preferred criteria of real

world connections and color to the five mathematical aesthetic criteria, except in the way

all criteria are related to visual appeal. Units and Alternation are mathematically

important themes, but they are mathematically simple. They are therefore probably more

important to children, for whom they are still fresh, than to mathematicians, for whom

these ideas are well known and therefore less beautiful. It is not surprising that children

used themes of Units and Alternation far more often than mathematicians.

Children’s Creation Criteria

In addition to the verbal remarks made by students in the evaluation of tilings,

there were indications that students were also guided by aesthetics in the creation of their

tilings. This is the generative role of mathematical aesthetics. They rarely talked about

these driving forces, though this generative aesthetic sometimes became explicit when I

asked them why they preferred certain tilings they had just made.

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By far the most important apparent aesthetic in the creation of children‘s tilings

was visual symmetry. Because symmetry is both an aesthetic criterion and a

mathematical (cognitive) property, I discuss it later. (See Role of Symmetry on page 210.)

Aside from symmetry, there is evidence that children also used criteria of complexity,

uniqueness, alternation, and units in creating their tilings. All four of these qualities were

also themes expressed verbally during the evaluation tasks.

Students had no control over the color of the tiles in their creation tasks. However,

in the Regular Polygons task, the more prototiles they used, the greater the variety of

color. Students mentioned this as a reason for preferring certain regular polygon creations

over others.

Michelle seemed to prefer radial patterns when she could make them. This may

have been a strategic choice or an aesthetic choice. Other students seemed to prefer

making periodic patterns, but they sometimes made radial patterns. Students‘ strategic

choice to structure tilings in rows and columns may also have been aesthetic.

In fact, any of the children‘s strategic choices can be seen as aesthetic as well as

strategic. As I discussed in the theoretical framework, our aesthetic cognition is what

drives us to make our mathematical choices. This aesthetic is developed by our

mathematical experience. However, it is certain that complexity, uniqueness, alternation,

and units were considered aesthetic by the students because these qualities were also

expressed in their tessellation evaluations. I will discuss these four qualities in more

depth.

In addition to these criteria which apply to infinite tessellations, students also

seemed to favor convexity when creating finite patterns.

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Complexity. Students tried to make tilings that were neither too simple nor too

complicated. Moses began with the standard tiling for rhombi with all the tiles in the

same orientation. When he saw that it was possible to complicate this pattern by

alternating the orientation of the rows, he immediately changed his structure, even

thought this created difficulties for his perceived border problem. The O-tetromino is

simply a square and all the students knew it could be tiled as a regular 44 tiling. Kelsey

decided to complicate her tiling by staggering the squares in a brick wall type tiling,

which she explicitly said she preferred. A clear example where a student chose moderate

complexity is in Kobe‘s three I-tromino tilings in Figure 64 below. The first one was the

one at the bottom left—the standard edge-to-edge tiling. The second one was the one at

the top left with a moderately complex 3×4 unit laid out in alternating orientations. The

third one was the one on the right, which has a 7×12 unit and is so complex that he could

only create one unit. Kobe envisioned repeating this 7×12 unit without alternating the

orientation. Kobe explicitly preferred the second tiling saying the first was ―too simple,‖

and the third was ―too hard to make.‖ There were few instances where students preferred

creating random tilings. If they knew how to make a non-random tiling they usually did.

This may be partly because randomness is complex. One example of creating a random

tiling despite an awareness of a periodic tessellation would be Kelsey‘s attempts to make

a rhombus tiling. She seems to describe the standard tessellation and attempts to draw it,

though her drawing is not clear. (See Figure 65 below.) This partial understanding leads

her to attempt the standard tiling twice, but each time she creates a semi-random pattern

instead, apparently because of focus on local fit rather than global pattern.

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Figure 64: Three I-tromino tilings of increasing complexity by Kobe

Figure 65: Kelsey‘s rhombus tessellation drawing

Uniqueness. Students deliberately tried to create tilings that were different from

what they had seen or created before. Especially during the open-ended Regular Polygons

task, students shied away from familiar tilings and tried to create something new. For

example, early in the Regular Polygons task, Moses put two octagons and a square

together. This could have been the beginning of the semi-regular 4.82 tiling, and in fact

Moses said, ―Yeah I see a pattern that I can make with that.‖ But rather than pursuing this

203

pattern, he tried to create something with greater diversity of tiles. For the next 18

minutes, Moses struggled unsuccessfully to create a more complex pattern, before finally

creating the 4.82 tiling just as time was up. The reason for avoiding this tiling may have

been its simplicity or its familiarity. Children from both schools confirmed during the

sixth interview that this was a familiar tiling. Kelsey‘s creation of a brick wall tiling with

the O-tetromino was probably also partially motivated by familiarity with the regular 44

tiling. Marie made two parallelogram tilings, which were identical except for orientation.

She preferred the one with tiles oriented vertically because it was more unfamiliar and

harder to realize in the computer environment.

Alternation. On several occasions, students took pleasure in creating alternating

patterns, perhaps as a way to make the tilings unique or more complex. Kelsey preferred

her alternating row tiling of acute isosceles triangles over the tilings she created using

rectangle or parallelogram units for the other two triangles in Set E. (See Figure 66

below.) She thought it was ―cool‖ how the triangles pointed up and down, ―like one

upwards and then one down and one up and one down.‖ Moses explicitly preferred

alternating the orientation in his rows of rhombi.

Figure 66: Kelsey‘s tilings of the acute isosceles, right scalene, and obtuse scalene

triangles

204

Units. When students noticed certain finite patterns appear in their construction,

they would deliberately create more of those patterns, sometimes constructing the entire

tiling with the unit they discovered, as Kelsey did in the two tilings on the right in Figure

66 above. When students saw hexagons in their evolving triangle tessellations, they

would frequently point them out and begin creating more hexagons to extend the pattern.

In the case of the triangle tessellation, unitizing was not needed to create the tiling; the

units appear to have been used mainly for aesthetic reasons. Even if their tiling was

complete, students sometimes looked for unit patterns in their completed tilings,

especially if the tilings were random. Marie placed the eight triangles in the second Finite

Tiling problem without unitizing. But when she was finished, she reanalyzed the tiling as

four diamonds. See also Rachel‘s analysis of her semi-random tiling in Figure 70 on page

212. Unitizing is related to the aesthetics of fit, which all students evidenced. After initial

hesitation at being chosen to do interviews that had something to do with mathematics,

Marie decided that she thoroughly enjoyed making tilings. They reminded her of puzzles,

which she liked. Composition and decomposition of shapes into other shapes is not

simply a cognitive skill children need to learn; it is also an aesthetic process that they

enjoy. This includes putting tiles together to make larger pleasing units as well as

breaking them down into smaller subunits, as a few students did. This generative

aesthetic of units (as well as fit) would be an example of the motivational role of

aesthetics as well.

Convexity. Tilings are infinite and are therefore trivially convex.36 But children

must first create finite patterns as they explore how to create their tessellations. Students

seemed to prefer convex patterns when creating finite patterns. Units used to create

36 A set is convex if the line segment joining any two points of the set always lies entirely within the set. A

tessellation occupies the entire plane, which is trivially convex.

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tessellations were always convex. When students sought a stopping point to explain their

tessellations, they would often complete a convex pattern, if possible, though there is no

particular mathematical reason to do so. Such patterns are easier to relate to the infinite

tessellation. I would frequently push them to extend their tessellations beyond these

convex patterns in order to test their vision of how the tessellation continues.

Real world connections and color have less empirical support from my data as

creation criteria, but are nevertheless likely factors in children‘s creation criteria because

they were such important factors in children‘s evaluation criteria. The real world

connection of honeycombs was noted by four of the children for the hexagon tiling and

may have had aesthetic value. Children could not color, but the aesthetic appeal of having

many colors motivated some children to include as many polygons as possible in the

Regular Polygons task, which increased the complexity of the patterns.

Non-tessellation aesthetic events. On a few occasions, children began aesthetic

tasks that were not tessellations. Kobe, towards the end of one interview, decided to make

a man with the polygons at his disposal. He knew this was not a tessellation. Michelle put

all five tetromino pieces together to see if she could make a convex shape with them.

This was a finite tiling and a mathematical task, as well as an aesthetic task, though it was

not part of the task of creating monohedral infinite tessellations with tetrominoes.

INTERACTION OF THE TWO PERSPECTIVES

Some analyses must be carried out from the cognitive and aesthetic viewpoints

simultaneously. (1) I look at how the tool influences the motivating aesthetic factors of

the task and also how it both helps and constrains the cognitive possibilities. (2) I look at

symmetry results, which can be understood both from a visual, aesthetic viewpoint and

from a mathematical, cognitive viewpoint. Through tools and symmetry, we see how the

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aesthetic and cognitive viewpoints combine for a more complete understanding of

children‘s understanding of tessellations. (3) I also look at how the previously discussed

aesthetic creation criteria relate to the cognitive perspective.

Tasks and Tools

The tool and other aspects of the task inevitably both limit and help the aesthetic

and cognitive possibilities for tiling in various ways. By imposing the constraint of

making a mathematical tessellation, the child‘s aesthetic was partially constrained by the

mathematical aesthetic of the researcher. I write ―partially‖ because students sometimes

disregarded these constraints. The children‘s cognitive abilities were influenced by the

tool, which sometimes limited them from actions they may have wanted to explore, and

sometimes helped the child to see possibilities they may not have seen otherwise.

Some tools made certain aspects of tiling easier and other aspects more difficult.

At least one student wanted to select and copy units on the computer, but this was not a

possibility in the software environment that we used. This limitation imposed a constraint

on the complexity of possible tilings. Another example of a limitation was color. No tool

allowed students to choose the color of their tilings, and therefore questions of color

rarely entered into consideration when students created their tilings. The only exception

was the Regular Polygons task, where each prototile was a different color. Some students

tried to use as many prototiles as possible in order to make the tiling colorful.

Students frequently gave different responses according to which tool they were

using. For example, when asked if she could tile with the obtuse scalene triangle, Kelsey

at first responded yes. She vaguely described how the tiles could be matched up with

congruent sides touching. When asked to draw her vision, she began the drawing in

Figure 67 below, which at first she thought would work. But then as she continued to

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draw and erase, she decided the tiling would not work. Finally, when she attempted the

tiling in the computer environment, she began by placing two triangles together in such a

way that they formed a parallelogram. She immediately saw she could continue such a

pattern and created the standard parallelogram unit tiling in Figure 68 below. This is not a

case of tiles pre-structuring the tessellation because there are many ways of putting these

triangles together. However the way the software environment worked did help Kelsey to

see structural possibilities that she probably would never have seen in the pencil and

paper environment.

Figure 67: Kelsey‘s attempt to draw a tiling with an obtuse scalene triangle

Figure 68: Kelsey‘s tiling of an obtuse scalene triangle in the software environment

When Michelle was asked if she could tile with the L-tromino, she described the

creation of rectangular units. However, when she drew the tiling on dot paper, she made a

random tessellation. When asked if there was another way to tile, she demonstrated the

beginning of a five-fold symmetric radial pattern using blank paper instead of dot paper.

(See Figure 42 on page 152.) Each tool seemed to solicit a different type of tiling. The

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variety may also have been due to an attempt to create unique answers, but the five-fold

symmetric radial pattern was only possible on the blank paper and she deliberately chose

the blank paper because she knew she could not realize her vision on the dot paper.

The nature of the tool influenced the type of discoveries children were likely to

make. In a pattern block environment, children always matched up sides. In the computer

environment, the tiles behaved as if the corners were magnetized, and therefore students

sometimes attached them at the corners rather than the sides. This behavior in the

software environment led Rachel in Set G to create a row of hexagons touching only at

the corners, from which she constructed the 2-isogonal 32.6

2/3.6.3.6 tiling in Figure 69

below by filling the gaps with triangles. The environment influences more than just the

result; it also influences thinking. Rachel stated during this task that the tiling must be

edge-to-edge, and she explained why, even though it was the fifth interview and she had

seen and created non-edge-to-edge tilings in the previous interviews. It is likely that the

magnetic behavior of the tiles in the computer program was influencing her current

thinking.

Figure 69: 2-isogonal tiling creation by Rachel in the software environment

In the software environment, rotating a tile was easy, but not nearly as easy as it

was for pattern blocks. In the software environment, students had to make a deliberate

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decision to rotate a tile. Pattern blocks spun so easily that students were constantly

turning them whether they needed to or not. This almost surely had an impact on the

results. For example, most of the students did not recognize the right angle in the Set E

right scalene triangle task because the prototile was at an unusual orientation on the

computer screen. As a result, most students did not realize they could create rectangular

units until they began playing with the computer tiles and accidentally created a

rectangle. It is quite likely that this discovery would have happened more quickly if the

tiles had been made of wood or plastic. The original non-standard orientation would not

have been kept because the students would have been constantly turning the prototile to

study it. No student failed to recognize the right angles in the polyomino tasks because

these were always introduced as plastic tiles. As in the software environment, the

prototile was static during the oral and drawing phases of the finite tiling tasks and had to

be rotated mentally. The constraint of doing this rotation in their heads during the oral

and drawing phases created difficulties for the students that they were able to overcome

when they could handle a pattern block during the final construction phase.

During the oral descriptions, students were limited by their inability to use the

environment to help their thinking. As a result, patterns described orally were simple and

local, with little or no concern for symmetry. When students could draw or construct,

they developed more complex tilings, sometimes in an attempt to be unique and creative.

Outhred and Mitchelmore had noticed that ―drawings may be seen as reflecting, or at

least as being guided by, students‘ mental images of the array structure‖ (2000, p. 146).

However, the act of drawing may sometimes bring students quite far from their initial

ideas. Students sometimes even said they saw new ideas as they drew their tessellations.

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Oral descriptions, though limited, may be much closer to students‘ initial mental models

of tessellations than drawings.

There were many other ways the tool shaped student thinking. The squares in the

plastic polyomino models seemed to suggest to some students that the task was one of

tiling units of squares rather than a single shape. The complexity of the Polydrons

prevented one student from understanding that the mathematical question was one of

tiling pentagons and that the tabs therefore had no mathematical importance. Dot paper

constrained the positioning of the tiles in such a way that small gaps and overlaps were

impossible; tiles might leave a large hole, but not a small gap. The software had an

automatic zoom which reinforced the endless, borderless nature of the plane. Every tool

shaped thinking in some way.

Context. The task shaped the children‘s thinking. The context of the task also

influenced the children‘s aesthetic choices. This was particularly evident during the sixth

interview where the context of tiling certain rooms of the house influenced which tiling

they chose. At both schools, the choice for kitchen floor was swift and unanimous for the

semi-regular 4.82 tiling. Both groups felt that this type of tiling was appropriate in the

context of kitchens. During another task for the sixth interview without the house context,

I asked students which of two tilings they preferred. Michelle answered, ―Um. It depends

for what. What.‖ In some ways aesthetics, like cognition, is situated.

Role of Symmetry

There is evidence that students used symmetry in their creations, usually in the

visual, aesthetic sense, which understands symmetry as a mathematical object, but also in

the transformational sense, which understands symmetry as a mathematical process.

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A coding analysis was made of the symmetry of all finite patterns made with

more than six tiles. Analysis was limited to finite patterns because it was not always clear

exactly what infinite tessellations students intended by their patterns. Analysis was

limited to patterns of more than six tiles because small patterns frequently have

unintended, accidental symmetry. It was found that 77% of all students‘ patterns with

more than six tiles were symmetric.

The word ―symmetry‖ was used by one student on the pre-test, but was never

used in the interviews. However, other terms such as ―pattern‖ were sometimes used to

refer to visual symmetry, and words like ―copy‖ were used to describe transformational

symmetry. Explicit descriptions of transformational symmetry were fairly rare. Moses

was one of the few to describe how he achieved reflection symmetry, noting that he was

not sure if the dihedral squares and triangles tessellation could work if he did not impose

reflection symmetry.

Scott: Now how do you know when to put a triangle and when to put a

square?

Moses: Um. I can see, where I should do it and then I, and then, so, then I

copy that on the other side …

Scott: Oh.

Moses: … to get a pattern.

Scott: [echoing] Copy it on the other side to make a pattern.

Moses: So like, I do it on one side then, and I copy it on the other.

Scott: Ok. Now why do you do that?

Moses: Because I, I like more like patterns, I‘m (good) ...

Scott: Because you like patterns?

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Moses: Yeah I like patterns better than, regular sh-, [unintelligible

mumbling]

Scott: Would it work if you didn‘t copy the pattern? If you didn‘t copy

the same thing on both sides would it still work?

Moses: I don‘t think, don‘t, know.

Scott: You don‘t think so?

Moses: I don‘t know.

Students rarely created random tilings if they could find a way to make a

symmetric tiling. If they did make a random tiling, they often tried to identify patterns

contained in the tiling. Michelle first made the semi-random rhombus tessellation in the

photo on the left in Figure 70 below. She then explained some of the patterns she

perceived in it and separated the tiling into four parts, three of which had their own

patterns (middle photo). She then identified the top left pattern as her favorite and

showed how it could be extended to make a symmetric tiling, in the photo on the right.

This final symmetric tiling was described as ―more easy‖ and ―you could do this forever

and ever.‖

Figure 70: How Michelle extracted a symmetric tiling from a semi-random one

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Once students obtained a symmetric pattern with a significant number of tiles,

they usually kept the pattern symmetric as it grew, which demonstrates intent to use

symmetry. It was rare for a student to lose any form of symmetry once it was achieved in

a sizable pattern.

All students used a variety of symmetry in their patterns. Analysis of symmetry

codes revealed no clear preference for any one type of symmetry. All students created

patterns using reflection symmetry, rotation symmetry and combinations of reflection and

rotation symmetry. All students created patterns at some point with 6-fold symmetry,

with order-2 rotation symmetry, and with vertical reflection symmetry. Half the students

also used horizontal reflection symmetry. There was only one instance of a student using

oblique reflection symmetry.

The type of symmetry appearing in children‘s patterns was often constrained by

the tile. The parallelogram naturally led to patterns with order-2 rotation symmetry.

Equilateral triangles, with or without squares, led to 6-fold symmetric patterns more often

than other shapes. Vertical reflection symmetry was usually achieved with isosceles or

equilateral triangles.

An attempt was made to analyze whether students preferred tilings with greater

symmetry over tilings with less symmetry in the aesthetic evaluation tasks, but the results

were inconclusive. See Appendix G for details.

Symmetry and confidence. Students expressed confidence about most of their

tilings. But students were even more confident of tilings that incorporated symmetry. For

example, one of the reasons Michelle preferred a six-fold symmetric pattern of squares

and triangles over a random tiling of squares and triangles was because she was more

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confident the six-fold symmetric pattern would work. This result agrees with the similar

finding in my pilot study (Eberle, 2010).

Codes for symmetry and confidence were compared in the data for this study. Of

the 88 times students created tilings with some form of symmetry, they displayed strong

confidence in their tilings 85 times (97% of the time). Of the 11 times students expressed

some form of doubt about their created tilings, the tilings lacked symmetry 8 times (73%

of the time).

Note that confidence did not correlate so highly with mathematical accuracy. For

example, oral and drawing phases were coded according to the accuracy of the student‘s

understanding of how polygons fit together. (Saying that only four triangles can fit

around a common vertex is not mathematically accurate, as students discovered during

the construction phase.) Of the 110 accurate descriptions of fit, students showed strong

confidence of their answer only 75 times (68% of the time). Of the 65 times students

expressed doubt over their answer, their answers were mathematically inaccurate only 30

times (40% of the time). Similar percentages can be found for other measures of

accuracy. Symmetry was a much greater predictor of confidence than mathematical

correctness.

The Cognitive Role of Other Aesthetic Criteria

As discussed in Children’s Creation Criteria (page 199), children‘s aesthetic

creation criteria played an important role in their creation of tessellations. These criteria

were clearly aesthetic. Most of them also seemed to play a cognitive role in the children‘s

work. We have seen how symmetry is simultaneously an aesthetic and cognitive factor in

tessellations. Units are also both aesthetic and cognitive. Units played a clear aesthetic

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role, as demonstrated by their use as an aesthetic evaluation theme. Units also played a

cognitive role, as demonstrated by their use as a strategy for creating tessellations.

The aesthetic theme of Uniqueness had the cognitive role of seeking new

mathematical structures. If students had not sought unique structures, they would have

only repeated familiar cognitive structures rather than explored ideas they had not yet

assimilated. By pursuing unique patterns, students created a wider variety of tessellation

structures than they would have otherwise.

The aesthetic factor of complexity is related to that of uniqueness. Complexity led

students to create tessellations that pushed their cognitive limits without pushing so far

they could no longer grasp the mathematical structure of their creations. Kobe‘s I-

tromino tilings in Figure 64 on page 202 are an example of this. Kobe had already

demonstrated an understanding of the structure of square arrays and parallelogram arrays

in earlier tasks. The simplest I-tromino tiling, which was a simple rectangle tiling,

presented no new mathematical structure for his cognition. His third I-tromino tiling was

so complex that it was probably difficult for him to visualize as an infinite tiling. The

second tiling, which he found the most pleasing, presented a rich variety of mathematical

structures which were within his cognitive grasp: unitizing, alternation, row and column

structure, and so on. His ability to visualize this tiling was clear from his extended

drawing.

Alternation contributed to the complexity of the patterns. Rhombus patterns with

alternate orientations of the rows were more mathematically complex than those in which

the tiles were all in the same orientation. Such patterns were therefore somewhat more

cognitively demanding. In the case of triangles, the students had no choice concerning

alternation; simple rows of triangles can be formed only by alternating the orientations of

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the triangles. The children found this alternating orientation striking. Kelsey, Marie, and

Rachel commented on this alternating feature of isosceles or equilateral triangle

tessellations as a necessary part of their structure.

The aesthetic factor of surprise, of course, sometimes served the important

cognitive role of introducing cognitive dissonance, leading to new insights concerning the

structure of the tessellations. Moses, for example, was clearly surprised when he

discovered that six triangles fit around a vertex, forming a hexagon unit. He went on from

this discovery to use hexagonal structures in his triangle tessellation.

Validity has clear mathematical importance and is therefore mainly a cognitive

factor. Children were attempting to make mathematically valid tessellations that

respected the three criteria laid down at the beginning of the interview. And yet validity

was also an aesthetic factor, as evidenced by its occurrence as an aesthetic theme during

the evaluation tasks.

Connectedness was another aesthetic theme that served a cognitive role during

tessellation creation. For example, Michelle used the connection between her regular

polygon tiling and her triangle and square tiling to demonstrate the validity of the former.

(See Figure 52 on page 178.) The aesthetic quality of this connection was probably

evident in the excitement Michelle showed when she made the discovery of this

connection.

Even seemingly non-mathematical aesthetic criteria such as real world

connections and color may have occasionally played a cognitive role in the children‘s

work. The real world connection with honeycombs served as a confirmation of the

mathematical validity of children‘s hexagon tiling. Color may have been helpful to the

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children during the Regular Polygons task in order to see symmetry and structure more

easily.

DIFFERENCES BETWEEN SCHOOLS OR ACROSS TIME

I analyzed the coded data to discern any special differences between the two

schools or between earlier and later interviews.

Differences Between Schools

I was able to discover very few differences between the types of responses at the

two schools in this study. Only two students traced around the prototile to create precise

drawings, and both were from the private school. They did this 15 times. Students at the

private school showed more awareness of the fact that freehand drawings are only a

model of the tessellation and may not be accurate or to scale. These school differences

may be due to the greater emphasis on precision in geometry in the French curriculum, an

emphasis made even in the early grades. However, both schools expressed equal concern

for accuracy in their final results.

No other differences were found.

Differences Across Time

The format of this study is a clinical interview. It was not intended to be a

teaching experiment. It is therefore important to check that no major changes in thinking

were observed during the interviews. It was expected that some changes would occur as

the students got used to the format of the tasks, and some learning was inevitable, but

there were no expectations that large changes in cognitive or aesthetic understanding of

tessellations would happen. Consistency across the interviews can also serve as evidence

for robustness of thinking.

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Another concern was that the final interview was in an entirely different format

from the first five. Students may express aesthetic values differently simply because of

the new format, but these differences were not expected to be great. Students could have

expressed different aesthetics because the sixth interview focused on evaluation, whereas

the other five interviews focused on creation. However, I found that the aesthetics

expressed during the final interview appear to be fairly similar to the aesthetics expressed

during the other interviews, as described below.

Cognitive perspective. Responses to the dihedral squares and triangles tasks

were compared between the second and fifth interviews. Very little change in thinking

was noted. In fact, other than a greater readiness to abandon tilings that were not

working, each child‘s thinking was remarkably similar during both interviews.

Codes related to the cognitive perspective were analyzed for change across the

first five interviews. There was very little change other than certain tendencies that were

expected as the children got comfortable with tiling. Some changes were inevitable

simply because different tasks elicited different types of thinking. See Appendix F for

details.

The only clearly discernible changes were in codes that concerned accuracy. The

two children who traced tiles during the drawing phase did so only in the first interview.

(Marie used a tile to help her draw in the fourth interview without actually tracing around

it.) Children voiced questions about accuracy and how imprecision in drawing might not

reflect mathematical reality 28 times in the first interview, twice in the second interview,

7 times in the third interview, and not at all in the fourth or fifth interviews. The last two

interviews were almost entirely on dot paper and in the software environment, so there

were not many opportunities to discuss accuracy in drawings. This change in tools,

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together with increasing comfort with the clinical interview process, may account for

most of the change in expressions about accuracy.

There was no significant change in codes that measured various aspects of

understanding, confidence, or use of symmetry. The examples of children‘s finite,

process-oriented understanding of tilings all came from Set A‘s rhombus and triangle

tasks. (See pages 159 to 168.) Otherwise, there was no significant change in codes related

to types of reasoning or strategy.

Aesthetic perspective. Because the sixth interview was devoted entirely to

aesthetic evaluation, most of the aesthetic data came from that interview. But aesthetic

questions were also asked during Interviews 2 to 5, and students sometimes made

unprompted aesthetic comments throughout all interviews. It is important to ask if there

is evidence of aesthetic change between the cognitively oriented interviews, which

focused on creation of tessellations, and the final sixth interview, which focused on

evaluation of tessellations. If there is a difference, then the sixth interview may be

reflecting aesthetic values learned during the first five interviews, or aesthetic values

related to evaluation that do not apply to creation tasks.

The graph in Figure 71 below shows the number of codes in each of the 18

aesthetic themes from three sets of data: (1) mathematicians, (2) children during the first

five interviews, and (3) children during the sixth interview. With few exceptions, the

differences were as expected. Color was much more important during the sixth interview.

This was expected because the children had almost no control over color during the

creation tasks. The Interconnectedness theme was expressed much more often during the

sixth interview. This is expected because several of the tilings in the sixth interview did

have a complex path-like appearance. Such tilings are complicated and not likely to be

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created by children during their creation tasks, though some expressed this aesthetic for

tilings made with the S-tetromino. Units, Symmetry, and Alternation were mentioned

more frequently during the creation tasks, probably because these were important

strategies used in the creation of children‘s tessellations. It is nevertheless noteworthy

that children did not explicitly mention symmetry during the sixth interview, a theme

which the mathematicians believed to be very important. The only result in Figure 71 that

seems unexpected is the more frequent mention of the Culture theme during the creation

tasks.

Figure 71: Comparison of mathematicians‘ and children‘s aesthetic themes

I calculated the correlations between the three sets of data of the number of

occurrences of each code. The correlation between the creation tasks and the sixth

interview was .44. This seems to me a very high correlation considering the very different

nature of the interviews and the expected differences discussed above. It is in fact much

higher than the correlation between codes for mathematicians and the sixth interview

(.04) even though both groups evaluated the same tilings. In other words, children‘s

expression of aesthetic values seems to have been moderately consistent across the

interviews, whereas mathematicians‘ aesthetic values were quite different from the

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children‘s, especially during evaluation tasks. In fact the correlation of .04 between

mathematicians and children for the evaluation tasks suggests that mathematicians and

children approached the evaluation tasks in entirely different ways, even though they

used largely the same aesthetic themes. It is noteworthy that mathematicians‘ evaluation

aesthetic has a moderate correlation with the aesthetics expressed in children‘s creation

tasks (.22). This may suggest that children in the creation tasks were more focused on

mathematical criteria that mathematicians notice even when simply evaluating others‘

tessellations. Correlations with mathematicians‘ data must be used with caution because

there were only two interviews with mathematicians and they were of an informal nature.

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Chapter 5: Conclusions

There are few geometric topics as rich or as appropriate as tessellations for the

teaching of the geometry curriculum in late elementary school. This exploratory study

supports this claim and lays a foundation for further study of how children think and

learn about mathematical tessellations.

DISCUSSION OF THE RESULTS

Children‘s thinking is complex. However, we can describe children‘s

mathematical understandings of tessellations by identifying certain tendencies in their

early understanding, or certain phenomena that are true for many children and which we

should therefore be aware of. This description becomes more complete if we come at it

from more than one direction. In this study, I have analyzed children‘s understanding of

tessellations through a cognitive lens and through an aesthetic lens. The latter viewpoint

has been generally neglected in the literature, but in this study we see how the two

viewpoints together create a clearer picture of children‘s mathematical thinking.

Children’s Cognitive Understanding of Tessellations

The first part of the research question for this study is as follows:

What are children’s cognitive understandings of tessellations? What strategies

and types of reasoning do they use to create tessellations? What mathematical

concepts are elicited?

In Chapter 4, I identified some of the main components of children‘s initial

cognitive understanding of tessellations. Students used a variety of strategies to create

tessellations. These strategies were somewhat constrained by the shape of the prototile.

They also used a variety of reasoning to justify their tessellations. I look at some of the

mathematical concepts elicited on page 239.

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Among the more important conclusions are those concerning children‘s ontology

and epistemology of tessellations, and their understanding of the infinite space in which

tessellations are embedded. Ontologically, children see tilings as both an object and a

process. Epistemologically, children come to know tilings partly through their initial

ideas, and partly by subsequent constructions. Students understand the consequences of

tessellations‘ infinite extension in different ways.

Students sometimes see a tiling as a mathematical or concrete object and sometimes as

a process or action.

Some children did not understand that different finite tilings can represent the

same infinite tessellation. This seems to be partly true because students were focusing on

the process by which they made the tessellations. Students who described tessellations

primarily in terms of their own actions tended to see their tilings through the lens of the

actions that had created them. It was not obvious to these students, even after reflection,

that different actions could create the same object. For them, the tiling was understood as

a process or action, rather than as a mathematical object. Other students, who used more

descriptive terminology to explain their tessellations, tended to see their tilings as objects

independent of the actions they used to create them. Both types of students used both

active and descriptive wording in describing their tessellations, but when it came to

understanding their results, two students seemed to understand them primarily in terms of

actions and two others primarily in terms of final objects.

It is important for students to reach the point where they can understand a

mathematical concept in terms of either object or process, as needed. The evidence from

this study suggests that children who preferred the process understanding could not yet

conceptualize their creation as an infinite mathematical object.

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The situation is similar to that of other geometric constructions, which can be

conceptualized either as a series of actions or as the resultant constructed object. Both

ways of understanding a construction are necessary. A complete understanding of

geometric constructions or tessellations requires understanding the physical construction

process as well as the abstracted mathematical object that the actions represent.

The study of tessellations makes an ideal environment for children to construct

the idea that mathematical results may be independent of the processes that create them.

Tilings are a combination of initial ideas and further discovery.

Children‘s epistemology of tessellations is not simple. There rarely appears to be

any simple mental model in the students‘ minds prior to the construction of a tessellation.

Students have ideas concerning the tiling, and these ideas may have various levels of

clarity. The child may even believe that she knows the structure and appearance of the

tessellation. But the tessellation is not really known until it is realized in sufficient detail

via some construction tool. By the time this realization occurs, the initial idea the child

had has probably been modified. Children mentioned seeing new ideas as they made their

constructions. Drawings were inevitably more detailed than oral descriptions. Physical

constructions were more detailed still, and frequently different from the oral description.

The final construction is not independent of the initial idea, however. The child‘s tiling is

neither a pre-existing idea, nor a construction ex nihilo. It is a combination of certain

initial ideas about the structure and appearance of the tessellation and the final realization

constructed with tools. Sometimes children may initially have only a very hazy idea of

what the tessellation will be like, and then construct it until the pattern is clear. Such

constructions are perhaps better understood as the creation of a new understanding. Other

times children may have a very clear idea and the final construction is faithful to their

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idea. Even in this latter case, however, new structure becomes evident as the child sees

the actual appearance of the tessellation. There is no clear dividing line between these

two extreme cases of creation of a new idea and construction of a pre-existing idea. In

most cases, children had partial ideas that were then fleshed out through the construction

process.

In a certain sense, there is no point in time at which we can say, ―This is the

student‘s tiling.‖ Children‘s tilings are as much a process as they are a final construction

because the actual mathematical tilings are infinite. The tiling is never fully constructed;

the displayed finite construction can really only serve as an aid to describe what the

infinite mathematical object looks like. Even after construction of the pattern that

represents the tiling, the student may still not have a clear idea of the structure of the

infinite tiling. Consider for example Michelle‘s regular polygon tiling on the right in

Figure 52 on page 178. After she explained that the tiling was the same as the one on the

left, but with some triangles replaced by hexagons, it was still not clear to me which

triangles were to be replaced by hexagons, nor did it appear to be completely clear to her.

She assumed it would become obvious if she continued the construction.

Even if the student‘s ideas are clear, the finite pattern may or may not clearly

suggest to other people the structure of the infinite mathematical object. The pattern

displayed on the table or computer may be accompanied by verbal explanations of how

the student intends to extend the pattern to fill the plane. (For adult mathematicians, the

patterns may be accompanied by descriptions of the mathematical rules that will extend

the pattern.) Simple periodic tilings with small units will probably be clear to all with

little or no explanation. Radial tilings such as Michelle‘s, or tilings with large complex

units, may be interpreted differently by different people. As such, children‘s tilings are in

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some sense a social construct. Even more than other geometric objects, they cannot be

precisely realized in the real world. A tiling is an infinite mathematical concept that can

be communicated to others through visual and verbal description. The tiling that the child

displays is in one sense just a finite pattern and in another sense a representation of the

child‘s vision of how the mathematical tessellation can be constructed.

The child‘s initial understanding of the appearance and structure of the

tessellation may be vague or clear. If the understanding is clear and yet the constructed

tiling does not match pre-conceived ideas, then the child will experience cognitive

dissonance. Moses was the only one to show surprise that six triangles fit around a

common vertex. Other students did not notice this contradiction to their oral descriptions

of four triangles around a common vertex, even with prompting. It is likely that most of

the students had not thought through clearly exactly how triangles fit around a vertex.

Rachel was certain that equilateral triangles placed on the sides of a central triangle

would point left and right. When asked to demonstrate, she was surprised to find that they

formed a trapezoid in precisely the way she said they would not. These moments of

cognitive dissonance were rare during the interviews, partly because this was not a

teaching experiment and partly because these moments can only occur when the child has

a definite idea of what she expects to find.

In order to build up their intuition of what tessellations will look like before they

actually construct them, students need to practice, to play, to explore. By building up

their mental models, tilings can become more deliberate and initial ideas can be closer to

the constructed result. But construction will always be needed. Tiling is a verb as well as

a mathematical object. Even research articles on tessellations written by mathematicians

inevitably contain drawings to aid visualization. Hilbert succeeded in setting geometry

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free from the real world, but subjects such as tessellations are still very much grounded in

visual space.

Tessellations for children will probably never be fully interiorized as, say, mental

arithmetic might be. Tessellations will always need to maintain a tight contact with the

physical realizations from which students take their understanding. Children‘s

understanding of tessellations is intimately related to the action of tiling.

Students have limited understanding of infinite space.

Studies of children‘s understanding of infinity have been few and studies of

children‘s understanding of infinite space have been essentially non-existent. Though it

was not intended to be, this appears to be the first study of children‘s understanding of

infinite space at the concrete operational stage.

Previous studies of infinity (Fischbein et al., 1979; Monaghan, 2001) have shown

that students hold contradictory ideas of infinity because their ideas are based on

schemata constructed through experiences in a finite world. Students‘ ideas about infinity

change easily and are based more on a process understanding than on an object

understanding. These studies have focused on ideas such as numerical infinity and

continuity, but not spatial infinity.

I have shown that the observation that some children do not distinguish between

finite and infinite patterns is partly explained by process-oriented thinking. This

observation also shows that these students do not grasp that their finite patterns are

intended to represent infinite mathematical constructions. This failure to understand the

consequences of infinity limits these students‘ ability to the study of mere patterns, rather

than periodic tessellations. We already knew before this study that younger children at

the pre-operational stage cannot appreciate the infinite extension of tessellations. It would

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appear that at the concrete operational stage, students can appreciate that tessellations

continue indefinitely, but for some this understanding is not complete.

Other indications that children are struggling with the idea of infinite space are:

forgetting the infinity constraint; expressing completion with a finite pattern; and

showing discomfort over tiling in an empty plane without walls or reference points. Not

every piece of evidence applied to every student, but every student struggled with the

infinite space of the tessellations in some way.

Infinity is a fundamental concept and, like other concepts in mathematics, there is

no reason to postpone students‘ exposure to important mathematical concepts until they

reach calculus (Stroup, 2005). The infinite plane is an important aspect of Euclidean

geometry and tessellations are an ideal environment for allowing students to begin

exploring this idea for the first time.

Children’s Aesthetic Understanding of Tessellations

The second part of the research question for this study is:

What aesthetic considerations guide students’ thinking about tessellations?

How do they evaluate tessellations made by themselves and by others?

What mathematical value is there in their aesthetics of tessellations?

The aesthetic viewpoint has been largely neglected in the literature on children‘s

mathematical understanding. This study adds to the research that indicates that aesthetics

is not some optional epiphenomenon, but is an integral part of how mathematics is done,

including children‘s mathematics. Only the most rote, elementary mathematics can be

done without aesthetics, whether we are consciously aware of it or not.

I identified 18 aesthetic themes which categorized most of the verbal expressions

of aesthetics in this study. Nearly all of these themes were used by both children and

mathematicians. However, the children and the mathematicians used these themes in

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different ways and placed emphasis on them very differently. Mathematicians tended to

prefer themes that aligned with aesthetic qualities identified by previous researchers as

defining a mathematical aesthetic. Children tended to prefer other themes, principally

color and real world connections. Mathematicians‘ aesthetics are therefore not something

entirely foreign to children‘s experience, but they do reflect different awareness of what

is mathematically important. I conclude from this that mathematicians‘ aesthetics are not

something that children cannot appreciate, but are something children can learn as they

gain experience in the mathematical domain.

Children used aesthetics not only to evaluate tessellations, but also to create them.

As children created tessellations, they were guided in their creations by several aesthetic

factors, especially that of visual symmetry.

There is an implicit use of symmetry in most of the students’ construction and

evaluation of tilings.

In the study of tessellations, symmetry can be either a visual, aesthetic object or a

mathematical, physical process (Schattschneider, 1978a). Children tended to use

symmetry to guide their tiling creations. This was seen by the fact that once a pattern

attained significant symmetry, children took care to keep it symmetric as they expanded

it. In all, 77% of their finite patterns were symmetric. Children rarely created non-

symmetric patterns if they knew how to create symmetric ones.

When students create symmetric patterns, they are more confident that their

tilings are valid. This confidence is mathematically justifiable because the validity of

symmetric patterns is much easier to verify.

The mathematical role of symmetry in tessellations suggests a probable

mechanism for the role of aesthetic cognition in mathematical exploration. The aesthetic

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value of symmetry probably derives from the innate appeal of patterns. Mathematics

itself being the study of patterns, it seems natural that symmetry should be an essential

part of mathematics. Children, however, are not aware of this deep connection. At first,

they simply enjoy the pleasing quality the symmetry brings to their designs. However, as

they work with their designs they probably notice, at least subconsciously, that patterns

with symmetry are easier to analyze, hence the increase in their confidence that the

tessellations are valid. This increases their motivation to apply symmetry to future

tessellations.

Children also have a strong aesthetic preference for color. Certain colors are

appealing and a variety of colors makes the pattern more interesting. Color, however, has

little mathematical role in any of the tasks from this study. Mathematicians realized this

in their interviews. Mathematicians mentioned color as having aesthetic importance, but

rarely used it to evaluate the tessellations. If children work with tessellations beyond what

this introductory study could examine, they will probably discover that color does not

help them much with their mathematical task and will eventually turn to other aesthetic

qualities to guide them. Lehrer et al. (1998) found that children who studied the

mathematics of quilt patterns initially were attracted to the colors of quilts, but as they

grew in mathematical sophistication they increasingly expressed aesthetic appreciation

for symmetry, shape complexity, transformations, and the aesthetic constraints of certain

pattern choices. Mathematical aesthetics is not some vague force, but is a way of thinking

that is increasingly perfected as it is honed by our mathematical experiences. Aesthetics

guides our thinking, and our thinking about our mathematical experiences, in turn, shapes

our aesthetics.

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Conclusions about the types of symmetry children valued are difficult to confirm,

but I did notice that no tiling shocked them in the sense of Le Lionnais‘s romantic

aesthetic. This was not unexpected. Ideas about tiling cannot be disturbing if children are

just beginning to develop and formalize their ideas about tiling. I observed no cases of

cognitive dissonance as children observed some of the unusual tilings in the sixth

interview, such as the tilings in Figure 72 below. A few noticed the dilation symmetry of

Ls in the tiling on the left; others did not. If they did notice the unusual pattern, they

simply delighted in it. They never showed dismay or confusion over any tiling to be

evaluated.

Figure 72: Two tilings with unusual symmetry in the aesthetic evaluation tasks

Symmetry and transformations are foundational to the study of modern geometry.

As this study shows, symmetry is also a fundamental aesthetic quality that guides

children as well as mathematicians in the mathematical task of creating tessellations.

Symmetry is important mathematically both as an aesthetic object and as a

transformational process.

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Aesthetics had an influence on the way students tiled.

I have shown that aesthetic symmetry had a definite positive impact on children‘s

tessellation creations. I found other aesthetic qualities that drove children‘s mathematical

thinking about tessellations as well, including complexity, uniqueness, alternation, and

units. These qualities were clearly shown to be aesthetic in the children‘s thinking during

the evaluation tasks. They were also used by the children as they created their tilings.

Another aesthetic quality used by children for their finite patterns was the mathematical

concept of convexity. Real world connections and color may also have been aesthetic

factors in children‘s creation of tessellations. All of these aesthetic criteria, as well as

aesthetic criteria of surprise, validity, and connectedness, were shown to have cognitive

roles in addition to their aesthetic roles. (See The Cognitive Role of Other Aesthetic

Criteria on page 214.)

These qualities therefore highlight the deep connection between cognitive and

aesthetic understanding in the children‘s thinking. If the children could not grasp the

mathematics behind a particular concept used in a tiling, they could not see that concept

as beautiful. Aesthetic appreciation is linked to cognitive understanding. Students found

the tiling on the left in Figure 72 above beautiful if they noticed the dilation pattern of Ls.

Other students appreciated this tiling for its maze-like quality, but they only got really

excited if they saw the ever-increasing L patterns. One mathematician did not

aesthetically appreciate the tiling on the right in Figure 72 until he understood how it had

been made from a simpler radial pattern. His aesthetic appreciation was tied to his

cognitive understanding of the tessellation.

Mathematical understanding therefore has an impact on aesthetic appreciation.

Even more importantly, aesthetics guides and motivates students‘ mathematical

understanding. Aesthetics and understanding impact each other as in Figure 73 below.

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Mathematics is the study of patterns and the natural appeal of patterns motivates students

to investigate the mathematics behind these patterns. Aesthetics also guides students to

seek the most generative paths of mathematical inquiry. For example, we saw above that

the aesthetic quality of symmetry guided students in their search for mathematical

validity. Symmetry also served as a powerful motivating force to study the mathematical

structure of tessellations.

Figure 73: Mutual influence of aesthetics and mathematics

Aesthetic factors other than symmetry also drove students‘ investigation of

mathematics. The aesthetic quality of uniqueness, for example, drove students to seek

new patterns and new ways of understanding tessellations. Instead of simply making the

same tessellations they had seen before, students usually preferred trying to make a

tessellation that they were not yet familiar with. They enjoyed exploring the space of

possibilities and deliberately used strategies and shapes that they had not yet tried. The

aesthetic quality of units drove students to explore mathematical concepts of fit,

composition, and unitizing. The aesthetic quality of complexity drove students to explore

patterns that were richer than the simplest patterns and yet still within their cognitive

grasp. The aesthetic quality of alternation helped add an element of complexity. These

aesthetic qualities served as a force to drive students to explore their zone of proximal

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development—the space of ideas they are ready to understand with others, but have not

yet learned. American classroom instruction is often organized around teacher- or

curriculum-imposed exercises. This study would suggest that students should be allowed

more often to let their own aesthetics motivate them towards the concepts they need to

learn.

Even seemingly non-mathematical aesthetic qualities probably play an important

role in driving children‘s study of mathematics. Real world connections, for example,

may inspire the study of patterns known from experience. Children were immediately

comfortable with the regular hexagon tiling, even if they were constructing it for the first

time, because they had frequently seen it as a honeycomb pattern.

Aesthetics and mathematics have a mutual impact on each other, but aesthetics

comes before the mathematics. This is seen in the simple fact that students had an

immediate aesthetic attraction to various aspects of tiling, such as symmetry and

dimensionality, without yet understanding the mathematics to which these aesthetic

qualities will lead. Current mathematical curricula assume that aesthetic appreciation is

an optional application to mathematical study. But aesthetics is actually at the very

foundation of mathematics. Therefore, aesthetics should be understood as the initial

driving force towards mathematical understanding. Children are drawn towards

numerical and geometric patterns and this aesthetic force leads them to study and

understand mathematics. The consideration of aesthetics cannot be postponed to the role

of educational afterthought. It is the driving force behind the study of mathematics and its

role in learning should be considered explicitly in education.

INTERACTION OF THE TWO PERSPECTIVES

The third part of the research question is:

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How do the cognitive and aesthetic viewpoints inform each other for a more

complete understanding of children’s conceptualizations of tessellations?

The two perspectives together are needed for a more complete understanding of

children‘s mathematical thinking. In this study I found the two perspectives together

particularly useful for exploring the role symmetry played in children‘s understanding of

tessellations and how their aesthetic values influenced their mathematical results. (See

above.) It was also useful for understanding how tools influenced thinking. Also, an

essential part of this study was considering the cognitive role of the children‘s aesthetic

criteria. (See above.)

The tilings students created depended on the task given to the students and the tools

used to accomplish the task.

The task given to the students and the tools students had to accomplish the task

determine the space of possibilities for the result. Students were never given options of

color (independent of the shape), so color almost never entered as a mathematical aspect

of children‘s tilings. The software environment did not have the capacity to select and

copy units of tiles, so children were somewhat limited in the complexity they could

achieve. The magnetized behavior of the polygon corners in the software environment

influenced some students to create certain types of tessellations that they might not have

created with another tool. The relative difficulty of rotating a computer tile compared to

spinning a plastic tile meant that few students recognized the right angle in the right

scalene triangle task, but all recognized the right angles in the polyomino tasks. The task

of describing a tiling orally without construction tools requires students to visualize the

structure mentally, so their oral descriptions tended to be simple and local. Dot paper

helps students avoid overlaps and small gaps. The automatic zoom in the software

highlighted the endless, borderless nature of the tiling plane.

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Outhred and Mitchelmore (2000) point out that drawing tools get closer to

children‘s mental models of tessellations than physical construction tools do. My study

shows that oral descriptions get even closer to children‘s initial ideas about the structure

and appearance of tessellations. Drawings produce a combination of initial ideas and

fresh discoveries.

It seems reasonable that the heavy influence of tools should lead us to conclude

that a variety of tools is needed for exploring tessellations. Instruction that is limited to a

single tool will inevitably limit thinking along the lines constrained by the tool. One

reason the teaching experiment of Upitis, Phillips, and Higginson (1997) was so powerful

is that the teachers used a wide variety of tools and tasks to explore the mathematical

concept of tiling.

IMPLICATIONS FOR FUTURE RESEARCH

Tessellations have been neglected in the research literature, in spite of their

importance. My research is exploratory in nature. As such, it is intended to be the first

step in Clement‘s (2000) cycle of research extending from generative, exploratory studies

to convergent, confirmatory studies. All aspects of this research need to be followed up.

Tessellations are a broad field and I have touched on several different aspects of this

topic. Different types of tessellations use different types of reasoning and geometric

properties. Studies are needed which treat each kind of tessellation task in greater depth.

Children‘s understanding in each area can be pushed further to test which beliefs are

robust and which are easily changed.

There are many important mathematical concepts elicited by tessellations, as I

discuss on page 239. Studies are needed which explore the role of tessellations for each

of these concepts. How do students understand each of these mathematical concepts in

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the context of tessellations, and how can they best extend their understanding? What are

children‘s understandings of isotropic space with respect to tessellations? How does the

orientation of the figure affect children‘s ability to perceive acute, right, and obtuse

angles and see possibilities of fit? What are children‘s understandings of symmetry with

respect to infinite tessellations and how do children learn about symmetry in the context

of tessellations?

There is essentially no research on children‘s understanding of spatial infinity.

This study shows that this important topic can be studied in the context of tessellations.

Previous studies suggested that students‘ understanding of infinity was impervious to

instruction (Fischbein et al., 1979; Monaghan, 2001). Is this true in the case of children‘s

understanding of spatial infinity? If not, how can children learn to appreciate the

consequences of tiling in an infinite plane? How can students be taught to distinguish

when their pattern is sufficiently described to make a well-defined tessellation? Upitis et

al. (1997) encouraged children to invent notations to describe their patterns. Can this be

extended to help them understand that their finite patterns must be extended throughout

infinite space in a way that others will understand? How can such activities connect the

children‘s understandings of transformations and symmetry?

More research needs to be done on children‘s ontological conceptions. I have

presented evidence that some children fail to understand that two different finite creations

make the same infinite tessellation partly because their understanding of infinite space is

incomplete and partly because they are using a process-oriented ontology. This needs to

be confirmed and understood at greater depth.

This study also shows that tessellations can be used for the study of children‘s

aesthetics and the impact their aesthetics have on their mathematical choices. How do the

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aesthetic themes identified here compare with the aesthetics of other geometric tasks?

How do children‘s aesthetics evolve as they deepen their understanding of tessellations?

Most importantly, this study serves only to identify children‘s initial thinking.

Therefore, teaching experiments are needed which build on the results presented here.

How can students best be taught to understand the mathematical structure of

tessellations? How can we best teach angle properties and angle perceptions at various

orientations? How can teaching help students to attain a deeper understanding of

symmetry and the transformations that create tessellation structure? At present, symmetry

at this grade level is usually limited to identifying simple reflections and rotations. This is

clearly insufficient if students need to understand, say, the isogonal nature of the regular

pentagon. Students need to be able to identify richer forms of symmetry. Students also

need to be able to extend their reasoning beyond the simple empirical reasoning they

often used in this study. In what ways can this be done?

IMPLICATIONS FOR CURRICULUM AND INSTRUCTION

Many recent studies in geometry education attempt to identify levels or learning

progressions or trajectories that map out the steps students go through as they learn the

subject matter. Such an approach seems inappropriate for the results of this study for two

reasons. First, this is not a teaching experiment, so learning progressions or trajectories

cannot be reliably identified. Second, I do not feel it would be right to try to assign levels

to children‘s understandings of tessellations because their thinking in this area has so

many dimensions. Take for example Michelle. At times she came up with very

sophisticated results, such as drawing a two-dimensional parallelogram grid with a ruler,

or creating two different tessellations with six-fold symmetry and then correctly

analyzing the isometry between their structures. And yet she gave several answers that

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suggest she is still sometimes reasoning at Piaget‘s intrafigural stage. Or consider Kelsey.

She was one of only two students who found successful tilings for all three triangles in

Set E, using mathematically valid unitizing and row strategies, and yet she did not

demonstrate understanding of the row and column structure of simple square arrays.

Children‘s thinking is complex, but I have nevertheless identified several aspects

of their thinking which are true for all six students. I have also noted a few of the

immediate implications from these conclusions for curriculum and instruction. I now

return to my contention that there are few topics in geometry that can combine so many

powerful and important geometric ideas that this age group of children needs to be

studying.

What Students Learn With Tessellations

Students learn a wide variety of geometric concepts when studying tessellations.

Most importantly, they learn these concepts in the context of a genuine mathematical

task. Instead of studying concepts in isolation through exercises that either have no

connection to students‘ needs or are couched in highly artificial ―real world‖ contexts,

problems such as the tessellation tasks in this study provide an environment where

students can explore and discover the need for the geometric properties that they are

learning in a motivating context that naturally highlights children‘s mathematical

aesthetics.

Mathematical Concepts Elicited by the Tiling Tasks

A number of mathematical concepts were elicited during the tessellation creation

tasks in this study. The following is a brief summary of those concepts.

Fit. Students needed to be able to fit polygons together both as tiles and as units

of tiles in order to create interesting tilings. This requires coordination of sides and angles

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in a way that was not always obvious to the students. Students sometimes struggled with

fitting tetrominoes together in a way that did not leave holes. Misjudging the sizes of

sides and angles made certain tasks, such as the sixth finite tiling problem, difficult.

Angles. Students needed to work with angles in order to understand fit.

Sometimes students referred to angles as ―corners,‖ though their understanding of this

term was not explored. The word ―angle‖ was not used often. When it was, it was usually

used to mean ―orientation‖ or ―triangle.‖ An understanding of angles was needed to

analyze the pentagon task and to see the correct number of polygons that could go around

a common vertex. Additivity of angles, along with independence of order, is needed to

see that the order of regular polygons can be permuted—a fact that would have been very

helpful for the square and triangle task had the students realized it.37 Basic angle facts

were needed, such as four right angles or six equilateral triangles can fit around a vertex

precisely, facts that were not well known by the students. The abilities to estimate angles,

draw approximately congruent angles, and spot right angles at non-orthogonal

orientations were needed for several tiling tasks. Because most students did not recognize

a right angle at a non-standard orientation, they failed to see a simple solution to the right

scalene triangle task until they explored with the physical tiles. They also frequently

failed to judge if a gap angle was acute, right, or obtuse during the Regular Polygons

task. However, students did seem to realize that the prototiles had different sized angles,

so if one angle did not fit, they could choose a polygon with a smaller or larger angle.

37 Contrast the way additivity of angles can be explored in tessellations with the less meaningful way

complementary and supplementary angles are often taught by simply memorizing that they must add up to

90° or 180° without much real understanding of what the importance of these sums might be. These

concepts are usually introduced later than fourth grade. Do students understand the additivity of angles

when they study these concepts? They apparently do not at the fourth grade level.

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Sides. In addition to angles, students must pay attention to the length of polygon

sides when deciding which tiles can fit where. In the case of polyominoes, which are

usually not edge-to-edge, children must also implicitly work with the sums of the edges

in order to align two or more polyominoes against a given side. The fact that some

students expected the parallelograms not to match up in the standard tessellation shows

that they were not thinking through the consequences of the parallelogram‘s congruent

sides.

Congruence. Concepts of congruence are implicit in the concept of the prototiles,

which are congruent with the tiles in the tessellation. Students seemed to notice that the

sides of the regular polygons were all congruent and therefore they correctly focused on

the problem of getting the various angles to fit together in the Regular Polygons task.

Students realized in the Triangles tasks of Set E that it was important to match congruent

sides together. This sometimes proved to be difficult during the drawing phase and

students sometimes erased and redrew as they struggled to coordinate sides and

orientation.

Transformations. Transformations and symmetry are foundational tools for the

study of tessellations. But it is not easy to envision that the entire plane is transformed by

a mathematical transformation. Tessellations require such understanding and offer an

environment where such understanding can be made concrete.

Students sometimes misused reflections in their tilings. By flipping some tiles in

the computer environment and not remembering that some were flipped, they sometimes

made it impossible to create a successful tiling. In order to make certain tilings, students

need to be aware that a tile may have two forms related by reflection. Shapes with

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reflection symmetry have only one form, and therefore flipping them does not change

their shape. Michelle wanted to flip such tiles anyway in one of the tasks.

Symmetry. Students explicitly created tilings with reflection symmetry, though

they almost never used the word symmetry. Moses talked about copying what he did on

one side to the other side. Students seem to have also used visual rotation symmetry,

though they did not express this. Reflections and rotations are needed to create the

symmetry that students desired and that made tilings easier to create. Students explicitly

used translation symmetry to defend the infiniteness of some of their tilings. They could

show that a tiling continued forever by showing how a unit could be translated and still fit

into the pattern. It is difficult to justify that a tiling will be endless without translation

symmetry.

Dilations are another type of transformation that can be studied in the context of

tessellations. Dilations were noticed by the students in this study for some of the tilings.

Some students created rhombus tessellations by using a dilation strategy. During the

aesthetic evaluation tasks, some students noticed that one of the tilings had a

sophisticated dilation symmetry.

Orientation. Students needed to be aware of polygons and angles in non-typical

orientations. Failure to mentally rotate a tile to a more standard orientation created a

failure to see tiling possibilities, as was the case with the right scalene triangle and in

some of the Set B finite tiling tasks.

Infinity. Students were more successful at arriving at a mathematical

understanding of their tessellations if they could work comfortably in an infinite, empty

plane. Tasks of fitting needed to be carried out without any final finite shape as a goal.

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Students needed to understand and visualize how their tiling would look extended

indefinitely in all directions.

Dimensions. Students need to understand that planes have two dimensions and

that tiling translations in the plane need to go in two independent directions. Periodic

tilings can be created by extending first in one dimension (as a row) and then in the other

dimension (stacking rows). Tilings with 6-fold symmetry generally have some sort of

rows in three different directions, a fact students did not notice.

Systematic counting. One or two students had difficulty counting the sides of

polygons. Students need counting strategies for counting items arranged in a circuit, as

well as counting shapes contained in a unit that has not been visually delineated.

Inductive reasoning. In order to evaluate the validity of a tiling, students need to

use mathematical reasoning, eventually in the form of deductive reasoning. They need to

understand the limits of the inductive reasoning they frequently used. It is often pointed

out that teaching the limits of inductive reasoning is difficult because presenting good

examples is not easy. Tiling provides an accessible activity for exploring these limits.

Deductive reasoning. Students usually understood the role of counterexamples in

deductive reasoning. A failed tiling was not a valid counterexample to the hypothesis that

a shape could tile. A successful tiling was a valid counterexample to the hypothesis that a

shape could not tile. The fact that the regular pentagon could not tile does not imply that

all pentagons cannot tile.

Composition. Composing and decomposing shapes is a basic skill for geometry.

Students need to learn to unitize and to subdivide shapes into other shapes, both of which

were necessary skills for the tessellation tasks.

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Visualization. Tiling requires and aids two-dimensional visualization skills,

including the recognition of two-dimensional patterns, symmetry, and transformations.

Tessellations with polyominoes require visualization on a square grid. Students

sometimes failed to visualize polyominoes correctly with respect to the grid, resulting in

mathematically invalid tilings.

Parallelograms. Students need to learn that all parallelograms tile like squares,

with rows and columns. They need eventually to understand the properties of

parallelograms shared by squares and rectangles (themselves special parallelograms).

These properties are learned naturally in the context of tiling and students in my study

noticed the similarities.

In addition to the mathematical concepts elicited by the tasks in this study, the

study of tessellations can also be used to explore classification of polygons (e.g., squares

and rectangles share the properties of all parallelograms), properties of parallel lines (e.g.,

van Hiele-Geldof, 1957), polygon vocabulary (e.g., Upitis et al., 1997), and of course

area concepts (e.g., Owens & Outhred, 1998).

The Teaching of Tessellations

This was not a teaching experiment, but we can see certain factors that will

probably need to play a role in geometry instruction that uses tessellations. Most

especially, we can see the importance of allowing children to have a rich set of

tessellation experiences. Owens and Outhred (1998) pointed out that students did poorly

with finite tilings of shapes they were not familiar with. In order for students to gain the

aesthetic and cognitive knowledge they need for geometric conjecture and insight, they

need ample opportunity to explore and gain familiarity with geometric figures. Moses

noted that some of his insights came from having played with pattern blocks in previous

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years; he was already aware of the possibilities of combining them together. Students

need multiple experiences to learn geometric properties, preferably set in contexts that

highlight the usefulness of those properties. Tessellations provide an environment for

such experiences, provided students can explore possibilities and study why some things

work and other do not. The successful teaching experiments of van Hiele-Geldof (1957)

and Upitis et al. (1997) allowed students to conduct such guided explorations. More

teaching experiments are needed to determine the best ways for students to learn from

tessellations. What elements help students to learn? What elements detract from learning?

For example, color was very important to the children. The color pink stirred great

passions among the children at the private school, usually negative. Will allowing

students to choose their own colors increase the aesthetic and motivational factors for

creating tessellations? Will it allow a deeper investigation of symmetry (see page 84)? Or

will it distract from the more important properties that need to be the focus? These are

issues that need further research.

Teaching should also take into account mathematical aesthetics. Children

generally share the same basic aesthetic categories as mathematicians, but they use these

aesthetics in different ways. In some ways this difference is appropriate. Children need

the aesthetic qualities of alternation and units to guide them in the creation of

tessellations. In other ways, their aesthetics needs to be molded by geometric experience

that helps them learn which aesthetic criteria are most useful for the task of tiling and

which criteria are less useful. Color and real world connections may not be

mathematically helpful; complexity, connectedness, and surprise will guide students to

richer geometric learning.

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Importance of the Study of Tessellations

Tessellations provide a space for the geometric experience and knowledge

students need. Tessellations cover much of the geometry curriculum around this grade

level. They also provide an environment where students can explore and be guided by

appropriate mathematical aesthetics, which will increase motivation and can help them

focus on what is most important for them to learn at this grade level in geometry.

This claim that tessellations provide an appropriate space for this age level is not

new, though the theoretical base that demonstrates how children initially understand

tessellations has never been thoroughly investigated before. Teaching experiments have

already shown that tessellations are a powerful topic for the learning of geometry. The

study of tessellations has, in fact, already been mandated between third and seventh grade

(often in several grades) in some states, provinces, and countries, including many that are

among the highest ranking in comparative tests of mathematics, such as Japan,

Singapore, Ontario, Quebec, and Massachusetts. The NCTM Focal Points (2006)

recommends tessellations for fourth grade as a context for the study of transformations.

Unfortunately, tessellations are not required in the Texas Essential Knowledge

and Skills, nor in the new Common Core State Standards for Mathematics (CCSS-M;

Common Core State Standards Initiative, 2010). The CCSS-M are intended only to be

standards from which states must devise curricula that specify how the standards are to be

implemented. States may or may not decide to include tessellations in their curricula in

order to cover the content of the standards. Even if tessellations are not included in the

curriculum, teachers can use tessellations to cover the mandated material. The question in

today‘s American climate of intense testing pressure is whether teachers will feel the

freedom to introduce a significant unit on tessellations if it is not included in the

curriculum. These are details that remain to be worked out at the policy level. However, I

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strongly recommend that teachers and curriculum writers consider the rich possibilities

inherent in the study of tessellations for fourth and fifth grade children.

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Appendix A: Pre-test

This appendix contains a reduced copy of the English pre-test.

Figure 74: First page of the pre-test, shown half size

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Figure 75: Second page of the pre-test, shown half size

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Figure 76: Third page of the pre-test, shown half size

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Appendix B: Protocols

This appendix contains the original protocols for the six interviews. After the first

interview, Set B was moved to the third interview, so it has been deleted from the original

protocol for Interview #1. A brief two-page outline was made for each of the six

interview protocols to remind the interviewer of the tasks during the actual interviews.

INTERVIEW #1 PROTOCOL

Preparation

(Materials: video camera, batteries, and cord; tripod)

Prepare the materials for today‘s tasks and the master key. Position the video

camera in such a way that it will need to be moved as little as possible and yet can

capture the work space and students‘ actions. The camera must capture pattern block

tilings first, and then the computer screen later. (Check the equipment in advance to make

sure the microphone can capture all dialog.) Check which students will be participating

and prepare software codes for them.

Introduction

(Purpose: To set the ground rules and introduce the main task)

(Materials: blank paper, marker, square pattern blocks; pre-test solution)

Information – Collect the assent form if it has not been previously collected.

Return the main part of the consent form if submitted. Record the child‘s first name, age,

pseudonym (a two letter code written on all papers for later identification), gender, and

school on the master key.

Introduction – The interviewer introduces himself and explains that he is

studying how children create patterns with shapes. Confirm that the child understands

why a video recording is being made. Start the camera and say today‘s date, interview

number, and student pseudonym.

Ground rules – Attempt to make a point of connection which invites the student

to enjoy the tasks by asking students how they feel about math, geometry, art, patterns,

floor tiles, and so on. Discuss that the interviewer will be giving some problems to do and

that there are no right or wrong answers to the problems. In fact, there will be many good

ways of doing most of the problems and we will explore together possible ways to do

them. There is no grade attached to the work and the child‘s teacher will not know about

their work. We are doing these tasks ―for the fun of it.‖ Again, the reason we are here is

because the interviewer is studying how children think while making patterns with

shapes, so it is important to think out loud and explain all work as much as possible.

Explain that you will be asking the student how they know their answer works.

Initial problem – The problem we will be exploring is that of tiling a floor.

Discuss floor tiling including the facts that there can be no gaps or overlaps, and the tiling

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must be able to go on and on. We do not care what happens when the tiling reaches the

walls. Let‘s imagine it goes on forever. Ask the child what it means for the pattern to go

on forever.

Show the student his or her answer to the first task of the pre-test (see the

preliminary task of Set B) and ask if it could go on forever. Next, ask the student to make

the tiling with square pattern blocks and discuss any discrepancies with their oral

description, as well as apparent gaps or overlaps in the drawing and whether the

inaccuracies in the freehand sketch are important or not (Owens & Outhred, 1998). If

necessary, discuss again the constraints against gaps and overlaps and the condition of an

infinite pattern. If there is a square tiling on the floor, point it out.

Set A: Basic Tilings

(Purpose: To see how students perceive and construct basic monohedral

tessellations and to introduce the computer software)

(Materials: blank paper; marker [preferably] or pencil [if the student keeps starting

over]; triangle, hexagon, and rhombus pattern blocks; pre-test solution; computer

with software and mouse)

Rhombi – Give the student a single blue rhombus pattern block. Ask if it would

be possible to tile the floor with such a diamond. How certain is the student?

Ask the student to explain their answer orally, using a tile for support. (If needed,

the child can use a second tile.) Based on the child‘s answer, ask the child to explain on

paper (starting in the center of the page) either why the tiling cannot work or what the

tiling will look like. (If the student wants to trace the shape, allow this.) If the child

creates a tessellation, discuss how it works—the patterns, how it continues, or any

unusual features. Discuss any discrepancies with the oral explanation. Finally, allow the

child to create the tiling with the pattern blocks, or else demonstrate with the pattern

blocks why the shape cannot tile. Again, discuss any apparent differences with earlier

descriptions. Also, if the tessellation is not standard, discuss any consequences of unusual

features of the tiling. At some point in comparing the sub-tasks there may be cognitive

dissonance. If the cognitive dissonance is resolved during the interview, this might lead

the child to change his or her initial ideas. Help clarify in what ways these ideas changed.

Ask the student how certain they are that the tiling will or will not work. If the child

believes at this point that it is impossible to tile with this shape, end the task here.

Are there other ways to tile with rhombi? If so, how? How are these other tilings

different? How are they the same? Which one does the student like the best? Why?

Label the student‘s work with their pseudonym code and the principal orientation

of the paper with respect to the student and file away.

Triangles and hexagons – Repeat the above procedure first with hexagons, then

with triangles.

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Parallelograms – Repeat the above procedure for parallelograms. In place of the

pattern blocks, display a 60°/120° 1×2 parallelogram on the computer screen using the

software with the student‘s name. Ask if it would be possible to tile the floor with such a

shape. How certain is the student?

Ask the student to explain their answer orally. Discuss the child‘s solution on the

pre-test. Discuss any discrepancies with the oral explanation. Conclude by showing the

student how the software works and allow the student to construct the tiling with the

software (or to show why the tiling cannot work). Again discuss any apparent differences

with earlier descriptions. Finally, show the student his or her answer from the pre-test.

Discuss any differences. If any of the final tessellations are not standard, discuss any

consequences of unusual features of the tiling. At some point in comparing the sub-tasks

there may be cognitive dissonance. If the cognitive dissonance is resolved during the

interview, this might lead the child to change his or her initial ideas. Help clarify in what

ways these ideas changed. Ask the student how certain they are that the tiling will or will

not work. If the child believes at this point that it is impossible to tile with this shape, end

the task here.

Are there other ways to tile with parallelograms? If so, how? How are these other

tilings different? How are they the same? Which one does the student like the best? Why?

Final task – If the student created a tessellation for each of the four shapes, finish

this set by asking if there is a shape that cannot tile the floor. If the student believes there

is such a shape, give the student paper to draw such a shape.

Set B: Tiling Completion

(This set was moved to the third interview.)

The interviewer will thank the student for her or his participation and say he is

looking forward to next week‘s interview.

Guidelines for Interviews

The clinical interview will be carried out according to the guidelines explained in

Entering the Child’s Mind (Ginsburg, 1997). The interviewer will do his best to assure

the interviews are conducted with a fun and supportive atmosphere. It is important to give

ample time and material for the child to explore each task in searching for an answer that

clearly communicates the child‘s ideas. When possible, allow the student to continue the

tiling for as long as they like. For example, encouraging the child to continue the tiling is

better than asking ―And what goes here?‖

All responses by the child are to be accepted as honest attempts to explain their

thinking and occasional affirmation (―Ok,‖ ―Cool,‖ ―Good,‖ ―Nice,‖ ―Interesting,‖

―Wow, none of the other students have made that pattern‖) will be given equally to all

task results without regard to the interviewer‘s own aesthetic preference. Attempts will be

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made to modify or skip any tasks which appear unduly frustrating to the child. Clinical

interviews require the interviewer to hypothesize during the interview, invent appropriate

questions on the spot to test these hypotheses, and occasionally even create new tasks

during the interview, though this will usually be done as a follow-up during the last two

interviews.

Coercion and seduction – The interviewer will make every effort not to bias the

child‘s thinking by asking leading questions unless he is deliberately attempting to test

the robustness of the child‘s thinking. Answers are not to be judged right or wrong; the

only goal is to understand the child‘s thinking. Each week, the interviewer will review the

videos and note any instances of coercion according to diSessa‘s (2007) four criteria: (1)

Is there independent evidence of conviction on the part of the subject? (2) Is it possible

that the viewpoint was prompted? (3) Were the responses systematic (e.g. always

changing one‘s mind after a prompt) or were they natural? (4) Is there converging

evidence? ―Seduction,‖ the deliberate attempt to draw a student into areas that are likely

to reveal interesting thinking, is not the same as coercion and is acceptable as long as

there are no leading questions that would bias answers.


Preparation

(Materials: blank paper; marker; pencil; computer and mouse; square, pentagon,

and hexagon Polydrons; square and triangle pattern blocks)

Review each student‘s answer for the hexagon tiling. Prepare generating sets in

the software for regular pentagons and for squares and triangles.

Assemble a sample square tiling and a sample hexagon tiling with Polydrons.

Check camera positions and sound settings.

Set C: Pentagons

(Purpose: To investigate a shape that cannot tessellate)

(Materials: blank paper; marker; pencil; software; square, pentagon, and hexagon

Polydrons)

Show the students a square tiling and a hexagon tiling with Polydrons. Remind

students of the three constraints for making a tessellation.

Oral: Give them a single pentagon Polydron and ask if it is possible to tile with

that shape. Are they certain? (If necessary, discuss the difference between pentagons and

hexagons. Compare their answer to the answer they gave for the hexagon tiling.)

Written: After students attempt to justify their answer orally, ask them to

describe their thinking with paper, beginning in the center of the page. Discuss

similarities and differences with their oral answer.

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With either task, if they believe the tiling is impossible, ask why hexagons can

tile, but not pentagons.

Polydrons: Ask students to attempt to make the tiling with Polydrons. Discuss

any differences in the result with their oral and written descriptions. (If students have

difficult manipulating the Polydrons, move to the computer environment.)

Set D: Squares and Triangles

(Purpose: To explore a rich, dihedral tessellation task which can reveal both

conceptual and aesthetic thinking)

(Materials: paper, marker, software, square and triangle pattern blocks)

Show student square and triangle pattern blocks. Ask if it is possible to make a

tiling with both together. Are they certain?

Oral and written: Ask students briefly to justify their answer orally and with

paper.

Pattern blocks: Give students all the triangles and squares and ask them to make

a tiling. Will their tiling continue forever? How? Are they certain? After they complete

one tiling, ask them if they can make a different tiling. Which of their tilings do they like

best? Why? What are the similarities and differences?

Software: After two tilings with pattern blocks, ask students to make further

tilings with the computer software as time allows. Will their tiling continue forever?

How? Are they certain? Compare the different tilings they have made. How are they alike

or different? Which do they like best and why?


Preparation

(Materials: pre-printed sheets [see figures on page 257]; marker; pencil; triangle

and square pattern blocks; computer software)

Prepare the three triangles in the triangle set in the computer software—acute

isosceles, right scalene, and obtuse scalene triangles.

Analyze the students‘ responses to the first task of the pre-test. Refer to the article

by Battista, Clements, Arnoff, Battista, and Borrow (1998). Prepare questions to ask the

students concerning their response to this task.

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Set B: Tiling Completion

(Purpose: To identify mathematical structures in the context of a finite tiling; to

investigate the ability to visualize basic decompositions; and to provide a context for

exploring the constraints against gaps and overlaps)

(Materials: pre-printed sheets [see figures on page 257]; marker; pencil; triangle

and square pattern blocks)

This set was originally planned for Interview #1. However, Set A took more time

than anticipated, so Set B was moved to Interview #3.

The problems for this set are loosely based on the problems in Owens and

Outhred (1998).

Preliminary problem – Discuss the student‘s response to the first problem in the

pre-test (see Figure 77 below), which was based on a problem by Battista, Clements,

Arnoff, Battista, and Borrow (1998). Students were asked how many squares were

originally in the rectangle. The details of the protocol for this problem will be determined

based on the individual response on the pre-test and prepared in advance of the interview.

(The researcher also anticipates drawing from the responses to the tasks in Set A.)

Questions will be prepared to explore other aspects of thinking described in Battista et al.,

as well as exploring how much help a student needs to visualize the complete row- and

column-structure.

Figure 77: Pre-test Item 1

Problem 1 – Students will be shown the first sheet (on the left in Figure 78

below). Instead of a big, infinite room, we are now tiling a little room. Ask whether the

orange square can be used to tile the rectangle.

If the answer is no, students will be asked to justify their thinking.

If the answer is yes, students will be asked how many orange squares it will take

to complete the tiling. After oral discussion, or if the student is stuck, the student

will be asked to sketch the squares in the rectangle.

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Problems 2–6 – Problem 1 will be repeated with the five other sheets (figures

below). For Problem 2, note that lines indicate where someone has started to show the

tiling. If needed, students will be offered the chance at the end of Problems 1–4 to verify

their answer with pattern blocks. Label each paper with the student‘s pseudonym.

Figure 78: Set B, Problems 1 and 2


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The interviewer will thank the student for her or his participation and say he is

looking forward to next week‘s interview.

Set E: Triangles

(Purpose: To explore children’s beliefs and conceptualizations of non-equilateral

triangles)

(Materials: Computer software with triangle set—acute isosceles, right scalene,

obtuse scalene)

The interviewer will present the triangle set in the computer environment and ask

students if it is possible to tile first with the acute isosceles, then the right scalene, and

finally the obtuse scalene triangles. As in past interviews, students will be asked first to

describe the possibility or impossibility orally, and how certain they are. Next they will

describe their answer by sketching on blank paper. Finally students will attempt to make

the tiling in the computer environment. After creating a tiling, students will be asked if it

is possible to use the same triangle to tile in a different way. Which way does the student

prefer, and why? This process will be repeated for each of the three triangles. Finally,

students will be asked to compare all the triangle tessellations they have made. If

necessary, access the snapshots created by the program. Which do they prefer, and why?

Is the preference a factor of the tile selected, or the way they tile? What makes a beautiful

or interesting tiling?

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Preparation

(Materials: Plastic tromino and tetromino tiles; dot paper; pencil (no marker);

software with tromino and tetromino sets; a pre-drawn example of a non-edge-to-

edge tiling with the I-tromino)

Prepare a dot paper example of a non-edge-to-edge tiling of the I-tromino (a

―brick wall‖ tiling). Prepare the tromino and tetromino tiling sets in the software. Also

prepare polyiamonds tilings sets in case there is extra time at the end. (See Set H.)

Set F: Polyominoes

(Purpose: To study how students tile with non-convex, orthogonal shapes, and how

students choose which subtask to work on.)

(Materials: Plastic tromino and tetromino tiles; dot paper; pencil (no marker);

software with tromino and tetromino sets; a pre-drawn example of a non-edge-to-

edge tiling with the I-tromino)

Trominoes: Show the student the I-tromino plastic piece and ask if it is possible

to make a tiling with that piece. Explore the child‘s reasoning orally first. Is the child

certain of the answer? Is there more than one way to tile with this piece? Then, give the

child dot paper to sketch the solutions with. Which tiling does the child like the best? If

the child found only the edge-to-edge tiling, show the child a non-edge-to-edge tiling and

probe the reaction. Is this tiling acceptable? Which tiling is best?

Next show the student the L-tromino plastic piece and ask if it is possible to tile

with that shape. If not, why not? If it is possible, is there more than one way to tile? Is the

child certain? Give the child dot paper to sketch the solutions. Which tiling does the child

like best?

Tetrominoes: Show the student all five tetromino pieces. Which ones can be used

to tile the floor? How certain is the student? Ask for brief oral justifications of any tile

that the child is certain about.

Allow the student to choose any tile the student wants to create a tiling using dot

paper. Is that the only tiling possible? Why did the student choose that tile first? (If the

student asks if a tile can be flipped over, allow the child to decide if that should be

allowed or not.) After a couple of tilings, ask the student which tile the student would like

to use next, and so on for the remainder of the half hour. After a few tilings, or when the

child is stuck or tired, move to the computer environment and continue creating tilings

there.

Compare all the polyomino tilings made by the student. Which does the student

like best? Why? What makes a good tiling?

260


Preparation

(Materials: computer software)

Prepare the regular polygon set with 15° rotations.

Set G: Regular Polygons

(Purpose: To observe the aesthetic choices of an open-ended tessellation task.)

(Materials: computer software with regular polygon tiling set)

Remind students of the task—to create infinite, gapless tilings.

Triangles and squares – Students will be asked to create one more square and

triangle tessellation.

Regular polygons – Show the student the regular polygon tiling set. Students will

be asked to continue making tilings using any combination of regular polygons.

After a few tessellations, ask if it is necessary that the vertices matched. Should

the snapping magnets be turned off? What kind of tiling is best—with or without

snapping? Why?

After each tessellation, ask the student‘s opinion of the tiling. Is it beautiful? Why

or why not? What makes a good tiling? Why did the student choose the polygons used?

At the end of the interview, show the student all the tilings made during this

interview and ask which tiling is the favorite. Why? What makes a beautiful tiling?


Preparation

(Materials: Selected tilings from subjects, public domain, the Alhambra, and

Escher; four sheets marked with names of rooms; sticky notes)

Select student tilings from first five interviews using aesthetic criteria from

theoretical framework and interviews to create as wide a space of tilings as possible. Print

them on card stock and sort into three diverse sets. (Put similar tilings in different sets.)

Prepare four sheets of paper with names of four rooms in a house (kitchen,

bathroom, etc.).

Use the aesthetic criteria to create more tilings. Draw also from tilings in the

public domain. Print these out on card stock. Prepare a list of pairs (or small groups) of

tilings to be compared according to specific aesthetic criteria.

Prepare a selection of postcards from the Alhambra.

Mark with sticky notes selected pages from books on Escher.

261

Prepare sticky notes with ―1st‖, ―2

nd‖, and ―3

rd‖.

Aesthetic Tasks

(Purpose: To observe the aesthetic choices of tilings.)

(Materials: Selected tilings from subjects, public domain, the Alhambra, and

Escher; four sheets marked with names of rooms; sticky notes)

Remind students of tessellation criteria: infinite, gapless, and non-overlapping.

There are four separate tasks in this interview.

Student tilings

Give each student one of the three packets of student tilings. Explain that these

examples are from their own answers to the tiling tasks in earlier interviews. Lay out the

four sheets of paper. Ask the students to select nice tilings for each of the four rooms.

The cards are meant to be the pattern one would show to the person who has to lay the

tiles.

Next, ask the students to work together as a group to choose the ―best‖ tiling for

each room. Encourage them to discuss and explain their reasons.

(See Figure 81 below for students‘ tilings. The prototiles on the left were removed

from each card before the task, leaving only the tiling.)

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Figure 81: Students‘ tilings for the first aesthetic task

263

Other tilings (Student Sorting, Comparisons in Pairs, Professional Preferences)

(1) Spread out the cards representing tilings created by the interviewer and others.

(See Figure 82 below for the tilings used in these tasks. P was a blue and green 4.82

tiling.) Ask the students to put them in groups according to any criteria they want. Ask

which group is the best and why. Which tiling in that group is the best tiling? Ask the

same question of the other groups.

(2) Go through the list of prepared pairs of tilings (A-N, C-H, M-I, B-I, D-O, C-F,

M-R, D-I-G-O-P, and C-Q-L-N). For each pair (or set), ask the group which tiling is best.

Encourage the group to come to a consensus on each decision and to explain their

reasoning to each other. Do not allow them to spend too much time on any one decision.

(3) Which tiling do they believe an artist would prefer? Why?

Which tiling do they believe a mathematician would prefer? Why?

Figure 82: Most of the tilings for three of the aesthetic tasks

Professional tilings: Alhambra

Show the students the postcards from the Alhambra one at a time. Ask them to

rank them in order from ―best‖ to ―worst.‖ Encourage them to explain why the one or two

best tilings are their favorite.

Which tiling do they believe an artist would prefer? Why?

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Which tiling do they believe a mathematician would prefer? Why?

(See Figure 83 below for the seven Alhambra tilings used in this task.)

Figure 83: The seven Alhambra tilings

Professional tilings: Escher

Begin with the collection of tiling types drawn by Escher. Ask which one they

favor.

Explain that Escher is an artist who created tilings with animal shapes. They are

to be the judges of his work and award certain tilings first, second, and third prize. Flip

through the selected Escher drawings. Whenever the students say they really like a

certain tiling, mark it with a sticky note and encourage them to explain their reasons.

Once all the tilings have been looked at, have them discuss which of the tilings marked

with their sticky notes should win first, second, and third prize. Have them explain their

reasons for each choice.

The tilings used in this task are in Table 8 below. Most were prepared in advance.

A few were noticed by the children as I turned the pages and they stopped to comment on

them.

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Table 8: Escher drawings used in the sixth interview

Title (description) Book38 Page

Poster of eight tessellations used by Escher in lectures 1 32

“Fundamental Forms of Regular Division of the Plane” (Second illustration used by Escher in lectures)

1 33

“Regular Division of the Plane Drawing #6” (camels) 1 120

“Regular Division of the Plane Drawing #7” (squirrels) 1 121

“Regular Division of the Plane Drawing #58” (see Figure 57 on page 185)

1 161

“Regular Division of the Plane Drawing #67” (horsemen) 1 169

“Regular Division of the Plane Drawing #66” (see Figure 63 on page 195)

1 168

“Regular Division of the Plane Drawing #128” (black and white birds)

1 221

“Regular Division of the Plane Drawing #127” (angular black and white birds)

1 220

“Regular Division of the Plane Drawing #132” (red and blue flowers)

1 225

“Regular Division of the Plane Drawing #131” (black and white flowers)

1 224

“Regular Division of the Plane Drawing #34B” (birds and fish) 2 51

“Regular Division of the Plane Drawing #106” (birds) 2 89

“Regular Division of the Plane Drawing #104” (black and white lizards)

2 88

38 The three books used are:

1: M. C. Escher: Visions of Symmetry (Schattschneider & Escher, 2004)

2: M. C. Escher’s Universe of Mind Play (Escher & Seibu, 1983)

3: The Magic of M.C. Escher (Escher & Locher, 2000)

All authors use Escher‘s numbering system, but give the sketches somewhat different names. I am

following the titles given by Escher & Locher.

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Table 8. cont.

Title (description) Book Page

“Regular Division of the Plane Drawing #118” (lizards) 2 98

“Regular Division of the Plane Drawing #119” (flying fish) 2 99

“Regular Division of the Plane Drawing #6” (extended) 3 Inside cover

“Regular Division of the Plane Drawing #21” (basis for “Cycle”)

3 53

“Cycle” 3 55

“Regular Division of the Plane Drawing #69” (detail) 3 62

“Regular Division of the Plane Drawing #22” (red birds, white fish)

3 63

“Regular Division of the Plane Drawing #20” (extended) 3 90

“Reptiles” 3 175

An assortment of 35 regular division of the plane drawings, numbers 7, 10, 12, 13, 14, 16, 20, 24, 25, 28, 34, 37, 39, 42, 46, 54, 55, 65, 67, 71, 72, 76, 89, 97, 99, 111, 112, 118, 120, 121, 122, 123, 124, 128, 129

3 74–75

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Appendix C: Software

In this appendix I explain some of the design decisions I made for the software

used in this study.

One problem with all existing software is the relative complexity of rotating a tile.

A child typically needs to rotate a tile several times while placing it. However, some

existing software requires several mouse clicks each time the tile is rotated. With some

software, the user must release the tile, click on a rotation button, click and drag the tile

to rotate it, reclick the rotation button, and finally resume moving the tile. If the rotation

is not right, these steps must be repeated several times. With more sophisticated graphics

software, instead of a separate button, the user can often just click on the shape itself to

enter rotation mode. But such software tends to have an interface which is overwhelming

for children, and the number of clicks is sometimes still the same.

I know from experience that children adapt to this complexity, but it slows the

process down and is not nearly as intuitive as the placing of physical tiles on a table. I

wanted the software to be as simple and intuitive as possible so that the child could focus

on the tessellation rather than the tool. Some tessellation software designed for children

take these needs into consideration by allowing the child to click on the corner of a tile

and to begin immediately rotating it without having to click again, as if the mouse were

grabbing the corner of the tile and turning it. Unfortunately, the tile still does not usually

line up in the correct orientation until after several attempts of alternately rotating and

sliding the tile into position.

Some software (e.g., Tessellation Creator, NCTM, 2011) ―helps‖ the child by

automatically aligning the polygon in an edge-to-edge manner against any nearby

previously placed tile. This can make for quick tiling, but has two drawbacks with respect

to this research: (1) the software may help the child too much, placing the tile in positions

that the child had not thought of, thereby helping the child to create a tiling the child had

not conceived; and (2) the child in some cases may not want to make an edge-to-edge

tiling.

Other difficulties that I faced included: (1) children‘s software typically comes

with a restricted set of prototiles; (2) I needed frequent automatic screen captures to

document the actions of the children; and (3) I needed a flexible environment, e.g. one

that could make either edge-to-edge or non-edge-to-edge tilings, as needed.

I decided therefore to design my own software, ―Polygon Tiler,‖ using Visual

Basic (Microsoft, 2003). The advantage to creating my own software was that I could

tailor it very precisely for this research. In order to make both rotations and edge-to-edge

placements intuitive, the tile behaves as if the vertices were magnetized. The child begins

by simply dragging a tile to its destination. The tile initially moves in a simple translation

mode. Once a vertex of the moving tile comes close to the vertex of a previously placed

tile, the moving vertex ―snaps‖ to the already placed vertex and the moving tile then

268

begins to rotate about the snapped point until it reaches the desired position. If a second

vertex comes near another vertex, the tile snaps solidly into place and stops moving

unless the mouse pointer moves too far from the center of the tile, in which case the

moving tile unsnaps and can be moved freely again. Once the mouse button is released,

the tile stays put. The entire process of moving and rotating the tile from its initial

position to its final position is accomplished with a single click-and-drag in an intuitive

manner. A special algorithm prevents last-second slipping of the tile when the mouse

button is released. If the child desires to place the tiles in a non-edge-to-edge manner, the

magnets can be turned off either temporarily or permanently. If needed, a tile can also be

rotated by clicking on its vertex instead of the interior of the tile, as with other children‘s

software. These two rotation methods give the child a choice of intuitive methods for

placing the tile.

Flipping of tiles is much more rare. Tiles with reflection symmetry never need to

be flipped at all and other tiles never more than once. Flipping in Polygon Tiler therefore

simply relies on the traditional method of clicking a ―Flip‖ button (which is hidden if not

needed).

Because I designed my own software, I could tailor the program to the precise

needs for this research, including the need to take snapshots of all steps in each

construction and file these data by pseudonym and date. I included many other elements

in the program, some of which I included just in case I saw a need for them during the

interviews when I no longer had time to program. Some of the elements included in the

software are:

A spare, full screen with no more than three unobtrusive buttons;

An optional flip button which can be hidden or displayed as needed;

A hidden toolbar for seldom-used options, just in case;

Keyboard shortcuts for these options, including a shortcut for temporarily

disabling ―magnetization‖ when it occasionally interferes with tile placement in a

complex tessellation;

Automatic zooming and centering of the tessellation if it grows too close to the

edge of the screen, using a smooth animation;

The ability to zoom in or out and center manually, if needed;

The ability to easily disable snapping entirely if a non edge-to-edge tiling is

desired;

Control of snapping sensitivity, zooming factors, and graphics animation in order

to adjust to the child‘s dexterity and the computer‘s graphics capability;

Automatic sizing of the tiles in the initial generating set according to screen size;

An algorithm to prevent slipping of the tile when the mouse button is released;

Automatic screenshots of each tile placement organized by pseudonym, date, and

tiling;

Management of names and pseudonyms in a password protected environment;

269

Rejection of seriously overlapping tiles using animation (which the students found

amusing);

Various rotation modes to constrain the allowable rotations (usually I allowed

only 15° rotations);

The ability to change the color of the tiles (this was not used);

Automatic saves and the ability to resume a previous tiling;

Four different graphic methods for quickly defining any polygon needed for this

research;

The ability to define generating sets ready for specific interviews, including the

kinds of flipping and rotation allowed for each set;

The ability to edit polygons and generating sets;

The ability to choose file locations for tile definitions and screenshots;

And a password lock to prevent children from altering sensitive settings without

permission.

After using the software for the interviews, I only noticed two significant changes

that needed to be made. (1) The setting for the anti-slipping function needs to be

adjustable. This parameter needed to be increased for most of the students and there was

no way to adjust it quickly on the fly. The only aspect of the software that seemed to

frustrate the students was making fine adjustments without the mouse slipping at the last

moment, which might cause the tile to overlap and fall. Nevertheless, they found the

falling tiles to be amusing and sometimes enjoyed making the tiles overlap and fall on

purpose. (2) The automatic zooming function should be a bit less sensitive, or else

adjustable on the fly. The idea for the automatic zooming was to reinforce the idea that

there was no boundary to the tiling space. If tiles were placed close to opposite screen

borders, the program zoomed out automatically. Most students did not mind the

automatic zooming function, but a few did. Fortunately the automatic zooming was not

that important and I had allowed for an easy way to disable the automatic zooming

completely, so this did not present a problem. Aside from these two difficulties, the

software worked well and the children seemed to enjoy working in this environment.

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Appendix D: Analysis Codes

Table 9 below contains the list of codes that emerged from the analysis of my

data. Not all codes proved useful in identifying patterns in the data. Codes in italics are

just headings for other codes; no episodes were coded with these headings.

Table 9: Codes used in the analysis of the data

Category

Codes and sub-codes Description

Aesthetics Type of aesthetics expressed, if any. Instances where students could not express a reason were not coded.

Mathematical Aesthetics promoted by mathematicians

Connectedness Appealing because it relates otherwise unrelated ideas

Significance Appealing because it is mathematically important

Simplicity Simple, elegant, economical, controlled, or ingenuous

Surprise Surprising, unexpected, shocking, new, or insightful

Validity A tiling is appealing only if it meets the 3 criteria for a tiling

Visual appeal Appeals to the visual sense

Negative appeal Category for aesthetic qualities that are unattractive

Bad color Color is unappealing.

Bad real-world object Reminds students of a repulsive or boring real-world object

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Table 9. cont.

Category


Bad tiles Poor choice of prototile

Boring Tiling is too simple.

Can’t get a box unit Kobe disliked one tiling because he could never close it off straight.

Femininity Some tilings are considered “girl stuff.”

Finite Cannot be extended indefinitely

Insufficient color Needs more color

Invalid Not a valid tiling

Lack of pattern Too random

Not unique Tiling is not sufficiently different from other tilings.

One dimension Has only one interesting dimension

Overly complicated Too complex

Radial is bad Less interesting than periodic tilings

Semi-random is bad Having some order, but not completely patterned

Single orientation All tiles are in one orientation.

Too simple Even a child could create such tilings.

Other All other positive aesthetic criteria—these were used to describe appealing tilings

3D Tiling has three-dimensionality, whether real or illusion

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Table 9. cont.

Category


Alternating Tiling is cool because of alternation in the lengths or orientations, or makes a zigzag pattern.

Animal used An Escher tiling uses an animal which students like or find awesome.

Appropriateness Tiling is appropriate (frequently used) for a certain real world tiling task, such as tiling a kitchen floor. It is appropriate for the given or an imagined context.

Color Colors of tiling are appealing.

Complexity Variety of shapes, or the way shapes within shapes are used (such as L-tromino dilation tiling).

Craziness Tiling is appealing because it is “crazy” or “weird.”

Cultural style Tiling is appealing because it appears to be in a certain foreign style, such as “Asian,” “African,” or “Fantasy World.”

Diagonal Tiles or units appear in a diagonal pattern.

Energy Tiling is appealing because it is energetic or suggests explosions.

Folding Tiling suggests it could be folded in surprising ways.

Fun Tiling is considered fun.

Futuristic Pattern suggests a “future look.”

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Table 9. cont.

Category


Imagined or familiar context Tiling is appealing because of what person imagined could be done with it, or because of imagined similarity with real-world example.

Induction Mathematicians like the induction suggested by Tiling N. It is a “rep-tile.”

Intricacy Tilings have a labyrinthine, knotted, puzzle, or interconnected quality.

Masculinity Tilings that are “boyish” are more appealing than those that are “girlish.”

Movement Tiling suggests movement.

Multi-colored Tiling is colorful.

Multiple possibilities Mathematicians liked tilings that could be interpreted in multiple ways.

Non-tiling element A photo was appealing because of some elements (such as grass or “earthy”) that had nothing to do with the tiling itself.

Optical illusion Tiling suggests an optical illusion.

Proportions Appealing proportions

Randomness Randomness is appealing.

Realistic In one case, a photo was preferred over a drawing because it was “realistic.”

Real-world suggestion Tiling is appealing because it reminds people of a real-world object, symbol, or style (such as “Easter”).

Relaxing Tiling has relaxing quality.

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Table 9. cont.

Category


Semi-random Tiling has elements of order and randomness.

Shapes contained Tiling is appealing because it contains certain shape elements, such as flowers, arrows, zig-zags, or stars.

Shapey The private school described one tiling as “shapey,” having curves and lines and looking good even without color, but I’m not sure what they mean.

Size Rachel said she preferred the L-tromino to the L-tetromino because she likes “little things.”

Symmetry Tiling is better when it includes symmetry, sometimes described as having patterns or being organized or “tidy.”

Thought-provoking Mathematicians liked tilings that were thought provoking.

Tiles used Tiling is appealing because of the specific shapes used or the diversity of shapes used to compose it.

Touch of assymetry A tiling is appealing if it is symmetric, but contains a small element that destroys the symmetry.

Unique-creative Describes tilings that are different, not like the others. Tiling is “creative.”

Confidence Level of confidence in answer, especially in validity of tiling

No confidence Student claims to be guessing or very uncertain

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Table 9. cont.

Category


Strong confidence Student expresses strong confidence in voice and answer, e.g. “Are you sure? Yeah,” or “I’m pretty sure,” or “I think it’s gonna work.”

Weak confidence Student expresses hesitation or some uncertainty, e.g. “I’m not 100% sure,” or “Pretty probably maybe.”

Confidence at symmetry Confidence was coded for each episode where symmetry was coded using nearby statements of confidence so that symmetry and confidence could be compared.

Doubt tiling will work (at symmetry) Suspects tiling will not work, e.g. “Kind of think it won’t work,” but has not yet decided.

No confidence (at symmetry) Same as “No confidence”

Strong confidence (at symmetry) Same as “Strong confidence”

Weak confidence (at symmetry) Same as “Weak confidence”

Polygon Which polygon is the student working with?

Acute isosceles triangle Acute isosceles triangle

Obtuse scalene triangle Obtuse scalene triangle

Parallelogram Parallelogram

Pattern blocks Pattern blocks

Equilateral triangle Green triangle

Hexagon Yellow hexagon

Rhombus Blue rhombus

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Table 9. cont.

Category


Square Used in the introduction; not recorded for all students

Squares and triangles Dihedral tilings of squares and triangles

Pentagon Polydron pentagon

Polyominoes Polyominoes

Domino Domino (used in Finite Tiling task)

I-tetromino I-tetromino

I-tromino I-tromino

L-tetromino L-tetromino

L-tromino L-tromino

O-tetromino O-tetromino

S-tetromino S-tetromino

T-tetromino T-tetromino

Regular polygons Tilings made in Interview #5, generally non-monohedral and frequently changing the generating set

Right isosceles triangle Right isosceles triangle (Finite Tiling task)

Right scalene triangle Right scalene triangle

Possibility Which tiles can tile? Are there tiles that cannot tile?

All polygons can tile States that any polygon can tile

All shapes can tile Expresses the view that all shapes can tile

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Table 9. cont.

Category


Impossibility reason Why will the shape not work?

Curves or no flat sides make tiling impossible

Curves or no flat sides make tiling impossible

Empirical—can’t find a tiling that works

Empirical reason; cannot find a tiling that works

Equilateral not right kind of triangle for square

Squares can only tile with right isosceles triangles

Gaps make tiling impossible The shape cannot work because it will leave gaps.

Known impossible shape makes tiling impossible

Kobe thought that because squares and triangles together made pentagons, which were known not to tile, squares and triangles cannot tile.

Neither round nor sharp enough to fit

Hexagons are round enough to tile; squares are sharp enough. Pentagons are neither.

Not a square-type tile, so cannot tile

Only square-type tiles can tile.

Overlap makes tiling impossible

Circles cannot tile because they must overlap.

Some pentagons don’t work (others will)

Some pentagons don’t work. (Others will.)

Impossibility shape Which shape cannot tile?

Bizarre shapes can’t tile Bizarre shapes cannot tile.

Circles can’t tile Circles cannot tile.

Curved edges can’t tile Shapes with curved edges cannot tile.

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Table 9. cont.

Category


Non-polygons can’t tile Non-polygons cannot tile.

Pentagons can’t tile Pentagons cannot tile.

Rhombi can’t tile When attempting to sketch, student concludes the rhombus will not tessellate.

Squares and triangles can’t tile Dihedral tilings with squares and triangles are impossible.

Trapezoids can’t tile The right-angled trapezoid of the pre-test cannot tile.

Reasoning Type of reasoning displayed

Infinity Type of reasoning about infinity of tiling

Functional infinity Understands consequences of limitless tiling, but prefers working with finite boundaries to create tiling

Limited infinity Displays lack of understanding of consequences of limitless tiling, such as failing to see two patterns make same infinite tiling

Mathematical infinity Displays sound reasoning about the consequences of limitless tiling

Tiling justification Reasoning used to justify the validity of the tiling or explanation of how they know where to put the tiles

Empirical justification Uses empirical or inductive evidence

Experience justification Uses previous experience with tiling or similar tiling

Gapless justification Notes that gaps are always filled

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Table 9. cont.

Category


Known unit justification Explicitly uses another known tiling by making units of known tiling, e.g. 2 L-trominoes make a rectangle

Similar to known tiling justification

Notes resemblance to another known tiling

Subunit justification Says a tiling can be made with a shape because it can be decomposed into smaller shapes known to tile

Symmetric justification Demonstrates an (unstated) use of symmetry in the justification

Transformational justification Makes an implied or explicit reference to a transformation, usually translation in one direction only

Sorting criteria What criteria were used in sorting exercises during Interview #6?

Sort by certain attribute Sets created based on a certain aesthetic attribute such as “mazy,” “crazy,” or 3D

Sort by color Sets were created using color, either certain colors or the attribute of being colorful.

Sort by context Sets created according to where the tiling would be used, such as “patio” or “outdoors”

Sort by media Sets created according to media used, such as stones

Sort by multiple criteria Sets were created with more than one criterion, such as “gray and stony.”

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Table 9. cont.

Category


Sort by real-world resemblance Sets created because they reminded students of a certain category of real-world objects, such as “buildings”

Sort by shapes Sorting uses tile shapes.

Sort by simplicity vs complexity Sorting considers whether a tiling is basic or something unusual not often seen.

Strategy Strategy used to make tiling

Complex rows strategy Student makes complex rows. (If possible, use one of the three subcodes.)

Alternating rows strategy Makes rows with alternating orientations

Brick wall strategy Staggered, non-edge-to-edge rows

Units in rows strategy Student makes complex rows made of units. This occurs together with “Units strategy” coding.

Complex strategy Makes a sophisticated pattern using neither rows nor units

Fit one at a time strategy Student places one tile at a time resulting in apparent pattern or standard tiling, but without expressing or implying any organizing strategy such as rows or units.

Grid strategy Student sees square, rectangle, or parallelogram grid, simultaneously in two directions.

Growing strategy Makes a growing pattern, such as a tower, dilation, or radial pattern

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Table 9. cont.

Category


Random strategy Makes apparently random tiling

Semi-random strategy Makes apparently random pattern, though there are strong elements of order in parts of the tiling

Random units strategy Creates units, but then combines units in apparently random way

Simple rows strategy Student makes simple rows; this includes alternating square rows and triangle rows.

Spiral strategy Student sees tiles as expanding in a spiral.

Units strategy Student makes units. If the units are placed in rows, this is also coded as “Units in rows strategy.”

Symmetry Type of symmetry created by student, as observed during drawing, physical tiles, or computer creation; only patterns with more than 6 tiles are considered (significant progress towards actual tiling, not just exploring fit). Codes were applied when largest such symmetry was created (or noted in transcript) for entire tiling (not local symmetries). If there was no symmetry, but the infinite tiling was clearly intended to be symmetric (e.g. Moses’ parallelograms), I left the tiling uncoded.

2-fold symmetry 2-fold symmetry




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Table 9. cont.

Category




No symmetry No symmetry (other than necessary translation symmetry) was observed throughout the tiling creation, nor did the student try to create symmetry beyond first few tiles. (Coded at end of tiling.)

No symmetry at end Some significant symmetry was observed during the tiling creation, but the final product was distinctly not symmetric.

Order-2 rotation symmetry No reflection symmetry (e.g. standard parallelogram tiling)

Reflection symmetry Reflection symmetry (coded by axis orientation)

Horizontal reflection symmetry

Simple horizontal symmetry (horizontal axis)

Oblique reflection symmetry There was one case of an oblique axis.

Vertical reflection symmetry Simple vertical symmetry (vertical axis)

Tool With what tool was the tiling created or envisioned?

Blank paper drawing Blank paper drawing

Computer program Computer program

Dot paper drawing Dot paper drawing

Oral description Oral description

Pattern blocks Pattern blocks

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Table 9.cont.

Category


Polydrons Polydrons

Pre-test discussion Pre-test discussion

Understanding How well does the student understand how the tiles fit together to form a tiling? Is the description a mathematically valid way of making a tessellation?

Fit What is the level of understanding for how the polygon fits together around each vertex?

Accurate fit Describes accurately how tiles fit together

Inaccurate fit Does not put the correct number of polygons around a vertex or otherwise displays lack of understanding of how polygons fit together

Tiling Is the tiling mathematically valid?

Accurate tiling Accurately describes a valid tessellation

Apparently accurate tiling Tiling with errors, which student explains as being drawing imprecision that will not appear in physical tiling, or finite tilings that would be correct except for the size of the tile

Embodied tiling Student lacks a mental model for the tiling, but can construct it on paper. (The student knows it works, but cannot describe it.)

Inaccurate tiling Does not give a mathematically valid description that will work with the physical tiles

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Table 9.cont.

Category


Incomplete tiling The student sees how the tiles fit together, but did not draw enough to show how it would repeat in all directions; student possibly lacks an overall repeating pattern for forming an indefinite tessellation or possibly just was not pushed to extend tiling.

Partially accurate tiling Displays partly valid tessellation, with some errors

Wording Analysis of active or passive wording in first interview

Active wording Student uses active wording to describe tiling, e.g. “Put this one right here like that and then do it again.”

Descriptive wording Student uses passive description, e.g. “One line’s gonna be facing one way, the other, the next line’s gonna be facing the other way.”

(Uncategorized)

Abstractness of drawing Demonstrates understanding that a freehand drawing may not be accurate, but is only a representation of geometric ideal

Change of center As student makes a growing, symmetric tiling starting with a central core, the center of the tiling shifts to a new point.

Concern for accuracy Expresses concern over inaccuracy of drawing

Reference to symmetry Student makes an explicit reference to symmetry (not necessarily using that word).

Traces physical tile Uses pattern block to trace tiles in drawing

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Appendix E: Results by Task

This appendix provides greater detail about the specific results from each task.

References to the ―standard tiling‖ mean the simplest, edge-to-edge tiling. See Appendix

B for protocol details.

SET A: BASIC TILINGS

Students were asked about tiling various pattern blocks. The initial phase for each

shape was an oral phase where students were given one pattern block and asked to

describe how the tiling would work. This was followed by a drawing phase (with marker

and paper) and a construction phase (with a set of pattern blocks of the given shape).

Moses was also allowed to use the computer for the construction phase of the triangle.

All students used the computer for the parallelogram, which had no pattern block.

Rhombus

All six students thought from the beginning that it would be possible to tile with

the rhombus. Some were initially concerned about walls where tiles would have to be cut.

With the exception of Moses, each student described the tiling by describing how other

rhombi would fit around the first tile. Moses described how there would be rows of

rhombi going off in all four directions from the sides of the initial tile. This was a rare

instance of explicit awareness of structure in two different directions. After describing the

initial fit, Michelle described how the tiling would continue to grow in all directions. The

other four did not mention how the tiling would work beyond the tiles next to the initial

tile, limiting their thinking to how the tiles fit together. Describing how tiles fit around

the initial tile tended to be the primary way of describing a tiling during the oral phase of

each task.

All students, including Moses, sketched their ideas of how the tiling would work

one tile at a time. Three students drew the standard tiling with all tiles in the same

orientation. Kelsey also drew all tiles in the same orientation, but without any clear

structure and with one tile drawn as a large triangle. The other two students drew random

tilings.

When students were given pattern blocks, three created random tilings, two

created the standard tiling (often by making large rhombi), and one created a tiling with

alternating rows. Kelsey attempted a second tiling which she seemed to describe as the

standard tiling, ―just turn them sideways like that ... on and on and on.‖ But with the

eighth tile, the tiling turned random. She afterward described patterns in the random tiling

which she used to create portions of the tiling. This occurred with several students and

seems to reflect a focus on local fit, rather than the global structure of the tiling, at least

when the child is engaged in the process of creating the tiling. Kelsey‘s second tiling had

smaller sections of tiles with the same orientation, which Kelsey saw as aesthetically

preferable.

286

Michelle made a random tiling at first, but then separated out part to make a

double row with opposite orientations, which she said would be both easier and better

aesthetically. (See Figure 70 on page 212.) The two students who made the standard

tiling maintained that the only way to tile was the standard tiling with all rhombi in the

same orientation.

Moses had described the standard tiling in his oral description and drawing, but

partway through his construction with pattern blocks he realized he could make a tiling

with ―zigzagging‖ rows, which he felt was a much better tiling. At one point, Moses

began to focus entirely on the border of his pattern, despite frequent reminders that there

were no walls in this infinite room. (See Figure 48 on page 164.) He felt sure the sides

would get closer and closer at the top and spent much time trying to build up the border

with an empty interior before finally abandoning his idea when prompted to consider

whether tiles would actually fit inside his long curving borders.

Hexagon

All students thought the hexagon would tile, though Marie and Kobe were

doubtful until they actually tried to draw it. All were fairly successful in sketching a small

section of the tiling, some by tracing the pattern block. Because there is only one possible

tiling, all students successfully made it, though they used slightly different strategies to

tile. Some emphasized symmetry more than others. Most took a one-tile-at-a-time

approach, focusing on how the tiles fit together.

Moses focused on vertical columns that were being formed. Four students

recognized the tiling as a honeycomb tiling and Moses recalled previous experience with

pattern blocks. Kobe built up his tiling as diamonds with two-fold symmetry. Marie

found she could make the pattern by either focusing on rows like a sidewalk, or using a

growing pattern like a flower. She felt the latter was more creative. Kelsey made a

―flower‖ of seven hexagons and said she was done until prompted to tile further. She

carefully explained the two ways that hexagons fit together using sides and angles.

Michelle recalled that she could make the hexagon out of triangles (perhaps

meaning the rhombi) and therefore the hexagon should tile too. This subunit reasoning

occurred a couple of times to different students: If a shape can be divided into subunits

that can tile, then the shape itself should be able to tile. Moses used this strategy by

remembering that two trapezoids (with which he apparently was more familiar) could be

used to make a hexagon.

Triangle

During the oral phase of this task students described how other tiles would fit

around the initial triangle. No one saw that six triangles could fit around a common

vertex. In fact, most children thought that once three triangles had been placed on the

sides of the initial tile, a single triangle would be needed to complete the fit at each vertex

of the initial tile, as with the rhombi. Moses expressed surprise when he drew the tiling

287

and discovered that six tiles would actually fit together, ―kinda like a hexagon.‖ The other

children never realized that their constructed tilings contradicted their initial description,

even with prompting. Two students saw that two triangles would make a rhombus and

therefore the triangle would also tile.

All students drew and constructed the regular 36 triangle tiling. When drawing,

three students occasionally put five triangles around a single vertex. (See Figure 46 on

page 158.) Kelsey organized her drawings by rows, as did Moses for his pattern blocks.

Moses used a growing strategy in his drawing and on the computer, adding layers of

triangles around the initial hexagon. Moses also saw he could make a large triangle and

use that as a unit to tile with—a dilation strategy, though he did not mention further

levels. Many students organized their tiling using a variety of patterns without any clear

single way to structure the tiling. Kobe noted, ―With the triangle you can get into any of

the patterns, diamonds, octagons, hexagons.‖ Rachel kept noticing the alternating

orientations of the triangles. Students usually preferred creating a convex pattern.

Two students believed that each tiling was different, though if extended all would

result in the same regular tiling. Two students even identified certain tilings as their

favorite, though mathematically they were all equivalent. Kelsey believed there were

more hexagons in one pattern than another. Two students saw that all the tilings were

really the same. Rachel thought they were the different, though she showed some

hesitation and at the end changed her mind. Kobe was not sure.

Parallelogram

Five of the students answered on the pre-test that it was possible to tile with the

parallelogram. Kobe understood the problem to be whether the parallelogram could tile to

exactly fill the sheet of paper, which it cannot without cutting. Other students expressed

hesitation orally concerning the problem of walls. Kobe changed his answer once it was

explained that the tiling was to be infinite.

Students tended to structure the tiling in either rows or columns. Michelle noticed

both and drew the tiling on her pre-test by using a ruler to make a grid. (See Figure 53 on

page 180.) Nearly all students created the standard tiling on the computer, which many

saw as three-dimensional, as though it were a rectangular floor tiling observed from an

angle.

Kelsey drew two columns in her pre-test, but during the interview she made two

rows, which started edge-to-edge but did not stay together. (See Figure 34 on page 143.)

She talked about how the rows ―could be shorter and longer,‖ which would look cool. On

the computer she made the standard tiling.

Kobe used a subunit reasoning. The parallelogram could be divided into four

triangles, therefore it must tile the plane. He focused entirely on the triangles until

prompted to show how the parallelograms would actually tile, in which case he gave the

standard tiling.

288

Nearly all students made the standard tiling, several in two orientations, which

some recognized as basically the same. Many claimed these were the only possible ways

to tile with the parallelogram. Kelsey and Rachel also offered the brick wall tiling and

Moses added a tiling with rows of alternating orientations.

Impossible Shapes

Students were asked if there is a shape which cannot be used to tile a floor.

Most students mentioned the circle as an example of a shape that could not tile.

Students said either circles would have to overlap or else they would leave gaps. Moses

said that the circle lacked flat sides. Rachel said that non-polygons could not tile and

recited criteria she had learned for defining a polygon.

Kobe had maintained earlier in the interview that all shapes could tile, mentioning

that the only exception would be if we moved to other dimensions. (Kobe frequently

mentioned things that showed he was being exposed to some very deep mathematical

ideas.) Marie believed the right-angled trapezoid on the pre-test could not tile the floor.

She did not see that she could rotate the trapezoid. (Rachel had also thought the pre-test

trapezoid could not tile because it was not symmetric, which is the only instance of a

student using this word. During the interview, however, she only mentioned circles and

non-polygons as impossible.)

Note: Set B was done after Set D.

SET C: PENTAGON

Students were shown Polydron tilings of squares and hexagons to remind them of

tilings from the first interview. (See Figure 84 below.) They were then asked if it was

possible to tile with the regular pentagon.

Figure 84: Mo and I discussing square and hexagon tilings with Polydrons

All students believed that the regular pentagon could tile, some with hesitation,

until they tried to do it with the Polydrons, after which they concluded it was impossible.

289

Most seemed to think the tiling would look somewhat like the hexagon tiling and drew

something to that effect, with three or four pentagons around each vertex. Michelle drew

a tiling with five pentagons around a common vertex. (See Figure 35 on page 144.)

Students were not sure what the difference was between the pentagon and the

other regular polygons to account for the fact that the pentagon could not tile. In the end,

three students said the pentagon would not work simply because it always left gaps.

Moses saw from his drawing that other pentagons could tile, namely ―skinnier‖

pentagons. He also noticed the Polydrons could be bent into a three dimensional shape

(the beginning of a dodecahedron). Michelle thought the sides and angles of the pentagon

were different, accounting for the impossibility of tiling; the hexagon could tile because it

was rounder, and the square could tile because it was sharper.

Kelsey seemed confused by the Polydrons themselves. She focused on the hinges

even in her drawing (see Figure 85 below) and did not seem to understand how the

Polydrons worked until she actually tried to construct the tiling. Even then, she seemed

confused by the Polydrons.

Figure 85: Kelsey‘s drawing of a pentagon tiling with Polydrons

Kobe reiterated his belief from the previous interview that ―you can tile with

almost any shape.‖ He wrestled for over four minutes with the Polydrons before finally

deciding the tiling would not work, and even then kept trying. This seemed to make a big

impression on him and during the next task (squares and triangles), he focused on making

tilings with pentagons composed of one square and one triangle, in order to show that

some pentagon tilings were possible.

Students saw the hexagon as more symmetric than the pentagon. They noticed the

bilateral symmetry of the pentagon, but not its five-fold symmetry. The pentagon was

observed to be a ―house‖ or a square with a triangle on top. It was not obvious that the

base could be any side. The hexagon was more easily seen to be the same from any side.

The hexagon was also seen to be composed of six triangles.

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Michelle had some unusual difficulty counting the sides of pentagons and

hexagons and persisted in believing for a long time that the pentagon had at least as many

sides as the hexagon.

Students never saw that it was sufficient to study how pentagons fit around a

single vertex. Though most saw that three pentagons created a gap that could not be

filled, they continued to try to create a larger tiling in hopes that somehow something

would eventually work.

SET D: SQUARES AND TRIANGLES (ALSO SET G)

The squares and triangles task was visited twice—once during the second

interview and again during the fifth interview. During the second interview, all students

described tiling possibilities in oral and drawing phases and then attempted at least one

tiling using pattern blocks. All but Michelle also attempted a tiling using the computer.

For the fifth interview, students used only the computer.

Approaches to the squares and triangles task varied considerably. Every response

during the oral phase was unique. Most students believed during the oral phase that a

tiling would be possible, though Michelle expressed some hesitation. Rachel believed

that a tiling was not possible because the triangle needed to be a right isosceles triangle

(half square) in order to go with the square. However Rachel seemed to believe that some

sort of overlapping was being asked for in the task and when this misunderstanding was

cleared up during the physical construction phase, she changed her mind.

During the oral phase Kobe focused on the pentagon formed by a square and

triangle, seeing how they could make both a 33.4

2 pattern, which would work, and a

triangle surrounded by squares, which he believed would not work. Moses thought the

latter pattern could work by placing a single triangle at each corner of the central triangle,

making a large hexagon. (See the drawing on the right in Figure 46 on page 158. During

the construction phase, he saw he needed two triangles at each corner, which did not form

a hexagon but still led to a valid tiling.) Marie saw how two squares and three triangles

would fit together in a 33.4

2 pattern. Michelle described a hexagon of six triangles

surrounded by squares, which she decided during the drawing phase (where she actually

drew seven triangles) could not work. See Figure 86 below.

Figure 86: One of Michelle‘s attempts at making a tiling with squares and triangles

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A few students began with a central triangle, which usually did not lead to

problems. However, three students tried to create a tiling centered about a square (or

larger square made of square tiles), and this usually failed because they believed the

square should be entirely surrounded by triangles. It did not occur to them that another

square could be placed at each corner of the central square. In each case the students

struggled to fit five triangles around the corner of a square, shifting the triangles so that

they were no longer edge-to-edge with the square in order to minimize the size of the

inevitable gaps that formed. (See, for example, Figure 38 on page 148.) During the

second interview, students usually believed that such a tiling would eventually be

successful. Michelle, for example, thought the small gaps were the result of the

randomness of her tiling. During the fifth interview, Kelsey and Michelle again tried to

surround a square corner with triangles. This time, Kelsey decided that squares and

triangles could not tile and Michelle eventually found a different strategy. Kobe was the

only student to successfully create a tiling centered around squares, making a complex

tiling having four-fold and translation symmetry with rectangles made of six squares at

the center of each unit.

With both the pattern blocks and the computer, Moses made patterns which began

around a triangle and had three-fold symmetry. Each time, as he built it, the center shifted

and he ended with bilateral symmetry. He described how he kept the symmetry by

copying one side to the other because he liked patterns. Moses believed his second

pattern could be used as a large unit to tile the floor.

Kobe and Michelle found strategies based on ―houses‖—pentagons formed by a

square and a triangle. Kobe remembered from the previous task that the regular pentagon

was not possible and initially believed this tiling would not work either, preferring at first

to focus on a row strategy instead. Later when he successfully made a more complex

pattern, he pointed out all the pentagons that made it up and how this was different from

the failure of the pentagon task. Michelle used houses to create a radial pattern. Her

strategy eventually abandoned using houses alone, but she still saw houses throughout

her pattern.

The most common strategy, used at some point by four different students, was to

use row patterns—nearly always the 33.4

2 pattern with horizontal rows. Kobe also made a

tiling with double square rows. In the fifth interview Marie made a tiling based on a

tower with a double row of triangles at the top. (See Figure 51 on page 175.) She

eventually decided the double row of triangles were to be used along the walls of the

room. Unlike the pilot study where some students saw the tower as a finite unit, Marie

indicated that the rows were infinite. No one else used a tower strategy.

Three students made patterns with six-fold symmetry. Kobe worked out the 2-

isogonal 36/3

2.4.3.4 tiling and described its translation symmetry. (See Figure 37 on page

148.) The other two students created more complex radial tilings without translation

symmetry.

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Three students used different methods to create more or less random patterns.

Michelle‘s and Kelsey‘s patterns had small gaps. Michelle explained the gaps as a

necessary consequence of the randomness of her pattern. In the second interview, Kelsey

thought her random tiling would eventually work, ―As long as you can do it right.‖

Most students did not seem to recall for Interview #5 how they did the task for

Interview #2. And yet answers were similar for both interviews for most students. Kobe

came up with complex symmetric patterns both times. (He also made a row pattern the

first time.) Rachel made row patterns both times, after struggling to fit triangles around a

square corner the first time. Marie made the 33.4

2 tiling both times and both times said it

reminded her of a 3D farm. The second time, her row pattern evolved into a tower

pattern. Kelsey tried the same unsuccessful strategy of fitting triangles around a square

corner both times. She believed it worked the first time, but not the second. Michelle

struggled to put triangles around a square corner both times. The second time she

discovered a six-fold radial pattern after struggling with the square corner. Moses made

complex patterns both times that began with trilateral symmetry and changed to either

bilateral or six-fold symmetry.

SET B: FINITE TILINGS

Comparison of my students‘ initial answers with answers given by students in

Owens and Outhred‘s (1998) study are very comparable, except my students tended to

get the correct answer more often, partly because Owens and Outhred‘s study also

included second graders, but also perhaps because of the interview format. (Owens &

Outhred had noticed that some students gave the wrong answer on their written test, but

then showed correct understanding on a follow-up interview.) Both my students‘ and

Owens and Outhred‘s tended to answer that the tiling was not possible more often for

problems B3 and B6. (B5 was not in Owens & Outhred‘s test.)

Pre-test

The first pre-test item showed a 5×7 grid with most of the interior erased.

Students were asked how many squares were in the original rectangle and how they

knew. This item was adapted from a problem by Battista, Clements, Arnoff, Battista, and

Borrow (1998). This item was discussed with students as an introduction to finite tilings.

Kelsey appeared to be at Battista et al.‘s Level 1—complete lack of row and

column structuring. (See Figure 39 on page 149.) Though there was some awareness of

alignment of squares, the awareness was local rather than global and squares appeared in

different sizes. The other five students used the row and column structure to deduce that

there were 35 squares in the original rectangle. All but Kobe redrew the squares, the

students from the French school using a ruler. Half the students used multiplication to

find the final answer and half counted the squares.

For the remaining finite tiling exercises, students were shown a shape to be tiled

and a prototile and then asked if the tiling would work and if so, how many tiles it would

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take. There was an oral phase, a drawing phase, and for the first few problems a

construction phase to check their answer.

Many students had a tendency to underestimate the size of the prototiles, usually

resulting in an overestimate of the number of tiles that would fit, both in the oral phase

and in the drawing phase. Their judgment often got better in the later problems. Marie,

for example, drew half-size tiles for problems B1, B2, and B3, and full-size tiles for the

remaining problems.

When students had pattern blocks (problems B1 – B4), they were always

successful at creating the standard physical tiling during the construction phase. No tiles

were available for the last two problems.

Problem B1

Students were asked about tiling a 2×3 rectangle (without any interior markings to

guide them) with a tilted square the size of a pattern block.

Most students saw that the square could be rotated and used to make the standard

square array tiling of the rectangle. Initial estimates for the number of tiles required

ranged from 4 to 12. After drawing, two students correctly saw there would be 6 tiles;

other answers after drawing included 9, 18 and 19. Kelsey revised her answer from 19 to

5 when she saw the actual pattern block. Apparently seeing the physical block enabled

her to do a more accurate estimate than seeing only a drawing of the prototile.

Michelle eventually described an unusual mental model where the tiles were left

in their original tilted orientation for the left and middle columns and turned to the

standard orientation for the right column, resulting in a failure to sketch a tiling. (See

Figure 87 below.) She was successful with the physical pattern blocks, after which she

explained what her initial mental model had been.

Figure 87: Michelle‘s attempt to solve problem B1

Kelsey drew different-sized squares and did not impose a global row and column

structure on her drawing. (See Figure 39 on page 149.)

Problem B2

Students were asked about tiling a 2×2 rhombus with a unit triangle. The rhombus

had markings on the side to suggest the standard tiling.

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All students thought the large rhombus could be tiled with triangles. Initial

estimates for the number of tiles ranged from 6 to 12. Four students were successful in

using the guide marks to sketch the standard tiling of eight tiles.

Marie drew small triangles with various angles; she abandoned her attempt to

draw a tiling about halfway through. She also struggled a bit with the physical tiling. In

the end, she used symmetry to find the positions of the last few tiles. She then analyzed

the tiling as composed of diamonds (small rhombi).

Kelsey left gaps between the triangles in the center of her drawing, but explained

she had just messed up a little. She explained that the gaps would not appear in the

physical tiling.

Problem B3

Students were asked about tiling an isosceles trapezoid with a unit triangle. The

triangle was turned to an orientation that did not match the trapezoid.

All students thought the trapezoid could be tiled with triangles. Five students were

certain the tiling could be made with three triangles and were successful at sketching the

standard tiling. Kelsey made two attempts because she drew the triangles too small. Her

final drawing did not fill the trapezoid, but she still felt sure that three tiles would work.

Michelle initially doubted that a tiling was possible, but after measuring the prototiles

with her fingers, she decided it could be done with three triangles. Marie thought perhaps

7 tiles were needed and drew 8 slightly overlapping triangles of different sizes.

All students were successful with the physical tiling, at which time Marie realized

her triangles had been drawn too small.

Problem B4

Students were asked about tiling a large equilateral triangle (side of length 2) with

a unit triangle turned to an orientation that did not match the large triangle.

Initial estimates for the number of tiles ranged from 3 to 4. Most were successful

at sketching the standard four-triangle tiling, though some struggled a bit.

Marie believed the tiling could be made with four tiles, but when she attempted to

draw it, she could not find the correct positions for the lines and claimed to no longer see

her initial idea.

Kelsey made two attempts, again drawing the triangles too small. Her final

drawing did not fill the large triangle, which left her uncertain if the tiling could work.

All were successful at finding the standard tiling with the physical pattern blocks.

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Problem B5

Students were asked about tiling an L-tromino with a domino. No tiles were

available to check their answers.

Four students saw immediately that the tiling was not possible, because a second

domino could not fit in the square that would be left after placing the first domino, or

because two tiles would necessarily overlap.

Michelle initially said the tiling was possible, apparently believing it would be

acceptable to complete the tiling with a square. She then showed the tiling was

impossible by showing how the dimensions of the domino matched the dimensions of the

L-tromino and that only room for a half-tile would be left.

Moses was also certain at first that the tiling was possible, but after a few seconds

of thinking and using his finger to imagine rotating and moving the prototile, he changed

his mind. The tiling would only work with squares.

Problem B6

Students were asked about tiling a right-angled trapezoid (a triabolo) with a unit

right isosceles triangle. The trapezoid can be tiled with three triangles in two different

ways. The triangle was in the correct orientation for one of the tiles of one method, but

not in the orientation that would fit the acute angle of the trapezoid. Students found this

task to be the most difficult and gave a variety of answers.

Moses and Kobe saw and sketched two different valid tilings. Marie thought it

could be done with three tiles, but was not sure. Her sketch seems to show the middle tile

turned with the hypotenuse at the bottom:

Figure 88: Marie‘s attempt to solve Problem B6

Michelle was sure the tiling could be done with two tiles:

Figure 89: Michelle‘s attempt to solve Problem B6

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Rachel thought a tiling would be impossible. She tried to insert the triangle in the

left part at the wrong orientation, making the rest of the tiling impossible. (See Figure 90

below.) Kelsey gave a similar response.

Figure 90: Rachel‘s attempt to solve Problem B6

SET E: TRIANGLES

Students were asked about tiling three triangles shown on the computer. There

was an oral phase, a drawing phase, and a computer construction phase for each triangle.

Acute Isosceles Triangle

The acute isosceles triangle had angles of 40°, 70°, and 70°. All students believed

it was possible to tile with this triangle.

Four students envisioned, sketched, and constructed the standard tiling by using

rows of triangles. Michelle thought two rows looked like teeth because of the opposing

orientations. Rachel organized her rows as trapezoids (sets of three triangles at a time),

two of which could combine to make a hexagon. In addition to a row pattern, Marie also

attempted a random pattern, which she abandoned as impossible.

Kobe saw that the triangles could be put together at their apex to form a nonagon,

which he believed could then tile a floor. After sketching, Kobe began to have some

doubts. After an unsuccessful attempt to create the tiling on the computer, Kobe switched

to a row tiling. Michelle thought the triangles would make an octagon and sketched a

radial pattern (with inconsistent numbers of triangles in the second layer) based on this

idea. On the computer, Michelle constructed a successful radial pattern starting with a

nonagon.

Right Scalene Triangle

Students were asked about tiling a right triangle with legs of lengths 2 units and 3

units. The orientation of the triangle was such that no side was vertical or horizontal.

Because of the unusual orientation, students did not easily see that two triangles

could be combined to create a rectangle, but they still believed a tiling was possible. Only

Kobe realized he could make rectangles during the initial oral and drawing phases. Other

students tried a variety of strategies, including fan patterns and radial patterns. Most

students eventually discovered rectangle units as they explored tiling possibilities.

Michelle struggled with the task. She eventually discovered rectangle units, but

was not successful in arranging them in a clear pattern. (See Figure 40 on page 150.)

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Moses envisioned and sketched a fan pattern. He made a complex version of this

pattern on the computer with a central rectangle (―mail with wings‖) and believed this

unit could be used to tile a floor. (See Figure 91 below.) Kelsey also sketched a fan

pattern, but abandoned it and eventually tiled with rectangle units.

Figure 91: Moses‘ ―mail with wings‖

After discovering the rectangle unit tiling, Kobe thought he saw a second way to

tile and created a parallelogram unit, which he quickly rejected as the basis for a new

tiling.

Obtuse Scalene Triangle

Students were asked about tiling an obtuse triangle. Responses were often similar

to responses for the right triangle. All students initially thought a tiling was possible.

Kelsey and Kobe struggled with sketches, deciding that a tiling was not possible

after all. (See Kelsey‘s drawing in Figure 67 on page 207.) On the computer they both

discovered the tiling with parallelogram units.

Time ran out before Marie and Michelle could sketch or construct a tiling, but

they both seemed to describe a parallelogram unit tiling.

Moses tried to make an ―arrow‖ unit (non-convex kite), but was not successful in

expanding this unit to a tiling. He decided the obtuse triangle could not tile the floor.

Rachel sketched a tiling with kites. Under prompting she showed how each kite

could be divided into two obtuse scalene triangles, but it is not clear if she initially

thought her kites were units or tiles. On the computer she began a complex tiling with

both parallelogram and kite units. (See Figure 58 on page 191.) She placed one unit in a

non-edge-to-edge position at the end, which created problems, but she still thought her

tiling could eventually work.

SET F: POLYOMINOES

Trominoes

Students were shown a single plastic tromino (first an I-tromino and later an L-

tromino) and asked if it could be used to tile a floor. There was an initial oral phase and

then a drawing phase on dot paper.

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Kelsey, Michelle, and Rachel drew their trominoes subdivided into squares,

which meant that it was impossible to see where the outlines of the trominoes were in

their tiling drawings. (See, for example, Figure 41 on page 151.) I quietly attempted to

suggest the shape be drawn without squares simply by drawing the first shape for them

without squares, but they still subdivided my drawing. (I did not otherwise prompt them

not to use squares.) I used the video recordings to reconstruct their tilings. The plastic tile

was itself divided into squares, which appears to be the source of the idea to subdivide

the tiles. (In the pilot study no plastic tile was shown and no student thought to subdivide

the shapes.) Wheatley and Reynolds (1996) did not report this phenomenon in their

study.

I-tromino

Students had no difficulty tiling with the 3×1 rectangle, most making several

possible tilings. Most students made standard tilings. Three students drew at least one

complex tiling with units and one student made a ―brick wall‖ tiling. (See Figure 64 on

page 202 for Kobe‘s complex unit tilings.)

At the end students were shown a brick wall tiling with the I-tromino and asked if

such a non-edge-to-edge tiling was possible. All thought it was a valid tiling.

Other than changing the orientation of the standard tiling, three students believed

that no other ways of tiling with the I-tromino were possible. After seeing the brick wall

tiling, they thought the brick wall and standard tilings were the only two possible tilings.

L-tromino

All students believed a tiling was possible with the L-tromino.

Four students drew random patterns but their initial explanations were different.

Kelsey envisioned stacking the tiles like arrows, but sketched a random tiling. Kobe

made rectangle units at first, but then his tiling became random. Marie struggled to draw

the L-tromino at first. She succeeded by drawing a 2×2 square and erasing one corner.

She explained her tiling with reference to walls (which she knew were not really there),

showing how the tiles could line up against two walls, but then she drew a random

pattern. Michelle described a tiling with rectangle units, but did not use them in making

her random pattern. Students who made random patterns were always certain they would

work, sometimes expressing empirical reasoning.

Only Moses and Rachel envisioned and drew the standard tiling using rectangle

units. Moses explained how the units would line up in rows and columns. Rachel

believed this was the only possible way to tile with the L-tromino.

Michelle had previously tried unsuccessfully to make a tiling with I-trominoes at

a slant across the dot paper. When asked if her random L-tromino tiling was the only way

to tile, Michelle made a sketch on blank paper showing how she believed L-trominoes

could be arranged in a radial pattern if she did not have to respect the dot grid. (See

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Figure 42 on page 152.) She had some doubts because her drawing was not accurate.

When asked if she was sure it would work, she replied, ―From the drawing, no, but in my

head, yes.‖ She later expressed further doubt, saying, ―‗cause in my head most of the time

I can make anything work.‖

Tetrominoes

For this more open-ended task, students were shown plastic tiles for each of the

five tetrominoes and asked which ones could tile. They then explored one or two tilings

of their choosing on dot paper, and further tilings on the computer.

Students generally believed that most or all of the tetrominoes could be used to

tile a floor. Rachel and Marie were not sure concerning the T-tetromino, but Marie

eventually decided the T could tile. Michelle was not sure about the L-tetromino. Moses

did not believe either the L or the T could be used to tile; he drew small random tilings to

demonstrate why they would not tile. Rachel thought the S-tetromino would not tile.

Whenever a student attempted to make a tiling, they almost always eventually

concluded it could tile, even if they initially believed otherwise, expressed doubts during

the tiling, or failed to make a valid tiling. This was true even if their tiling was random.

Few students had trouble drawing the shapes. Moses stretched all of his S-

tetrominoes into hexominoes, or even longer. Some of his L-tetrominoes were

pentominoes. At one point, Rachel began drawing S-tetrominoes as two dominoes

touching only at a vertex. Students who subdivided their polyominoes, making it hard to

see the boundaries between the tiles, occasionally slipped an extra square into their tiling.

There are two symmetric ways to tile using the S-tetromino if all tiles keep the

same orientation. Four students constructed the more common row pattern which has

straight lines separating the rows. Kelsey began making the less common tiling, which

has no such lines, though she only placed five tiles and did not describe the pattern, so

she may have had in mind a random tiling. Kobe made an interesting pattern with a single

glide reflection axis. (See Figure 45 on page 156.) Two students created random patterns

with the S. Most students expressed a preference for the S-tetromino, believing its shape

to be more ―cool‖ than the others.

Three students made random patterns with the L-tetromino. Kobe created a tiling

of diagonal rows. Rachel was the only one to create rectangle units, which she used to

create the standard rectangle unit tiling.

Students found the O- and I-tetrominoes very easy to tile with. Rachel created

both the standard I tiling and a complex tiling with a mix of both orientations. Rachel

created the standard tiling with the O by making a large rectangle. Kelsey created a brick

wall tiling with the O, saying this was more ―cool‖ than the standard square array.

Rachel created the unique tiling with the T-tetromino that keeps all tiles in the

same orientation. She believed this was the only way to tile, ―unless you did it upside

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down of course.‖ Three students tried to make random tilings with the T-tetromino, but

only Marie believed she was successful.

SET G: REGULAR POLYGONS

This final open-ended task presented students with five regular polygons (triangle,

square, pentagon, hexagon, and octagon) on the computer and asked them to make tilings

any way they wanted.

The presence of the pentagon proved an irresistible temptation to all students. No

student remembered the second interview where they learned the pentagon cannot tile by

itself. Indeed, it cannot tile with other regular polygons either, but some students focused

the majority of their attention on unsuccessfully trying to create a tiling with pentagons.

Of the 28 attempted tilings, 18 included pentagons and failed. Another 3 attempts did not

use pentagons and also failed. The other 7 tilings were successful.

Kelsey and Rachel created tilings using hexagons and triangles. Kelsey‘s pattern

was random. (See Figure 43 on page 154.) Rachel created the 2-isogonal 32.6

2/3.6.3.6

tiling. (See Figure 69 on page 208.) Kobe made the regular 63 hexagon tiling.

Three students found the semi-regular 4.82 tiling using octagons and squares.

Michelle created a radial pattern using triangles, squares, and hexagons. At one

point she realized it was identical to her last tiling using triangles and squares (see

above), except groups of six triangles had been replaced with hexagons. (See Figure 52

on page 178.)

Marie was the only student not to find a valid tiling, though she did create a large

radial pattern centered on an octagon using successive layers of triangles, pentagons, and

then squares. The tiles did not quite fit exactly and she could not find a way to add a fifth

layer. She made many other tiling attempts and remained optimistic that there was a

solution somewhere.

Sets H and I were optional and were never used.

SIXTH INTERVIEW (SET J)

The sixth interview consisted of six aesthetic evaluation tasks. These tasks were

done in two groups of three students, one group from each school. See Appendix B for

illustrations of the tilings used.

Student Tilings

Students had to choose four tilings for four rooms of a house. The charter school

chose a pattern with slightly overlapping tiles for the bathroom because it looked fun and

resembled a sun. The private school students almost picked this pattern as well, but

rejected it at Kobe‘s insistence because of the overlapping. It is also flawed because it is

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not perfectly symmetric and cannot be continued beyond the finite pattern shown, but this

generally went unnoticed by the children.

Both schools chose the 4.82 tiling for the kitchen. The charter school believed this

was typical of kitchens and that kitchens should be basic. Both schools chose one random

pattern and one rich six-fold pattern.

Other Tilings

For the next three tasks, I showed students 18 tilings ―done by adults.‖

Student Sorting

First, students had to sort the tilings, and then choose their favorite group and

tiling. The preferred groups all had to do with colors, except for Kobe‘s ―Mazy‖ group

using the Interconnectedness aesthetic theme. There was no coordination of criteria in

making the sorted groups. The groups students formed, roughly in order of preference,

along with descriptions and favorite tiling in the group, are summarized in Table 10

below.

Table 10: Groups formed by student sorting

School Group Description Favorite in group

Private H J O P Blues H

Private E N Mazy N

Private K M R Crazy K

Private I L Q They go together I

Private B G Windows G

Private A C D F Gray and stony D

Charter K M R Colorful R

Charter J P Same colors J

Charter E N Maze N

Charter B Q They go together Q

Charter L O Basic shapes and matching colors O

Charter H I Basic shapes H

Charter A C D F G Outdoors A

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Comparisons in Pairs

Next, students had to express preferences in pairs (or larger groups) of tilings I

showed them. Both schools chose R (the random pattern) over M, explicitly for the

colors, though M was also colorful. M was said by the private school to have bad

patterns. (It had only translation symmetry). See Table 7 on page 196 for other

unanimous results from this task.

Professional Preferences

Lastly, students were asked which tilings a mathematician or an artist would

prefer. Both schools thought the artist would like R (the random pattern). They also chose

J, K, and D. Both schools thought the mathematician would pick K (because of its

illusion of 3D cubes). The private school also suggested N and Q for the mathematician.

Professional Tilings: Alhambra

Students were shown seven postcards of Alhambra tilings. Both schools preferred

tiling #7. The private school also chose tiling #5. Tilings #5 and #7 are radial patterns

which they compared to stars and many other things. They liked the colors of tiling #7.

Their least favorite tilings were #2 and #6. The private school also included #4 among

least favorite tilings. These were all considered boring and too simple.

Professional Tilings: Escher

Students were shown a variety of tilings by Escher and asked to award prizes to

the best three. From Escher‘s first set of basic tilings, both groups liked #3, though for

different reasons. For the second set of basic tilings, both groups picked the rhombi

because they looked three-dimensional, though a majority at the private school actually

preferred the parallelograms because they looked like home.

Escher‘s ―Regular Division of the Plane #66‖ was chosen by the private school

for first place and by the charter school for second place. (See Figure 63 on page 195.)

They cited the colors and fantasy or mythology element as their reasons. Both groups

were impressed with the way the different colored lions went in opposite directions and

fit together perfectly.

The charter school chose Escher‘s ―Regular Division of the Plane #12,‖ a tiling of

blue and white butterflies, for first prize, citing themes of Color, Energy, and Real World

Connection. The private school chose Escher‘s ―Regular Division of the Plane #58‖ for

second prize claiming it had an Asian style. (See Figure 57 on page 185.) The charter

school chose Escher‘s ―Regular Division of the Plane #132‖ (blue and red flowers in a

complex pentagonal pattern) for the third prize. The private school chose Escher‘s

―Regular Division of the Plane #97‖ (black and white dogs) for third prize.

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Appendix F: Analysis of Change

All of the most important cognitive codes were analyzed for change across the

interviews. Change in aesthetic codes was analyzed by comparing the first five interviews

with the sixth interview. (See page 219 for the analysis of aesthetic change. There were

insufficient aesthetic codes to analyze aesthetic change across the first five interviews.)

This appendix contains the details of the analysis of the change in cognitive codes. All

important non-aesthetic codes were totaled for each of the five interviews. Codes that

were task related, such as Possibility, Polygon, and Tool, were not under the control of

the students and were not analyzed. Certain codes, such as Wording and Attribute Focus,

were only coded for certain tasks; these were not analyzed either. Table 11 below shows

the code totals for each interview.

Table 11: Analysis of code changes across first five interviews

Code Interview #

1 2 3 4 5

Uncategorized codes Abstractness of drawing 7 0 2 0 0 Change of center 0 1 0 0 0 Concern for accuracy 6 0 5 0 0 Reference to symmetry 0 1 0 0 0 Traces physical tile 15 0 0 0 0

Understanding Fit

Accurate fit 41 4 29 10 0 Inaccurate fit 15 11 17 1 1

Tiling Accurate tiling 44 1 27 23 0 Apparently accurate tiling 1 0 7 1 0 Embodied tiling 4 0 0 0 0 Inaccurate tiling 12 13 20 12 0 Incomplete tiling 3 0 2 4 0 Multiple tiling 0 0 0 0 0 Partially accurate tiling 3 1 1 1 0

Confidence No confidence 7 4 5 2 0 Strong confidence 54 11 18 27 2 Weak confidence 23 9 9 5 1 Confidence at symmetry 0 0 0 0 0 Doubt tiling will work (at symmetry) 0 0 2 0 2 No confidence (at symmetry) 0 1 1 0 0 Strong confidence (at symmetry) 37 17 19 21 13

304

Table 11. cont.

Weak confidence (at symmetry) 2 2 1 0 0 Symmetry

2-fold symmetry 7 5 1 3 2 3-fold symmetry 3 3 0 0 0 4-fold symmetry 1 1 0 0 3 6-fold symmetry 5 3 0 0 5 8-fold symmetry 0 0 1 0 3 9-fold symmetry 0 0 1 0 0 No symmetry 10 2 2 10 2 No symmetry at end 2 3 0 0 1 Order-2 rotation symmetry 10 1 8 4 0 Reflection symmetry 1 0 0 0 0 Horizontal reflection symmetry 3 0 2 0 0 Oblique reflection symmetry 0 0 1 0 0 Vertical reflection symmetry 6 6 8 4 2

Reasoning Infinity 0 0 0 0 0

Functional infinity 3 3 2 2 2 Limited infinity 8 0 0 0 0 Mathematical infinity 2 0 1 0 0

Tiling justification Empirical justification 2 1 0 6 1 Experience justification 9 0 0 3 1 Gapless justification 0 0 0 0 0 Known unit justification 3 0 5 1 0 Similar to known tiling justification 3 0 0 0 0 Subunit justification 1 0 0 0 1 Symmetric justification 0 0 0 0 0 Transformational justification 0 1 0 0 1

Strategy Complex rows strategy 0 0 0 1 1 Alternating rows strategy 9 0 5 1 0 Brick wall strategy 2 0 0 1 0 Units in rows strategy 0 0 1 1 0 Complex strategy 1 1 3 3 1 Fit one at a time strategy 21 1 0 3 4 Grid strategy 1 0 0 0 0 Growing strategy 8 6 2 3 5 Random strategy 1 2 1 12 0 Semi-random strategy 10 6 0 2 1 Random units strategy 1 0 1 0 0 Simple rows strategy 9 9 4 8 3

305

Table 11. cont.

Spiral strategy 1 0 2 0 0 Units strategy 5 3 14 5 0

Uncategorized codes did show some change. The two students who traced tiles

both did so only in the first interview. There are more instances of ―Abstractness of

drawing‖ and ―Concern for accuracy‖ in Interviews #1 and #3. These would indicate

more awareness of the limitations of hand drawings in those interviews. This is likely

task related because Interviews #2, #4, and #5 used less freehand drawing.

Understanding (accuracy of fit and tiling) showed a small decrease in accuracy

in the later tasks, probably due to the more complex nature of the those tasks. There is no

clear trend towards greater inaccuracy as the interviews progress. There are more codes

of all kinds for the first interview, which used four different basic shapes. Most codes

were done during the drawing phases, so there are few codes for the tasks that did not use

paper and pencil, such as in the fifth interview.

Confidence was more heavily coded in the first interview. There were few

expressions of confidence (strong or weak) during the fifth interview. There is no clear

tendency towards greater or weaker confidence across the interviews, only fewer

expressions of either kind.

Symmetry is closely related to the prototile and therefore depends on the task.

(See page 213.) For example, the parallelogram in Interview #1 naturally created rotation

symmetry of order 2. The fifth interview was more complex and resulted in some higher

order symmetry. Codes for ―No symmetry‖ tended to occur during the first interview

(Basic Tilings) and the fourth interview (Polyominoes). There was no tendency to use

more or less symmetry as the interviews progress.

Reasoning (infinity and tiling justifications) did not change across the

interviews, with the possible exception of ―Limited infinity,‖ which was only evident

during discussion of the basic tilings in the first interview. This may indicate that students

began to accept the idea of an infinite tiling after the first interview, or it may indicate

that limited conceptions of infinity were only clearly revealed in simple tilings. In fact, all

of the data concerning students‘ limited ideas of infinity came from only two tasks: the

equilateral triangle task and the rhombus task, both in the first interview. In other tasks,

students‘ actions were sometimes coded as ―functional infinity,‖ which meant that they

recognized the infinite nature of the tiling, but still struggled with it in some way, such as

Marie‘s creation of temporary walls to scaffold her tiling. Other high frequency codes

seem related to the task. For example, many students were familiar with the basic tilings

in Interview #1, but not the tiling tasks in the other interviews. The codes for ―Experience

justification‖ are therefore all in the first interview.

306

Strategy codes showed almost no change across the interviews. Codes that seem

to show change are probably all task related. For example, the ―Fit one at a time‖ strategy

is closely related to the ―Random‖ strategy; it is applied only when students begin placing

tiles like a puzzle in hopes that a non-random pattern will quickly emerge. This mostly

occurred for the hexagon task in the first interview and the regular polygon task in the

fifth interview. Alternating rows was a common strategy for triangles, and was therefore

used mainly in the first and third interviews.

In conclusion, with very few exceptions, I can find no evidence for significant

change in children‘s thinking across the first five interviews.

307

Appendix G: Analyzing Greater Symmetry

In order to analyze whether children preferred tessellations with greater

symmetry, I analyzed the symmetry of the tessellations in the Other Tilings tasks of the

sixth interview and then studied children‘s preferences between pairs of tilings. There is

not a single clear way to analyze the symmetry of these tilings. The most rigorously

mathematical method would be to analyze their symmetry groups.

The set of symmetries in a pattern, together with the operation of composition,

creates a symmetry group for any pattern, finite or infinite. We can therefore analyze the

symmetry of any tessellation by describing its symmetry group. There are currently

several different notations for describing the various symmetry groups of tessellations,

the most useful of which is orbifold notation. Very briefly and at the risk of

oversimplifying slightly, orbifold notation lists all orders of rotation symmetry about

distinct points (without reflection symmetry), followed by a ―star‖ (*) if there is

reflection symmetry, followed by orders of symmetry that have reflection and rotation

combined at a distinct point. By ―distinct‖ point I mean that we do not count points

equivalent by a translation symmetry more than once. Glide reflection symmetry is

indicated by a ―cross‖ (×). If the pattern has no translation symmetry, so that all

symmetries fix a point, the point symbol (•) is added. For frieze groups with translation

symmetry in one direction only, the symmetry order is considered infinite and ∞ is used.

For periodic patterns, no further symbols are needed, unless the tessellation has

translation symmetry only, in which case a ―wonder-ring‖ (o) is used to describe the

symmetries.

For example, the infinite tessellation in Figure 92 below has the orbifold notation

*632 because it has reflection and rotation symmetries of orders 6, 3, and 2. The notation

is such that it describes the symmetries of any pattern completely. I will not go into

further detail because orbifold notation did not provide a conclusive analysis for this

study. For more details, and much greater mathematical depth behind the notation, see

(Conway, Burgiel, & Goodman-Strauss, 2008).

Figure 92: Tessellation with orbifold notation *632

308

Did students prefer tilings with greater symmetry? In order to carry out such an

analysis, it was necessary to analyze each tiling for symmetry so that they could be

compared. There is not one clear way to do this. Three methods were tried:

1. Tilings were analyzed according to orbifold notation. Numbers were paired

between tilings to be compared as far as possible. Larger numbers represented

richer symmetry; numbers after a star represented more symmetry than numbers

before a star. If one tiling had more symmetry for each symbol than another tiling,

then it was considered to have richer symmetry. For example, *442 is richer in

symmetry than 442; 2*22 is richer than 22*; and so on. *632 and *442 are not

comparable because 6 > 4, but 3 < 4. Dilation symmetry could not be compared

with other symmetries using this method.

2. Tilings were analyzed by simply listing all types of symmetry: reflection, rotation,

dilation, glide reflection, etc. A tiling with more types of symmetry, or higher

order symmetry, than another tiling was considered to have richer symmetry.

Only the highest order rotation or reflection symmetry was considered.

3. Tilings were analyzed according to simple reflection and rotation symmetry only,

under the idea that these are the most important symmetries to children and

perhaps the only ones they noticed. A tiling with n-fold symmetry was considered

to have richer symmetry than a tiling with order-n rotation symmetry because n-

fold symmetry contained both reflection and rotation symmetry. Only the highest

order symmetry was considered.

In any method that was tried, there was the problem that many tilings were not

clearly comparable. Dilation symmetry is neither richer nor poorer in symmetry than

other types of symmetry. Simple reflection symmetry cannot be directly compared to

simple rotation symmetry. And so on. A further problem is that these methods can only

be carried out for the Other Tilings task, in which data were probably insufficient for

drawing any conclusions.

Each of the three methods revealed a slight or moderate (but not overwhelming)

preference for more symmetric tilings. Other aesthetic criteria frequently intervened to

lead children to choose the less symmetric tiling. Children also frequently failed to agree

among themselves concerning which was the favorite tiling as each child applied

different aesthetic criteria to the evaluation. It would appear that children sometimes

preferred greater symmetry and sometimes turned to other aesthetic criteria, but no firm

conclusions can be reached from these data.

309

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Vita

In 1981 Robert Scott Eberle received the degree of Bachelor of Arts in French and

mathematics from Rice University in Houston, Texas. He received the degree of Master

of Arts in mathematics from Columbia University in New York in 1987 and the degree of

Master of Education in mathematics education from the University of Texas at Austin in

1989. He has worked as a teacher at the secondary school or college level in Texas, South

Carolina, Zaïre, and Niger. He has served as a Christian missionary with SIM in Niger

since 1992. In August, 2006, he entered the Graduate School at the University of Texas at

Austin.

Permanent email address: [email protected]

This dissertation was typed by the author.

Children's Mathematical Understandings of Tessellations

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