Teaching Math to Diverse Adolescent Learners ... Diversity Guide.… · Teaching Math to Diverse...

Teaching Math to Diverse Adolescent Learners: Instructional Equity Guide

West Texas Middle School Math Science Partnership Zenaida Aguirre-Muñoz, Ph.D.

Texas Tech University

Summer 2010

Contents OVERVIEW ........................................................................................................ 1

West Texas Middle School Math Science Project ............................................. 1

Goals of the Workshop ...................................................................................... 1

Expectations for Follow-Up Activites ................................................................. 1

Sources of Content & Materials ......................................................................... 2

Disclaimer .......................................................................................................... 2

DESIGN INSTRUCTION AROUND BIG IDEAS .................................................. 3

The Challenge ................................................................................................... 3

Use Big Ideas to Develop Conceptual Understanding ....................................... 3

Earth Science Example ..................................................................................... 4

Mathematics Example ....................................................................................... 5

Identifying Big Ideas in the Math TEKS/C-SCOPE ............................................ 7

Critical Elements of Conceptually-Based Instruction ......................................... 8

CREATE A CONTEXT FOR MAXIMIZING GROWTH POTENTIAL .................. 13

The Challenge ................................................................................................. 13

Use Language To “Think Together” ................................................................. 13

Learning as Change in Participation Over Time .............................................. 14

Getting In The “Zone” ...................................................................................... 15

PLAN TO SCAFFOLD & DIFFERENTIATE ...................................................... 17

The Challenge ................................................................................................. 17

Scaffolding De-mystified .................................................................................. 17

Essential Elements of Scaffolding ................................................................... 18

Tasks that Promote Autonomy (Structure of Scaffolding) ................................ 18

Teacher Interactions that Promote Autonomy (Scaffolding Process) ............... 19

Student Interactions that Scaffold Learning ..................................................... 21

Features to Strive for in Pedagogical Scaffolding ............................................ 21

HELP STUDENTS REASON MATHEMATICALLY ........................................... 23

The Challenge ................................................................................................. 23

Pose “Thinking” Questions .............................................................................. 23

Investigate Your Questioning to Support Conceptual Understanding .............. 25

Questioning and Classroom Assessment ........................................................ 26

Use ‘Teacher Talk’ to Model Ways of Thinking about Mathematics ................. 29

Types of Language Moves in the Math Classroom .......................................... 29

Engage Students in the Discourse Community ................................................ 31

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Contents (continued) Investigate Your Language Moves in the Classroom........................................ 32

DRAW ON STUDENTS’ LANGUAGE AND CULTURE ..................................... 35

The Challenge ................................................................................................. 35

Build on Students’ Linguistc & Cultural Knowledge ......................................... 35

Build Students’ Confidence & Trust ................................................................. 36

Cultural Links to Mathematics Content & Processes ....................................... 36

GLOSSARY ..................................................................................................... 37

A - L ................................................................................................................. 37

L - Z ................................................................................................................. 38

REFERENCES ................................................................................................. 39

HANDOUTS ..................................................................................................... 41

Handout 1: Identifying Concepts & Skills ......................................................... 43

Handout 2: Verbs for Bloom’s Taxonomy ........................................................ 44

Handout 3: Cognitive Processing Matrix .......................................................... 45

Handout 4: Strategy Instruction Case Study ..................................................... 46

Handout 5: Concept Lesson Brainstorm .......................................................... 47

Handout 6: Analyzing Questions ..................................................................... 48

Handout 7: Investigatign Your Questioning ..................................................... 50

Handout 8: Language Moves Case Study ....................................................... 51

Handout 9: Language Moves Worksheet ......................................................... 54

Handout 10: Insights from Language Self-Analysis ......................................... 55 Handouts 4, 11-13 will be handed out.

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OVERVIEW

WEST TEXAS MIDDLE SCHOOL MATH SCIENCE PARTNERSHIP PROGRAM

This document is part of a series of teacher manuals developed by the West Texas

Middle School Math Science Project (WTMSMSP) to meet programmatic

requirements of the National Science Foundation Math Science Partnerships

Program Award No. 0831420. In accordance with the goals of this project, this report

integrates findings from research pertaining to content area instruction of culturally

and linguistically diverse (CALD) students.

GOALS OF THE WORKSHOP

This document serves as a companion to the WTMSMSP workshop on developing

middle-school teachers’ understanding of math instruction that enhances the

academic achievement and learning of CALD students.

This document is organized around instructional strategies that have documented

evidence for improving the learning and achievement of all learners, particularly

those who struggle to make grade level expectations. What unifies the set of

approaches described in this manual is that they all support the development of

CALD students’ conceptual understanding and mathematical reasoning. These

approaches also provide teachers with additional tools addressing the needs of

English language learners (ELLs; students whose first language is not English and

who are not yet proficient in English) as their linguistic needs pose a unique set of

challenges for teachers.

For each of the five components of this workshop, connections to conceptual

learning are highlighted. Classroom scenarios, applications and case studies are

used to demonstrate sound practices as well as serve as targets of focused

discussion.

EXPECTATIONS FOR FOLLOW-UP ACTIVITES

In order to help teachers support students’ development of mathematical reasoning

and understanding, workshop participants will be trained to record their ‘language

moves’ during math lessons. In order to help us understand how to continue

tosupport middle-school math teachers in this process, an electronic form will be

provided to document efforts in this area. Participants will be expected to complete

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these forms during two scheduled sessions in the fall semester and at least one

additional time in the spring semester.

SOURCES OF CONTENT & MATERIALS

The majority of the content contained in this guidebook is summaries of primary and

secondary sources. Citations to this work are included when summaries are very

close to the original text as well as when direct quotes are used. Many of the ideas

and examples that comprise this guidebook were not originally conceived by the

author of this guidebook. Although citations were provided to give credit to all of the

authors, occasionally a citation was omitted to facilitate reading flow and wherever an

extended summary was provided. Since this is intended strictly for educational

purposes, this action was deemed appropriate. Three sources were extensively

summarized. These sources include: (1) Scaffolding the Academic Success of

Adolescent English Language Learners by A. Walqui & L. van Lier, 2010; (2)

Responding to Diversity: Grades 6-8 by the National Council of Teachers of

Mathematics, 2008; and (3) Effective Teaching Strategies that Accommodate Diverse

Learners by E. J. Kame’enui, D. W. Carnine, R. C. Dixon, D. C. Simmons, & M. D.

Coyne, 2002.

DISCLAIMER

The content or opinions expressed in this document do not reflect the opinions or

beliefs of the National Science Foundation.

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DESIGN INSTRUCTION AROUND BIG IDEAS

THE CHALLENGE

The proliferation of objectives, standards, and performance goals at local, state, and

national levels in recent years can make it difficult to know what content should be

emphasized. At one extreme, teachers feel compelled to cover so much content that

they are forced to teach “for exposure.” Only the highest-performing students are likely

to learn, understand, and apply knowledge in a learning environment that merely

“exposes” them to content topics. At the opposite extreme, one finds teachers who

abandon objectives in favor of an approach that assumes a rich learning environment

will result in learning, even if no particular set of skills is the target of instruction. This

approach also favors the highest performing students. “Little evidence suggests that

either approach serves the goal of empowering all students with the knowledge

required to solve complex problems” (Kame’enui, et al., 2002, pg. 8).

USE BIG IDEAS TO DEVELOP CONCEPTUAL UNDERSTANDING

A response to the challenge posed above is instruction that focuses on the big ideas of

a content area. Big ideas are highly selective concepts and principles that facilitate the

most efficient and broadest acquisition of knowledge. Big ideas serve to link several

different smaller ideas and strategies together within a content domain such as

mathematics. They can be conceived as keys that unlock a content area for a broad

range of diverse learners and are best demonstrated through examples alongside

definitions. Big ideas facilitate student understanding by guiding student attention to

key concepts and principles. The linkages and connections between math concepts

are made explicit by linking previously learned big ideas to new concepts and problem

solving situations. By emphasizing the big ideas in each lesson, teachers can build

students' acquisition and use of key conceptual knowledge across content.

To clarify the nature of big ideas and why they are a unique structural feature of

content area teaching, two illustrations are provided next. The earth science example

is adapted from Kami’enui et. al., (2002) and the math example was developed for this

project.

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EARTH SCIENCE EXAMPLE

In science, the principle of convection represents a dynamic natural phenomenon in

several domains, such as geology, oceanography, and meteorology, which are

usually associated with earth science. Convection represents a big idea because it

reveals for the learner how concepts that appear to be very different (e.g., solids,

liquids, gases) operate in similar ways across different domains (e.g., geology,

oceanography). In simple terms, the principle of convection refers to a specific

pattern of cause-and-effect relations involving phenomena that range from a pot of

boiling water to ocean currents to earthquakes. Convection is a big idea because it

reveals how these different natural phenomena all follow the same flow of matter or

energy in a manner that represents a rectangular figure (See Figure 1).

Figure 1: The Big Idea of Convection and Simple Visual Maps of Some of Its Applications

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The convection principle can be used to demonstrate how complex concepts (e.g.,

density, heating and cooling, force, pressure) across different domains such as

geology and meteorology operate in the same or similar ways.

MATHEMATICS EXAMPLE

In middle school mathematics, a comparable example is size in the measurement of

geometric shapes (Figure 2). Size of a geometric object refers to a numerical quality of

an object that we attribute to its ‘bigness.’ How size is depicted depends on the

dimensionality of the geometric object we are describing, specifically: (a) number of

objects in a set of zero dimension, (b) length of a set of one dimension, (c) area of a

two dimensional set, and (d) volume of a three dimensional set. To be meaningful

however, we must be able to measure the size of an object in order to unambiguously

communicate that size to others. Formulas for computing the numerical measure of

the size of objects and standard units of measure are used to describe the size of

objects unambiguously.

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Figure 2: The Big Idea of Size for the Measurement of Geometric Shapes.

S I Z E

M e a s u r e

0-dimension Number

1-dimension Length

2-dimensions Area

3-dimensions Volume

Unit: u

Unit: 2u

Unit: 3u

Unit: Name

Counting Distance formula

Area formulas

Volume formulas

U n i t A n a l y s i s

A zero dimensional geometric object is simply a finite set of points. The basic

concept of size in one dimension is the length of a line segment measured by

comparing it with line segments of some agreed upon standard unit of length: inch,

foot, meter, centimeter, etc. The basic concept of size in two dimensions is the area

of a region swept out by a line segment that is translated for a given distance in a

direction perpendicular to itself. The measure of the area is defined to be the product

of the length of the line segment times the length of the translation. The units of area

(Figure 3) are square units based on the units used for the length.

Figure 4: Volume (𝛺) = Area (R) *4 = (3cm × 2cm) * 4cm = 24cm3.

Figure 3: Area (B) = 2in x 5in = 5in x 2in = 10in2.

5 in 5 in 5 in 5 in

2 in 2 in

2 in 2 in

B B

Area Units

Cubic Units

4cm

R

R

3cm

2cm

𝛀

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The basic concept of size in three dimensions is the volume of a region swept out by a

two dimensional region that is translated for a given distance in a direction

perpendicular to itself. The measure of the volume is defined to be the product of the

two dimensional area times the length of the translation. The unit of area is cubic units

based on the units of the region and the length of the translation (Figure 4).

IDENTIFYING BIG IDEAS IN THE MATH TEKS/C-SCOPE

When educators analyze each standard of a content area and identify its essential

concepts and skills, the result is more effective instructional planning, assessment,

and learning. Follow the “unwrapping” process outlined below when analyzing the

math TEKS (Ainsworth, 2003a). If you are using C-Scope, this process can also be

used to confirm that the identified concepts targeted by the lesson are aligned with

the targeted TEKS. It can also help teachers identify the big idea(s) associated with

C-Scope lessons, as C-Scope materials do not make explicit this aspect of good

curriculum design.

1. Underline content nouns and circle verbs as demonstrated in Figure 5. The nouns represent the concepts (what students need to know) and the verbs correspond to skills (what students need be able to do).

2. Using Handout 1, list concepts and skills identified in the targeted standards for a specific strand and grade level.

3. Use Handout 2 and Handout 3, to determine the intended level of cognitive processing implied by the standard. This information can assist you in identifying the big idea(s) to which to link the lesson (Ainsworth, 2003b).

SUMMARY OF BIG IDEAS

Math big ideas are key math concepts that can be continually used to teach a variety of math

strategies/processes and skills.

Provides referential starting points for students when learning new math concepts and strategies.

Examples include "objects and groups", place-value, size, proportion (part-whole relationships),

estimations, etc.

Math big ideas are explicitly described and modeled by the teacher.

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4. You can also use additional information to determine big idea(s) to organize the concepts you list when unwrapping standards. This information has to do with the relative endurance, readiness, and leverage of the concept. • Endurance - Is the essential knowledge and skills in order to be literate in

this area beyond a single test (i.e. is there life-long value to this)? • Readiness -Is the knowledge and skills needed in order for the student to be

successful and achieve not only in this grade level, but in subsequent grades?

• Leverage - Will the knowledge and skills prove to be valuable in learning essential content in other academic areas?

Figure 5: Sample Analysis of a Fifth Grade Math Standard.

Standard Expectations

(5.10) Measurement. The student

applies measurement concepts

involving length (including

perimeter), area, capacity/volume,

and weight/

The student is expected to:

mass to solve

problems.

(A) perform simple conversions within

the same measurement system

(B) connect

(SI

(metric or customary);

models for perimeter, area,

and volume with their respective

formulas; and (C) select and use

appropriate units and formulas to

measure length, perimeter, area,

and volume

.

CRITICAL ELEMENTS OF CONCEPTUALLY-BASED INSTRUCTION

Identify math key concepts, “big ideas” that can be used within and across knowledge strands.

Although there are many concerns with C-Scope (the instructional framework

adopted by the majority of districts in Texas) this system identifies conceptual ideas

that can be used as starting points. The task for the teacher is to identify the broader

concept(s) that bridges those identified by this instructional tool. In other words, the

learning goal should be to assist students to understand the relationships between

and among these important concepts. Choose math big ideas that are foundational

to the lesson and that represent understanding that can be applied across lessons

(e.g., area).

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Design tasks that focus student attention on these key math big ideas.

If the goal of instruction is to focus student attention on big ideas, it is difficult to

achieve this instructional goal if the tasks that are used during instruction do not

reflect the kind of reasoning that underlies the conceptual relationships corresponding

to the targeted big idea. One strategy for focusing instruction on key math big ideas is

by designing or selecting tasks aligned to these big ideas and that generate higher

level thinking and reasoning. Table 1 presents examples of tasks that generate either

lower-level or higher-level reasoning. The highest of these is labeled as “doing math.”

Table 1 Lower-Level vs. Higher-Level Task Approaches to Representations of Fractional Quantities.*

Lower-Level Reasoning Higher-Level Reasoning 1. Memorization: 3. Procedures with Connections:

What are the decimal and percent equivalents for the fractions 1

2 and 1

4?

Using a 10 × 10 grid, identify the decimal and percent equivalents of 3

5.

Expected Student Response: Expected Student Response 12 = .5 = 50%

14 = .25 = 25%

Fraction Decimal Percent

60100 =

35

60100 = .60 . 60 = 60%

2. Procedures without Connections: 4. Doing Math: Convert the fraction 3

8 into a decimal and

percent.

Shade 6 small squares in a 4 × 10 rectangle. Using the rectangle explain how to determine each of the following: (a) the percent of the area that is shaded, (b) the decimal part of the area that is shaded, and (c) the fractional part of the area that is shaded.

Expected Student Response: One Possible Student Response: Fraction Decimal Percent

(a)One column will be 10% since there are 10 columns. So four squares is 10%. Then 2 squares is half a column and half of 10% is 5%. So the 6 shaded blocks equal 10% plus 5% OR 15%.

(b)One column will be .10 since there are 10 columns. The second column has only 2 squares shaded so that would be one half of .10 which is .05. so the 6 shaded blocks equal .1 plus .05 which equals .15.

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.375 8√3.000 24 60 56 40 40

. 375 = 37.5%

(c)Six shaded squares out of 40

squares is 640� which

reduces to 320� .

*Adapted from: Stein, M. K., Schwan Smith, M., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction. NCTM, Teachers College Press, NY, NY.

While it is tempting to focus on lower level skills with students who are one or more

years behind in content understanding, the research repeatedly shows that when

instruction is focused on developing deep conceptual knowledge, students

outperform their peers enrolled in courses that focus on lower-level reasoning. For

example, Guiton and Oakes (1995) found that regardless of students’ initial

achievement level, those who were placed in lower level courses obtained smaller

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gains over time than students of comparable achievement who were placed in higher

level courses.

Teach directly and systematically the accompanying strategies using explicit teaching techniques, then move towards more authentic activities.

When students develop multiple concepts in some manner, they are performing a

strategy. Any routine that leads to both the acquisition and use of knowledge can be

considered a strategy (Prawat, 1989). While the ultimate purpose of a strategy is

meaningful application, acquisition is most reliable for diverse learners when initial

instruction explicitly focuses on the strategy itself, rather than its meaningful

application. (Read and discuss Handout 4.)

The challenge for the teacher is to determine at what level of specificity to target the

strategy. If the strategy is too specific and narrow, they are little more than rote

sequences for solving a particular problem or a very small set of very similar

problems. This trend is particularly evident when instruction and practice is directed

at small ideas or procedures, such as cross multiplying to solve problems.

At another extreme are strategies that are so general that they are little more than a

broad set of guidelines. While they are better than nothing, they do not dependably

lead most students to solutions for most problems. Examples of such strategies are

“draw a picture” or “read, analyze, plan, and solve.”

Effective strategy interventions should be intermediate in generality, falling between

the extremes described above. In order to accomplish this instructional goal, follow

the Law of Parsimony: The best strategy results in the greatest number of students

successfully solving the greatest number of problems or completing the broadest

range of tasks by applying the fewest possible strategic steps.

The whole purpose of developing strategy instruction is to illuminate expert cognitive

processes so that they become visible to the novice learners.

The section titled, “Help Students Reason Mathematically” describes two key

strategies that aid teachers in making expert cognitive processes explicit to CALD

students.

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Demonstrate frequently and deliberately linkages between the math big idea(s), previously learned information, and targeted math strategies and skills.

The use of visual maps or models to present and illustrate big ideas helps make the

links evident for CALD students. In the convection example, this included the map

that illustrates the movement of air currents in a room in a rectangular pattern as well

as a visual scheme to represent the movement of water in a heated pan. These maps

make distinct and obvious the connections that are important to the big idea(s). It is

important to continue to refer back to these links both during the instruction and in the

feedback to students as they work through problems. Also important are the words

that are used to call attention to the different features of the big ideas and how

different situations link back to the big idea. Past research provides extensive

evidence that all students and in particular diverse learners benefit from having good

strategies made explicit, if and only if, the strategies are designed to result in

transferable knowledge of their application.

One approach to accomplish this instructional goal is to explicitly teach the targeted

math strategy within the context of the math big idea. During a lesson on the

measurement of geometric figures, for example, focusing the students’ attention on

the dimensionality of the object as the first step to determining the unit analysis is a

strategy that teachers can explicitly model to students. Use Handout 5 to

outline/brainstorm how this might be done.

Apply conceptual understanding and procedural mastery to new content.

The notion of promoting higher-level thinking is universally endorsed, yet instructional

practice rarely provides opportunities to engage the content at a conceptual level. Too

often mathematical procedures take precedence over conceptual understanding. The

goal of instructional design should be to introduce and combine information in ways

that result in new and more complex knowledge. In designing a teaching sequence

for the big idea of size several concepts must be integrated, such as dimensionality

and unit analysis. Because these are complex concepts, the integration of each

concept should be designed with care. For example, the concept of dimensionality

should be taught first, then scaffolding is removed and unscaffolded practice is

provided in the context of teaching about unit analysis.

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Likewise, when the formulas for measuring geometric shapes is introduced,

unscaffolded practice with previously taught concepts of dimensionality and unit

analysis is provided in the context of learning the relationships between

dimensionality and unit analysis to determine size. These concepts are further

reviewed when integrated in the basic size strategy.

Instruction that sets the groundwork for conceptual understanding, should involve:

• Providing multiple meaningful practice opportunities for students using the

math big idea with the new math strategy/skill you taught.

• Appling the math big idea to the target math strategy/skill using a variety of

problem solving situations.

• Pairing a visual cue with each math big idea/concept (e.g. a picture of an

array for the concept of "size").

• Posting the visual cue along with one sentence describing why the big idea is

important.

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Classroom Scenario 1

After a recent review of your math TAKS scores, you notice that 30% of your

students scored significantly lower on the measurement items of the test.

Design a higher-level reasoning task involving the big idea of size. Include the

following information in the description of the design:

• What visual aids would you provide?

• What strategy would you introduce?

• How would you model the strategy and its connection to the big idea?

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CREATE A CONTEXT FOR MAXIMIZING GROWTH POTENTIAL

THE CHALLENGE

In an increasingly more diverse student population, the need to prepare teachers for

the increasing instructional demands of CALD students has never before been

approached with such urgency. In the decade between 1995 and 2005 ELLs grew by

56% (Batalova, Fix, & Murray, 2007), compared to the K-12 US population growth of

less than 3% during this same time period. Further, the greatest increases were

observed in regions that have traditionally had small to virtually non-existent ELL

populations in the past. Despite this growing trend, both teacher training and

curriculum materials and programs designed to meet the instructional needs of

CALD middle and secondary students are in short supply. This section summarizes

learning theory principles, presented by Walqui & van Lier (2010) that are

particularly relevant for developing higher-order thinking in ELLs, though they apply

to the broader CALD student population as well. In addition, scaffolding is described

as the principle framework for developing tasks that engage CALD students.

USE LANGUAGE TO “THINK TOGETHER”

Students develop higher order functions as they engage in activity that requires

them to use language. Higher-order functions refers to the mental processes

involved in higher-order thinking. Vygotsky emphasized the primacy of linguistic

mediation in the development of higher mental processes; he contended that

language is the main vehicle of thought and that all language use is based on social

interaction (dialogical).

Even when we have gained a great deal of knowledge and expertise in a given area,

language continues to support thinking. When we are faced with a difficult task that

requires much thought and concentration, we will often make our inner speech overt.

This private speech is audible but it is not directed at others. For example, a learner

who is struggling with algebraic functions engages in this audible private speech:

“Oops, that can’t be right…Maybe I should start by making a function

table…Ah, good! I see why that relationship is off.”

In this example, we see language and thought intimately connected as the learner

attempts to organize resources and control the task. The main argument from a

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Vygotskian perspective is that if social interaction is the basis for language and

learning, the notion of consciousness, the development of identity, and physical and

mental skills all emerge from and in interaction. According to Vygotsky, “every

function in the child’s development appears twice, on two levels. First, on the social,

and later on the psychological level; first between people as an interpsychological

category, and then inside the child as an intrapsychological category”

(Vygotsky,1978, p. 128).

LEARNING AS TRANSFORMATION

Children internalize what they learn in social interactions not by “copying and pasting”

what they see and hear but through a process of transformation involving

appropriation and reconstruction. Thus, all knowledge arises in social activity, and all

learning is co-constructed, with the learner transforming the social learning into

psychological, or individual, learning over time. Such learning takes place in the

learner’s zone of proximal development (ZPD).

While the original concept emphasized the importance of learning between a learner

and a more knowledgeable other, it is now recognized that interactions between

peers with essentially equal knowledge can also result in learning (Donato, 1994;

Gibbons, 2002; Mercer, 1995; Rogoff, 1995). The latter form of learning is referred to

as joint construction of knowledge.

This kind of interaction does not refer to just any kind of ‘assistance’ or ‘helping,’ or

any kind of group interaction. In order for learning to occur in interaction, the learners

need to have agency (i.e., active involvement, initiative, and autonomy). Thus, the

teacher should strive to set up tasks that invite learner agency.

LEARNING AS CHANGE IN PARTICIPATION OVER TIME

Collaborative activity around meaningful and contextualized tasks supports the type

of interaction that leads to joint construction of knowledge. Hence, the learning tasks

need also to support the development of conceptual knowledge. Participation in

activity progresses from apprenticeship to appropriation (from the social to the

individual plane). As learners engage in collaborative activity just beyond their

individual ability, they apprentice the ways of “doing it right,” consistent with the

patterns of behavior valued in the learning environment. As they work through

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scaffolded activity with their peers, their understanding increases. Over time, students

appropriate the ways of thinking, acting, and interacting valued in the learning

environment. From this perspective, learning can be viewed as qualitative changes in

participation over time. Therefore, “it is more revealing to observe students’

participation in academic activity over time, to see how their potential is gradually

realized” (Walqui & van Lier, 2010. pp. 12).

GETTING IN THE “ZONE”

Central to instructional practice aligned with this theoretical approach to learning is

the explicit planning and incorporation of supports or scaffolds that enable students to

take advantage of learning opportunities. Scaffolding, as described here, is different

from simply helping students complete tasks they cannot do independently. It is

creating the contexts and supports that allow students to interact in their ZPD.

By definition, every student has a unique ZPD which is constantly changing as their

knowledge and skills develop. Thus, in order for teachers to maximize a child’s

growth potential, scaffolding entails routinely differentiating the scaffolds provided to

individual students across topics and tasks and to continue to do so over time.

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PLAN TO SCAFFOLD & DIFFERENTIATE

THE CHALLENGE

Scaffolding is widely used in education. Unfortunately it is used imprecisely.

Scaffolding enables differentiation to occur and can be a transformative teaching tool.

Scaffolding refers to both a special, supportive way of interacting and to a temporary

structure that assists learning. Conceived as both an instructional process and a

structural instructional element, scaffolding can be described as the pedagogical way

in which the ZPD is established and learning takes place (Walqui & van Lier, 2010).

SCAFFOLDING DE-MYSTIFIED

To successfully scaffold instruction, one needs to have a good sense of both the

predictable and unpredictable aspects of the instructional context. The structure of

instruction is predictable, whereas, the process of carrying out instructional events

and activities—the moment-by-moment words and actions—is much less predictable. A

scaffold as originally concieved, refers to the teacher’s responsiveness to the

students unexpected actions that occur in instructional processes. In the education

context, scaffolding refers both to the planning and setting up of the task or activity as

well as how a teacher responds to student actions. Thus, both task design and

feedback to students comprise important aspects of scaffolding.

Scaffolding happens when new and unpredictable behaviours emerge and the

teacher channels and stimulates the student’s ongoing responses and behaviours,

making it possible for the student to be a “head taller than himself,” (Vygotsky, 1978).

A teacher who scaffolds successfully can identify signs of an emerging new skill, such

as a word, behavior, or expression, and use it to provide an opportunity to engage the

student in higher-level functioning.

As mentioned previously, scaffolding is not simply helping students to complete tasks;

it is assistance that allows the student to take increasing control of the thinking in the

activity. That is, gradually, actions and initiatives carried out by the teacher are taken

over by the student. Not until the student shows signs of being able to take over some

piece of the action/activity/skill, does the teacher scaffolds the ‘takeover,’ handing

over that particular action and guiding the student towards accomplishing it.

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For such development to be possible, control must be shared. Thus, an additional

challenge for the teacher is structuring tasks and activities that entice the student to

take as much initiative as possible. The teacher should be willing to relinquish some

control in these tasks.

ESSENTIAL ELEMENTS OF SCAFFOLDING

Past research and theory describe essential elements of scaffolding as follows:

• Recruiting interest in the task;

• Simplifying the task;

• Maintaining pursuit of the goal;

• Marking critical features of discrepancies between what has been produced

(or uttered) and the ideal solution (utterance);

• Controlling frustration and risk during problem solving;

• Demonstrating an idealized version of the act to be produced (uttered);

• Actively engaging the learner (increasing agency and autonomy); and

• Intervention and/or feedback to student should accommodate (be modified)

according to level of understanding/skill demonstrated.

TASKS THAT PROMOTE AUTONOMY (STRUCTURE OF SCAFFOLDING)

It is important to remember that the essence of scaffolding is not something planned

into the design of a lesson or curriculum scope and sequence. Recall that by

definition, scaffolding is unplanned; it is something that happens on the spot when a

learner says or does something that foreshadows a new development or a promising

direction. This conception of scaffolding does not mean that design is not important.

It is the expert (but flexible and adaptable) design that makes possible a student’s

innovative actions and budding development.

Consider the following description (taken from Walqui & van Lier, 2010) of the

scaffold metaphor:

The builders put a scaffold around a building that needs to be

renovated, but the scaffold itself is only useful to the extent that it

facilitates the work to be done. The scaffold is constantly

changed, dismantled, extended, and adapted in accordance with

the needs of the workers. In itself, it has no value.

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Thus, scaffolding begins with tasks that allow for learner autonomy and initiative.

These tasks neither lead to chaos nor stifle the learner’s development. The structures

are not rigid but they are supportive and robust. The purpose of the structure is to

facilitate the process. The process refers to how the teacher handles the

unpredictable, as introduced by the students’ thinking, acting, and communicating

behaviors. Thus, one needs to examine the kinds and quality of interactions that are

needed to promote autonomy.

TEACHER INTERACTIONS THAT PROMOTE AUTONOMY (SCAFFODLING

PROCESS)

From the perspective of scaffolding (as defined here), the teacher’s role in the

classroom is not to control the learner but to support and encourage the learner’s

emergent autonomy. Consider the two teacher-student dialogues represented in

Table 2. Can you identify an instance of scaffolding? Who has control of the direction

of the interaction?

Table 2. Teacher Decision Making in a Practice Dialogue: Dialogues 1 & 2

Dialogue 1 (IRE)* Dialogue 2 (IRF)**

Teacher: Excuse me.

Student: Yes?

Teacher: Can you tell me how I can get to Highway 1 from here?

Student: No problem! You go straight that way and see traffic light. When traffic light, you…left, then go, eh, go more….straight and then the Highway 1, you will see it.

Teacher: Okay. Listen. Go straight TO the traffic light, turn left, and go straight ahead UNTIL you see the sign for Highway 1.

Student: Ehm…go straight TO traffic light…(etc)

Teacher: Excuse me.

Student: Yes?

Teacher: Can you tell me how I can get to Highway 1 from here?

Student: No problem! You go straight that way and see traffic light. When traffic light, you…left, then go, eh, go more….straight and then the Highway 1, you will see it.

Teacher: Thanks!

Student: You welcome!

*IRE = Initiation-Response-Evaluation; **IRF = Initiation-Response-Feedback; Both represent common questioning patterns in teacher-student dialogue.

If you were not able to find an instance of scaffolding, you are on the right track in

terms of thinking about scaffolding. Although both of these response patterns have

been associated with scaffolding in the past, neither can be called an instance of

scaffolding. Let’s consider why they do not reflect basic principles of scaffolding.

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Even though the dialogue in the left column (IRE) the teacher offers ‘assistance’ to

the learner, the student has no initiative or autonomy and no control over what is

being said, by whom, or when. The student is merely designated to answer the

question and must rely on the teacher’s approval to know if s/he completed the task

satisfactorily. Thus, the interaction, even the student’s utterances ‘belongs’ to the

teacher, who controls the agenda. In this case, it is difficult to see opportunities for

the student to takeover or for the teacher to hand over control of learning.

In the dialogue in the right column (IRF), although the teacher affirms the student’s

agency, the teacher did not capitalize on the student’s response to develop new

understanding of the language of directions. Not only were meaningful

handover/takeover possibilities absent, the interaction did not lead to new

understandings or initiative on the part of the learner.

Compare the interactions captured in Table 2 with the dialogues presented in Table

3. Who has control of the direction of the interaction? What are the instances of

scaffolding?

Table 3. Teacher Decision Making in a Practice Dialogue: Dialogues 3 & 4

Dialogue 3 Dialogue 4

Teacher: What season comes after fall?

Student: Winter.

Teacher: Good girl.

Student: It’s like everybody should get the same rights and protection, no matter like race, religion.

Teacher: Yeah. Everybody.

Student: No matter if they are a citizen or illegal, they should get the same protection.

Teacher: I agree with you, but why do you say that with confidence?

Student: Because it [the 14th Amendment] says that.

Teacher: Because it says that?

Student: Also, because it says it should not deny any person of the right to life, liberty, and property without due process.

Teacher: Okay, not any citizen?

Student: Any person.

Teacher: Okay, so is the 14th Amendment helpful to you?

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In Dialogue 3, the teacher knows the answer and is checking to see whether the

student does. When the student answers, the teacher evaluates and approves the

student’s answer. Thus, no scaffolding is present. In contrast, in Dialogue 4, the

teacher prompts the learner to elaborate his response and thinking and then by

connecting the student’s learning to her experiences as an immigrant. The student

has greater control of the interaction, and the teacher uses the student’s responses to

lead the student toward more ideal responses that are more precise and present a

clearer argument. Thus, the teacher scaffolds the student’s disciplinary knowledge

and language learning simultaneously. These deliberate language moves by the

teacher are useful for all students, not just ELLs or CALD students.

STUDENT INTERACTIONS THAT SCAFFOLD LEARNING

In traditional descriptions of scaffolding and ZPD, the focus is on expert (teacher)-

novice (student) interactions. This conception stems from the original contexts of

scaffolding where the objects of discussion involved interactions between mother and

child and between a tutor and child. Indeed, a very common belief in a classroom-

based learning context is that in order for learning to occur knowledge must be

transmitted from a more knowledgeable person to a less knowledgeable person.

Yet many accounts of learning involve interactions between two or more novices. This

line of research has demonstrated that scaffolding can occur in interactions between

individuals of equal knowledge working on a shared task (see Mercer, 1995). In these

circumstances, ideas emerge in interaction, are shared with peers, and are further

developed in interaction. In their explanation of the profound power of collaborative

problem solving, Walgui & van Lier (2010) state:

Students found that one half-idea invites completion, one

thought leads to another, and one tentative step opens up a

direction to explore. (pp. 30)

FEATURES TO STRIVE FOR IN PEDAGOGICAL SCAFFOLDING

The features of pedagogical scaffolding represent a consolidation of points made

earlier in this section; learner agency (autonomy and initiative), scaffolding, and

interactions. The features outlined in Figure 6 reflect that learning can and does

happen in all sorts of different ways, some predictable, many not; that it flows from

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planned to improvised, from ‘designed-in’ to ‘interactional’, from the macro level of a

curriculum to the micro level of moment-by-moment interaction.

More Planned

More Improvised

Continuity and Coherence

task repetition with variation; connecting tasks and activities; project-based or action-based learning

Supportive Environment

environment of safety and trust; experiential links and bridges

Intersubjectivity

mutual engagement; being ‘in tune’ with each other

Flow

student skills and learning challenges in balance; students fully engaged

Contingency

task procedures and task progress dependent on actions of learners

Emergence, or Handover/Takeover

increasing importance of learner agency

Figure 6: Features of Pedagogical Scaffolding.

Taken together the features of pedagogical scaffolding serve as guidelines for

constructing a learning environment that is both controlled and free, or moves

between letting go and pulling back. The structures that allow for scaffolding are

integrated with the processes that are scaffolding. Thus, scaffolding necessarily

involves a continuum that begins with structural features of tasks and activities

(continuity and coherence) and moves to the procedural feature of handover/

takeover.

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HELP STUDENTS REASON MATHEMATICALLY

THE CHALLENGE

Past research reveals that the prevailing form of questioning in math and science is

low-level fill in the blank questions and that instruction is focused on getting students

to say the right things instead of helping students to make sense of the underlying

mathematical ideas (Weis et. al., 2003). Focusing instruction on big ideas and

providing visual cues is necessary but insufficient to promoting conceptual

understanding. It also requires ongoing monitoring of student understanding of those

big ideas. Questioning and modelling of expert cognitive processes (through

language moves) are two strategies for monitoring and fostering this type of

understanding.

POSE “THINKING” QUESTIONS

Teachers have many reasons for asking questions during mathematics instruction.

Thinking about important purposes for questions can help teachers refocus their

instructional planning so that students learn more. Using different kinds of questions

for different purposes can help differentiate instruction by tailoring instruction to the

specific needs of students.

This section discusses three types of questions that are particularly suited for both

monitoring conceptual understanding and increasing the proportion of students who

remain engaged in conversations about important math ideas. These question types

include: (1) engaging questions, (2) refocusing questions, and (3) clarifying questions

(Bright & Joyner, 2004). Figure 7 illustrates a context in which these kinds of

questions can be posed to students.

Engaging questions invite students into discussion, keep them engaged in

conversation, invite them to share their work, or get answers “on the table.” Usually

directed at the whole class at the start of a discussion, engaging questions are open-

ended with multiple acceptable answers. They can also be used to re-engage

students who may have “tuned-out.” Students with low math self-efficacy might

benefit most by being invited into discussions that reward multiple solutions based on

alternative, accurate math reasoning.

Refocusing questions serve to get students back on track or to move away from a

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dead-end strategy. For long-term learning it is generally ineffective to simply tell

students what to do differently when they are going in the wrong direction. Doing so

constitutes substituting the teacher’s thinking for students’ thinking, which does not

help students understand why the strategy is ineffective or reorganize their thinking

so that the use of the ineffective strategy is minimized. Refocusing questions remind

students about some important aspect of a problem they may be overlooking.

Clarifying (probing) questions help students explain their thinking or help the teacher

understand their thinking. Clarifying questions can be used when: (a) a teacher is

fairly certain that a student understands an idea but the language used to explain that

thinking is not clear or precise; or (b) a teacher needs to reveal more about a

student’s thinking to make sense of it.

In the first case, a clarifying question can simply be posed as “What does ‘it’ refer

to?" In the second case, questions become critically important for a teacher to

understand a student’s thinking and may expose misunderstanding or a

misconception. A common clarifying question is “how did you get that answer?”

Two Similar Rectangles

Engaging Question: How can we decide what value the question mark stands for?

Refocusing Question: What does it mean for two rectangles to be similar?

Clarifying Question: (In response to a student who says that the answer is 5) How did you get 5?

Figure 7: Examples of Question Types in Math Context.

Clarifying questions can reveal the logic that students are using to solve problems or

organize concepts and their interrelatedness. This logic is likely to reflect both their

academic backgrounds (i.e., what they have previously learned) and their “real world”

backgrounds (e.g., their life outside of school). The language that students use may

give cues about what motivates them and what they see as an acceptable

contribution in a discussion.

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INVESTIGATE YOUR QUESTIONING TO SUPPORT CONCEPTUAL UNDERSTANDING

When you go back to your classrooms in the fall, we expect you will try these

questioning techniques to support your students’ conceptual understanding. The

following process can be used to record and reflect on your questioning skills.

• Select a class that you believe posses some challenges in terms of prior

knowledge limitations, engagement, self-efficacy, or motivation.

• Select a day in early fall to focus on questioning and use an audio or video

recorder to capture the questions you pose in dialogues with your students.

• Listen to the recorded interaction, attending carefully to the questions you pose to

your students. Use Handout 7 to document these types of questions.


Suppose the learning target for a lesson is to distinguish area from perimeter. In a class

discussion a student says the area is 50 centimeters. If the teacher wants to refocus the

student to the general math idea (unit analysis), what question can be posed? If the

teacher wants to call attention to (clarify) the student’s response what question can be

posed?

Analyzing Questions in a Classroom Context

The conversation in Handout 6 occurs when the teacher stops to talk with two students

who have been playing a game (based on Fraction Tracks). Considering how the

questions are phrased, what do you think the purposes of the following questions are?

• How could you move your pieces across to the other side if your card was 810

?

• Could you go 45 and then

810

?

• Can I move the whole 45 now?

25

• Identify what you like about your questioning as well as instances where you

could have used a different question type.

• Record these insights in Handout 7.

• Select another question focus day to add question types to your interactions

with your students. As you plan your lesson, anticipate the multiple solutions

strategies your students might offer or possible areas of misconception or

knowledge gaps. Use this information to decide what kinds of questions you

can asks students to help them think critically about the bid ideas targeted in

your lesson. Audio tape or videotape this lesson.

• Repeat the reflection process to examine your question types until you believe

that engaging, refocusing and clarifying questions are a regular part of your

instructional repertoire.

QUESTIONING AND CLASSROOM ASSESSMENT

Classroom assessment is the process of gathering information about what students

know and can do. Inferences are then made from that evidence about students’ math

understanding. Based on these inferences, instructional decisions are made that are

aligned with student’s math knowledge needs. Questioning can be a powerful tool to

gather this kind of information that better align instruction with students’ needs.

To summarize, an emphasis on appropriate questioning to help students make sense

of math concepts is an important instructional technique for developing students’

conceptual understanding, in revealing and monitoring what they know and

understand, as well as in helping them become more comfortable with multiple ways

of using important math ideas. In this way, asking better questions has implications

for the overall quality of instruction.

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USE ‘TEACHER TALK’ TO MODEL WAYS OF THINKING ABOUT MATHEMATICS

A significant challenge in teaching mathematics is developing in students a sense of

intellectual significance of “doing math.” Past research demonstrates that students

need to go beyond thinking about math as a set of procedures and instead to think

about math as a reasoning (thinking) process.

One strategy for developing this understanding in students is to use language to

explicitly model ways of thinking and reasoning about mathematics (Herbel-

Eisenmann, 2000). Engaging students in this way enables them to adopt ways of

“talking mathematics” that are valued by, and important to, the mathematical

community (Herbel-Eisenmann & Schleppegrell, 2008). Developing norms for talking

mathematics in valued ways in turn affects students’ beliefs about math activity

(Cobb, Yackel, & Wood, 1993).

Although all students come to school with discourse practices that serve them well in

their homes and communities, some children need assistance in taking up the

patterns of language that are valued in classrooms.


Suppose a teacher asks students what number goes in the box (below) to make a true

number sentence.

152 + 230 = + 240

A student replies “382.” The teacher then asks a clarifying question “How did you get

382?” To which the student replies “I added 152 and 230.” As the student replies, she

notices that another student wrote “= 622” after 240. What fundamental misconception

do these responses represent?

Recognizing this misconception, the teacher organizes a mini-lesson around analyzing a

variety of mathematically equivalent equations, such as what is illustrated in Figure 8.

What type of questions should the teacher use to introduce this lesson? Provide

Examples where designated.

Set 1

Question(s):

152 + 230 = + 240

152 + 230 = 240 +

+ 240 = 152 + 230

240 + = 152 + 230

Set 2

Question(s):

241 + X = 376

241 + T = 376

241 + A = 376

Figure 8: Example of a Mini Lesson Targeting Student Misconceptions.

27

USE ‘TEACHER TALK’ TO MODEL WAYS OF THINKING ABOUT MATHEMATICS

A significant challenge in teaching mathematics is developing a sense of significance

in “doing math.” Past research demonstrates the value of helping students

conceptualize math as more than a set of procedures; students need to understand

that math is a thinking and reasoning process rather than a set of steps to go through

to get the right answer.

For CALD students, especially ELLs, a strategy that is gaining in popularity is using

language to explicitly model ways of thinking and reasoning about mathematics

(Herbel-Eisenmann, 2000). This type of modelling enables CALD students to adopt

ways of “talking mathematics” that are valued by, and important to, the mathematical

community (Herbel-Eisenmann & Schleppegrell, 2008). In essence, a focus on

language use enables teachers to develop and reinforce norms for talking

mathematics in valued ways which, in turn, affects students’ math beliefs and self-

efficacy (Cobb, Yackel, & Wood, 1993).

When classroom language construes mathematics as a process of

reasoning, students are engaged in learning mathematics in ways that

define mathematical activity as more than a set of procedures to be

followed, thus influencing their beliefs about what it means to know and do

mathematics. (Herbel-Eisenmann & Schleppegrell, 2008 pp. 23)

CALD students tend to need additional support in appropriating patterns of language

that are valued in schools. The challenge for teachers is to use language in ways that

explicitly demonstrates the kind of reasoning that is valued in different subjects to

support all students’ conceptual understanding. The activities in this section are

designed to help you become better language models of math reasoning for all your

students.

TYPES OF LANGUAGE MOVES IN THE MATH CLASSROOM

Math reasoning is difficult for most students. There is growing consensus that

students need to hear ways of explaining, justifying, and interpreting that are valued

by the math community. Hence, teachers are critical to exposing students to

modelling that directs students’ attention to valued ways of talking and thinking about

math principles and the relationships between and among concepts. Researchers

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have identified two important ways teachers model such reasoning: (1) stepping out,

(2) revoicing.

Stepping out refers to more explicit language moves that include reflection on math

actions; talking about math. For example, while solving a math problem, a teacher

might stop to comment on a math process more explicitly. In an example from

Herbel-Eisenmann & Schleppegrell, 2008, the teacher says:

That’s a great example of the kind of explanation I’m looking for. It’s

important that you not only give your answer but that you also explain what

you did and why you did it. I want you to explain the process you went

through, not just give an answer. (pp. 25)

The teacher is not just talking about the problem at hand, she momentarily ‘steps out’

of the discussion of the problem’s solution to explicitly state her expectations for an

appropriate mathematical explanation.

Revoicing refers to less explicit language moves that allow the teacher to reformulate

a student’s response by clarifying or extending what a student has said in an effort to

help other students understand the math significance of the contribution. Examples of

revoicing include teacher’s recasting of student’s verbal contributions in more

technical terminology with slight changes so as to move the discussion forward,

leading to more conceptually-based explanations that originated with the student’s

contribution. Revoicing is used to clarify statements, make connections, or fill in

missing elements of an explanation.

“By helping students articulate their understanding, teachers provide opportunities for

students to agree or disagree with the reformulated version, teaching them to explain

their reasoning.” (emphasis added, pp. 25)

In sum, stepping out and revoicing are effective strategies for making language

transparent to students so that they learn appropriate ways of talking about math.

The process of making language transparent is a complex task especially for

teachers who work with CALD students. Accomplishing this instructional goal in a

manner that does not lead to teacher talk that overwhelms students is not easy. Too

much teacher talk is also not ideal, as it indicates that students are not provided

sufficient opportunities to practice the language of the discipline or to take control of

the task/skill, activity and thus their own learning.

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Thus, it is important to bear in mind that teacher talk should strive to encourage

further attempts by the student to engage in math reasoning.

ENGAGE STUDENTS IN A MATH DISCOURSE COMMUNITY

When language moves are a regular feature of teaching, the teacher sends the

message that math is flexible, makes sense, has meaning, requires reasons for its

procedures, and requires particular kinds of explanations. Regularly providing

opportunities for students to talk about math understanding engages them in the math

discourse community. Students with fewer opportunities with this kind of academic

language development need repeated experiences and engagement with the ways of

interacting about math that are valued in school. A study conducted by Huang,

Normandia and Greer (2005) found that even when students are provided

opportunities to practice technical language, students had difficulty taking up the

teacher’s ways of math reasoning. They argued that students need explicit instruction

in articulating principles and not just focusing on the description, sequence, and

choice that are more practical (lower level) aspects of math knowledge.

The type of language moves presented here differs from the instruction presented in

Huang, et al., 2005 in that it represents the kind of modelling that shows students how

to be effective participants in a math discourse community. Figure 9 shows how these

language moves are tied to classroom norms and messages about math reflected in

the interactions.


Handout 8 presents an eighth-grade math class interaction that illustrates ways the

teacher used the language moves of stepping out and revoicing to make math content

and processes more explicit to students, moving from doing math to reflecting in

systematic ways. This teacher’s practice illustrates both (a) how effective use of

language, and (b) a push to engage students in language use enrich the math

classroom by making the ways of thinking that underlie math transparent for all

students.

Read the interaction with a focus on what and how the teacher moves students to

deeper understanding of the content. Can you identify instances of stepping out and

revoicing?

31 HH H

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Figure 9: Relationship between Language Moves and Classroom Norms and Messages about Math.

INVESTIGATE YOUR LANGUAGE MOVES IN THE CLASSROOM

Developing reasoning about math takes time and teachers at all levels can help

students begin to do so by modelling ways to talk about math, reasoning about the

activities they are engaged in, and responding to students’ contributions in ways that

extend and support their development of academic language.

When you go back to your classrooms in the fall we expect you to try stepping out

and revoicing and to reflect on the effect these moves have on your students. The

following processes can be used to record and reflect on your classroom interactions.

• Select a class that you believe poses some challenges in terms of student

prior knowledge, engagement, self-efficacy, or motivation.

• Select a day in early fall in which to focus on language use and use an audio

or video recorder to capture the conversation that takes place.

• Listen to the recorded interaction, attending carefully to the language moves

you make. Focus on the ways you ask students to link action with reflection,

articulate a meaning behind a procedure, and respond to students’

explanations. Use Handout 9 to document these language moves. Identify

what you like about the moves you make as well as instances where you may

have been able to use these moves more effectively.

• Think of specific language related to the goals you have for helping students

reason mathematically that you could have used to respond to their

comments. Record these insights in Handout 10.

• Select another language focus day to add language moves or change some of

the ways you use language. As you plan your lesson, anticipate the multiple

solution strategies your students might offer as well as identify the language

Language Moves

Using language moves, the teacher is able to:

• Request multiple solutions • Amplify solutions • Revoice to make math process-

es clearer and more precise • Make students aware of math

thinking and relationships • Help students develop ways of

thinking and talking about math

Class Norms

Students positioned to be thinkers and explainers within math community…

Transmitted Message

Math:

• Is flexible • Is about meaning • Makes sense • Has reasons for its pro-

cedure • Requires particular ways of

reasoning and explaining

32 HH H

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insights you recorded in Handout 10 and potential sources of confusion. Use

that information to decide what language moves might help students focus on

reasoning, explaining, and reflecting. Audio tape or videotape the lesson again

on this focus day.

• Repeat the reflection process to examine your language moves until you

believe that stepping out and revoicing have become integrated into your

practice in ways that support students’ use of academic language in reasoning

mathematically (at least 3 times in the fall semester).

Planning for classroom interaction is a way to offer all students opportunities to

observe math reasoning in action and to develop their own abilities with math

reasoning. Attending to language moves in the classroom that both reveals your own

thinking processes and clarifies those of your students is a step toward constructing

more meaningful math learning for all students, particularly CALD students who may

not have had a lot of opportunities to engage in such conceptually-rich interactions.

33 HH H

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DRAW ON STUDENTS’ LANGUAGE AND CULTURE

THE CHALLENGE

All of the major content area teaching and research organizations, such as the

National Council of Teachers of Mathematics (NCTM) and National Association for

Research on Science Teaching (NARST), aggressively promote the need to provide

all students with opportunities to learn the content area knowledge and skills. Yet,

according to NCTM, ELLs are the most likely to fall victim to low expectations. By

definition, ELLs are learning both the content area and the language. Thus, ELLs face

additional challenges as they learn English, negotiate the U.S. culture, and learn the

academic content of the curriculum in all subject areas (Abedi & Gándara, 2006).

Several studies document the misperceptions and lack of knowledge teachers have

regarding best practices for ELLs, and CALD students in general.

BUILD ON STUDENTS’ LINGUISTIC AND CULTURAL KNOWLEDGE

In addition to the approaches and strategies summarized in this guidebook, there is

general consensus for the need to draw on CALD students’ language and culture.

This instructional goal entails more than celebrating national holidays. The first step

towards building students’ agency and engagement in math activities is to recognize

the shared responsibility of all teachers to improve ELLs’ language learning. That is,

the role of the math teacher is both to teach the content and to teach English. The

strategies presented thus far will move many students toward greater agency,

however, many other students may not benefit from those approaches if they do not

feel connected with the curriculum.

Based on Handout 11 (to be handed out), the following are additional strategies

teachers can use to help students make connections with the curriculum:

• Draw on students’ identities and experiences.

• Use cognates to help ELLs transfer mathematical meaning from Spanish to

English. Cognates consist of words that are similar in spelling and meaning in

two languages (sum and suma).

• Use conceptually-based games to allow ELLs’ opportunities to communicate

math language in a nonthreatening environment.

35 DD D

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BUILD STUDENTS’ CONFIDENCE & TRUST

In order for teachers to be able to draw on students’ identities and experiences,

teachers need to develop a trusting relationship with students. For some students,

this is very difficult to accomplish. Read Handout 12 (to be handed out) that

describes how a teacher was able to build confidence and trust with her student,

Alfred, by engaging in regular, yet brief conversations with him.

In the fall, select one student who may be disruptive in class and use Handout 13 (to

be handed out) to document your interactions with him and your perceptions about

how these interactions impacted the student’s participation in your class.

CULTURAL LINKS TO MATHEMATICS CONTENT & PROCESSES

What should now be apparent is that this workshop is proposing that teachers should

make an effort to infuse CALD students in language-rich curricula as well as maintain

a focus on communication of conceptual understanding. This may pose additional

challenges for ELLs (a CALD subgroup). Many educators mistakenly assume that

math is a ‘culture free’ subject area, or that math is less linguistically demanding and

thus is easier for ELLs. This is a myth that this section will illustrate how teachers

need to be careful about the cultural references they introduce in math tasks,

explanations, or applications.

As presented in Handout 13, ELLs may be challenged by the need to interpret how

mathematics is presented and taught in the US (Kersaint, Thompson, & Petkova,

2009). Some of the topics that are interpreted differently in other countries include:

• Numerals: notations (e.g., commas and periods) are used differently

• Money: US coins don’t have numerals

• Fractions: some countries emphasize decimals over fractions

• Measurement: since the metric system is more regular as a base of 10, the US

system is often confusing for ELLs.

Thus, in addition to the language, ELLs have to adapt to methods of receiving and

conveying information in vastly different ways.

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36

GLOSSARY: A - L

Agency: Active involvement and the development of autonomy in individuals which serves

to motivate them into action.

Big ideas: Highly selected concepts, principles, rules and strategies that facilitate the most

efficient and broadest acquisition of knowledge because they serve to link several

different smaller ideas together within a content domain such as mathematics. pp. 7

Clarifying (probing) questions: Questions that help students explain their thinking or help

the teacher understand their thinking.

Culturally and linguistically diverse students: Also known as CALD students;

Endurance: The knowledge and skills essential to be literate in this area beyond a single

test.

Engaging questions: Questions that invite students into discussion, keep them engaged in

conversation, invite them to share their work, or get answers “on the table.”

English language learners: Also known as ELLs; students whose first language is not

English and who are not yet proficient in English pp. 5

Higher-level thinking: Thinking that leads to or demonstrates conceptual understanding.

Higher order functions: See also higher level functions. Mental processes involved in

higher-level thinking.

Joint construction of knowledge: Interactions between peers with essentially equal

knowledge can also result in learning

Law of Parsimony: The best strategy results in the greatest number of students successfully

solving the greatest number of problems or completing the broadest range of tasks

by applying the fewest possible strategic steps

Linguistic mediation: Refers to the explicit use of language to close gaps in cultural aspects

of language for the purpose of developing understanding.

Leverage: The knowledge and skill that prove to be valuable in learning essential content in

other academic areas

37

GLOSSARY: L - Z

Process (of instruction): The instructional events and activities that are less predictable and

involve the moment-by-moment words and actions between teachers and students

as well as between and among students.

Readiness: The knowledge and skills needed in order for the student to be successful and

achieve not only in this grade level, but in subsequent grades.

Refocusing questions: Questions that serve to get students back on track or to move away

from a dead-end strategy.

Revoicing: The less explicit language moves that allow the teacher to reformulate a

student’s response.

Scaffolding: The process of creating the contexts and supports that allow students to

interact in the zone of proximal development.

Stepping out: More explicit language moves that include reflection on math actions; talking

about math.

Strategy: Any routine that leads to both the acquisition and use of knowledge with the

ultimate purpose of meaningful application.

Structure (of instruction): The predictable elements of instruction which include the design

of the activity and the materials presented to students during instruction, in groups an

individually.

Zone of proximal development (ZPD): Refers to interaction between individuals during a

learning moment that helps the less knowledgeable individual to move to the next

level of understanding.

38

References Abedi, J. & Gándara, P. (2006). Performance of English language learners as a subgroup in

large-scale assessment: Interaction of research and policy. Educational

Measurement: Issues and Practice, 36-46.

Ainsworth, L. (2003a). “Unwrapping the Standards”: A Simple Process to Make Standards

Manageable. Advanced Learning Press, Denver: CO.

Ainsworth, L. (2003b). Power Standards: Identifying the Standards that Matter the Most.

Lead & Learn Press, Englewood: CO.

Batalova, J., Fix, M., & Murray, J. (2007). Measures of change: The demography and

literacy of adolescent English learners: A report to Carnegie Corporation of New

York. Washington, DC: Migration Policy Institute.

Bright, G. W., & Joyner, J. M. (2004). Dynamic Classroom assessment: Linking

mathematical understanding to instruction in middle grades and high school:

Core program: Faciliator’s Guide. Vernon Hills, ILL: ETA/Cuisenaire, 2004.

Cobb, P., Yackel, E., & Wood, T. (1993). “Theoretical Orientation.” In Rethinking

elementary school mathematics: Insights and issues, Journal for Research in

Mathematics Education Monograph No. 6 (pp. 21-32). Reston, VA: National

Council of Teachers of Mathematics.

Donato, R. (1994). Collective scaffolding. In J. Lantolf & G. Appel (Eds.), Vygotskian

approaches to second language research (pp. 33-56). Norwood: NJ: Albex

Publishers.

Gibbons, P. (2002). Scaffolding language scaffolding learning. Portsmouth, NH:

Heinemann.

Guiton and Oakes (1995). Opportunity to learn and conceptions of educational equality.

Educational Evaluation and Policy Analysis, 17(3), 323-336.

Herbel-Eisenmann, B. (2000). How discourse structures norms: A tale of two middle school

mathematics classrooms.” Doctoral dissertation, Michigan State University, East

Lansing, Mich.

39

Herbel-Eisenmann, B. & Schleppegrell, M. J. (2008). “What questions would I be asking

myslelf in my head?”: Helping all students reason mathematically. In

Huang, J., Normandia, B. & Greer, S. (2005). Communicating Mathematically: Comparison

of knowledge structures in teacher and student discourse in a secondary math

classroom. Communication and Education, 54(1), 34-51.

Kame’enui, E. J. , Carnine, D. W., Dixon, R. C., Simmons, D. C., & Coyne, M. D. (2002).

Effective Teaching Strategies that Accommodate Diverse Learners: Second

Edition. Columbus: OH: Merrill, Prentice Hall.

Kersaint, G., Thompson, D. R., & Petkova, M. (2009). Teaching mathematics to English

language learners. New York, NY: Routledge.

Mercer, N. (1995). The guided construction of knowledge: Talk between teachers and

learners in the classroom. Clevedon, UK: Mutlilingual Matters.

Prawat, R. S. (1989). Promoting access to knowledge, strategy and dispositions in

students: A research synthesis. Review of Educational Research, 59(1), 1-41.

Rogoff, B. (1995). Observing sociocultural activity on three planes: Participatory

appropriation, guided participation, and apprenticeship. In J. Wertsch, P. de Rio,

& A. Alvarez (Eds.), Sociocultural studies of the mind (pp. 139-164). New York:

Cambridge University Press.

Stein, M. K., Schwan Smith, M., Henningsen, M. A., & Silver, E. A. (2000). Implementing

standards-based mathematics instruction. NCTM, NY, NY: Teachers College

Press.

Vygotsky, L. (1978). Mind in Society. Cambridge, MA: Harvard University Press.

Walqui, A. & van Lier, L. (2010). Scaffolding the academic success of adolescent English

language learners: A pedagogy of promise. San Francisco, CA: WestED.

Weis, I. R., Pasley, J. D., Smith, S. P., Balinower, E. R., & Heck, D. J., (2003). Looking

inside the classroom: A study of K-12 Mathematics and science education in the

United States. Chapel Hill, NC: Horizon Research.

40

Handouts

41

Handout 1: Identifying Concepts & Skills Math Strand:

Concepts (nouns) Skills (verbs) Potential Big Idea(s)

43

HANDOUT 2: VERBS FOR BLOOMS TAXONOMY

Knowledge: defines, describes, identifies, knows, labels, lists, matches, names, outlines,

recalls, recognizes, reproduces, selects, states

Comprehension: comprehends, converts, defends, distinguishes, estimates, explains,

extends, generalizes, gives examples, infers, interprets, paraphrases, predicts,

rewrites, summarizes, translates

Application: applies, changes, computes, constructs, demonstrates, discovers,

manipulates, modifies, operates, predicts, prepares, produces, relates, shows,

solves, uses

Analysis: analyzes, breaks down, compares, contrasts, diagrams, deconstructs,

differentiates, discriminates, distinguishes, identifies, illustrates, infers, outlines,

relates, selects, separates

Synthesis: categorizes, combines, compiles, composes, creates, devises, designs,

explains, generates, modifies, organizes, plans, rearranges, reconstructs, relates,

reorganizes, revises, rewrites, summarizes, tells, writes

Evaluation: appraises, compares, concludes, contrasts, criticizes, critiques, defends,

describes, discriminates, evaluates, explains, interprets, justifies, relates,

summarizes, supports.

33

44

Handout 3: Cognitive Processing Matrix Math Standard:

Grade Level:

Skill (Verbs)

Know

ledge

Com

prehension

Application

Analysis

Synthesis

Evaluation

Type of “Thinking Question”

45

HANDOUT 4: STRATEGY INSTRUCTION CASE STUDY

----WILL BE HANDED OUT---

46

HANDOUT 5: CONCEPT LESSON BRAINSTORM

Big Idea: Size

Other Concepts:

Ideas for Visuals:

Problems/Tasks:

Sequencing Content/Strategies:

47

HANDOUT 6: ANALYZING QUESTIONS—FRACTION TRACKS

Students have a set of fraction lines for halves, thirds, fourths, sixths, eighths, and so on. Each fraction line is a segment from 0 to 1 for the appropriate fraction.

INSTRUCTIONS TO STUDENTS

Put a chip on 0 in each number line. The object is to move chips from the 0 to 1 on each fraction line.

Students draw a fraction card. They may move one or more pieces a total distance determined by the fraction on the card. For example, if a student draws a card with 3

4� the student may move :

one chip 3 4� of the way across the fraction line for fourths;

one chip 1 4� of the way across the fraction line for fourths

and one chip 1 2� of the across the fraction line for halves;

one chip 1 4� of the way across the fraction line for fourths

and one chip 3 6� of the way across the fraction line for sixths; OR one chip 2 8� of the way across the fraction line for eighths

and one chip 3 6� of the way across the fraction line for sixths.

The total move must be equal to the length represented by the fraction on the card drawn.

0

0

0

0

0

0

0

1

1

1

1

1

1

1

See the NCTM Illuminations Website for an interactive applet of Fraction Tracks. Illuminations.nctm.org/LessonDetail.aspx?ID=L411

48

HANDOUT 6: ANALYZING QUESTIONS—FRACTION TRACKS (CONTINUED)

Dialogue recorded during teacher monitoring of game activity.

Teacher-Student Dialogue Purpose of Question

Teacher: How could you move your pieces across to the other side if our card was 8/10?

Sue: Instead of 8/10, you could use 2/5—no, 4/5 —and you could go to 4/5 or 8/10.

Teacher: Could you go 4/5 and then go 8/10?

Sue: No. Only one of them.

Teacher: Why can you go 4/5?

Sue: Because 8/10 and 4/5 are the same. Both of them would work. Just use the row you need to finish.

Teacher: Is there a way I could move two pieces rather than one?

Joe: You can first go to 4/10. That would be half way, and then you can go to 4/5.

Teacher: Tell me how you’re thinking about that.

Joe: They’re both equivalent, so when you go half way…. (hesitates)

Teacher: Where did you move to first?

Joe: 4/10

Teacher: Okay. Can I move the whole 4/5 now?

Joe: 2/5?

Teacher: Does 2/5 work?

Joe: 2/5 plus 4/10 would equal 8/10.

Teacher: How do you know?

Joe: 2/5 equals 4/10, and 4/10 plus 4/10 is 8/10.

49

HANDOUT 7: INVESTIGATING YOUR QUESTIONING

This form can be used to document your questioning as you listen to the recording of your lesson. When you hear yourself asking questions, note when it occurs, how students respond, and how you follow up. Number each occurrence so you can follow it on the third column.

Questioning Types When It Occurs* How Students Respond How You Followed Up**

Engaging Questions HOW TO IDENTIFY: Listen for how you started your lesson, when you re-directed students who were not engaged in the discussion.

Refocusing Questions HOW TO IDENTIFY: Listen for times when you asked students about the strategy they used.

Clarifying Questions HOW TO IDENTIFY: Listen for times when you ask students to give reasons for their answers or to explain what they are doing or thinking.

50

HANDOUT 8: LANGUAGE MOVES CASE STUDY

Examples of Jackie’s Language Moves

001 Jackie: … Now, #2. I’m doing a division problem, so what question would I be

002 asking myself in my head as I start that problem? Tammy.

003 Tammy: How many halves in four?

004 Jackie: How many halves are in four? Okay, and when I ask that question, it is

005 pretty obvious that the answer is. How many?

006 Students: Eight.

007 Jackie: Eight. Okay. What if I’m going to write my intermediate step? How would

008 I write that out? Sachin, what would I write out here? Four divided by a half is

009 the same as…

010 Sachin: Times two

011 Jackie: Times two. And I get eight… Why is that true? Sandy

012 Sandy: Because with fractions dividing by a fraction is the same as multiplying

013 by a reciprocal.

014 Jackie: Okay, good. Now, yesterday, a sixth-grade teacher asked me how do I

015 explain to my students why that’s true. We know that there is a procedure, but

016 why is it true that dividing by a reciprocal is the same as multiplying by, [that is]

017 dividing by a fraction is the same as multiplying by the reciprocal? How can you

51

A few minutes into the class period, Jackie told students that she had put some problems on the board that

she wanted them to work on. As students work at their desks, Jackie moved around the room to assess

students’ work informally, answer questions, and ask students to explain their thinking.

1. Solve x – 7 = 15x + 21 2. Compute 4 ÷1

2 = ____

3. I have c coins. One-fourth are dimes. a. How can I represent the number of dimes? b. How can I represent the value of those dimes?

Jackie then proceeded to work through the problems with the whole class. The second problem,

presented below, provided an opportunity for students to talk about the division of fractions.

4 ÷ 12 ,

018 explain that to a sixth grader so they would understand why that’s true? Does

019 anybody have any suggestions for that teacher? Raise your hand if you have an

020 idea.

021 Kelly: It just is.

022 Jackie: “Just is” isn’t a good explanation. And “just is” is not an explanation

023 that helps me understand. (laughs) Huh?

024 Ms: (inaudible)

025 Jackie: What’s the sixth grader? But they want to know why, they want to

026 understand. Alisa, do you have a suggestion?

027 Alisa: For that problem, there’s two halves in every one, because four wholes, so

028 multiply four by two.

029 Jackie: So you can relate it to a particular instance where it can make sense in

030 that particular problem. Okay? Think about if you have any other suggestions

031 because it’s kind of a hard question, and that what teachers have to think about.

032 Kelly: Is this like a real person?

033 Jackie: Yes, this is a true, true story. No, I’m not making it up. True story. That’s

034 what I’ve been thinking about what, so I had some ideas like you did. Take

035 somewhere we can see the answer pretty easily and work from there. But I didn’t

036 have a really good answer to that, that I was really happy with. So, that’s why

037 I’m asking you. So if you think of something, let me know. Okay. Yeah, Carl?

038 Carl: Is it divided by half and since multiplication is opposite, can do opposite

039 with the opposite.

040 Jackie: Okay, that’s another way you can think about it. You kind of do the opposite of opposite, so it’s

the same (laugh). Okay, that’s true.

041 …. Evan, can you tell me what you think about when you see a question

042 like the first one.

043 Evan: Can I read? It’s one-third of y equals twelve.

044 Jackie: Okay, one-third of Y equals twelve. If you read it that way, it help you

Two days later, Jackie starts with the following warm up activity:

Solve for Y: 13

Y = 12

52

045 right there. So what did you think about when you did this one?

046 Evan: Um, I thought that the twelve is, whatever. Like whatever one-third Y is,

047 it’s twelve, so I just went and did like twelve times two – twelve times three--

048 and I got thirty-six.

049 Jackie: Okay.

050 Evan: And twelve is [inaudible]

051 Jackie: Okay, and you can check your answer because one-third of thirty-six is

052 twelve. Okay? Monica, when you did it, what were you thinking? You were

053 thinking a little differently [than Evan]. What did you think about to solve this?

054 Monica: Um, I just, um, divided by one-third.

055 Jackie: Divided both sides by one-third. What’s one-third divided by one-third?

056 One, right? Something divided by itself. When I divide twelve by one-third,

057 dividing by a fraction is the same as multiplying by its reciprocal. So twelve

058 times three would give me thirty-six. Sandy, what did you think about when you

059 did it?....

060 Sandy: I did times three on both sides

061 Jackie: Okay. Because three times a third is?....

062 Sandy: Oh, one.

063 Jackie: One. Okay. So one Y is what we want, and so twelve times three gives us the

064 thirty-six. So there’s a couple of different ways people thought about this.

065 One was just reasoning through it, one was just dividing by the coefficient of Y,

066 and one was thinking. What do I need to do to get one Y? Multiply by three. Ok, so

067 you had some different strategies, but they all got you the same answer, which is a

068 good thing. And we check that that answer worked because a third of thirty-six

069 is twelve.

53

HANDOUT 9: LANGUAGE MOVES WORKSHEET

This form can be used to document your language moves as you listen to the recording of your lesson. When you hear yourself making one of these moves, note when it occurs, how students respond, and how you follow up. Number each occurrence so you can match it with Handout 9.

Language Moves When It Occurs* How Students Respond**

Moving from what students did to what they were thinking HOW TO IDENTIFY: Listen for times when students say what they are doing and you ask them about their thinking.

Asking students to articulate the meaning behind the procedure HOW TO IDENTIFY: Listen for times when you ask students to give reasons for their answers or to explain what they are doing or thinking.

Defining “valued” explanations in mathematics HOW TO IDENTIFY: Listen for times when you “revoice” a student’s wording, ask students to explain in a different way, give reasons why an explana-tion is effective, or ask for multiple solutions.

*Note what is happening in the lesson at this moment; **Write down the exact words you use and then those the students use so you can think about the kind of mathematical knowledge development that you have supported.

54

HANDOUT 10: INSIGHTS FROM LANGUAGE SELF-ANALYSIS WORKSHEET

This form can be used to document your insights about your attempts to use language moves to support students’ development of mathematical reasoning and understanding. Listen to your recording again, this time try to identify other moments in the lesson when you could have used a language move that might have supported students’ development of mathematical reasoning and understanding.

Language Moves When It Occurs* How You Follow Up**

Moving from what students did to what they were thinking HOW TO IDENTIFY: Listen for times when students say what they are doing and you ask them about their thinking.

Asking students to articulate the meaning behind the procedure HOW TO IDENTIFY: Listen for times when you ask students to give reasons for their answers or to explain what they are doing or thinking.

Defining “valued” explanations in mathematics HOW TO IDENTIFY: Listen for times when you “revoice” a student’s wording, ask students to explain in a different way, give reasons why an explanation is effective, or ask for multiple solutions.

*Note the number corresponding to those recorded in Handout 8; **Write down the kind of language move you could have made to further support the development mathematical knowledge.

55

An example of a conspicuous strategy for the volume formula follows. Note that the first step prompts the connection with a more concrete representation of volume in which students can count the cubes in a figure. Step 2 introduces the strategy. In Step 3, the teacher does not assist the students because they have already been taught to compute the area of a rectangle. In contrast, Step 4 calls for a new calculation, so the teacher is more directive.

l. Linkage to prior knowledge: "Touch box A. You know how to figure out its volume. Count the cubes and write the volume. What did you write? Yes, 50 cubic meters."

2. Introduction of new strategy: "Touch box B. You're going to learn how to calculate the volume by multiplying the area of the base times the height. "

3. Computing the area of the base: "First calculate the area of the base for box B."

4. Computing the volume: "To figure out the volume of the box, you multiply the area of the base times the height. What are the two numbers you will multiply? Yes,6 X 7."

5. Writing the complete answer: "Write the answer with the appropriate unit. What did you write? Yes, 42 cubic inches."

a. Count the cubes: b. Multiply the area of the base times the height:

/ / / / / /' / /

V /./

/ / 7 inches

/ /v V /v / / / V / / V / 2 inches

Meters 3 inches

The applicability of the big idea for volume with variations of a Single strategy for three-dimensional figures is obvious. In contrast, it is not at all obvious how a single big idea with variations of a strategy could link the following six problems:

l. Five packages of punch mix make 4 gallons. How many gallons of punch can Juan make for the party with 15 packages?

2. How long will it take a train to go 480 miles to Paris if it travels at 120 mph?

Handout 4 Strategy Instruction Case Study 1

From Kame'enui, et. al., (2002). Effective Strategies that Accommodate Diverse Learners (2nd Edition), pp. 131-138. Columbus Ohio, Merrill Prentice-Hall.

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Critical Elements of Conceptually-Based Instruction: Teach Strategy First

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3. What is the average rate of a car that goes 450 miles in 9 hours? 4. How many pounds is 8 kilograms? 5. The oil transferred from the storage area has filled 44 tanks. There are 50

tanks. What percentage of the tanks are full? 6. There are 52 cards in a deck. Thirteen of them are hearts. The rest are not

hearts. If you took trials (drew a card and then replaced it) until you drew 26 hearts, about how many trials would you expect to take?

However, it is with such seemingly unrelated problem types that a strategy based on a big idea is most valuable, particularly with learners for whom such connections usually remain elusive. The big idea that connects these different problem types is proportions. The strategy for proportions must be applied to each problem type in a systematic manner to make clear that the same big idea underlies these very different problems . The application of proportions is most obvious in the first problem:

Five packages qf punch mix make 4 gallons. How many gallons of punch can Juan make with f!fteen packages?

A medium-level strategy for proportions might first have students map the units:

packages gallons

Next, students insert the relevant information:

packages 5 15 gallons;';: D

Finally, students solve for the missing quantity: 12 gallons

Rate problems, which are not typically viewed as proportion problems, also can be solved through a proportion strategy. Note that the key to setting up rate problems as proportions is realizing that the ratio in the proportion is a number of dis

units over a single unit of time. This principle is applicable to solving the sec-ond problem: .

<~,.y\ ." How long will it tak'E//:t'train to go 480 miles to Paris if it travels at 120 mph?

First, map the units. The abbreviation mph can be represented as:

miles hour



Next, insert the relevant information:

miles 120 480 hour 1 D

Finally, solve for the answer: 4 hours

In the next rate problem, students solve for the average rate:

What is the average rate of a car that goes 450 miles in 9 hours?

Map the units:

miles hour

Insert the relevant information:

miles D 450 hour 1 9

Solve for the answer: 50 miles per hour

Another application of proportions occurs with measurement eqUivalences. The key to this problem type is that students set up a ratio between the two units involved in the equivalence.

How many pounds is 8 kilograms?

Map the units:

pounds kilograms


pounds 2.2 D kilograms 1 8

Solve for the answer: 17.6 pounds

Similarly, percent problems can be set up as proportions. For percents, the key to treating them as proportions is labeling the second ratio as telling about the percentage and pointing out that the denominator of the percentage ratio, which is almost always unstated, is 100.



The oil traniferred from the storage area has filled 44 tanks. There are 50 tanks. What percentage of the tanks are full?

Map the units:

filled tanks total tanks


filled tanks 44 total tanks 55

Solve for the answer: 88 percent

percent D

100

With the following, more difficult percentage problem, the proportion strategy makes the problem quite manageable, even for students with learning difficulties .

The oil transferred from the storage area filled 44 tanks. So far, 88% of the oil in the storage area has been transferred into tanks. How many tanks will be filled when all the oil is transferred from the storage area?

Map the units:




44 D

Solve for the answer: 50 tanks

percent 88 100

The next problem type, illustrating odds and probability, also has a key for tying it to proportions: Setting up a ratio of one type of member to another type or to the total number of members. In the example that follows, the one type of winning trial is related to the total trials.

There are 52 cards in a deck. Thirteen of them are hearts. The rest are not hearts. if you took trials (drew a card and then replaced iO until you drew 26 hearts, about how many trials would you expect to take?

Map the units:




hearts trials

hearts 13 26 trials 52 D

Solve for the answer: 104 trials

The next connection to be illustrated with proportions involves the coordinate system. Proportions can link simple proportion problems-rate, measurement equivalence, percentage, and probability- to the coordinate system. This linkage is illustrated in the graphs for each problem type in Figure 5- 2. The concept of functions is also apparent in Figure 5-2 . A function table accompanies each graph in the figure .

Finally, let's revisit the milk-ordering problem discussed previously. In this problem, students integrate the advanced propo11ion strategy with data gathering and probability statistics strategies (see Figure 5-3). The students apply data gathering strategies to determine the ratio of chocolate milk to white milk for their class and to find out the total enrollment of the school. The students then invoke the advanced proportions strategy: mapping the relevant information. The fifth graders would need to assume that the preference for types of milk in their class represents the whole school's preference, which entails applying the concept of sampling from statistics and probability. Finally, the concept of missing addends is invoked to solve for the number for white milk cartons.

Many subtle variations are possible with a problem like this, but all are accommodated through the integrated strategies illustrated in Figure 5-3. One variation might be to use average attendance instead of total enrollment, which would cut down on milk ordered (and wasted). In another variation, students might have reason to believe that the preferences in their own class would not be representative of the entire school. The students could gather data from different classes, work the problem, compare the results, and discuss variations in solutions based on different samples. As a final variation, the students could compare results with actual figures on milk ordered to predict shortages or excesses of each type of milk.

The milk-ordering problem and its variations illustrate how goals of the NCTM-working together to enhance understanding, engage in conjecture and invention, and connect mathematical ideas- can be effectively met for diverse learners through conspicuous instruction in medium-level strategies based on big ideas. As students discuss their options for selecting a sample group, they are working together to enhance their understanding of mathematics. As they weigh the relative

authentic activities is far more important than starting out with



Five packages of punch mix make 4 gallons. Fifteen packages would make how many gallons of punch?

packages 5 15 gallons 4 D

15,---------------------~_,

14 13 12 11

g; 10

~ 9 "'"' 8 g 7 0.. 6

5 4

3 2 1

packages gallons

1 2 3 4 5 6 7 8 9 10 11 12 13 gallons

_p ___ ~ _3 _ _ _ 4_ ~ .8 x P .8 1 .6 2.4 3.2 4.0

How many pounds is 5 kilograms?

pounds 2.2 _ D kilograms 1 - "5

(IJ

U c ~ 0 0..

18 ,-____________ -.

17

16

15 14 13 12 11 10

9 8 7 6 5 4 3 2 1

12345678

kilograms

pounds 2.2 x k 2.2 4.4 6.6 8.8 11 kilograms - k- --1- --2- --3- --4- - -5-

FIGURE 5-2

(IJ

How long will it take a train to go 480 miles to Rome if it travels at 120 mph?

miles 120 _ 480 hour -1- - 0

500 -,---------..,--------, 450 400 350 300

~ 250 E 200

150 100 50

1234567 hours

miles 120xh 120 240 360 480 - - - ----- - -hour h 1 2 3 4

There are 52 cards in a deck. Thirteen of these are hearts. The rest are not hearts. If you took trials (drew a card and then replaced it) until you drew 26 hearts, about how many trials would you expect to take?

hearts 13 26 trials 52 D

27 24 21

(IJ 18 1ij 15 ~ 12

9 6 3

0 12 24 36 48 60 72 84 96108

trials

hearts h 3 6 9 12 15 18 21 24 27 -- -------------------trials 4 x h 12 24 36 48 60 72 84 96 10c

Using Proportions to Link Multiple Concepts to the Coordinate System



F/GURE5-3 Data Gathering, Advanced Proportions, and ProbabilityStatistics Strategies

Step 1: Data Gathering The students conduct a survey in their class to determine the preferences for white and

There are 32 students in the class; 22 prefer chocolate milk and the rest prefer white.

chocolate milk. The students also find out from the office the There are 479 students in the total enrollment for the school. school.

Step 2: Advanced Proportions Fifth-grade Entire The students map the units for Class school the advanced proportions Chocolate ~ CJ strategy and insert the relevant

White [QJ CJ information. - --Total ~ 479

Step 3: Probability and Statistics The students solve a proportion to Chocolate 22 1329 1 estimate the number of chocolate

White CJ milk cartons to purchase for the 10 - - -

entire school: Total 32 479

22 13291 ----32 479

Step 4: Missing Addends The students determine the Chocolate 22 329 estimate for white milk using their White 10 [1]QJ knowledge of missing addends: - --

479 - 329=150 Total 32 479

merits of using total enrollment versus average attendance, they are engaging in conjecture and invention. As they link their understandings of various strategies, they are clearly learning to connect mathematical ideas, solve problems, and apply mathematics broadly.

The application of the proportion big idea with variations of a strategy for these problem contexts will deepen the student's understanding not only of proportions but also of rate, measurement eqUivalencies, percentage, probability, the coordinate system, and functions. One of the most important ways to develop this



understanding is through learning how various concepts are linked by a single strategy. In other words , a deep understanding of proportions is constructed by applying the strategy across many contexts. For this reason, the application of a strategy can be thought of as more. important in developing understanding and proficiency than how the meaning of a strategy is initially constructed. These applications do far more to develop deep understanding than allowing students initially to construct their own meaning for proportions in authentic activities. Becoming proficient at authentic activities is far more important than starting out with authentic activities.



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