Central Connecticut State University - Math Department

402 Teaching Children Mathematics / March 2008

Traditional school geometry often introduces the angle concept to students by having teach-ers show visual examples on the blackboard.

We propose a teaching sequence that breaks away from this “ostensive” method. We hypothesize that the angle concept may emerge from solving real-life physical problems, and that such activities are motivating and meaningful for children.

Teaching GeometryTeaching instructions regarding geometry often differ widely from actual practice. Principles and Standards for School Mathematics suggests that teachers encourage students to use visualization, spatial reasoning, and geometric modeling to solve problems (NCTM 2000). In the same way, the Offi-cial Bulletin of French National Education (2002) highlights the importance of spatial knowledge: “The principal objective is to enable students to

improve their view of space,” particularly in order to solve problems in usual space. However, Berth-elot and Salin (1998) showed that in actual French elementary and middle school (grades 1–7), space and geometry are taught mainly via the “ostensive” presentation of geometric knowledge; when it exists at all, the reference to space is still only evoked. The task of establishing the proper relationships between perceptual space and the concepts taught is thus left up to the student.

NCTM’s Principles and Standards also suggests that “students should have the opportunity to apply geometric ideas and relationships to other areas of mathematics, to other disciplines, and to problems that arise from their everyday experiences” (2000, p. 169). In this article, we document a spatial experiment based on a physical situation, to lead students not only to use but also to grasp a geomet-ric concept: the angle. We hypothesize that it could help root the learning process in a spatial reality that is meaningful for students and, thereby, enable them to create an empirical referent.

Acquiring Angle ConceptAngle concept acquisition was initially explored from a developmental perspective by Piaget, Inhelder, and Szeminska (1960). Piagetian research suggests that children’s concepts of angle develop

Valérie Munier, [email protected], and Hélène Merle, [email protected], are colleagues at the Institut Universitaire de Formation des Maîtres (IUFM, or University Institute for Teacher Training) of Montpellier, France. They teach physics, do research, and are interested in the connection between mathematical and physical knowledge. Claude Devichi, [email protected], is a researcher in developmental psychology at the University of Montpellier and teaches at the University of Nîmes (France). She is inter-ested in cognitive development and educational psychology.

By Valérie Munier, Claude Devichi, and Hélène Merle

AAs a Way to

Physical Situation

Teach Angle

Copyright © 2008 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Teaching Children Mathematics / March 2008 403

gradually, over an extended period of time. For example, the authors asked children to reproduce a geometrical figure including two supplementary angles (see fig. 1) and showed that this figure is viewed by children as an angles system only after nine years of age (third Piagetian stage).

Another theory describing the development of geometric thinking as a succession of discrete stages is the van Hiele model (Burger and Shaugh-nessy 1986). The van Hieles define five levels: Students first (1) identify and operate on geometric shapes, and then (2) discover the properties of a class of shapes, followed by (3) interrelating the properties of shapes. At the higher level, they are able to (4) understand the role of axioms and theo-rems. Finally, students are able to (5) establish new theorems to analyze and compare.

Other studies criticize the strict hierarchy of the van Hiele model (for a review, see Owens 1996) because of the difficulty in classifying students precisely and the great variability of performances observed at a given level. As Wilson and Adams (1992) report, “The van Hieles’ research leads to an important point: Students need good activities designed to help them to explore angles and their properties and relation-ships.” More recently, Mitchelmore and White (1995) and Mitchelmore (1997) investigated how children from seven to twelve years of age abstract this con-cept from everyday physical situations. They describe three steps in the abstraction of a general angle con-cept: familiarity (children collect physical angle situ-ations on the basis of their superficial characteristics); similarity (similar situations are collected into physi-cal angle contexts); and reification (a general angle concept includes all contexts).

Difficulty Appropriating the Concept Several studies point out an erroneous angle repre-sentation that resists traditional teaching methods: “The angle is conceived of as the fact of having two segments with a common end and different medi-ums” (Balacheff 1988). Students think the length of the arms is related to the size of the angle (Berth-elot and Salin 1998, Wilson and Adams 1992), and this misconception seems to be very hard to overcome. For example, Lehrer, Jenkins, and Osana (1998) conducted a longitudinal study (reported by Sutherland et al. 2001) involving children in grades 1–3 who were followed through grades 3, 4, and 5. Their results show that “the length of the seg-ments used in the angle had a strong influence on

the students’ choices, and this fact did not change throughout the three years of the study.”

A Playground ExperienceIn this study, we investigated how children abstract the angle concept from their informal knowledge about a physical situation, based on the notion of visual field. In the presence of an obstacle, the visual field is delineated by an angle whose vertex is located at the observer and whose sides go through the extremities of the obstacle. This situation is spe-cifically designed to focus on the idea of direction to invalidate the misconception of angle as two seg-ments. Students were able to experimentally verify that angle value does not depend on length of sides.

The teaching sequence was proposed to the students by their regular classroom teacher in third- and fourth-grade classes. (Eighteen third-grade students and eighteen fourth-grade students took part in both the teaching sequence and the pre- and postassessment investigations.)

Session 1: Elaborating the problem and setting hypothesesThe teacher introduced the teaching sequence with a road safety picture (see fig. 1) and asked students if the children in the picture are in danger when they cross the road. The whole-class discussion led to the question, “What can we see when we stand in front of an obstacle?” The teacher then proposed an exercise during which the students color the hidden area. Four types of answers resulted: 31

The teaching sequence is introduced with this road safety picture.

Figure 1


percent of the children answered correctly (see fig. 2a), 27 percent thought the child could not see the “strip” perpendicular to the screen (see fig. 2b), 18 percent drew “oblique” lines that went through the endpoints of the screen but did not go through the observer (see fig. 2c), and 24 percent proposed other solutions. All the solutions were recorded on the blackboard, and a debate was organized. The children concluded that they needed to check their conjectures experimentally on the playground.

Session 2: Playground experienceThe teacher put an opaque screen and a chair on the playground. Students separated into several small groups. One group took the observer positions: one student from their group on a chair in front of the screen and the others—standing—close to the seated student. (During the experiment, each group took a turn in the observer positions.)

Students from the other groups, carrying skittles (traffic cones or “ninepins,” see fig. 3a) lined up behind the screen, where they could not be seen by the observers. One by one, the students behind the screen moved sideways and put their skittles on the floor as soon as they were visible to the group that was observing. They noticed that the skittles were in straight lines. To check the align-To check the align-ment, they held ropes between the skittles. The students also noticed that the lines were “slanted” or “diagonal,” allowing them to rule out the strip hypothesis from figure 2b. One pupil recalled the hypothesis that the boundary lines would pass through the observer. They verified the hypothesis experimentally by moving the ropes over to the observer, first repeating the experiment with two longer ropes, and then two more times, varying the observer position (see fig. 3). Students could see that the boundaries of the visible area were two half lines that went through the screen edges and

Children hypothesize about the shape of the hidden area.

Exercise (a) Correct answer (b) “strip” (c) “oblique”

A child is behind a screen. Color the area that the child cannot see.

–––––––––– Screen

• Child

Figure 2

Students conduct a playground experiment.

(a) (b) (c)

Figure 3

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by

Pat

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Bet

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; all

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d


the observer, and that the lines could be extended as far back as desired. They also noted that the hidden area increased or decreased in size as the observer stood closer to or farther from the screen. Back in the classroom, the teacher gave them an individual exercise in which they had to draw the hidden area for various positions of the observer, as in the playground experience (see fig. 4).

Session 3: Delimiting the hidden areaThe teacher put the previous exercise solution on the board and asked, “Is the hidden area the same everywhere?” For the two first diagrams (see figs. 4a and b) with the observer in the same posi-tion), some students said the hidden area would be the same, but not everyone agreed. For some students, the fact that the colored area was bigger or smaller in the different cases meant that the hidden area changed in size. Others thought that the boundaries could extend as far as desired and the hidden area, therefore, would be the same. The teacher gave students a set of four correct diagrams, asking them to think of a technique for comparing the colored areas. Some students superimposed the diagrams against a window pane or used tracing paper. They concluded that the hidden area was the same for a given observer position, no matter how long the ropes were. The teacher asked those students, “What delimits the hidden area?” Most of them replied, “a triangle,” or “a V-shape.” After a few minutes of discussion, the term angle came up: The hidden areas were delineated by the same angle. When comparing the last two diagrams (see figs. 4c and d) with the observer in different positions, students concluded that the hidden area varies with the observer position. The teacher then led them to verbalize—employing the term angle—what happens when the observer is standing closer to or farther away from the screen.

Session 4: Mathematics sessionThe last session was a “classical lesson” of mathematics. In accordance with French official directives, no formal definition of the angle was given. The teacher introduced only the geometri-cal vocabulary (vertex, arms). He then assigned an angle-copying exercise and asked the students to devise other techniques for comparing and reproducing angles. The students quickly devised several techniques for making templates, and some of them proposed using a pair of scissors laid on the figure. The teacher then quite naturally

introduced an angle model: the bevel-square (see fig. 5), opening its branches little by little while asking how the angle would change (see stage 1 in fig. 5). Students answered that it would be big-ger (greater), some using the phrase more open. The teacher continued to open the branches until he modeled a right angle (see stage 2), which the students quickly noticed. At stage 3, students com-mented that “the angle is becoming increasingly big or open.” When the angle reached 180° (stage 4), some students said that “there is no more angle.” Others commented, “When you open it up, there’s still an angle; it doesn’t disappear.” A debate set in, after which the teacher confirmed that the angle still existed and was called a flat angle. When the angle became greater than 180° (stage 5), several students exclaimed, “The angle has gone below. Now it’s more closed. From one side, you open, and from the other, you close.” The debate ended with the idea that for two half lines, there are two angles, a closed angle and an open angle, and that “the ‘closed’ one is the area that cannot be seen (except for the small triangle in front), and the ‘open’ one is the part that can be seen.” Students remarked on the necessity of

Students color the hidden area.

(a) (b) (c) (d)

Color the area you cannot see.

xx

x

x

Figure 4

Bevel-square used as dynamical angle model

1

23

4

5

Figure 5


Example of comparison exercise

Color the figure with the greatest angle.

Figure 6

marking angles in order to distinguish them. The ses-sion ended with various exercises of comparison.

Evaluating the SequenceTo estimate the extent to which the students pro-gressed, each child was tested individually before and after the sequence. Two basic tasks, defining and drawing an angle, assessed student progress. For the definition task, incorrect definitions included “I don’t know” or “It’s a line, a side, or a corner.” We also considered answers such as “It’s a square” or “It’s something in a triangle” as reflecting inaccurate conceptions. Correct answers included “right angle” and “straight lines meeting at a point.” For the drawing task, drawings were classified in four categories hierarchically ordered: “no answer,” “line,” “geometric figure,” and “angle” (see table 1).

The results of pre- and postassessments were

analyzed and compared to evaluate the effectiveness of the experiment in helping children define and draw an angle. The analysis of student responses indicates some progress. In third grade, before the teaching sequence, nine students out of seventeen failed to both define and draw an angle, and five failed in one of the two tasks. Among the fourteen third graders who could progress (i.e., who failed in preassessment), five students progressed (two in the definition task, two in the drawing task, and one in both tasks). In fourth grade, before teaching, seven students out of seventeen failed to both define and draw an angle, and five failed in one task only. Of the twelve students who could progress, seven stu-dents progressed (two in the definition task, four in the drawing task, and one in both tasks).

These results are corroborated by mathematics evaluations in which students had to compare three pairs of angles designed to verify whether they had progressed in their ability to define and draw an angle (see fig. 6 for an example). The results of these comparisons indicate a good understanding of the fact that angle size is independent of the sides’ lengths. Another exercise about the physical situa-tion “field of view” shows that after teaching, the majority of the students were capable of mastering the functional relationship between the size of the angle and the observer’s location on the median or the width of the screen.

ConclusionThe results reported here validate our hypothesis. Using a physics framework to introduce angles in problem situations taken from perceptual space is feasible and pertinent in elementary school. The teaching sequence allowed students to progress in abstracting angle concept, observable in both defining and drawing tasks. From a mathematical viewpoint, the proposed physical situation enabled the students to invalidate the misconception that using length is an appropriate way to compare angles, which was one of the main reasons we introduced this concept using a problem in a large space, where the idea of “direction” stood out.

Moreover, physical situations create opportuni-ties for students to explore, conjecture, display, and clarify their understanding of angle concept. Unexpected viewpoints and misconceptions could surface and be discussed in a whole-class arena monitored by the teacher. Throughout this ses-sion, the classroom discussion reached such an enthusiasm that the teacher rarely interrupted the

Students’ Productions in the Drawing Task

Categories Examples

No answer

Line

Geometric figure

Angle

Table 1


students. Instead of an ostensive classroom lesson, (“mathematics acquired while sitting down in a group lesson” [Clements 2001]), we proposed motivational activities, breaking away from conventional teaching contexts and making the concept of angle meaningful for elementary children.

This article proposes an efficient sequence that can be easily accomplished in grades 3 and 4 with elementary material. The sequence leads students to connect physical and geometrical reasoning and “highlights relationships between mathematics and the sciences” (NCTM 2000, p. 169).

ReferencesBalacheff, Nicolas. “Une Étude des Processus de

Preuve en Mathematiques Chez les Élèves de Col-lège.” Thèse d’Etat, Université Grenoble I, pp. 395–464, 1988.

Berthelot, René, and Marie-Hélène Salin. “The Role of Pupils’ Spatial Knowledge in the Elementary Teaching of Geometry.” In Perspectives on the Teaching of Geometry for the 21st Century, edited by C. Mammana and V. Villani, pp. 71–82. Dor-drecht, The Netherlands: Kluwer Academic Pub-lishers, 1998.

Burger, William F., and J. Michael Shaughnessy. “Char-acterizing the van Hiele Levels of Development in Geometry.” Journal for Research in Mathematics Education 17 (1986): 31–48.

Clements, Douglas H. “Mathematics in the Pre-school.” Teaching Children Mathematics 7 (2001): 270–75.

Lehrer, Richard, Michael Jenkins, and Helen Osana. “A Longitudinal Study of Children’s Reasoning about Space and Geometry.” In Designing Learn-ing Environments for Developing Understanding of Geometry and Space, edited by Richard Lehrer and Daniel Chazan, pp. 137–67. Mahwah, NJ: Lawrence Erlbaum Associates, 1998.

Mitchelmore, Michael. “Children’s Informal Knowl-edge of Physical Angle Situations.” Learning and Instruction 7, no. 1 (1997): 1–19.

Mitchelmore, Michael, and Paul White. “Abstraction in Mathematics: Conflict, Resolution, and Application.” Mathematics Education Research Journal 7 (1995): 50–68.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

Official Bulletin of French National Education. Min-istère de l’Education Nationals–Ministère de la Recherche, 2002.

Owens, Kay. “Recent Research and a Critique of Theories of Early Geometry Learning: The Case of the Angle Concept.” Nordic Studies in Mathemat-ics Education 4 (2/3) (1996): 85–104.

Piaget, Jean, Barbel Inhelder, and Alina Szeminska.

The Child’s Conception of Geometry. London: Routledge and Kegan Paul, 1960.

Sutherland, Jamie, Deborah Trzinski-Becker, and Tser-ing Duggyal. “Teaching and Learning Geometry.” Cognition and Instruction 811 (2001).

Wilson, Patricia S., and Verna M. Adams. “A Dynamic Way to Teach Angle and Angle Measure.” Arithmetic Teacher 39, no. 5 (1992): 6–13. s

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