trace. Moreover, for the instructor, producing gestures help to preserve the role of bodily and kinesthetic experience in mathematical learning, as highlighted in various studies (e.g. Nemirovsky, 2003; Arzarello et al., 2009).

## Feedback

The constructor follows the instructor's instructions: he/she takes the chosen instrument, positions it as requested and draws the required line. When he/she detects ambiguities in the instructor's instructions, he acts in the "least likely" way to help him/her remove the implicit meaning. In this way, the constructor needs to pay attention to the validity of the instructions for obtaining a correct figure, i.e. one constructed without the "guesswork" of using the instrument (such as positioning the ruler perceptually to draw a right angle).

By working in pairs, dyspraxic student (in the role of the instructor) can experience the instrumented actions he/she is planning by observing them, without worrying about their manipulative characteristics. He/she also benefits from feedback that is consistent with his/her instructions, on which he/she can base his/her thinking.

In the following section, we illustrate this dyad work system by analyzing an implementation in a pencil-and-paper environment with two sixth-grade students (dyspraxic student and standard student).

## EXAMPLE OF AN IMPLEMENTATION

Marion is an 11-year-old student diagnosed with visuo-spatial dyspraxia. Her psychomotor assessment demonstrated an immature spatial organization and difficulties with digital untying and oculomotor co-ordination. She has always had difficulty writing and drawing geometric figures with instruments at school. Despite her many attempts, her attention and her efforts, she was never able to achieve the required precision. This is illustrated by the comments made by her mathematics teacher in the following extract from an assessment (Figure 2). The use of ruler and set square act as a barrier between what she knows, what she wants to draw and the act of drawing. She indeed knows how to draw the symmetry of a point or a straight line with respect to a straight line, but when she draws with the instruments, she places the set square imprecisely, the instrument moves when she draws, etc.

<figure><img src="image_82.png" /><figcaption>Figure 2: Marion’s work</figcaption></figure>

We report on the implementation of the dyadic work system that we tried out a month and a half later with Marion and Bonnie, a student in her class.
trace. Moreover, for the instructor, producing gestures help to preserve the role of bodily and kinesthetic experience in mathematical learning, as highlighted in various studies (e.g. Nemirovsky, 2003; Arzarello et al., 2009).

## Feedback

The constructor follows the instructor's instructions: he/she takes the chosen instrument, positions it as requested and draws the required line. When he/she detects ambiguities in the instructor's instructions, he acts in the "least likely" way to help him/her remove the implicit meaning. In this way, the constructor needs to pay attention to the validity of the instructions for obtaining a correct figure, i.e. one constructed without the "guesswork" of using the instrument (such as positioning the ruler perceptually to draw a right angle).

By working in pairs, dyspraxic student (in the role of the instructor) can experience the instrumented actions he/she is planning by observing them, without worrying about their manipulative characteristics. He/she also benefits from feedback that is consistent with his/her instructions, on which he/she can base his/her thinking.

In the following section, we illustrate this dyad work system by analyzing an implementation in a pencil-and-paper environment with two sixth-grade students (dyspraxic student and standard student).

## EXAMPLE OF AN IMPLEMENTATION

Marion is an 11-year-old student diagnosed with visuo-spatial dyspraxia. Her psychomotor assessment demonstrated an immature spatial organization and difficulties with digital untying and oculomotor co-ordination. She has always had difficulty writing and drawing geometric figures with instruments at school. Despite her many attempts, her attention and her efforts, she was never able to achieve the required precision. This is illustrated by the comments made by her mathematics teacher in the following extract from an assessment (Figure 2). The use of ruler and set square act as a barrier between what she knows, what she wants to draw and the act of drawing. She indeed knows how to draw the symmetry of a point or a straight line with respect to a straight line, but when she draws with the instruments, she places the set square imprecisely, the instrument moves when she draws, etc.

<figure><img src="image_88.png" /><figcaption>Figure 2: Marion’s work</figcaption></figure>

We report on the implementation of the dyadic work system that we tried out a month and a half later with Marion and Bonnie, a student in her class.
Given Marion's visuo-spatial difficulties, the instructions were given orally so as not to add a reading task to the proposed geometric task (Figure 3). In addition, the two straight lines were differentiated by color (the axis of symmetry was in red) and the starting figure was presented using gestures (browsing the straight lines, pointing to the points) to accompany the speech.

<figure><img src="image_83.png" /><figcaption>Figure 3: Instructions given to Bonnie and Marion</figcaption></figure>

Bonnie and Marion both immediately announced that they knew how to do it, as soon as the instructions had been given. Bonnie gave a drawing technique (Tb) that involved extending the line (Δ) to its point of intersection with the axis of symmetry and then drawing the line through this point and point A'. Marion reacted by challenging this technique: "But actually, Bonnie, you can draw your line wherever you like, you can put it like this, like this, like this, so that it touches the point", pointing with her hand to different directions of the line that pass through the point of intersection that Bonnie was proposing to construct. Marion then gives her drawing technique (Tm), which involves drawing the symmetrical line of a second point on the line (Δ) in order to draw the line that passes through this symmetrical point and the point A'.

Both drawing techniques are shown in Figure 4 with the speech and gestures produced by Bonnie and Marion during their interactions.

<figure><img src="image_84.png" /><figcaption>Figure 4: Two different construction techniques</figcaption></figure>

Neither Bonnie nor Marion mentions point A', but they both point to it and run their fingers over it as they trace the symmetrical line they are considering. In this way, the two students can reflect together
Given Marion's visuo-spatial difficulties, the instructions were given orally so as not to add a reading task to the proposed geometric task (Figure 3). In addition, the two straight lines were differentiated by color (the axis of symmetry was in red) and the starting figure was presented using gestures (browsing the straight lines, pointing to the points) to accompany the speech.

<figure><img src="image_89.png" /><figcaption>Figure 3: Instructions given to Bonnie and Marion</figcaption></figure>

Bonnie and Marion both immediately announced that they knew how to do it, as soon as the instructions had been given. Bonnie gave a drawing technique (Tb) that involved extending the line (Δ) to its point of intersection with the axis of symmetry and then drawing the line through this point and point A'. Marion reacted by challenging this technique: "But actually, Bonnie, you can draw your line wherever you like, you can put it like this, like this, like this, so that it touches the point", pointing with her hand to different directions of the line that pass through the point of intersection that Bonnie was proposing to construct. Marion then gives her drawing technique (Tm), which involves drawing the symmetrical line of a second point on the line (Δ) in order to draw the line that passes through this symmetrical point and the point A'.

Both drawing techniques are shown in Figure 4 with the speech and gestures produced by Bonnie and Marion during their interactions.

<figure><img src="image_90.png" /><figcaption>Figure 4: Two different construction techniques</figcaption></figure>

Neither Bonnie nor Marion mentions point A', but they both point to it and run their fingers over it as they trace the symmetrical line they are considering. In this way, the two students can reflect together
on the geometric problem they have been given, without any language barriers limiting their reasoning: it's difficult to focus on using geometric language properly while at the same time exploring and explaining possible solutions to a problem. The obstacles associated with the layout that Marion would encounter if the initial discussions to solve the problem were based on instrumented layouts were also ruled out, given her disability.

Both techniques (Tm and Tb) will be put to the test. The following four vignettes (Figure 5) trace the beginning of the interactions between Marion, the instructor, and Bonnie, the constructor.

<figure><img src="image_85.png" /><figcaption>Figure 5: Carrying out the construction</figcaption></figure>

Marion gestures to the part of the line (Δ) on which she wants Bonnie to place a point (Vignette 1). As well as the necessary indication of the line to which the requested point must belong, she is probably making sure that the point chosen is a long way from point A to allow precise drawing. In formulating her technique, she had in fact specified that the second point should be "at the bottom of the line" (Figure 4, Tm).

Marion then names the introduced point "B" (Vignette 2), and announces her intention: to make the symmetrical point of point B (Vignette 3). This information can help Bonnie to anticipate the drawings, if necessary correcting the position of point B on the line (Δ) so that its symmetrical point does not leave the support sheet. She can also help her to give Marion feedback on her instructions ("do the least likely thing"). After asking for the set square, Marion specifies its position: "one side of the right angle on the line and the other side of the right angle must go through B" (Vignette 4). The instruction was incomplete: Marion hadn't specified which straight line she was talking about (the axis of symmetry (d)) or pointed to it with her finger. Bonnie then suggests a position that corresponds to the instruction given, by choosing the line (Δ) and asks Marion if this is appropriate. Marion can then add the missing detail if this was not the position she had in mind.
on the geometric problem they have been given, without any language barriers limiting their reasoning: it's difficult to focus on using geometric language properly while at the same time exploring and explaining possible solutions to a problem. The obstacles associated with the layout that Marion would encounter if the initial discussions to solve the problem were based on instrumented layouts were also ruled out, given her disability.

Both techniques (Tm and Tb) will be put to the test. The following four vignettes (Figure 5) trace the beginning of the interactions between Marion, the instructor, and Bonnie, the constructor.

<figure><img src="image_91.png" /><figcaption>Figure 5: Carrying out the construction</figcaption></figure>

Marion gestures to the part of the line (Δ) on which she wants Bonnie to place a point (Vignette 1). As well as the necessary indication of the line to which the requested point must belong, she is probably making sure that the point chosen is a long way from point A to allow precise drawing. In formulating her technique, she had in fact specified that the second point should be "at the bottom of the line" (Figure 4, Tm).

Marion then names the introduced point "B" (Vignette 2), and announces her intention: to make the symmetrical point of point B (Vignette 3). This information can help Bonnie to anticipate the drawings, if necessary correcting the position of point B on the line (Δ) so that its symmetrical point does not leave the support sheet. She can also help her to give Marion feedback on her instructions ("do the least likely thing"). After asking for the set square, Marion specifies its position: "one side of the right angle on the line and the other side of the right angle must go through B" (Vignette 4). The instruction was incomplete: Marion hadn't specified which straight line she was talking about (the axis of symmetry (d)) or pointed to it with her finger. Bonnie then suggests a position that corresponds to the instruction given, by choosing the line (Δ) and asks Marion if this is appropriate. Marion can then add the missing detail if this was not the position she had in mind.
The presentation of this short extract from the interactions between Bonnie and Marion suggests a certain potential of the dyadic working system: the two students can get involved in solving an instrumented construction problem, applying their knowledge and developing it. For the dyspraxic student, the obstacle of handling instruments is removed, giving way to geometric work.

Following this initial experiment outside the classroom, we continued our research in collaboration with teachers by implementing the dyadic system in the context of the class, adapting it to its constraints, so that it produced the expected effects in terms of geometric learning for all the students. We observed that it was not easy for some of students in the role of instructor to use the technical language and that the feedback from some of students in the role of constructor was not always appropriate. This led us to imagine a dyadic work in the digital environment, using an avatar, so that the students can learn the technical language, the rules of the dyad and the correct use of the instruments. We give an overview in the next section.

## GIVING INSTRUCTIONS TO AN AVATAR

In collaboration with Fabien Emprin, we used the Virtual Training Suite<fnref n="15" /> software as a human interaction simulator in an exploratory study (Emprin & Petitfour, 2021). With this software, reality is rendered by the setting of a classroom in which the avatar is installed as if next to the student instructor (Figure 6, left), and by videos of the hands of a person carrying out the required actions (Figure 6, right).

<figure><img src="image_86.png" /><figcaption>Figure 6: The avatar in the human interaction simulator</figcaption></figure>

The simulator proposes five elementary instrumented actions (Figure 7) for which the instructor must get the avatar to produce the red line on the starting drawing. Each of the five drawings corresponds to an instrumented action, that is, an action that leads to the production of a line with an instrument (non-graduated ruler, set square or compass). A combination of several different types of actions allows to construct a complex figure.

<figure><img src="image_87.png" /><figcaption>Figure 7: Elementary instrumented actions</figcaption></figure>

<footnote n="15">¹⁵ https://seriousfactory.com/virtual-training-suite/</footnote>
The presentation of this short extract from the interactions between Bonnie and Marion suggests a certain potential of the dyadic working system: the two students can get involved in solving an instrumented construction problem, applying their knowledge and developing it. For the dyspraxic student, the obstacle of handling instruments is removed, giving way to geometric work.

Following this initial experiment outside the classroom, we continued our research in collaboration with teachers by implementing the dyadic system in the context of the class, adapting it to its constraints, so that it produced the expected effects in terms of geometric learning for all the students. We observed that it was not easy for some of students in the role of instructor to use the technical language and that the feedback from some of students in the role of constructor was not always appropriate. This led us to imagine a dyadic work in the digital environment, using an avatar, so that the students can learn the technical language, the rules of the dyad and the correct use of the instruments. We give an overview in the next section.

## GIVING INSTRUCTIONS TO AN AVATAR

In collaboration with Fabien Emprin, we used the Virtual Training Suite<fnref n="15" /> software as a human interaction simulator in an exploratory study (Emprin & Petitfour, 2021). With this software, reality is rendered by the setting of a classroom in which the avatar is installed as if next to the student instructor (Figure 6, left), and by videos of the hands of a person carrying out the required actions (Figure 6, right).

<figure><img src="image_92.png" /><figcaption>Figure 6: The avatar in the human interaction simulator</figcaption></figure>

The simulator proposes five elementary instrumented actions (Figure 7) for which the instructor must get the avatar to produce the red line on the starting drawing. Each of the five drawings corresponds to an instrumented action, that is, an action that leads to the production of a line with an instrument (non-graduated ruler, set square or compass). A combination of several different types of actions allows to construct a complex figure.

<figure><img src="image_93.png" /><figcaption>Figure 7: Elementary instrumented actions</figcaption></figure>

<footnote n="15">¹⁵ https://seriousfactory.com/virtual-training-suite/</footnote>
Once a drawing selected, the student has to choose which instrument to use among ruler, set square and compass, in order to obtain the red line from the starting figure (Figure 8, step 1). He/she can then see the hand of the avatar taking the requested instrument in video. Four or five positioning formulation of the chosen instrument in relation to the geometric objects present are then proposed (Figure 8, step 2). One is valid, and the others are incomplete or not precise enough.

<figure><img src="image_88.png" /><figcaption>Figure 8: Example of choices proposed by the simulator</figcaption></figure>

The student selects the instruction that seems to be appropriate, and then watches a video of the avatar's hands performing the requested positioning, while observing the “least likely” rule. The student can go back on his/her decision or confirm the avatar to carry out the drawing. In the latter case (step 3), he/she sees a video of the avatar's hands making a drawing from the position of the chosen instrument. The avatar compares the drawing produced with the expected drawing. If it is not correct, the student makes another proposal, if it is correct, the avatar formulates the drawing produced in geometric language (for example for the drawing 4 of Figure 7, the avatar says “Well done, you’ve correctly drawn the circle with center A and radius AB”).

Computer simulation removes certain aspects of human interaction, which can have positive effects. Indeed, the fact that the computer shows no impatience or judgement can help to instill confidence in the student, particularly if he/she has a disability. He/she can think at his/her own pace, and benefit from interactions with the avatar that are less emotionally charged than they might be with a peer. Moreover, because the computer does what it's told without over-interpreting, the student can better understand the need to be explicit in his/her instructions. There are a few drawbacks to using the simulator, however. We had to abandon the development of the gestural dimension for the instructor to accompany the language: gestures cannot be considered by the software used, only language instructions can. What's more, the student's freedom of expression is restricted, since he/she has to select from a finite list.

Our first experiment was carried out as follows. The teacher began with focusing the whole class on the functions of the set square and the compass, then to the technical and geometric terms to be used, associating them with gestural exploration (see Figure 1a). The students then each worked freely with the software (student-instructor and avatar-constructor dyad), while the teacher observed their work. At the end, the teacher consulted a summary of the activities for each student memorized by the software: the number of (in)correct choices of formulation and instrument, the number of times the student (not) correctly anticipated that the positioning of the instrument was (in)valid; the number of times the student correctly identified that he/she had reached (or not) the goal. Next, a discussion led
Once a drawing selected, the student has to choose which instrument to use among ruler, set square and compass, in order to obtain the red line from the starting figure (Figure 8, step 1). He/she can then see the hand of the avatar taking the requested instrument in video. Four or five positioning formulation of the chosen instrument in relation to the geometric objects present are then proposed (Figure 8, step 2). One is valid, and the others are incomplete or not precise enough.

<figure><img src="image_94.png" /><figcaption>Figure 8: Example of choices proposed by the simulator</figcaption></figure>

The student selects the instruction that seems to be appropriate, and then watches a video of the avatar's hands performing the requested positioning, while observing the “least likely” rule. The student can go back on his/her decision or confirm the avatar to carry out the drawing. In the latter case (step 3), he/she sees a video of the avatar's hands making a drawing from the position of the chosen instrument. The avatar compares the drawing produced with the expected drawing. If it is not correct, the student makes another proposal, if it is correct, the avatar formulates the drawing produced in geometric language (for example for the drawing 4 of Figure 7, the avatar says “Well done, you’ve correctly drawn the circle with center A and radius AB”).

Computer simulation removes certain aspects of human interaction, which can have positive effects. Indeed, the fact that the computer shows no impatience or judgement can help to instill confidence in the student, particularly if he/she has a disability. He/she can think at his/her own pace, and benefit from interactions with the avatar that are less emotionally charged than they might be with a peer. Moreover, because the computer does what it's told without over-interpreting, the student can better understand the need to be explicit in his/her instructions. There are a few drawbacks to using the simulator, however. We had to abandon the development of the gestural dimension for the instructor to accompany the language: gestures cannot be considered by the software used, only language instructions can. What's more, the student's freedom of expression is restricted, since he/she has to select from a finite list.

Our first experiment was carried out as follows. The teacher began with focusing the whole class on the functions of the set square and the compass, then to the technical and geometric terms to be used, associating them with gestural exploration (see Figure 1a). The students then each worked freely with the software (student-instructor and avatar-constructor dyad), while the teacher observed their work. At the end, the teacher consulted a summary of the activities for each student memorized by the software: the number of (in)correct choices of formulation and instrument, the number of times the student (not) correctly anticipated that the positioning of the instrument was (in)valid; the number of times the student correctly identified that he/she had reached (or not) the goal. Next, a discussion led
by the teacher highlighted the rules of the dyad work system (role of the avatar as constructor and language to be used by the instructor). The students then practiced the five elementary instrumented actions (Figure 7) in the pencil-and-paper environment, alternating between the roles of instructor and constructor. They then worked on other geometric construction, taking the avatar's responses, and some actions, as a model for their own behavior. An exploratory study of the contribution of the human interaction simulator thus designed to teach geometry to dyspraxic students led us to establish a proof of concept (Emprin & Petitfour, 2021, 2023). Our initial experimental results indicate that alternating student- avatar and student-student dyads is of interest for teaching geometry to dyspraxic students, but also for any student. These results still need to be confirmed by further experiments.

## References

Arzarello, F., Paola, D., Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom. *Educational Studies in Mathematics*, 70(2), 97-109.

Elbasan, B., & Kayihan, H. (2012). Motor performance and activities of daily living in children with developmental coordination disorder. *Journal of Novel Physiotherapies*, 2(2) (#1000107).

Emprin, F., & Petitfour, E (2021). Using a Simulator to Help Students with Dyspraxia Learn Geometry. *Digital Experiences in Mathematics Education*, 7, 99-121.

Emprin, F., & Petitfour, E. (2023). Contribution of a human interaction simulator to teach geometry to dyspraxic pupils, in C. Guille-Biel Winder, T. Assude (coord.). *Articulations between tangible space, graphical space and geometrical space. Resources, practices and training* (pp. 129-153), Iste Science Publishing.

McNeill, D. (1992). *Hand and mind: what gestures reveal about thought*. University of Chicago Press.

Nemirovsky, R. (2003). Three conjectures concerning the relationship between body activity and understanding mathematics. In N. Pateman, B. Dougherty & J. Zillox (Eds), *Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 1, pp. 105–109). PME.

Petitfour, E. (2016). Teaching geometry to visual-spatial dyspraxic pupils. Paper presented at the *13th International Congress on Mathematical Education*. Germany: Hamburg (https://hal.archives-ouvertes.fr/hal-02119848/document).

Searle, J. (1983). *Intentionality. An essay in the philosophy of mind*. Cambridge University Press

Vygotsky, L. (1978). *Mind in society: the development of higher psychological processes*. Harvard University Press.
by the teacher highlighted the rules of the dyad work system (role of the avatar as constructor and language to be used by the instructor). The students then practiced the five elementary instrumented actions (Figure 7) in the pencil-and-paper environment, alternating between the roles of instructor and constructor. They then worked on other geometric construction, taking the avatar's responses, and some actions, as a model for their own behavior. An exploratory study of the contribution of the human interaction simulator thus designed to teach geometry to dyspraxic students led us to establish a proof of concept (Emprin & Petitfour, 2021, 2023). Our initial experimental results indicate that alternating student- avatar and student-student dyads is of interest for teaching geometry to dyspraxic students, but also for any student. These results still need to be confirmed by further experiments.

## References

Arzarello, F., Paola, D., Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom. *Educational Studies in Mathematics, 70*(2), 97-109.

Elbasan, B., & Kayihan, H. (2012). Motor performance and activities of daily living in children with developmental coordination disorder. *Journal of Novel Physiotherapies, 2*(2) (#1000107).

Emprin, F., & Petitfour, E (2021). Using a Simulator to Help Students with Dyspraxia Learn Geometry. *Digital Experiences in Mathematics Education, 7*, 99-121.

Emprin, F., & Petitfour, E. (2023). Contribution of a human interaction simulator to teach geometry to dyspraxic pupils, in C. Guille-Biel Winder, T. Assude (coord.). *Articulations between tangible space, graphical space and geometrical space. Resources, practices and training* (pp. 129-153), Iste Science Publishing.

McNeill, D. (1992). *Hand and mind: what gestures reveal about thought*. University of Chicago Press.

Nemirovsky, R. (2003). Three conjectures concerning the relationship between body activity and understanding mathematics. In N. Pateman, B. Dougherty & J. Zillox (Eds), *Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 1, pp. 105–109). PME.

Petitfour, E. (2016). Teaching geometry to visual-spatial dyspraxic pupils. Paper presented at the *13th International Congress on Mathematical Education*. Germany: Hamburg (https://hal.archives-ouvertes.fr/hal-02119848/document).

Searle, J. (1983). *Intentionality. An essay in the philosophy of mind*. Cambridge University Press

Vygotsky, L. (1978). *Mind in society: the development of higher psychological processes*. Harvard University Press.
# LANGUAGE AND MATHEMATICS: A CASE OF GEOMETRIC SHAPE IDENTIFICATION

Patrick Pieng¹, Yukari Okamoto², Lisa Weckbacher³

¹ Sacramento State University, ² University of California Santa Barbara, ³ Thousand Oaks

This study was motivated by earlier work exploring the language and mathematics hypothesis. Extending the earlier studies that focused on the relation between spoken number words and speakers' understanding of 2-digit numbers, the current study investigated potential influences of geometric shape names on American and Japanese preschoolers' shape understanding. Japanese geometric shape names include the number of angles in the shapes – a defining characteristic of a shape. We wondered, given the transparent shape names, if Japanese preschoolers are able to more quickly recognize complex geometric shapes than their US counterparts. We briefly taught the participants each shape name and its corresponding number of angles. This should benefit Japanese preschoolers who speak transparent shape names. As expected, we found that the country was the strongest predictor of children's shape knowledge. What is more interesting is that Japanese children appeared able to apply knowledge of simple angles as a defining characteristic of shapes to complex shapes. The implications of the current findings are discussed.

## BACKGROUND

The relation between mathematics language and mathematics achievement has been of interest to mathematic education researchers around the world. One area that has received much attention was an investigation into the influence of spoken number words on children's grasp of the base-10 numeration system. The initial work that sparked much interest was a comparison of Japanese and American children's understandings of two-digit numbers (Miura, 1987). In short, Miura found that Japanese-speaking first graders thought of two-digit numbers as consisting of tens and ones whereas English-speaking counterparts did so as collections of ones.

These findings were explained in terms of how Arabic numbers are spoken in English and Japanese. The Japanese spoken number words, beyond ten ("ju"), follow the rule of the traditional base-10 numeration system (Table 1). That is, a number word for any given two-digit number can be generated from a set of base-10 rules and a base sequence of spoken names. For example, 11, 12, and 13 are spoken as "ju-ichi" (i.e., ten-one), "ju-ni" (i.e., ten-two), and "ju-san" (i.e., ten-three). Compare these with how numbers are spoken in English: 11 (eleven) and 12 (twelve) are new words to young English speakers. The number, 13 (thirteen), is complex: instead of ten-three, the sound that hints 3 is pronounced first, followed by the base word of ten.

Miura's (1987) initial findings were later confirmed in a series of studies by Miura and her colleagues (e.g., Miura, Okamoto, Kim, Steere, Fayol, 1993) as well as those testing the language-mathematics hypothesis in other languages (e.g., Dowker, Bala, & Lloyd, 2008; Helmreich, Zuber, Pixner, Kaufman, Nuerk, & Moeller 2011). The consensus appears to be one that language does influence its speakers' mathematical thinking – at least in the area of two-digit numbers.
# LANGUAGE AND MATHEMATICS: A CASE OF GEOMETRIC SHAPE IDENTIFICATION

Patrick Pieng¹, Yukari Okamoto², Lisa Weckbacher³

¹ Sacramento State University, ² University of California Santa Barbara, ³ Thousand Oaks

This study was motivated by earlier work exploring the language and mathematics hypothesis. Extending the earlier studies that focused on the relation between spoken number words and speakers' understanding of 2-digit numbers, the current study investigated potential influences of geometric shape names on American and Japanese preschoolers' shape understanding. Japanese geometric shape names include the number of angles in the shapes – a defining characteristic of a shape. We wondered, given the transparent shape names, if Japanese preschoolers are able to more quickly recognize complex geometric shapes than their US counterparts. We briefly taught the participants each shape name and its corresponding number of angles. This should benefit Japanese preschoolers who speak transparent shape names. As expected, we found that the country was the strongest predictor of children's shape knowledge. What is more interesting is that Japanese children appeared able to apply knowledge of simple angles as a defining characteristic of shapes to complex shapes. The implications of the current findings are discussed.

## BACKGROUND

The relation between mathematics language and mathematics achievement has been of interest to mathematic education researchers around the world. One area that has received much attention was an investigation into the influence of spoken number words on children's grasp of the base-10 numeration system. The initial work that sparked much interest was a comparison of Japanese and American children's understandings of two-digit numbers (Miura, 1987). In short, Miura found that Japanese-speaking first graders thought of two-digit numbers as consisting of tens and ones whereas English-speaking counterparts did so as collections of ones.

These findings were explained in terms of how Arabic numbers are spoken in English and Japanese. The Japanese spoken number words, beyond ten ("ju"), follow the rule of the traditional base-10 numeration system (Table 1). That is, a number word for any given two-digit number can be generated from a set of base-10 rules and a base sequence of spoken names. For example, 11, 12, and 13 are spoken as "ju-ichi" (i.e., ten-one), "ju-ni" (i.e., ten-two), and "ju-san" (i.e., ten-three). Compare these with how numbers are spoken in English: 11 (eleven) and 12 (twelve) are new words to young English speakers. The number, 13 (thirteen), is complex: instead of ten-three, the sound that hints 3 is pronounced first, followed by the base word of ten.

Miura's (1987) initial findings were later confirmed in a series of studies by Miura and her colleagues (e.g., Miura, Okamoto, Kim, Steere, Fayol, 1993) as well as those testing the language-mathematics hypothesis in other languages (e.g., Dowker, Bala, & Lloyd, 2008; Helmreich, Zuber, Pixner, Kaufman, Nuerk, & Moeller 2011). The consensus appears to be one that language does influence its speakers' mathematical thinking – at least in the area of two-digit numbers.
<table><caption>Table 1: Comparison of English and Japanese spoken words</caption><thead><tr><th>Language</th><th colspan="10">Spoken Number Words</th></tr></thead><tbody><tr><td>English 1-10</td><td>one</td><td>two</td><td>three</td><td>four</td><td>five</td><td>six</td><td>even</td><td>eight</td><td>nine</td><td>ten</td></tr><tr><td>Japanese 1-10</td><td>ichi</td><td>ni</td><td>san</td><td>shi</td><td>go</td><td>roku</td><td>shichi</td><td>hachi</td><td>ku</td><td>ju</td></tr><tr><td>English 11-19</td><td>eleven</td><td>twelve</td><td>thirteen</td><td>fourteen</td><td>fifteen</td><td>sixteen</td><td>seventeen</td><td>eighteen</td><td>nineteen</td><td></td></tr><tr><td>Japanese 11-19</td><td>ju-ichi</td><td>ju-ni</td><td>ju-san</td><td>ju-shi</td><td>ju-go</td><td>ju-roku</td><td>ju-shichi</td><td>ju-hachi</td><td>ju-ku</td><td></td></tr></tbody></table>

The current study is an attempt to extend this language-mathematics hypothesis by examining children's early conception of geometric shape. Do shape names influence children's recognition of geometric shapes? This is the overarching question explored in this study. We want to emphasize that this is our first attempt at this question and to the best of our knowledge, no other studies exist at the time of writing this proposal.

Japanese shape names directly correspond to the exact number of angles each shape contains, with the exception of circle and rectangle. This naming system applies to all other two-dimensional (2D) shapes. Thus when one becomes aware of this feature, it is easy to name any 2D shape by counting the number of angles. As the literature shows (see the next section), young children are developing shape names and knowledge during preschool years. Beyond circles and squares, however, their understanding has yet to develop prior to entering kindergarten.

The question of interest is: would Japanese children speaking shape names that contain the number of angles – even if they are not familiar with complex shapes beyond rectangle – benefit from a brief instruction on the role of angles in defining shapes? Of course, angles are not the only defining feature of geometric shape. However, compared to English speaking children who need to memorize complex shape names that do not directly communicate shapes' defining characteristics, it seems likely that Japanese children would be the beneficiary of such language characteristics. Thus the specific question asked in this study is: With a brief exposure to the shape defining feature of angles, would Japanese children be able to apply this knowledge of simple shapes to complex shapes more so than American children would? In answering this question, we included age, gender, counting skills, and spatial thinking to determine the factor(s) that would most likely to explain the outcome.

<table><caption>Table 2: Geometric shape names by languages used in this study</caption><thead><tr><th>Language</th><th colspan="3">Shape names</th></tr></thead><tbody><tr><td>English</td><td>triangle</td><td>square rectangle</td><td>rectangle</td></tr><tr><td>Japanese</td><td>three-angle shape</td><td>canonical four-angle shape</td><td>long-side shape</td></tr><tr><td>English</td><td>pentagon</td><td>hexagon</td><td>octagon</td></tr><tr><td>Japanese</td><td>five-angle shape</td><td>six-angle shape</td><td>eight-angle shape</td></tr></tbody></table>

## EARLY KNOWLEDGE OF GEOMETRIC SHAPES

Knowledge of geometric shapes takes root early in life. From infancy, children are introduced to basic geometric concepts through manipulative toys, books, and even forms of educational and interactive
<table><caption>Table 1: Comparison of English and Japanese spoken words</caption><thead><tr><th>Language</th><th colspan="10">Spoken Number Words</th></tr></thead><tbody><tr><td>English 1-10</td><td>one</td><td>two</td><td>three</td><td>four</td><td>five</td><td>six</td><td>even</td><td>eight</td><td>nine</td><td>ten</td></tr><tr><td>Japanese 1-10</td><td>ichi</td><td>ni</td><td>san</td><td>shi</td><td>go</td><td>roku</td><td>shichi</td><td>hachi</td><td>ku</td><td>ju</td></tr><tr><td>English 11-19</td><td>eleven</td><td>twelve</td><td>thirteen</td><td>fourteen</td><td>fifteen</td><td>sixteen</td><td>seventeen</td><td>eighteen</td><td>nineteen</td><td></td></tr><tr><td>Japanese 11-19</td><td>ju-ichi</td><td>ju-ni</td><td>ju-san</td><td>ju-shi</td><td>ju-go</td><td>ju-roku</td><td>ju-shichi</td><td>ju-hachi</td><td>ju-ku</td><td></td></tr></tbody></table>

The current study is an attempt to extend this language-mathematics hypothesis by examining children's early conception of geometric shape. Do shape names influence children's recognition of geometric shapes? This is the overarching question explored in this study. We want to emphasize that this is our first attempt at this question and to the best of our knowledge, no other studies exist at the time of writing this proposal.

Japanese shape names directly correspond to the exact number of angles each shape contains, with the exception of circle and rectangle. This naming system applies to all other two-dimensional (2D) shapes. Thus when one becomes aware of this feature, it is easy to name any 2D shape by counting the number of angles. As the literature shows (see the next section), young children are developing shape names and knowledge during preschool years. Beyond circles and squares, however, their understanding has yet to develop prior to entering kindergarten.

The question of interest is: would Japanese children speaking shape names that contain the number of angles – even if they are not familiar with complex shapes beyond rectangle – benefit from a brief instruction on the role of angles in defining shapes? Of course, angles are not the only defining feature of geometric shape. However, compared to English speaking children who need to memorize complex shape names that do not directly communicate shapes' defining characteristics, it seems likely that Japanese children would be the beneficiary of such language characteristics. Thus the specific question asked in this study is: With a brief exposure to the shape defining feature of angles, would Japanese children be able to apply this knowledge of simple shapes to complex shapes more so than American children would? In answering this question, we included age, gender, counting skills, and spatial thinking to determine the factor(s) that would most likely to explain the outcome.

<table><caption>Table 2: Geometric shape names by languages used in this study</caption><thead><tr><th>Language</th><th colspan="3">Shape names</th></tr></thead><tbody><tr><td>English</td><td>triangle</td><td>square rectangle</td><td>rectangle</td></tr><tr><td>Japanese</td><td>three-angle shape</td><td>canonical four-angle shape</td><td>long-side shape</td></tr><tr><td>English</td><td>pentagon</td><td>hexagon</td><td>octagon</td></tr><tr><td>Japanese</td><td>five-angle shape</td><td>six-angle shape</td><td>eight-angle shape</td></tr></tbody></table>

## EARLY KNOWLEDGE OF GEOMETRIC SHAPES

Knowledge of geometric shapes takes root early in life. From infancy, children are introduced to basic geometric concepts through manipulative toys, books, and even forms of educational and interactive
media (Hannibal, 1999; Resnick et al., 2016). Both parents and preschool educators also play a role in terms of exposing children to types of shapes or comparing shapes in different categories, though the extent or the language that they may use in talking about shapes is not clear (Verdine et al., 2016) and perhaps not enough (Rudd et al., 2008). Environmental input aside, other evidence suggests that shape discrimination might even be innate to some extent considering newborns' capability to discriminate between classes of shapes, and forms of shapes within the same class (Quinn et al., 2010). Collectively, these factors might help explain why young children are equipped with some degree of geometric knowledge by the time they enter preschool, including the ability to label (Heibeck & Markman, 1987), to distinguish, and to recognize simple properties of familiar shapes (Clements et al., 1999). However, they tend to vary widely with respect to their understanding of shape concepts upon entering kindergarten well into elementary school (Satlow & Newcombe, 1998). This variation may in some way account for the underperformance of US students' knowledge of basic geometric concepts compared to students from other nations, such as China (e.g., Starkey et al., 1999; Zhou et al., 2005) and Japan (e.g., Gonzales et al., 2008; Stigler et al. 1990).

The developmental progression of shape knowledge neither reveal a consistent pattern relative to the identification of basic shapes nor the ability to distinguish between canonical versus noncanonical representations. According to Keil (1989), developing knowledge of an individual concept is a function of its complexity, or more specifically, its defining properties or characteristics. Accordingly, when concepts differ along one dimension, they tend to show the same developmental progression. In contrast, when concepts differ along multiple dimensions, development is more likely varied. Geometric shapes are good exemplars of concepts that can be considered simple or complex in terms of their defining properties, and therefore, the developing knowledge of such concepts should follow suit.

With the exception of circles, defining properties of shapes include closure, straight sides, and a particular number of sides and angles. Other non-defining properties can vary adding to the perceptual complexity of a shape. For example, the defining properties of a rectangle refer to the number of lines and parallelism; orientation and length of sides are non-defining properties that differ across representations of rectangles. In a similar vein, a triangle's orientation as well as the degree of its angles may also differ, as is the case with isosceles and scalene triangles. Consistent with Keil's (1989) line of reasoning, young children tend to demonstrate knowledge of circles, and even squares with relative success, as early as age 2 (Zambrzycka et al., 2017). This suggests that from an early age, children are quite capable of identifying simpler shapes based on defining properties that translate to "stronger and fewer prototypes" (Clements et al. 1999) with a lower number of non-defining attributes (Aslan & Arnas, 2007).

Findings pertaining to rectangles and triangles suggest otherwise (e.g., Hannibal, 1999; Satlow & Newcombe, 1998; Zambrzycka et al., 2017), particularly when non-defining properties (e.g., orientation or angularity) seem to interfere with an accurate identification of a shape. Children are more apt to experience failure when they do not pay attention to the defining properties of a shape that determine shape classification. Classification tends to be based on select visual information, or falling back on more familiar, spontaneous concepts (Hannibal, 1999). Perhaps perfect representations of shapes often depicted in books and puzzles may skew their interpretation of noncanonical representations (e.g., a scalene triangle is not a triangle because it is too "crooked").
media (Hannibal, 1999; Resnick et al., 2016). Both parents and preschool educators also play a role in terms of exposing children to types of shapes or comparing shapes in different categories, though the extent or the language that they may use in talking about shapes is not clear (Verdine et al., 2016) and perhaps not enough (Rudd et al., 2008). Environmental input aside, other evidence suggests that shape discrimination might even be innate to some extent considering newborns' capability to discriminate between classes of shapes, and forms of shapes within the same class (Quinn et al., 2010). Collectively, these factors might help explain why young children are equipped with some degree of geometric knowledge by the time they enter preschool, including the ability to label (Heibeck & Markman, 1987), to distinguish, and to recognize simple properties of familiar shapes (Clements et al., 1999). However, they tend to vary widely with respect to their understanding of shape concepts upon entering kindergarten well into elementary school (Satlow & Newcombe, 1998). This variation may in some way account for the underperformance of US students' knowledge of basic geometric concepts compared to students from other nations, such as China (e.g., Starkey et al., 1999; Zhou et al., 2005) and Japan (e.g., Gonzales et al., 2008; Stigler et al. 1990).

The developmental progression of shape knowledge neither reveal a consistent pattern relative to the identification of basic shapes nor the ability to distinguish between canonical versus noncanonical representations. According to Keil (1989), developing knowledge of an individual concept is a function of its complexity, or more specifically, its defining properties or characteristics. Accordingly, when concepts differ along one dimension, they tend to show the same developmental progression. In contrast, when concepts differ along multiple dimensions, development is more likely varied. Geometric shapes are good exemplars of concepts that can be considered simple or complex in terms of their defining properties, and therefore, the developing knowledge of such concepts should follow suit.

With the exception of circles, defining properties of shapes include closure, straight sides, and a particular number of sides and angles. Other non-defining properties can vary adding to the perceptual complexity of a shape. For example, the defining properties of a rectangle refer to the number of lines and parallelism; orientation and length of sides are non-defining properties that differ across representations of rectangles. In a similar vein, a triangle's orientation as well as the degree of its angles may also differ, as is the case with isosceles and scalene triangles. Consistent with Keil's (1989) line of reasoning, young children tend to demonstrate knowledge of circles, and even squares with relative success, as early as age 2 (Zambrzycka et al., 2017). This suggests that from an early age, children are quite capable of identifying simpler shapes based on defining properties that translate to "stronger and fewer prototypes" (Clements et al. 1999) with a lower number of non-defining attributes (Aslan & Arnas, 2007).

Findings pertaining to rectangles and triangles suggest otherwise (e.g., Hannibal, 1999; Satlow & Newcombe, 1998; Zambrzycka et al., 2017), particularly when non-defining properties (e.g., orientation or angularity) seem to interfere with an accurate identification of a shape. Children are more apt to experience failure when they do not pay attention to the defining properties of a shape that determine shape classification. Classification tends to be based on select visual information, or falling back on more familiar, spontaneous concepts (Hannibal, 1999). Perhaps perfect representations of shapes often depicted in books and puzzles may skew their interpretation of noncanonical representations (e.g., a scalene triangle is not a triangle because it is too "crooked").
With age, visual information continues to influence classification decisions but to a lesser degree for some children compared to others. This developmental shift partly explains why older children's reliance on defining properties versus visual information enables more correct classifications (Aslan & Arnas, 2007). To reiterate, the current study examined potential influences of shape names in English vs. Japanese on preschoolers' developing understanding of geometric shapes.

# BACKGROUND

## Participants

The participants were 46 Japanese children (24 girls, 22 boys, $M_{age} = 59.46$ months, $SD = 10.48$ months, age range: 42-75 months) and 38 US children (15 girls, 23 boys, $M_{age} = 50.18$ months, $SD = 12.37$ months, age range: 27-74 months). The Japanese children were recruited from one nursery school located in Yokohama, Kanagawa of Japan (the prefecture adjacent to Tokyo) and the US children came from three different preschools located in a metropolitan city in California, United States.

## Tasks and procedures

Children were seen individually in each nation by native speakers. They were first asked to count to 10 cubes. The highest number reached without any assistance was recorded. They were then given a brief lesson on geometric shapes. In the first part of the lesson, children saw three line drawings of an angle (45-degree acute angle, 90-degree right angle, and 135-degree obtuse angle) and were told, "This is an angle," as the interviewer pointed to the angle. In the second part of the lesson, they saw line drawings of triangle, rectangle, pentagon, hexagon, heptagon, and octagon and were told, for example, "This is a triangle. It has three angles." Following the brief lesson, children completed two tasks: spatial puzzle and shape identification tasks. For the spatial puzzle task, children solved three puzzles consisting of four, five, and six pieces of familiar objects (i.e., a cat, a dog, and a house). For example, they saw an outline of a cat and given pieces to fill in it (Figure 1). Children received 1 point for each correct solution for a maximum total of 3.

<figure><img src="image_89.png" /><figcaption>Figure 1: Outline of a cat used in the puzzle task</figcaption></figure>

For the Shape identification task, children saw line drawings of six shapes (i.e., triangle, rectangle, pentagon, hexagon, heptagon, and octagon) one shape type at a time. For each shape type (e.g., triangle), they saw four different drawings of triangles (two valid triangles and one non-enclosed and one missing-angel shapes that look like triangles - see Figure 2). Children were asked if each of the four shapes was a triangle and why or why not they thought so. Children received 1 point for responding "yes" to each of the valid forms and "no" to each of the non-valid forms for a maximum possible of 24 points.
With age, visual information continues to influence classification decisions but to a lesser degree for some children compared to others. This developmental shift partly explains why older children's reliance on defining properties versus visual information enables more correct classifications (Aslan & Arnas, 2007). To reiterate, the current study examined potential influences of shape names in English vs. Japanese on preschoolers' developing understanding of geometric shapes.

# BACKGROUND

## Participants

The participants were 46 Japanese children (24 girls, 22 boys, $M_{age} = 59.46$ months, $SD = 10.48$ months, age range: 42-75 months) and 38 US children (15 girls, 23 boys, $M_{age} = 50.18$ months, $SD = 12.37$ months, age range: 27-74 months). The Japanese children were recruited from one nursery school located in Yokohama, Kanagawa of Japan (the prefecture adjacent to Tokyo) and the US children came from three different preschools located in a metropolitan city in California, United States.

## Tasks and procedures

Children were seen individually in each nation by native speakers. They were first asked to count to 10 cubes. The highest number reached without any assistance was recorded. They were then given a brief lesson on geometric shapes. In the first part of the lesson, children saw three line drawings of an angle (45-degree acute angle, 90-degree right angle, and 135-degree obtuse angle) and were told, "This is an angle," as the interviewer pointed to the angle. In the second part of the lesson, they saw line drawings of triangle, rectangle, pentagon, hexagon, heptagon, and octagon and were told, for example, "This is a triangle. It has three angles." Following the brief lesson, children completed two tasks: spatial puzzle and shape identification tasks. For the spatial puzzle task, children solved three puzzles consisting of four, five, and six pieces of familiar objects (i.e., a cat, a dog, and a house). For example, they saw an outline of a cat and given pieces to fill in it (Figure 1). Children received 1 point for each correct solution for a maximum total of 3.

<figure><img src="image_95.png" /><figcaption>Figure 1: Outline of a cat used in the puzzle task</figcaption></figure>

For the Shape identification task, children saw line drawings of six shapes (i.e., triangle, rectangle, pentagon, hexagon, heptagon, and octagon) one shape type at a time. For each shape type (e.g., triangle), they saw four different drawings of triangles (two valid triangles and one non-enclosed and one missing-angel shapes that look like triangles - see Figure 2). Children were asked if each of the four shapes was a triangle and why or why not they thought so. Children received 1 point for responding "yes" to each of the valid forms and "no" to each of the non-valid forms for a maximum possible of 24 points.
<figure><img src="image_90.png" /><figcaption>Figure 2: Shape identification task example: triangle</figcaption></figure>

## RESULTS

The primary interest of this study was to see if children speaking English or Japanese respond differently to a brief lesson that emphasized one defining feature of geometric shapes. Because Japanese shape names include the number of angles, a brief exposure to pay attention to the number angles might benefit Japanese children more so than their US counterparts who do not have language support.

Descriptive statistics are presented in Table 3. A quick observation reveals that the two groups' mean counting scores were relatively similar. However, there seemed to be differences between the groups in the spatial puzzle and shape identification performance.

<table><caption>Table 3: Descriptive statistics of variables by country</caption><thead><tr><th rowspan="2">Variable (Possible Range)</th><th colspan="4">Japan (n = 46)</th><th colspan="4">US (n = 38)</th></tr><tr><th>M</th><th>SD</th><th>Min</th><th>Max</th><th>M</th><th>SD</th><th>Min</th><th>Max</th></tr></thead><tbody><tr><td>Counting Task (0-10)</td><td>9.09</td><td>2.84</td><td>0</td><td>10</td><td>8.97</td><td>2.06</td><td>0</td><td>10</td></tr><tr><td>Spatial Puzzle Task (0-3)</td><td>2.43</td><td>.83</td><td>0</td><td>3</td><td>1.56</td><td>1.01</td><td>0</td><td>3</td></tr><tr><td>Total Shape Score (0-24)</td><td>18.30</td><td>4.12</td><td>4</td><td>24</td><td>9.34</td><td>6.30</td><td>1</td><td>24</td></tr></tbody></table>

We then carried out an analysis of variance (Table 4). The dependent variable was the shape identification task and the primary independent variable was country (language groups). We also included as gender to see if any gender differences could be identified. Furthermore, we included age, counting, and spatial puzzle performance as covariates.

<table><caption>Table 4: Analysis of variance for shape knowledge by country and gender with age, counting to 10, and spatial puzzle as covariate</caption><thead><tr><th>Source</th><th>SS</th><th>df</th><th>MS</th><th>F</th><th>Sig.</th></tr></thead><tbody><tr><td>Corrected model</td><td>1854.90</td><td>6</td><td>309.15</td><td>11.62</td><td>.000</td></tr><tr><td>Intercept</td><td>114.95</td><td>1</td><td>114.95</td><td>4.32</td><td>.041</td></tr><tr><td>Country</td><td>908.14</td><td>1</td><td>908.14</td><td>34.13</td><td>.000</td></tr><tr><td>Gender</td><td>10.50</td><td>1</td><td>10.50</td><td>.395</td><td>.532</td></tr><tr><td>Age (mon)</td><td>126.12</td><td>1</td><td>126.12</td><td>4.73</td><td>.033</td></tr><tr><td>Counting10</td><td>.09</td><td>1</td><td>.09</td><td>.032</td><td>.953</td></tr><tr><td>Puzzle</td><td>2.50</td><td>1</td><td>2.50</td><td>.94</td><td>.760</td></tr><tr><td>Gender*Country</td><td>11.35</td><td>1</td><td>11.35</td><td>.43</td><td>.519</td></tr><tr><td>Error</td><td>2048.85</td><td>77</td><td>26.61</td><td></td><td></td></tr><tr><td>Total</td><td>20961.00</td><td>84</td><td></td><td></td><td></td></tr><tr><td>Corrected total</td><td>3903.75</td><td>83</td><td></td><td></td><td></td></tr></tbody></table>
<figure><img src="image_96.png" /><figcaption>Figure 2: Shape identification task example: triangle</figcaption></figure>

## RESULTS

The primary interest of this study was to see if children speaking English or Japanese respond differently to a brief lesson that emphasized one defining feature of geometric shapes. Because Japanese shape names include the number of angles, a brief exposure to pay attention to the number angles might benefit Japanese children more so than their US counterparts who do not have language support.

Descriptive statistics are presented in Table 3. A quick observation reveals that the two groups' mean counting scores were relatively similar. However, there seemed to be differences between the groups in the spatial puzzle and shape identification performance.

<table><caption>Table 3: Descriptive statistics of variables by country</caption><thead><tr><th rowspan="2">Variable (Possible Range)</th><th colspan="4">Japan (n = 46)</th><th colspan="4">US (n = 38)</th></tr><tr><th>M</th><th>SD</th><th>Min</th><th>Max</th><th>M</th><th>SD</th><th>Min</th><th>Max</th></tr></thead><tbody><tr><td>Counting Task (0-10)</td><td>9.09</td><td>2.84</td><td>0</td><td>10</td><td>8.97</td><td>2.06</td><td>0</td><td>10</td></tr><tr><td>Spatial Puzzle Task (0-3)</td><td>2.43</td><td>.83</td><td>0</td><td>3</td><td>1.56</td><td>1.01</td><td>0</td><td>3</td></tr><tr><td>Total Shape Score (0-24)</td><td>18.30</td><td>4.12</td><td>4</td><td>24</td><td>9.34</td><td>6.30</td><td>1</td><td>24</td></tr></tbody></table>

We then carried out an analysis of variance (Table 4). The dependent variable was the shape identification task and the primary independent variable was country (language groups). We also included as gender to see if any gender differences could be identified. Furthermore, we included age, counting, and spatial puzzle performance as covariates.

<table><caption>Table 4: Analysis of variance for shape knowledge by country and gender with age, counting to 10, and spatial puzzle as covariate</caption><thead><tr><th>Source</th><th>SS</th><th>df</th><th>MS</th><th>F</th><th>Sig.</th></tr></thead><tbody><tr><td>Corrected model</td><td>1854.90</td><td>6</td><td>309.15</td><td>11.62</td><td>.000</td></tr><tr><td>Intercept</td><td>114.95</td><td>1</td><td>114.95</td><td>4.32</td><td>.041</td></tr><tr><td>Country</td><td>908.14</td><td>1</td><td>908.14</td><td>34.13</td><td>.000</td></tr><tr><td>Gender</td><td>10.50</td><td>1</td><td>10.50</td><td>.395</td><td>.532</td></tr><tr><td>Age (mon)</td><td>126.12</td><td>1</td><td>126.12</td><td>4.73</td><td>.033</td></tr><tr><td>Counting10</td><td>.09</td><td>1</td><td>.09</td><td>.032</td><td>.953</td></tr><tr><td>Puzzle</td><td>2.50</td><td>1</td><td>2.50</td><td>.94</td><td>.760</td></tr><tr><td>Gender*Country</td><td>11.35</td><td>1</td><td>11.35</td><td>.43</td><td>.519</td></tr><tr><td>Error</td><td>2048.85</td><td>77</td><td>26.61</td><td></td><td></td></tr><tr><td>Total</td><td>20961.00</td><td>84</td><td></td><td></td><td></td></tr><tr><td>Corrected total</td><td>3903.75</td><td>83</td><td></td><td></td><td></td></tr></tbody></table>
As expected, country was the strongest predictor of the shape identification tasks, *F* (1, 77) = 34.13, *p* = .000. In addition, age was a significant variable, *F* (1, 77) = 4.74, *p* = .033. No other variables, including gender, were significant.

If the language-mathematics hypothesis is at work, we should see that Japanese children are able to accurately identify two canonical shapes as valid and one missing-angle shape to be non-valid. We carried out a MANOVA to see where children show strengths and weaknesses. For this analysis, we summed across all six canonical shapes, all six noncanonical shapes, all six missing-angle shapes, and all six non-enclosed shapes – all which were the dependent variables. The independent variable was country with age as a covariate (we did not include other variable as no other variables were significant in the previous analysis).

The results indicated that country was significant *F* (4, 78) = 14.18, *p* = .000: so was age *F* (4, 78) = 3.83, *p* = .007. Univariate analyses for country revealed all but noncanonical shapes turned out to be significant: canonical, *F* (1, 81) = 11.84, *p* = .001; noncanonical, *F* (1, 81) = .74, *p* = .391; missing angle, *F* (1, 81) = 58.20, *p* = .000; and non-enclosed, *F* (1, 81) = 47.24, *p* = .000. Similar trends were found for the age variable.

<table><thead><tr><th rowspan="2"></th><th colspan="2">Japan (n = 46)</th><th colspan="2">US (n = 38)</th></tr><tr><th>M</th><th>SD</th><th>M</th><th>SD</th></tr></thead><tbody><tr><td>Canonical</td><td>5.54</td><td>0.94</td><td>4.11</td><td>1.78</td></tr><tr><td>Noncanonical</td><td>2.30</td><td>1.98</td><td>2.00</td><td>1.68</td></tr><tr><td>Missing angle</td><td>5.32</td><td>1.49</td><td>1.66</td><td>2.21</td></tr><tr><td>Non-enclosed</td><td>5.13</td><td>1.73</td><td>1.58</td><td>1.68</td></tr></tbody></table>

Note: Possible range for scores is 0-6.

<caption>Table 5: Comparison between Japan and US</caption>

## DISCUSSION AND IMPLICATIONS

This study was motivated by earlier studies examining the language and mathematics hypothesis in the area of two-digit numbers. Earlier studies found transparency of Japanese number words that correspond to the rules of base-10 influenced first-graders' developing concepts of 2-digit numbers. The current study extended this language and mathematics hypothesis by examining a different area of mathematics, namely geometric shape understanding. Japanese geometric shape names incorporate defining characteristics of shapes – angles. Thus it would make sense to predict that Japanese preschoolers, once taught to realize how shape names are constructed, should be able to apply this knowledge even after a brief instruction on it in comparison to English speaking counterparts.

As expected, we found that the country (language group) was the strongest predictor of children's shape identification. Age was also a significant variable. The current results are in-line with the language and mathematics hypothesis. The only puzzling finding is the performance of the Japanese children on noncanonical shapes. If the language-mathematics hypothesis is at work, they should be able to identify noncanonical shapes as valid (though they were correct more than half the times). One possibility is the effect of age. Our analyses revealed that age was a significant covariate. Our participants, however, varied in age. It would be ideal to use methods such as stratified sampling to ensure that data can be analyzed by age groups.
As expected, country was the strongest predictor of the shape identification tasks, *F* (1, 77) = 34.13, *p* = .000. In addition, age was a significant variable, *F* (1, 77) = 4.74, *p* = .033. No other variables, including gender, were significant.

If the language-mathematics hypothesis is at work, we should see that Japanese children are able to accurately identify two canonical shapes as valid and one missing-angle shape to be non-valid. We carried out a MANOVA to see where children show strengths and weaknesses. For this analysis, we summed across all six canonical shapes, all six noncanonical shapes, all six missing-angle shapes, and all six non-enclosed shapes – all which were the dependent variables. The independent variable was country with age as a covariate (we did not include other variable as no other variables were significant in the previous analysis).

The results indicated that country was significant *F* (4, 78) = 14.18, *p* = .000: so was age *F* (4, 78) = 3.83, *p* = .007. Univariate analyses for country revealed all but noncanonical shapes turned out to be significant: canonical, *F* (1, 81) = 11.84, *p* = .001; noncanonical, *F* (1, 81) = .74, *p* = .391; missing angle, *F* (1, 81) = 58.20, *p* = .000; and non-enclosed, *F* (1, 81) = 47.24, *p* = .000. Similar trends were found for the age variable.

<table><thead><tr><th rowspan="2"></th><th colspan="2">Japan (n = 46)</th><th colspan="2">US (n = 38)</th></tr><tr><th>M</th><th>SD</th><th>M</th><th>SD</th></tr></thead><tbody><tr><td>Canonical</td><td>5.54</td><td>0.94</td><td>4.11</td><td>1.78</td></tr><tr><td>Noncanonical</td><td>2.30</td><td>1.98</td><td>2.00</td><td>1.68</td></tr><tr><td>Missing angle</td><td>5.32</td><td>1.49</td><td>1.66</td><td>2.21</td></tr><tr><td>Non-enclosed</td><td>5.13</td><td>1.73</td><td>1.58</td><td>1.68</td></tr></tbody></table>

*Note*: Possible range for scores is 0-6.

<caption>Table 5: Comparison between Japan and US</caption>

## DISCUSSION AND IMPLICATIONS

This study was motivated by earlier studies examining the language and mathematics hypothesis in the area of two-digit numbers. Earlier studies found transparency of Japanese number words that correspond to the rules of base-10 influenced first-graders' developing concepts of 2-digit numbers. The current study extended this language and mathematics hypothesis by examining a different area of mathematics, namely geometric shape understanding. Japanese geometric shape names incorporate defining characteristics of shapes – angles. Thus it would make sense to predict that Japanese preschoolers, once taught to realize how shape names are constructed, should be able to apply this knowledge even after a brief instruction on it in comparison to English speaking counterparts.

As expected, we found that the country (language group) was the strongest predictor of children's shape identification. Age was also a significant variable. The current results are in-line with the language and mathematics hypothesis. The only puzzling finding is the performance of the Japanese children on noncanonical shapes. If the language-mathematics hypothesis is at work, they should be able to identify noncanonical shapes as valid (though they were correct more than half the times). One possibility is the effect of age. Our analyses revealed that age was a significant covariate. Our participants, however, varied in age. It would be ideal to use methods such as stratified sampling to ensure that data can be analyzed by age groups.
Given this and other limitations, what do the current findings suggest? Theoretically speaking, the language and mathematics hypothesis appears to extend to young children's shape knowledge development. This contributes to the debate that specific aspects of language influence the corresponding cognitive mind.

In terms of applications, preschoolers should be encouraged to explore shapes that go beyond triangles and squares, and discuss defining features of geometric shapes. In English-speaking nations, it might be useful to introduce a triangle as a three-angle shape, etc. at early stages of learning. Once this feature is understood (along with other features), English terms might be introduced. As we delve further into our data, we hope to have clearer directions for future research and teaching.

## Acknowledgements

Please This research was supported by the Sacramento State Research and Creative Activity Faculty Awards Program and the University of California Santa Barbara Faculty Research Grants Program.

## References

Aslan, D., & Arnas, Y. A. (2007). Three-to-six-year-old children's recognition of geometric shapes. *International Journal of Early Years Education*, 15(1), 83-104.
https://doi.org/10.1080/09669760601106646

Clements, D. H., Swaminathan, S., Hannibal, M. A., & Sarama, J. (1999). Young children's concepts of geometric shapes. *Journal for Research in Mathematics Education*, 30, 192-212.
https://doi.org/10.2307/749610

Dowker, A., Bala, S., & Lloyd, D. (2008). Linguistics influence of mathematical development: How important is the transparency of the counting system? *Philosophical Psychology*, 21, 523-538.

Hannibal, M. A. (1999). Young children's developing understanding of geometric shapes. *Teaching Children Mathematics*, 5(6), 353-357.
https://doi.org/10.5951/TCM.5.6.0353

Heibeck, T. H., & Markman, E. M. (1987). Word learning in children: An examination of fast mapping. *Child Development*, 58(4), 1021-1034.
https://doi.org/10.2307/1130543

Helmreich, I., Zuber, J., Pixner, S., Kaufmann, L., Nuerk, H. C., & Moeller, K. (2011). Language effects on children's mental number line: How cross cultural differences in number word systems affect spatial mappings of numbers in a nonverbal task. *Journal of Cross Cultural Psychology*, 42, 598-613.

Keil, F. C. (1989). Concepts, kinds, and cognitive development. The MIT Press.

Miura, I. T. (1987). Mathematics achievement as a function of language. *Journal of Educational Psychology*, 79, 79-82.

Miura, I. T., Okamoto, Y., Kim, C. C., Steere, M., & Fayol, M. (1993). First graders' cognitive representation of number and understanding of place value: Cross-national comparisons--France, Japan, Korea, Sweden, and the United States. *Journal of Educational Psychology*, 85(1), 24-30.

Quinn, P. C., Slater, A. M., Brown, E., & Hayes, R. A. (2010). Developmental change in form categorization in early infancy. *British Journal of Developmental Psychology*, 19(2), 207-218.
https://doi.org/10.1348/026151001166038

Resnick, I., Verdine, B. N., Golinkoff, R., & Hirsh-Pasek, K. (2016). Geometric toys in the attic: A corpus analysis of early exposure to geometric shapes. *Early Childhood Research Quarterly*, 36, 358-365.
https://doi.org/10.1016/j.ecresq.2016.01.007

Rudd, L. C., Lambert, M., Satterwhite, M., & Zaier, A. (2008). Mathematical language in early childhood settings: What really counts? *Early Childhood Education Journal*, 36, 75-80.
https://doi.org/10.1007/s10643-008-0246-3
Given this and other limitations, what do the current findings suggest? Theoretically speaking, the language and mathematics hypothesis appears to extend to young children's shape knowledge development. This contributes to the debate that specific aspects of language influence the corresponding cognitive mind.

In terms of applications, preschoolers should be encouraged to explore shapes that go beyond triangles and squares, and discuss defining features of geometric shapes. In English-speaking nations, it might be useful to introduce a triangle as a three-angle shape, etc. at early stages of learning. Once this feature is understood (along with other features), English terms might be introduced. As we delve further into our data, we hope to have clearer directions for future research and teaching.

## Acknowledgements

Please This research was supported by the Sacramento State Research and Creative Activity Faculty Awards Program and the University of California Santa Barbara Faculty Research Grants Program.

## References

Aslan, D., & Arnas, Y. A. (2007). Three-to-six-year-old children's recognition of geometric shapes. *International Journal of Early Years Education, 15*(1), 83-104.
https://doi.org/10.1080/09669760601106646

Clements, D. H., Swaminathan, S., Hannibal, M. A., & Sarama, J. (1999). Young children's concepts of geometric shapes. *Journal for Research in Mathematics Education, 30*, 192-212.
https://doi.org/10.2307/749610

Dowker, A., Bala, S., & Lloyd, D. (2008). Linguistics influence of mathematical development: How important is the transparency of the counting system? *Philosophical Psychology, 21*, 523-538.

Hannibal, M. A. (1999). Young children's developing understanding of geometric shapes. *Teaching Children Mathematics, 5*(6), 353-357. https://doi.org/10.5951/TCM.5.6.0353

Heibeck, T. H., & Markman, E. M. (1987). Word learning in children: An examination of fast mapping. *Child Development, 58*(4), 1021-1034. https://doi.org/10.2307/1130543

Helmreich, I., Zuber, J., Pixner, S., Kaufmann, L., Nuerk, H. C., & Moeller, K. (2011). Language effects on children's mental number line: How cross cultural differences in number word systems affect spatial mappings of numbers in a nonverbal task. *Journal of Cross Cultural Psychology, 42*, 598-613.

Keil, F. C. (1989). Concepts, kinds, and cognitive development. The MIT Press.

Miura, I. T. (1987). Mathematics achievement as a function of language. *Journal of Educational Psychology, 79*, 79-82.

Miura, I. T., Okamoto, Y., Kim, C. C., Steere, M., & Fayol, M. (1993). First graders' cognitive representation of number and understanding of place value: Cross-national comparisons--France, Japan, Korea, Sweden, and the United States. *Journal of Educational Psychology, 85*(1), 24-30.

Quinn, P. C., Slater, A. M., Brown, E., & Hayes, R. A. (2010). Developmental change in form categorization in early infancy. *British Journal of Developmental Psychology, 19*(2), 207-218.
https://doi.org/10.1348/026151001166038

Resnick, I., Verdine, B. N., Golinkoff, R., & Hirsh-Pasek, K. (2016). Geometric toys in the attic: A corpus analysis of early exposure to geometric shapes. *Early Childhood Research Quarterly, 36*, 358-365.
https://doi.org/10.1016/j.ecresq.2016.01.007

Rudd, L. C., Lambert, M., Satterwhite, M., & Zaier, A. (2008). Mathematical language in early childhood settings: What really counts? *Early Childhood Education Journal, 36*, 75-80.
https://doi.org/10.1007/s10643-008-0246-3
Satlow, E., & Newcombe, N. (1998). When is a triangle not a triangle? Young children's conceptions of geometric shapes. *Cognitive Development, 13*, 547-559. https://doi.org/10.1016/S0885-2014(98)90006-5

Starkey, P., Klein, A., Chang, I., Qi, D., Lijuan, P., & Yang, Z. (1999). *Environmental supports for young children's mathematical development in China and the United States*. Paper presented at the meeting of the Society for Research in Child Development, Albuquerque, New Mexico.

Stigler, J. W., Lee, S.-Y., & Stevenson, H.W. (1990). *Mathematical knowledge of Japanese, Chinese, and American elementary school children*. National Council of Teachers of Mathematics, Reston, VA.

Verdine, B. N., Lucca, K. R., Golinkoff, R. M., Hirsh-Pasek, K. & Newcombe, N. (2016). The shape of things: The origin of young children's knowledge of the names and properties of geometric forms. *Journal of Cognition and Development, 17*(1), 142-161. https://doi.org/10.1080/15248372.2015.1016610

Zambrzycka, J., Kotsopoulos, D., & Lee, J. (2017). *Infant Behavior & Development, 46*, 144-157. https://dx.doi.org/10.1016/j.infbeh.2016.12.002

Zhou, Z., Peverly, S. T., & Lin, J. (2005). Understanding early mathematical competencies in American and Chinese children. *School Psychology International, 26*(4), 413-427. https://doi.org/10.1177/0143034305059018
Satlow, E., & Newcombe, N. (1998). When is a triangle not a triangle? Young children's conceptions of geometric shapes. *Cognitive Development, 13*, 547-559. https://doi.org/10.1016/S0885-2014(98)90006-5

Starkey, P., Klein, A., Chang, I., Qi, D., Lijuan, P., & Yang, Z. (1999). *Environmental supports for young children's mathematical development in China and the United States*. Paper presented at the meeting of the Society for Research in Child Development, Albuquerque, New Mexico.

Stigler, J. W., Lee, S.-Y., & Stevenson, H.W. (1990). *Mathematical knowledge of Japanese, Chinese, and American elementary school children*. National Council of Teachers of Mathematics, Reston, VA.

Verdine, B. N., Lucca, K. R., Golinkoff, R. M., Hirsh-Pasek, K. & Newcombe, N. (2016). The shape of things: The origin of young children's knowledge of the names and properties of geometric forms. *Journal of Cognition and Development, 17*(1), 142-161. https://doi.org/10.1080/15248372.2015.1016610

Zambrzycka, J., Kotsopoulos, D., & Lee, J. (2017). *Infant Behavior & Development, 46*, 144-157. https://dx.doi.org/10.1016/j.infbeh.2016.12.002

Zhou, Z., Peverly, S. T., & Lin, J. (2005). Understanding early mathematical competencies in American and Chinese children. *School Psychology International, 26*(4), 413-427. https://doi.org/10.1177/0143034305059018
# DESIGNING MEANINGFUL TASKS TO PROMOTE ARGUMENTATION SKILLS IN DGE – A CONCEPT FOR A PROFESSIONAL DEVELOPMENT PROGRAM

Elise Stroetmann, Ulrich Kortenkamp

University of Potsdam (Germany) and German Center for Teacher Education in Mathematics

In this article we describe our approach to a professional development (PD) program for teachers that focusses on ways to enhance the argumentation and proving skills of students. While research in technology-based geometry education has shown the positive effects of theory-based tasks in interactive of dynamic geometry environments (DGE), teachers struggle to implement such tasks in their own teaching appropriately. As a consequence, the PD program is using a twofold approach combining tasks that can be readily used in the classroom and the theory that has been used to build these tasks, such that teachers can use their newly acquired knowledge immediately and transfer it to other subject areas as well.

## INTRODUCTION

The introduction of interactive of dynamic geometry environments (DGEs) in the last decade of the last century has opened up a wealth of new tasks and activities that have the potential to improve mathematics instruction in the long term, and there has been lots of research demonstrating its effectiveness in small settings. Still, even teachers with a positive attitude to the use of technology are not always able to raise that potential, some are even are reluctant to use technology at all. (Thurm et al., 2017), so large-scale effects are missing despite the general availability of technology in schools. A key to remedy this situation can be professional development (PD), however, the effectiveness of PD is not guaranteed.

Starting from principles of good teaching in mathematics and theoretical results on proof and argumentation using DGE, we describe the design of a PD course that is being implemented within the framework of the German QuaMath program for developing quality for mathematics classrooms and mathematics professional development. The course focuses on improving argumentation skills of students using DGE and, while using a specific flagship example, is not tied to a particular topic. Through a scaffolding approach, teachers are enabled to design tasks that promote quality in mathematics teaching. We claim that our design is capable to bridge the gap between theory of DGE-based geometry education and the everyday practical jobs of teachers, in particular designing learning activities.

## THEORETICAL BASIS FOR THE TEACHER'S PD PROGRAM

### Principles for high-quality teaching

The ultimate goal of our PD program is to enable teachers to plan and execute mathematics lessons on a high standard. Since there is no universally accepted definition of good teaching, we first have to operationalize this goal. We follow Prediger et al. (2022), who identify five principles for high-
# DESIGNING MEANINGFUL TASKS TO PROMOTE ARGUMENTATION SKILLS IN DGE – A CONCEPT FOR A PROFESSIONAL DEVELOPMENT PROGRAM

Elise Stroetmann, Ulrich Kortenkamp

University of Potsdam (Germany) and German Center for Teacher Education in Mathematics

In this article we describe our approach to a professional development (PD) program for teachers that focusses on ways to enhance the argumentation and proving skills of students. While research in technology-based geometry education has shown the positive effects of theory-based tasks in interactive of dynamic geometry environments (DGE), teachers struggle to implement such tasks in their own teaching appropriately. As a consequence, the PD program is using a twofold approach combining tasks that can be readily used in the classroom and the theory that has been used to build these tasks, such that teachers can use their newly acquired knowledge immediately and transfer it to other subject areas as well.

## INTRODUCTION

The introduction of interactive of dynamic geometry environments (DGEs) in the last decade of the last century has opened up a wealth of new tasks and activities that have the potential to improve mathematics instruction in the long term, and there has been lots of research demonstrating its effectiveness in small settings. Still, even teachers with a positive attitude to the use of technology are not always able to raise that potential, some are even are reluctant to use technology at all. (Thurm et al., 2017), so large-scale effects are missing despite the general availability of technology in schools. A key to remedy this situation can be professional development (PD), however, the effectiveness of PD is not guaranteed.

Starting from principles of good teaching in mathematics and theoretical results on proof and argumentation using DGE, we describe the design of a PD course that is being implemented within the framework of the German QuaMath program for developing quality for mathematics classrooms and mathematics professional development. The course focuses on improving argumentation skills of students using DGE and, while using a specific flagship example, is not tied to a particular topic. Through a scaffolding approach, teachers are enabled to design tasks that promote quality in mathematics teaching. We claim that our design is capable to bridge the gap between theory of DGE-based geometry education and the everyday practical jobs of teachers, in particular designing learning activities.

## THEORETICAL BASIS FOR THE TEACHER’S PD PROGRAM

### Principles for high-quality teaching

The ultimate goal of our PD program is to enable teachers to plan and execute mathematics lessons on a high standard. Since there is no universally accepted definition of good teaching, we first have to operationalize this goal. We follow Prediger et al. (2022), who identify five principles for high-
quality teaching, which have been derived by combining normative, empirical, epistemological, and
pragmatic perspectives on instructional quality. These principles—*conceptual focus, cognitive
demand, student focus and adaptivity, longitudinal coherence, and enhanced communication*—are a
common guideline for teaching used in the larger context of the German nationwide teacher PD
initiative QuaMath, where they are considered in the context of typical situational demands for
teachers (Prediger, 2019). PD in QuaMath should enable teachers to set goals and plan teaching
progressions, to select and adapt activities and media, to diagnose students’ learning states and
processes, to support students in learning, and to moderate conversations.

Argumentation and proof in geometry education

Argumentation and proof characterize mathematics as a discipline (Heintz, 2000) and also play an
important role in mathematics education. In literature, the term *mathematical argumentation* is
defined and differentiated from the terms *proving* and *reasoning* in various ways (see for example
Brunner, 2014; Reid & Knipping, 2010). In this article, we align with an understanding of these terms
explicated by Reiss & Ufer (2009). According to their terminology, mathematical argumentation
generally aims at the formulation, investigation and validation of hypotheses and open questions. It
is a multifaceted activity that encompasses both exploratory activities, such as investigating and
discovering mathematical relationships and formulating corresponding conjectures, or testing the
plausibility of given conjectures, as well as substantiating a given and assumed plausible conjecture.
The latter aspect can be summarized as *reasoning* and in particular includes *proving* which is defined
quite consistently in literature. Proving refers to the validation of a claim through its purely deductive
derivation from known axiomatic definitions as well as from accepted derived statements. These
specific characteristics distinguish proving from other forms of mathematical reasoning like, for
instance, justifying one's own solution strategies for a mathematical task. In mathematical
argumentation, non-deductive inferences such as induction or abduction are also possible (Brunner,
2014; Reiss & Ufer, 2009). Although the conjecture generation phase has an exploratory character
and does not yet aim at formulating a deductive chain of arguments, there is a close cognitive link
between the conjecturing and the proving phase, as Boero et al. (1996) also describe in connection
with the concept of *cognitive unity*.

Geometry constitutes a part of the spiral curriculum from elementary to upper secondary school in
proving (Bescherer & Hoffkamp, 2022; Reiss & Ufer, 2009). Due to its axiomatic foundation since
Euclid, geometry is considered particularly suitable for learning argumentation, reasoning and
proving. This is also because of the different registers of representation in geometry, that allow for
various approaches and levels of argumentation for both conjecture generation and reasoning
(Wittmann, 2018). The advent of interactive and dynamic geometry environments (DGE) in the
1980s/90s expanded these possibilities by allowing interaction with dynamic representations, and
researchers have since been investigating the role that DGE can play in supporting argumentation
processes in mathematics education (e.g., Healy, 2000; Jones et al., 2000; Laborde, 2005).

DGE, the dragging mode and the role of invariants

Working with DGE to solve geometric problems offers a special affordance, the drag mode, which
allows to vary the position, size and shape of objects in DGE (seemingly) continuously. The fact that
dragging changes the drawing on the screen, but not the constructed figure (Laborde, 1993) makes
quality teaching, which have been derived by combining normative, empirical, epistemological, and
pragmatic perspectives on instructional quality. These principles—*conceptual focus, cognitive
demand, student focus and adaptivity, longitudinal coherence, and enhanced communication*—are a
common guideline for teaching used in the larger context of the German nationwide teacher PD
initiative QuaMath, where they are considered in the context of typical situational demands for
teachers (Prediger, 2019). PD in QuaMath should enable teachers to set goals and plan teaching
progressions, to select and adapt activities and media, to diagnose students’ learning states and
processes, to support students in learning, and to moderate conversations.

Argumentation and proof in geometry education

Argumentation and proof characterize mathematics as a discipline (Heintz, 2000) and also play an
important role in mathematics education. In literature, the term *mathematical argumentation* is
defined and differentiated from the terms *proving* and *reasoning* in various ways (see for example
Brunner, 2014; Reid & Knipping, 2010). In this article, we align with an understanding of these terms
explicated by Reiss & Ufer (2009). According to their terminology, mathematical argumentation
generally aims at the formulation, investigation and validation of hypotheses and open questions. It
is a multifaceted activity that encompasses both exploratory activities, such as investigating and
discovering mathematical relationships and formulating corresponding conjectures, or testing the
plausibility of given conjectures, as well as substantiating a given and assumed plausible conjecture.
The latter aspect can be summarized as *reasoning* and in particular includes *proving* which is defined
quite consistently in literature. Proving refers to the validation of a claim through its purely deductive
derivation from known axiomatic definitions as well as from accepted derived statements. These
specific characteristics distinguish proving from other forms of mathematical reasoning like, for
instance, justifying one's own solution strategies for a mathematical task. In mathematical
argumentation, non-deductive inferences such as induction or abduction are also possible (Brunner,
2014; Reiss & Ufer, 2009). Although the conjecture generation phase has an exploratory character
and does not yet aim at formulating a deductive chain of arguments, there is a close cognitive link
between the conjecturing and the proving phase, as Boero et al. (1996) also describe in connection
with the concept of *cognitive unity*.

Geometry constitutes a part of the spiral curriculum from elementary to upper secondary school in
proving (Bescherer & Hoffkamp, 2022; Reiss & Ufer, 2009). Due to its axiomatic foundation since
Euclid, geometry is considered particularly suitable for learning argumentation, reasoning and
proving. This is also because of the different registers of representation in geometry, that allow for
various approaches and levels of argumentation for both conjecture generation and reasoning
(Wittmann, 2018). The advent of interactive and dynamic geometry environments (DGE) in the
1980s/90s expanded these possibilities by allowing interaction with dynamic representations, and
researchers have since been investigating the role that DGE can play in supporting argumentation
processes in mathematics education (e.g., Healy, 2000; Jones et al., 2000; Laborde, 2005).

DGE, the dragging mode and the role of invariants

Working with DGE to solve geometric problems offers a special affordance, the drag mode, which
allows to vary the position, size and shape of objects in DGE (seemingly) continously. The fact that
dragging changes the drawing on the screen, but not the constructed figure (Laborde, 1993) makes
the drag mode a powerful tool to check the correctness of geometric constructions, and to initiate
mathematical argumentation processes. Figures in DGE are given by a system of relations, that are
either directly constructed or implications from constructed relations according to the Euclidean
theory (Kortenkamp, 1999). Those relations can be explored as motion invariant properties of the
figure during dragging. Moving parts of the figure and identifying invariants form the core of
generating conjectures about mathematical relations with DGE (Baccaglini-Frank, 2012; Baccaglini-
Frank & Mariotti, 2010; Bescherer & Hoffkamp, 2022; Healy, 2000; Hölzl, 1996; Laborde, 2005).

<figure><img src="image_91.png" /></figure>

The following task is a very typical example for a conjecture generating activity with DGE in a german textbook: "Use a DGE to draw a circle with center M and diameter AB. Place a point C on the circle and connect it to A and B. Measure $\gamma$ and drag C along the circle. What can you discover? Formulate a conjecture."

The first invariant in this first example is the position of point *C* on the circle. During dragging, a second invariant can be observed: $\gamma$ is a right angle. According to Healy (2000), both properties of the figure are *robust invariants* because they are invariant under any kind of dragging (Laborde, 2005). The position of *C* is called a *direct* robust invariant, because *C* is directly constructed to be located on the circle, whereas the right angle at point *C* is a consequence of the construction within the Euclidean theory (Thales’ Theorem), an *indirect* robust invariant (Baccaglini-Frank, 2012).

The following second example demonstrates the counterpart to “robust invariants”:

<figure><img src="image_92.png" /></figure>

"Use a DGE to draw a triangle ABC. Measure the angle $\gamma$ at point C. Drag Point C so that $\gamma$ is a right angle. Use the trace tool to describe the positions of C. What can you discover? Formulate a conjecture."

The right angle at point *C* is no permanent property of the figure. It must be induced and maintained by dragging *C* in a special way and is called a *soft invariant*. During this activity, students can observe that the *path* of the dragged point *C* has the shape of a circle with diameter *AB*. The *path* is an ideal, mental construct, which would be traced by *C* if the soft invariant could be maintained continuously (Baccaglini-Frank & Mariotti, 2009, p. 235). The property “*C* lies on the circle” is a second soft invariant (Healy, 2000). Since in this task, the right angle $\gamma$ is directly and intentionally established by dragging *C*, it is a *direct* soft invariant, whereas the circle shape of the path of *C* is a consequence of maintaining the size of $\gamma$ (according to the reversal of Thales’s Theorem), an *indirect* soft invariant. Baccaglini-Frank and her colleagues call the kind of dragging required in this task with the aim of maintaining the right-angle property *maintaining dragging* (Baccaglini-Frank, 2011, 2012; Baccaglini-Frank & Mariotti, 2010).

Recognizing invariants in DGE and formulating conjectures about the underlying mathematical relations, moving from the phenomenological experiential world of DGE to the formal axiomatic and static world of Euclidean geometry, is challenging for students (Baccaglini-Frank, 2010). In the case of the examples above, the observation of the invariants can lead to the following conjectures: "If *C* lies on the circle with diameter *AB*, the angle at point *C* of the triangle *ABC* is a right angle" (Ex. 1) and "If the angle at point *C* of the triangle *ABC* is a right angle, *C* lies on a circle with diameter *AB*" (Ex. 2). The *premises* of these conjectures correspond to the observed *direct* invariants and the *conclusions* to the *indirect* invariants. Concerning maintaining dragging, Baccaglini-Frank and her colleagues suggest that this interpretation of motion dependence between the two invariants as logical
the drag mode a powerful tool to check the correctness of geometric constructions, and to initiate
mathematical argumentation processes. Figures in DGE are given by a system of relations, that are
either directly constructed or implications from constructed relations according to the Euclidean
theory (Kortenkamp, 1999). Those relations can be explored as motion invariant properties of the
figure during dragging. Moving parts of the figure and identifying invariants form the core of
generating conjectures about mathematical relations with DGE (Baccaglini-Frank, 2012; Baccaglini-
Frank & Mariotti, 2010; Bescherer & Hoffkamp, 2022; Healy, 2000; Hölzl, 1996; Laborde, 2005).

<figure><img src="image_97.png" /></figure>

The following task is a very typical example for a conjecture generating activity with DGE in a german textbook: "Use a DGE to draw a circle with center M and diameter AB. Place a point C on the circle and connect it to A and B. Measure $\gamma$ and drag C along the circle. What can you discover? Formulate a conjecture."

The first invariant in this first example is the position of point *C* on the circle. During dragging, a second invariant can be observed: $\gamma$ is a right angle. According to Healy (2000), both properties of the figure are *robust invariants* because they are invariant under any kind of dragging (Laborde, 2005). The position of *C* is called a *direct* robust invariant, because *C* is directly constructed to be located on the circle, whereas the right angle at point *C* is a consequence of the construction within the Euclidean theory (Thales’ Theorem), an *indirect* robust invariant (Baccaglini-Frank, 2012).

The following second example demonstrates the counterpart to “robust invariants”:

<figure><img src="image_98.png" /></figure>

> "Use a DGE to draw a triangle ABC. Measure the angle $\gamma$ at point C. Drag Point C so that $\gamma$ is a right angle. Use the trace tool to describe the positions of C. What can you discover? Formulate a conjecture."

The right angle at point *C* is no permanent property of the figure. It must be induced and maintained by dragging *C* in a special way and is called a *soft invariant*. During this activity, students can observe that the *path* of the dragged point *C* has the shape of a circle with diameter *AB*. The *path* is an ideal, mental construct, which would be traced by *C* if the soft invariant could be maintained continuously (Baccaglini-Frank & Mariotti, 2009, p. 235). The property “*C* lies on the circle” is a second soft invariant (Healy, 2000). Since in this task, the right angle $\gamma$ is directly and intentionally established by dragging *C*, it is a *direct* soft invariant, whereas the circle shape of the path of *C* is a consequence of maintaining the size of $\gamma$ (according to the reversal of Thales’s Theorem), an *indirect* soft invariant. Baccaglini-Frank and her colleagues call the kind of dragging required in this task with the aim of maintaining the right-angle property *maintaining dragging* (Baccaglini-Frank, 2011, 2012; Baccaglini-Frank & Mariotti, 2010).

Recognizing invariants in DGE and formulating conjectures about the underlying mathematical relations, moving from the phenomenological experiential world of DGE to the formal axiomatic and static world of Euclidean geometry, is challenging for students (Baccaglini-Frank, 2010). In the case of the examples above, the observation of the invariants can lead to the following conjectures: “If *C* lies on the circle with diameter *AB*, the angle at point *C* of the triangle *ABC* is a right angle” (Ex. 1) and “If the angle at point *C* of the triangle *ABC* is a right angle, *C* lies on a circle with diameter *AB*” (Ex. 2). The *premises* of these conjectures correspond to the observed *direct* invariants and the *conclusions* to the *indirect* invariants. Concerning maintaining dragging, Baccaglini-Frank and her colleagues suggest that this interpretation of motion dependence between the two invariants as logical
dependence between two mathematical statements acts as a key element for conjecture generation processes with DGE and provides a bridge for the ‘experimental-theoretical gap’ (Lopez-Real & Leung, 2006) between the two mentioned worlds. Example 2 further shows how using the trace tool can help to find invariants (Baccaglini-Frank, 2012; Baccaglini-Frank & Mariotti, 2010).

## Moving from Dragging Exploration to Documentation

The discovery and formulation of mathematical relationships can also be supported by further features of DGE. Let us revisit Ex. 2: instead of continuously drawing the trace of point C during maintaining dragging, we introduce a *stamping tool* such that the location of C can be marked for only a finite number of sample triangles. Compared to maintaining dragging with an activated trace, there is an advantage in reduced demands on hand-eye coordination. Instead of permanently maintaining the right angle at C during dragging, one can first search for a triangle that satisfies this desired condition. Stamping then transitions from dynamically varying C on the screen to fixing a specific position of C on the drawing surface, moving from an experimental playground (*Spielraum*) to a document (*Dokument*) that conserves the experimental findings (Kortenkamp & Wollring, 2017). We will use the additional benefits of stamping for conjecture generation in DGE, particularly regarding mathematical equivalence statements, in our PD.

## Opportunities and Challenges of Using DGE

In the previous section, we already took a *theoretical perspective* on the opportunities offered by DGE through the construction and dynamic variation of geometric figures for making explicit the relations between the constructed objects, for free experimentation and the discovery of mathematical relationships, as well as for the formulation or refutation of conjectures (Bescherer & Hoffkamp, 2022). From an *empirical perspective* as well, the potential of DGE for mathematical argumentation is supported by numerous studies. For example, it has been shown that DGE have a positive impact on the sub-competency of conjecture generation in open problems (see, e.g. Baccaglini-Frank, 2010, 2011; Baccaglini-Frank & Mariotti, 2010). Building on (Arzarello et al., 2002), who analyzed and classified various dragging modalities in students' solutions to open problems, Baccaglini-Frank et al. put a focus on maintaining dragging and describe in detail how DGE can enhance cognitive processes during conjecture generation within the context of maintaining dragging. There is even a hint that maintaining dragging can lead to a *cognitive unity* (Boero et al., 1996) and thus can promote the construction of deductive proofs of the generated conjectures.

However, the use of DGE for argumentation also comes with pedagogical challenges. The transition from the *process*-oriented, empirical exploration in DGE and conjecture generation to the logical-deductive proof of these conjectures as a *product* of the exploration process is described in the literature as one of the major difficulties associated with argumentation in DGE (Bescherer & Hoffkamp, 2022; Healy, 2000). In addressing these challenges, teachers play a crucial role. They must not only provide students the opportunities for independent exploration and conjecture generation in DGE but also strengthen their students’ needs to prove their conjectures, as has been extensively and controversially discussed in the literature. In particular, there is the concern that the persuasive power of dynamic figures in DGE might diminish the need for proof (Olivero, 2003). They also have to assist the learners in developing supporting arguments and arranging them into a deductive proof, without providing too much guidance and reducing the cognitive demand of the
dependence between two mathematical statements acts as a key element for conjecture generation processes with DGE and provides a bridge for the ‘experimental-theoretical gap’ (Lopez-Real & Leung, 2006) between the two mentioned worlds. Example 2 further shows how using the trace tool can help to find invariants (Baccaglini-Frank, 2012; Baccaglini-Frank & Mariotti, 2010).

## Moving from Dragging Exploration to Documentation

The discovery and formulation of mathematical relationships can also be supported by further features of DGE. Let us revisit Ex. 2: instead of continuously drawing the trace of point C during maintaining dragging, we introduce a *stamping tool* such that the location of C can be marked for only a finite number of sample triangles. Compared to maintaining dragging with an activated trace, there is an advantage in reduced demands on hand-eye coordination. Instead of permanently maintaining the right angle at C during dragging, one can first search for a triangle that satisfies this desired condition. Stamping then transitions from dynamically varying C on the screen to fixing a specific position of C on the drawing surface, moving from an experimental playground (*Spielraum*) to a document (*Dokument*) that conserves the experimental findings (Kortenkamp & Wollring, 2017). We will use the additional benefits of stamping for conjecture generation in DGE, particularly regarding mathematical equivalence statements, in our PD.

## Opportunities and Challenges of Using DGE

In the previous section, we already took a *theoretical perspective* on the opportunities offered by DGE through the construction and dynamic variation of geometric figures for making explicit the relations between the constructed objects, for free experimentation and the discovery of mathematical relationships, as well as for the formulation or refutation of conjectures (Bescherer & Hoffkamp, 2022). From an *empirical perspective* as well, the potential of DGE for mathematical argumentation is supported by numerous studies. For example, it has been shown that DGE have a positive impact on the sub-competency of conjecture generation in open problems (see, e.g. Baccaglini-Frank, 2010, 2011; Baccaglini-Frank & Mariotti, 2010). Building on (Arzarello et al., 2002), who analyzed and classified various dragging modalities in students' solutions to open problems, Baccaglini-Frank et al. put a focus on maintaining dragging and describe in detail how DGE can enhance cognitive processes during conjecture generation within the context of maintaining dragging. There is even a hint that maintaining dragging can lead to a *cognitive unity* (Boero et al., 1996) and thus can promote the construction of deductive proofs of the generated conjectures.

However, the use of DGE for argumentation also comes with pedagogical challenges. The transition from the *process*-oriented, empirical exploration in DGE and conjecture generation to the logical-deductive proof of these conjectures as a *product* of the exploration process is described in the literature as one of the major difficulties associated with argumentation in DGE (Bescherer & Hoffkamp, 2022; Healy, 2000). In addressing these challenges, teachers play a crucial role. They must not only provide students the opportunities for independent exploration and conjecture generation in DGE but also strengthen their students’ needs to prove their conjectures, as has been extensively and controversially discussed in the literature. In particular, there is the concern that the persuasive power of dynamic figures in DGE might diminish the need for proof (Olivero, 2003). They also have to assist the learners in developing supporting arguments and arranging them into a deductive proof, without providing too much guidance and reducing the cognitive demand of the
learning activity. The selection and adaptation of meaningful tasks—as a situational demand of the teacher—is crucial for a rich learning activities in argumentation (Bescherer & Hoffkamp, 2022).

A condition for addressing these challenges is a sound professional knowledge of teachers regarding argumentation in DGE and, especially, the creation of effective DGE-supported tasks that stimulate argumentative processes. Our experience with teachers in Germany indicates that so far, only few teachers have incorporated rich DGE-supported activities for argumentation into their teaching. While there is still a lack of research with respect to open exploration in proving in school, the existing literature describes that there is currently less exploration than desirable in the classroom (Jahnke et al., 2023). Therefor, we will now show strategies to transform traditional robust construction tasks into meaningful, cognitively demanding, and discovery-oriented tasks for the process of conjecture generation.

## MEANINGFUL TASKS FOR ARGUMENTATION IN DGS IN TEACHERS' PD

Starting from a task related to a robust construction (Healy, 2000) of the mathematical relationship, meaningful tasks promoting argumentation skills can be generated by systematically relaxing and/or changing conditions. It becomes evident that the complexity of the tasks varies depending on how many conditions of the theorem's premise are passed to the DGS and mentioned in the task, providing different levels of space for students' own discoveries. We illustrate this approach to task construction using the already mentioned converse of Thales' theorem as an example: If the angle at point C of the triangle ABC is a right angle, C lies on a circle with diameter AB.

(1) **Robust Construction:** If each condition of the theorem's premise is robustly constructed in a DGE, a triangle with an invariant 90°-angle at Point C over segment AB (direct invariant) is obtained. The following task prompts students to discover the position of C on the Thales circle (robust indirect invariant) through dragging: *"The triangle ABC has a right angle at Point C. Drag C to change the shape of the triangle. Which path does Point C move on? Generate Conjectures."* A task like this, focusing on generating conjectures about relationships between robust invariants, relies on students' ability to differentiate between direct and indirect invariants (Baccaglini-Frank, 2012). Students can also perform the construction themselves, enhancing not only tool competencies but also an awareness of all required conditions in the theorem while getting direct feedback on the correctness. Tasks involving robust constructions are also suitable for demonstrating mathematical relationships by the teacher (Laborde, 2005).

(2a) **Relaxing/changing conditions:** As in Ex. 2 above, some conditions of the theorem's premise are omitted from the construction. Here, the condition $γ = 90°$ is mentioned but not constructed robustly. The position of point C on the circle is not obvious due to this soft construction, making it more challenging for students to discover the geometric path of Point C through maintaining dragging. This variation is more open compared to (1).

(2b) **Relaxing/changing conditions:** The condition $γ = 90°$ can be further modified to create room for additional interesting geometric discoveries. Instead of right angles, any angle $γ$ can be investigated. The instruction *"Investigate other angle sizes. What do you observe?"* can follow task 2a and leads to the converse of the Inscribed Angle Theorem. Students can discover that for various angles $γ$, point C lies on an circular arc. The Thales' relationship is a just special case of this.
learning activity. The selection and adaptation of meaningful tasks—as a situational demand of the teacher—is crucial for a rich learning activities in argumentation (Bescherer & Hoffkamp, 2022).

A condition for addressing these challenges is a sound professional knowledge of teachers regarding argumentation in DGE and, especially, the creation of effective DGE-supported tasks that stimulate argumentative processes. Our experience with teachers in Germany indicates that so far, only few teachers have incorporated rich DGE-supported activities for argumentation into their teaching. While there is still a lack of research with respect to open exploration in proving in school, the existing literature describes that there is currently less exploration than desirable in the classroom (Jahnke et al., 2023). Therefor, we will now show strategies to transform traditional robust construction tasks into meaningful, cognitively demanding, and discovery-oriented tasks for the process of conjecture generation.

## MEANINGFUL TASKS FOR ARGUMENTATION IN DGS IN TEACHERS' PD

Starting from a task related to a robust construction (Healy, 2000) of the mathematical relationship, meaningful tasks promoting argumentation skills can be generated by systematically relaxing and/or changing conditions. It becomes evident that the complexity of the tasks varies depending on how many conditions of the theorem's premise are passed to the DGS and mentioned in the task, providing different levels of space for students' own discoveries. We illustrate this approach to task construction using the already mentioned converse of Thales' theorem as an example: If the angle at point C of the triangle ABC is a right angle, C lies on a circle with diameter AB.

(1) **Robust Construction:** If each condition of the theorem's premise is robustly constructed in a DGE, a triangle with an invariant 90°-angle at Point C over segment AB (direct invariant) is obtained. The following task prompts students to discover the position of C on the Thales circle (robust indirect invariant) through dragging: *"The triangle ABC has a right angle at Point C. Drag C to change the shape of the triangle. Which path does Point C move on? Generate Conjectures."* A task like this, focusing on generating conjectures about relationships between robust invariants, relies on students' ability to differentiate between direct and indirect invariants (Baccaglini-Frank, 2012). Students can also perform the construction themselves, enhancing not only tool competencies but also an awareness of all required conditions in the theorem while getting direct feedback on the correctness. Tasks involving robust constructions are also suitable for demonstrating mathematical relationships by the teacher (Laborde, 2005).

(2a) **Relaxing/changing conditions:** As in Ex. 2 above, some conditions of the theorem's premise are omitted from the construction. Here, the condition $γ = 90°$ is mentioned but not constructed robustly. The position of point C on the circle is not obvious due to this soft construction, making it more challenging for students to discover the geometric path of Point C through maintaining dragging. This variation is more open compared to (1).

(2b) **Relaxing/changing conditions:** The condition $γ = 90°$ can be further modified to create room for additional interesting geometric discoveries. Instead of right angles, any angle $γ$ can be investigated. The instruction *"Investigate other angle sizes. What do you observe?"* can follow task 2a and leads to the converse of the Inscribed Angle Theorem. Students can discover that for various angles $γ$, point C lies on an circular arc. The Thales' relationship is a just special case of this.
(2c) Relaxing/changing conditions: Another task variation is: "*Draw a triangle ABC. Use the angle tool to measure all interior angles of the triangle. Move Point C and investigate special types of triangles (regarding angles). Describe the positions of Point C for each triangle type. Use the trace tool for Point C to assist. Verify your conjectures through construction.*" Through the request "*investigate special types of triangles*," the condition regarding the interior angle of triangle ABC is not only omitted in the DGS construction as in (2a), but also left out from the task description. This variation allows for considering all interior angles and is not restricted to right triangles. For instance, investigating isosceles triangles reveals that Point C moves along the perpendicular bisector of segment AB. For equilateral triangles ABC, the position of Point C is uniquely determined. Thus, this task is the one most open to discovery. Solving this task requires students to independently determine whether and which interior angles should be kept invariant. In other words, the soft invariants directly induced by maintaining dragging must now be determined freely by the students, demanding higher problem-solving skills than before.

The process of task construction demonstrated here can be applied to any geometric relationship and
can be combined with the *stamping* approach described earlier: Select one or more conditions from
the premise of the theorem to be discovered and relax these conditions in the DGS construction and
in the task statement. This creates room for additional (related) interesting geometric discoveries that
enhance the understanding of underlying concepts and increase students' activity. Baccaglini-Frank
(2012) also argues that tasks involving conjecture generation using soft invariants offer potential for
richer classroom discussions. In contrast to robust constructions, students must first identify the
conditions under which a figure exhibits specific properties or—speaking about motions of the
dragged point—the path on which point C must be dragged on.

The strategy for task construction can also be applied to other concepts, for example the angle bisector theorem: For angles up to 180°, if a point P has the same distance from both sides of the angle, then P lies on the angle bisector. For the robust variant, the observation and conjecturing task would be "Given a point A and two rays, g and h, starting at A. Point P is constructed in such a way that the distances from P to g and h are equal. Use the trace tool to describe the positions of point P.", while relaxed/changed variants using the soft construction of P could start with "Given a point A, two rays, g and h, starting at A, and a point P." and then (a) "Move P in a way that the distances to the two sides of the angle are equal. Use the trace tool to describe the positions of P." or (b) "Move P in a way that the distances to the two sides of the angle are in a ratio of 1:3. Use the trace tool to describe the positions of P." or (c) "Measure the distances $d_g$ and $d_h$ from P to g and h. Investigate different ratios of $d_g$ and $d_h$ by moving point P. On which line does point C lie in each case?"

Another example of transferring this task construction technique is the investigation of the area of a
triangle ABC: Here, the role of $\gamma = 90^\circ$ is taken by *area*(ABC) = 50 (or any other value), and the
locus of C is a straight line parallel to AB instead of a circle, and the tasks are reformulated
accordingly.

The implementation in the PD program is based on a sandwich model of in-person PD and self
conducted teaching experiments. Equipped with the technique of opening up proving and
argumentation tasks the teachers are enabled to use this technique for their own, current teaching,
even if the content is different. In fact, in the get-together phase after the teaching experiments, they
are able to report on different content that could be converted to theory-based lesson activities, and
(2c) Relaxing/changing conditions: Another task variation is: "*Draw a triangle ABC. Use the angle tool to measure all interior angles of the triangle. Move Point C and investigate special types of triangles (regarding angles). Describe the positions of Point C for each triangle type. Use the trace tool for Point C to assist. Verify your conjectures through construction.*" Through the request "*investigate special types of triangles*," the condition regarding the interior angle of triangle ABC is not only omitted in the DGS construction as in (2a), but also left out from the task description. This variation allows for considering all interior angles and is not restricted to right triangles. For instance, investigating isosceles triangles reveals that Point C moves along the perpendicular bisector of segment AB. For equilateral triangles ABC, the position of Point C is uniquely determined. Thus, this task is the one most open to discovery. Solving this task requires students to independently determine whether and which interior angles should be kept invariant. In other words, the soft invariants directly induced by maintaining dragging must now be determined freely by the students, demanding higher problem-solving skills than before.

The process of task construction demonstrated here can be applied to any geometric relationship and
can be combined with the *stamping* approach described earlier: Select one or more conditions from
the premise of the theorem to be discovered and relax these conditions in the DGS construction and
in the task statement. This creates room for additional (related) interesting geometric discoveries that
enhance the understanding of underlying concepts and increase students' activity. Baccaglini-Frank
(2012) also argues that tasks involving conjecture generation using soft invariants offer potential for
richer classroom discussions. In contrast to robust constructions, students must first identify the
conditions under which a figure exhibits specific properties or—speaking about motions of the
dragged point—the path on which point C must be dragged on.

The strategy for task construction can also be applied to other concepts, for example the angle bisector theorem: For angles up to 180°, if a point P has the same distance from both sides of the angle, then P lies on the angle bisector. For the robust variant, the observation and conjecturing task would be "Given a point A and two rays, g and h, starting at A. Point P is constructed in such a way that the distances from P to g and h are equal. Use the trace tool to describe the positions of point P.", while relaxed/changed variants using the soft construction of P could start with "Given a point A, two rays, g and h, starting at A, and a point P." and then (a) "Move P in a way that the distances to the two sides of the angle are equal. Use the trace tool to describe the positions of P." or (b) "Move P in a way that the distances to the two sides of the angle are in a ratio of 1:3. Use the trace tool to describe the positions of P." or (c) "Measure the distances $d_g$ and $d_h$ from P to g and h. Investigate different ratios of $d_g$ and $d_h$ by moving point P. On which line does point C lie in each case?"

Another example of transferring this task construction technique is the investigation of the area of a
triangle ABC: Here, the role of $\gamma = 90^\circ$ is taken by *area*(ABC) = 50 (or any other value), and the
locus of C is a straight line parallel to AB instead of a circle, and the tasks are reformulated
accordingly.

The implementation in the PD program is based on a sandwich model of in-person PD and self
conducted teaching experiments. Equipped with the technique of opening up proving and
argumentation tasks the teachers are enabled to use this technique for their own, current teaching,
even if the content is different. In fact, in the get-together phase after the teaching experiments, they
are able to report on different content that could be converted to theory-based lesson activities, and
compare and reflect their approaches. As the next step, there is still the challenge of transitioning
from *conjecturing* to *proving*. However, as the teaching content has been designed to follow the
principles of *cognitive demand* and *longitudinal coherence* (Prediger et al., 2022), teachers can build
on these. In our guiding Thales example, students will be able to argue using isosceles triangle and
their base angles after working with the soft constructions, and use this knowledge to build a proof
using an auxiliary line connecting the midpoint of AB to C.

CONCLUSION AND OUTLOOK

In this paper we describe a technique for opening up proving tasks by converting robust constructions into soft constructions. This technique is suitable for inclusion in practice-oriented PD, as it enables teachers to address their situational demands, in particular selection and adaptation of tasks and activities. We will use the results of the implementation in the next years within the QuaMath program for further refinement of this approach.

As a counterpart of the relaxation approach described, it is also possible to use DGE to add additional
constraints, for example through pseudo-objects (Baccaglini-Frank et al., 2018). Again, it will not
suffice to provide a single example of this proof technique, but we need to provide a general schema
to enable teachers to create substantial tasks and activities on their own.

References

Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. *Zentralblatt Für Didaktik Der Mathematik*, 34(3), 66–72.
https://doi.org/10.1007/BF02655708

Baccaglini-Frank, A. (2010). *Conjecturing in dynamic geometry: A model for conjecture-generation through maintaining dragging* [Doctoral Dissertation, University of New Hampshire].
https://scholars.unh.edu/dissertation/529

Baccaglini-Frank, A. (2011). Abduction in generating conjectures in dynamic geometry through maintaining dragging. In M. Pytlak, T. Rowland, & E. Swoboda (Hrsg.), *Proceedings of the 7th Congress of the European Society for Research in Mathematics Education*.

Baccaglini-Frank, A. (2012). Dragging and making sense of invariants in dynamic geometry. *Mathematics Teacher*, 105(8), 616–621.

Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. A. (2018). From pseudo-objects in dynamic explorations to proof by contradiction. *Digital Experiences in Mathematics Education, 4*(2–3), 87–109.
https://doi.org/10.1007/s40751-018-0039-2

Baccaglini-Frank, A., & Mariotti, M. A. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. *International Journal of Computers for Mathematical Learning, 15*(3), 225–253.
https://doi.org/10.1007/s10758-010-9169-3

Baccaglini-Frank, A., & Mariotti, M. A. (2009). Conjecturing and proving in dynamic geometry: The elaboration of some research hypotheses. *Proceedings of CERME 6*. CERME 6, Lyon.

Bescherer, C., & Hoffkamp, A. (2022). Argumentieren und Beweisen mit digitalen Werkzeugen. In G. Pinkernell, F. Reinhold, F. Schacht, & D. Walter (Hrsg.), *Digitales Lehren und Lernen von Mathematik in der Schule: Aktuelle Forschungsbefunde im Überblick*. Springer Berlin Heidelberg.
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Boero, P., Garuti, R., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems: A hypothesis about the cognitive unity of theorems. *Proceedings of the International Group for the Psychology of Mathematics Education PME-XX, 2*, 113–120.
compare and reflect their approaches. As the next step, there is still the challenge of transitioning
from *conjecturing* to *proving*. However, as the teaching content has been designed to follow the
principles of *cognitive demand* and *longitudinal coherence* (Prediger et al., 2022), teachers can build
on these. In our guiding Thales example, students will be able to argue using isosceles triangle and
their base angles after working with the soft constructions, and use this knowledge to build a proof
using an auxiliary line connecting the midpoint of AB to C.

CONCLUSION AND OUTLOOK

In this paper we describe a technique for opening up proving tasks by converting robust constructions into soft constructions. This technique is suitable for inclusion in practice-oriented PD, as it enables teachers to address their situational demands, in particular selection and adaptation of tasks and activities. We will use the results of the implementation in the next years within the QuaMath program for further refinement of this approach.

As a counterpart of the relaxation approach described, it is also possible to use DGE to add additional
constraints, for example through pseudo-objects (Baccaglini-Frank et al., 2018). Again, it will not
suffice to provide a single example of this proof technique, but we need to provide a general schema
to enable teachers to create substantial tasks and activities on their own.

References

Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. *Zentralblatt Für Didaktik Der Mathematik*, 34(3), 66–72.
https://doi.org/10.1007/BF02655708

Baccaglini-Frank, A. (2010). *Conjecturing in dynamic geometry: A model for conjecture-generation through maintaining dragging* [Doctoral Dissertation, University of New Hampshire].
https://scholars.unh.edu/dissertation/529

Baccaglini-Frank, A. (2011). Abduction in generating conjectures in dynamic geometry through maintaining dragging. In M. Pytlak, T. Rowland, & E. Swoboda (Hrsg.), *Proceedings of the 7th Congress of the European Society for Research in Mathematics Education*.

Baccaglini-Frank, A. (2012). Dragging and making sense of invariants in dynamic geometry. *Mathematics Teacher*, 105(8), 616–621.

Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. A. (2018). From pseudo-objects in dynamic explorations to proof by contradiction. *Digital Experiences in Mathematics Education, 4*(2–3), 87–109.
https://doi.org/10.1007/s40751-018-0039-2

Baccaglini-Frank, A., & Mariotti, M. A. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. *International Journal of Computers for Mathematical Learning, 15*(3), 225–253.
https://doi.org/10.1007/s10758-010-9169-3

Baccaglini-Frank, A., & Mariotti, M. A. (2009). Conjecturing and proving in dynamic geometry: The elaboration of some research hypotheses. *Proceedings of CERME 6*. CERME 6, Lyon.

Bescherer, C., & Hoffkamp, A. (2022). Argumentieren und Beweisen mit digitalen Werkzeugen. In G. Pinkernell, F. Reinhold, F. Schacht, & D. Walter (Hrsg.), *Digitales Lehren und Lernen von Mathematik in der Schule: Aktuelle Forschungsbefunde im Überblick*. Springer Berlin Heidelberg.
https://doi.org/10.1007/978-3-662-65281-7

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Kortenkamp, U., & Wollring, B. (2017). Raum und Form. In M. Abshagen, B. Barzel, J. Kramer, T. Riecke-Baulecke, B. Rösken-Winter, & C. Selter (Hrsg.), *Basiswissen Lehrerbildung: Mathematik unterrichten* (S. 99–112). Klett/Kallmeyer.

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Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry environments. *International Journal of Mathematical Education in Science and Technology, 37*(6), 665–679. https://doi.org/10.1080/00207390600712539

Olivero, F. (2003). *The proving process within a dynamic geometry environment* [Doctoral Thesis, University of Bristol]. https://telearn.archives-ouvertes.fr/hal-00190412

Prediger, S. (2019). Investigating and promoting teachers’ expertise for language-responsive mathematics teaching. *Mathematics Education Research Journal, 31*(4), 367–392. https://doi.org/10.1007/s13394-019-00258-1

Prediger, S., Götze, D., Holzäpfel, L., Rösken-Winter, B., & Selter, C. (2022). Five principles for high-quality mathematics teaching: Combining normative, epistemological, empirical, and pragmatic perspectives for specifying the content of professional development. *Frontiers in Education, 7*, 969212. https://doi.org/10.3389/feduc.2022.969212

Reid, D., & Knipping, C. (2010). *Proof in mathematics education: Research, learning and teaching*. https://doi.org/10.1163/9789460912467

Reiss, K., & Ufer, S. (2009). Was macht mathematisches Arbeiten aus? Empirische Ergebnisse zum Argumentieren, Begründen und Beweisen. *Jahresbericht der DMV, 4*, 155–177.

Thurm, D., Klinger, M., Barzel, B., & Rögler, P. (2017). Überzeugungen zum Technologieeinsatz im Mathematikunterricht: Entwicklung eines Messinstruments für Lehramtsstudierende und Lehrkräfte. *mathematica didactica, 40*(1), 19–35.

Wittmann, G. (2018). Beweisen und Argumentieren. In H.-G. Weigand, A. Filler, R. Hölzl, S. Kuntze, M. Ludwig, J. Roth, B. Schmidt-Thieme, & G. Wittmann (eds.), *Didaktik der Geometrie für die Sekundarstufe I* (S. 21–42). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-56217-8
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Hölzl, R. (1996). How does ‘dragging’ affect the learning of geometry. *International Journal of Computers for Mathematical Learning, 1*(2), 169–187. https://doi.org/10.1007/BF00571077

Jahnke, H. N., Sommerhoff, D., & Ufer, S. (2023). Argumentieren, Begründen und Beweisen. In R. Bruder, A. Büchter, H. Gasteiger, B. Schmidt-Thieme, & H.-G. Weigand (Hrsg.), *Handbuch der Mathematikdidaktik* (S. 369–398). Springer. https://doi.org/10.1007/978-3-662-66604-3_12

Jones, K., Gutiérrez, Á., & Mariotti, M. A. (Hrsg.). (2000). Proof in dynamic geometry environments: A PME Special Issue. *Educational Studies in Mathematics, 44*(1–3).

Kortenkamp, U. (1999). *Foundations of Dynamic Geometry* [Dissertation, ETH Zurich]. https://doi.org/10.3929/ETHZ-A-003876663

Kortenkamp, U., & Wollring, B. (2017). Raum und Form. In M. Abshagen, B. Barzel, J. Kramer, T. Riecke-Baulecke, B. Rösken-Winter, & C. Selter (Hrsg.), *Basiswissen Lehrerbildung: Mathematik unterrichten* (S. 99–112). Klett/Kallmeyer.

Laborde, C. (1993). The computer as part of the learning environment: The case of geometry. In C. Keitel & K. Ruthven (Hrsg.), *Learning from computers: Mathematics education and technology* (S. 48–67). Springer. https://doi.org/10.1007/978-3-642-78542-9_3

Laborde, C. (2005). Robust and soft constructions: Two sides of the use of dynamic geometry environments. *Proceedings of the 10th Asian Technology Conference in Mathematics, 22–35*.

Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry environments. *International Journal of Mathematical Education in Science and Technology, 37*(6), 665–679. https://doi.org/10.1080/00207390600712539

Olivero, F. (2003). *The proving process within a dynamic geometry environment* [Doctoral Thesis, University of Bristol]. https://telearn.archives-ouvertes.fr/hal-00190412

Prediger, S. (2019). Investigating and promoting teachers’ expertise for language-responsive mathematics teaching. *Mathematics Education Research Journal, 31*(4), 367–392. https://doi.org/10.1007/s13394-019-00258-1

Prediger, S., Götze, D., Holzäpfel, L., Rösken-Winter, B., & Selter, C. (2022). Five principles for high-quality mathematics teaching: Combining normative, epistemological, empirical, and pragmatic perspectives for specifying the content of professional development. *Frontiers in Education, 7*, 969212. https://doi.org/10.3389/feduc.2022.969212

Reid, D., & Knipping, C. (2010). *Proof in mathematics education: Research, learning and teaching*. https://doi.org/10.1163/9789460912467

Reiss, K., & Ufer, S. (2009). Was macht mathematisches Arbeiten aus? Empirische Ergebnisse zum Argumentieren, Begründen und Beweisen. *Jahresbericht der DMV, 4*, 155–177.

Thurm, D., Klinger, M., Barzel, B., & Rögler, P. (2017). Überzeugungen zum Technologieeinsatz im Mathematikunterricht: Entwicklung eines Messinstruments für Lehramtsstudierende und Lehrkräfte. *mathematica didactica, 40*(1), 19–35.

Wittmann, G. (2018). Beweisen und Argumentieren. In H.-G. Weigand, A. Filler, R. Hölzl, S. Kuntze, M. Ludwig, J. Roth, B. Schmidt-Thieme, & G. Wittmann (eds.), *Didaktik der Geometrie für die Sekundarstufe I* (S. 21–42). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-56217-8
# THE USE OF STRAIGHTEDGE, COMPASS, AND FIXED SHAPES TO ENHANCE THE CONTENT KNOWLEDGE OF MATHEMATICS TEACHERS IN THEIR PROFESSIONAL TRAINING IN GEOMETRY EDUCATION

Oleksiy Yevdokimov

University of Southern Queensland

The paper discusses a special approach to professional learning with a focus on building geometrical reasoning and argumentation through the extensive use of straightedge, compass and fixed shapes together with some special constructions. The novelty of this method is in combining the elements of content knowledge with specially designed rich geometry tasks, where the use of instruments takes the dominant role until the end of the solution for each task.

## INTRODUCTION AND THEORETICAL BACKGROUND

Despite the wide range of professional learning models related to geometry education that are available in literature (Jacobs et al, 2020; Smith et al, 2005) and known from practical implications (Seago et al, 2010; White et al, 2012), an appeal for new resources and educational tools that can be developed and used for professional training and development in this area remains high. The proposed model of teachers' professional development aims to answer these questions and outline further perspectives bringing attention to a special method of teacher training. The method is based on the use of special constructions from different topics of school geometry affiliated with meaningful tasks that require the active use of drawing instruments while making reasoning and argumentation. It is considered from the perspective of activity theory, where mathematical ideas are regarded as historical artifacts of human culture (Vygotsky, 1987). Artifacts may also include drawing instruments such as straightedge, compass or fixed shape(s) that may be used individually or collaboratively within a group to solve mathematical problems. The novelty of this method is in combining the elements of content knowledge with specially designed rich geometry tasks, where the use of instruments takes the dominant role until the end of the solution for each task. The active use of straightedge and compass for proof and problem-solving activities in geometry was never under a question mark in geometry education. However, its importance diminished within the last several decades. As a result of that, school graduates in Australia, even those who complete 'Specialist Mathematics' and 'Mathematical Methods' – the two most challenging mathematical subjects in Grades 11 and 12, can hardly use a straightedge and compass beyond drawing an initial figure for a geometry task. In other words, these instruments are mainly considered as drawing instruments that do not have any relation to reasoning and argumentation in geometry education. A very similar situation takes place with teachers. The drawing instruments provide unlimited opportunities in school geometry which are not explored often, unfortunately. They help to see appropriate steps towards solutions in a more logical way and, more importantly, help to connect different parts of geometric knowledge together and look at the same piece of content from different perspectives. There is a consensus among mathematics teachers and educators that geometrical facts and statements
# THE USE OF STRAIGHTEDGE, COMPASS, AND FIXED SHAPES TO ENHANCE THE CONTENT KNOWLEDGE OF MATHEMATICS TEACHERS IN THEIR PROFESSIONAL TRAINING IN GEOMETRY EDUCATION

Oleksiy Yevdokimov

University of Southern Queensland

The paper discusses a special approach to professional learning with a focus on building geometrical reasoning and argumentation through the extensive use of straightedge, compass and fixed shapes together with some special constructions. The novelty of this method is in combining the elements of content knowledge with specially designed rich geometry tasks, where the use of instruments takes the dominant role until the end of the solution for each task.

## INTRODUCTION AND THEORETICAL BACKGROUND

Despite the wide range of professional learning models related to geometry education that are available in literature (Jacobs et al, 2020; Smith et al, 2005) and known from practical implications (Seago et al, 2010; White et al, 2012), an appeal for new resources and educational tools that can be developed and used for professional training and development in this area remains high. The proposed model of teachers' professional development aims to answer these questions and outline further perspectives bringing attention to a special method of teacher training. The method is based on the use of special constructions from different topics of school geometry affiliated with meaningful tasks that require the active use of drawing instruments while making reasoning and argumentation. It is considered from the perspective of activity theory, where mathematical ideas are regarded as historical artifacts of human culture (Vygotsky, 1987). Artifacts may also include drawing instruments such as straightedge, compass or fixed shape(s) that may be used individually or collaboratively within a group to solve mathematical problems. The novelty of this method is in combining the elements of content knowledge with specially designed rich geometry tasks, where the use of instruments takes the dominant role until the end of the solution for each task. The active use of straightedge and compass for proof and problem-solving activities in geometry was never under a question mark in geometry education. However, its importance diminished within the last several decades. As a result of that, school graduates in Australia, even those who complete 'Specialist Mathematics' and 'Mathematical Methods' – the two most challenging mathematical subjects in Grades 11 and 12, can hardly use a straightedge and compass beyond drawing an initial figure for a geometry task. In other words, these instruments are mainly considered as drawing instruments that do not have any relation to reasoning and argumentation in geometry education. A very similar situation takes place with teachers. The drawing instruments provide unlimited opportunities in school geometry which are not explored often, unfortunately. They help to see appropriate steps towards solutions in a more logical way and, more importantly, help to connect different parts of geometric knowledge together and look at the same piece of content from different perspectives. There is a consensus among mathematics teachers and educators that geometrical facts and statements
that require additional constructions to be added to the initial figure for their proofs are regarded as
more challenging for students (Bartolini Bussi et al, 2009; Vondrova and Divisova, 2013; Gridos et
al, 2022). As a consequence, such situations are more difficult for teachers to manage in the classroom
and achieve desirable learning outcomes for students. This method of training aims to address these
issues so teachers can acquire practical skills in using different drawing instruments while making
reasoning which is expected to have positive effect on their content knowledge and enrich their
professional experience. Specifically, the paper addresses the following research question: How can
a program of professional development for mathematics teachers in geometry education integrate
geometrical constructions, meaningful tasks and the use of drawing instruments to enhance content
knowledge of mathematics teachers and their skills in geometrical reasoning?

THE METHOD OF PROFESSIONAL TRAINING AND DEVELOPMENT

A geometrical construction representing some special property or fact of Euclidean geometry is
assigned with a task to complete using a straightedge, or straightedge and compass, or fixed shape(s).
A straightedge can be used to construct a line or a line segment through two points. A straightedge
cannot be used for measuring any distance. It is also called a ruler (without scaling). A compass can
construct a circle or some arc of a circle. However, its main functionality is in measuring distances
which cannot be carried out with a straightedge. Fixed shapes represent special geometrical shapes
that can be used for drawing and remain unchanged. If a fixed shape contains a straight part of its
shape, it can be used for constructing line segments. For example, a fixed shape of a right-angled
triangle I discuss later can be used for constructing right angles or line segments up to the length of
its hypotenuse. For practical usage, fixed shapes can be made of cardboard or veneer. These
instruments play a central role in this method of professional training. A card with a special
construction is given to the participants of the training as well as a card with the task to work on. In
some cases, a proof (or justification) of the given construction is provided together with its statement,
in other cases participants are encouraged to prove some constructions on their own before moving
on to work on the tasks. Some cards may contain a set of constructions rather than just a single
construction as it happens for some topics, where a number of different constructions can be often
considered and used together. For example, many properties of circle geometry are worth to be
considered together. The solution to complete the task requires participants to use some of the
aforementioned instruments. It also requires to look for the ways of how a special construction
assigned to the task can be used as part of a solution for that task. While working on each task,
teachers refresh their knowledge of some constructions and make acquaintance with others. Whatever
their case is, they need to demonstrate the active use of the prescribed instrument(s) to complete the
task successfully. This gives participants the opportunity to be involved in constructing solutions to
the given tasks at the time when they are involved in studying some geometrical content and applying
it through their practical work. Modules containing constructions and their corresponding tasks within
a certain topic are called activities. In the next section I discuss several examples of activities set in
an increased order of difficulty. Within each activity I provide a brief analysis of what teachers were
asked to do, why that activity was important, how the use of instruments was organized to stimulate
teachers' thinking and enhance their content knowledge of the topic. Also, I identify directions for
potential questions for teachers to think and implement the tasks in learning environments and present
some participants' work and reflections on the selected tasks. The data was collected from task-based
interviews with small groups of participants and observations made in the training sessions. The
that require additional constructions to be added to the initial figure for their proofs are regarded as
more challenging for students (Bartolini Bussi et al, 2009; Vondrova and Divisova, 2013; Gridos et
al, 2022). As a consequence, such situations are more difficult for teachers to manage in the classroom
and achieve desirable learning outcomes for students. This method of training aims to address these
issues so teachers can acquire practical skills in using different drawing instruments while making
reasoning which is expected to have positive effect on their content knowledge and enrich their
professional experience. Specifically, the paper addresses the following research question: How can
a program of professional development for mathematics teachers in geometry education integrate
geometrical constructions, meaningful tasks and the use of drawing instruments to enhance content
knowledge of mathematics teachers and their skills in geometrical reasoning?

THE METHOD OF PROFESSIONAL TRAINING AND DEVELOPMENT

A geometrical construction representing some special property or fact of Euclidean geometry is
assigned with a task to complete using a straightedge, or straightedge and compass, or fixed shape(s).
A straightedge can be used to construct a line or a line segment through two points. A straightedge
cannot be used for measuring any distance. It is also called a ruler (without scaling). A compass can
construct a circle or some arc of a circle. However, its main functionality is in measuring distances
which cannot be carried out with a straightedge. Fixed shapes represent special geometrical shapes
that can be used for drawing and remain unchanged. If a fixed shape contains a straight part of its
shape, it can be used for constructing line segments. For example, a fixed shape of a right-angled
triangle I discuss later can be used for constructing right angles or line segments up to the length of
its hypotenuse. For practical usage, fixed shapes can be made of cardboard or veneer. These
instruments play a central role in this method of professional training. A card with a special
construction is given to the participants of the training as well as a card with the task to work on. In
some cases, a proof (or justification) of the given construction is provided together with its statement,
in other cases participants are encouraged to prove some constructions on their own before moving
on to work on the tasks. Some cards may contain a set of constructions rather than just a single
construction as it happens for some topics, where a number of different constructions can be often
considered and used together. For example, many properties of circle geometry are worth to be
considered together. The solution to complete the task requires participants to use some of the
aforementioned instruments. It also requires to look for the ways of how a special construction
assigned to the task can be used as part of a solution for that task. While working on each task,
teachers refresh their knowledge of some constructions and make acquaintance with others. Whatever
their case is, they need to demonstrate the active use of the prescribed instrument(s) to complete the
task successfully. This gives participants the opportunity to be involved in constructing solutions to
the given tasks at the time when they are involved in studying some geometrical content and applying
it through their practical work. Modules containing constructions and their corresponding tasks within
a certain topic are called activities. In the next section I discuss several examples of activities set in
an increased order of difficulty. Within each activity I provide a brief analysis of what teachers were
asked to do, why that activity was important, how the use of instruments was organized to stimulate
teachers' thinking and enhance their content knowledge of the topic. Also, I identify directions for
potential questions for teachers to think and implement the tasks in learning environments and present
some participants' work and reflections on the selected tasks. The data was collected from task-based
interviews with small groups of participants and observations made in the training sessions. The
teachers' tasks were analyzed at three levels: competence to solve or make progress in the proposed task, effective (or ineffective) use of the required instruments and skills to see connections between geometrical constructions assigned to the tasks and the tasks.

# THINKING GEOMETRICALLY WITH STRAIGHTEDGE, COMPASS, AND FIXED SHAPES

## Activity 1: A right angle inscribed in a circle

The first activity is based on a stand-alone, simple property of a right angle inscribed in a circle. An instrument for the first activity is a fixed shape which represents a right-angled triangle. One can use this instrument for constructing right angles or line segments of size up to the length of its hypotenuse.

## Construction

A right angle inscribed in a circle is subtended by the diameter of the circle (Figure 1).

<figure><img src="image_93.png" /><figcaption>Figure 1: Right angle inscribed in a circle</figcaption></figure>

## Task

A circle on the left side is drawn on paper and a fixed shape of the right-angled triangle on the right side is given as an instrument (Figure 2). The location of the centre of the circle is not known for participants. Using the fixed shape, construct a point which is the centre of the circle.

<figure><img src="image_94.png" /><figcaption>Figure 2: Task for Activity 1</figcaption></figure>

## Analysis

The main idea of this task is to stimulate teachers' awareness that each part of the meaningful task in geometry can be actively used in reasoning and argumentation. A geometrical object can be modified to serve the goals of the current task. The straightforward use of the construction above does not look
teachers' tasks were analyzed at three levels: competence to solve or make progress in the proposed task, effective (or ineffective) use of the required instruments and skills to see connections between geometrical constructions assigned to the tasks and the tasks.

# THINKING GEOMETRICALLY WITH STRAIGHTEDGE, COMPASS, AND FIXED SHAPES

## Activity 1: A right angle inscribed in a circle

The first activity is based on a stand-alone, simple property of a right angle inscribed in a circle. An instrument for the first activity is a fixed shape which represents a right-angled triangle. One can use this instrument for constructing right angles or line segments of size up to the length of its hypotenuse.

## Construction

A right angle inscribed in a circle is subtended by the diameter of the circle (Figure 1).

<figure><img src="image_99.png" /><figcaption>Figure 1: Right angle inscribed in a circle</figcaption></figure>

## Task

A circle on the left side is drawn on paper and a fixed shape of the right-angled triangle on the right side is given as an instrument (Figure 2). The location of the centre of the circle is not known for participants. Using the fixed shape, construct a point which is the centre of the circle.

<figure><img src="image_100.png" /><figcaption>Figure 2: Task for Activity 1</figcaption></figure>

## Analysis

The main idea of this task is to stimulate teachers' awareness that each part of the meaningful task in geometry can be actively used in reasoning and argumentation. A geometrical object can be modified to serve the goals of the current task. The straightforward use of the construction above does not look
to be workable here since the hypotenuse of the fixed shape even visually is longer than the diameter of the circle. However, the modified version of the fixed shape where the right angle only is important should work. Indeed, put the right angle vertex of the fixed shape on the circle. Then, the construction assigned with the task can be in use. Indeed, let the sides of the fixed shape intersect the circle at points A and B (Figure 3).

<figure><img src="image_95.png" /><figcaption>Figure 3: Solution</figcaption></figure>

According to the construction, *AB* is the diameter. If the right angle vertex of the fixed shape remains at the same point on the circle and the location of the fixed shape is changed by rotation, the sides of the fixed shape intersect the circle at other points, say *C* and *D*. Note that *CD* is also a diameter, distinct from *AB*. Hence, using the fixed shape, two diameters *AB* and *CD* can be constructed. Both contain the centre of the circle, so the intersection *O* of *AB* and *CD* is the centre of the circle.

This activity helps teachers to advance their knowledge of the simple property of the right angle to a higher level of understanding where they can use that property in the new teaching environment through meaningful tasks that support reasoning and proof building from basic initial ideas. For example, joining points *B* and *O* (Figure 1) contributes to potential questions for teachers to think and implement the above task to the concept of the central angle (∠BOC) and its relationship to ∠BAC, and to the concept of a tangent line at point *B*. One more potential question to problematize this task relates to constructing the midpoint of *AC* and *BD* (Figure 3) using a straightedge and compass. The transcript below shows reflections of two participants of the training about the task above.

Interviewer: Did you find a fixed shape helpful in this task?

Participant A: Yeah, it was. I see the benefits of the fixed shape of a right-angled triangle here. But, after this task, a straightedge and compass seem more meaningful to me than they were before and I can change this task in my class and go in the opposite direction. First, I can ask students to construct a circle using a compass. So, we know the centre. Then, construct a line through the centre using a straightedge. And by connecting each point of intersection the line and the circle have with a point on the circle, we get a right angle constructed.

Participant D: I think so. The solution is clear to me after discussion. A bit tricky though. I am trying to find another way to use the fixed shape here. What if we ask students to construct a circle that this fixed right-angled triangle is inscribed in?

## Activity 2: Corollary of Pythagoras Theorem

The second activity is based on the famous Pythagoras theorem. Instruments for the second activity are a straightedge and compass. This activity helps participants to look at Pythagoras theorem from a different point of view and develop more understanding of how useful Pythagoras theorem can be in situations where this famous property does not seem at first glance to be part of the solution.
to be workable here since the hypotenuse of the fixed shape even visually is longer than the diameter of the circle. However, the modified version of the fixed shape where the right angle only is important should work. Indeed, put the right angle vertex of the fixed shape on the circle. Then, the construction assigned with the task can be in use. Indeed, let the sides of the fixed shape intersect the circle at points A and B (Figure 3).

<figure><img src="image_101.png" /><figcaption>Figure 3: Solution</figcaption></figure>

According to the construction, *AB* is the diameter. If the right angle vertex of the fixed shape remains at the same point on the circle and the location of the fixed shape is changed by rotation, the sides of the fixed shape intersect the circle at other points, say *C* and *D*. Note that *CD* is also a diameter, distinct from *AB*. Hence, using the fixed shape, two diameters *AB* and *CD* can be constructed. Both contain the centre of the circle, so the intersection *O* of *AB* and *CD* is the centre of the circle.

This activity helps teachers to advance their knowledge of the simple property of the right angle to a higher level of understanding where they can use that property in the new teaching environment through meaningful tasks that support reasoning and proof building from basic initial ideas. For example, joining points *B* and *O* (Figure 1) contributes to potential questions for teachers to think and implement the above task to the concept of the central angle (∠BOC) and its relationship to ∠BAC, and to the concept of a tangent line at point *B*. One more potential question to problematize this task relates to constructing the midpoint of *AC* and *BD* (Figure 3) using a straightedge and compass. The transcript below shows reflections of two participants of the training about the task above.

Interviewer: Did you find a fixed shape helpful in this task?

Participant A: Yeah, it was. I see the benefits of the fixed shape of a right-angled triangle here. But, after this task, a straightedge and compass seem more meaningful to me than they were before and I can change this task in my class and go in the opposite direction. First, I can ask students to construct a circle using a compass. So, we know the centre. Then, construct a line through the centre using a straightedge. And by connecting each point of intersection the line and the circle have with a point on the circle, we get a right angle constructed.

Participant D: I think so. The solution is clear to me after discussion. A bit tricky though. I am trying to find another way to use the fixed shape here. What if we ask students to construct a circle that this fixed right-angled triangle is inscribed in?

## Activity 2: Corollary of Pythagoras Theorem

The second activity is based on the famous Pythagoras theorem. Instruments for the second activity are a straightedge and compass. This activity helps participants to look at Pythagoras theorem from a different point of view and develop more understanding of how useful Pythagoras theorem can be in situations where this famous property does not seem at first glance to be part of the solution.
Construction

In an acute-angled triangle *ABC* with altitude *AD* (Figure 4) the following property holds

$$AD^2 = AB^2 - BD^2 = AC^2 - CD^2.$$

<figure><img src="image_96.png" /><figcaption>Figure 4: Corollary of Pythagoras theorem</figcaption></figure>

Task

An acute-angled triangle *ABC* with altitude *AD* is given (Figure 4 again). Using a straightedge and compass, divide the triangle *ABC* into two triangles that have the equal sums of squares of their sides.

Analysis

The construction is not hidden in the figure for the task. In fact, both, construction and task, refer to the same Figure 4. However, to be connected with the task, this construction requires some preparatory work to be done before it can be used. It can be rewritten as $AB^2 + CD^2 = AC^2 + BD^2$. However, *AB* and *CD* as well as *AC* and *BD* cannot form triangles. Thus, if *CD* can be replaced with a congruent line segment emanating from *B*, say *BE*=*CD*, then the required property of the equal sums of squares could come to the scene (Figure 5).

<figure><img src="image_97.png" /><figcaption>Figure 5: Solution</figcaption></figure>

Distances can be measured with a compass, so one can construct such a point *E* on the side *BC* so that *BE*=*CD* which implies *BD*=*CE*. Thus, using a straightedge to draw *AE*, the two triangles, *ABE* and *ACE*, with the required property can be constructed. This activity helps teachers to gain confidence in using a compass for measuring equal distances when required, and constructing new points that have some special properties. It has connections with Activity 4. This task can be made more challenging for participants, if the altitude *AD* is not given in the task. The following transcript demonstrates participants' views in relation to Pythagoras theorem and the use of drawing instruments in the task above.
## Construction

In an acute-angled triangle ABC with altitude AD (Figure 4) the following property holds

$$AD^2 = AB^2 - BD^2 = AC^2 - CD^2.$$

<figure><img src="image_102.png" /><figcaption>Figure 4: Corollary of Pythagoras theorem</figcaption></figure>

## Task

An acute-angled triangle ABC with altitude AD is given (Figure 4 again). Using a straightedge and compass, divide the triangle ABC into two triangles that have the equal sums of squares of their sides.

## Analysis

The construction is not hidden in the figure for the task. In fact, both, construction and task, refer to the same Figure 4. However, to be connected with the task, this construction requires some preparatory work to be done before it can be used. It can be rewritten as $AB^2 + CD^2 = AC^2 + BD^2$. However, AB and CD as well as AC and BD cannot form triangles. Thus, if CD can be replaced with a congruent line segment emanating from B, say $BE=CD$, then the required property of the equal sums of squares could come to the scene (Figure 5).

<figure><img src="image_103.png" /><figcaption>Figure 5: Solution</figcaption></figure>

Distances can be measured with a compass, so one can construct such a point E on the side BC so that $BE=CD$ which implies $BD=CE$. Thus, using a straightedge to draw AE, the two triangles, ABE and ACE, with the required property can be constructed. This activity helps teachers to gain confidence in using a compass for measuring equal distances when required, and constructing new points that have some special properties. It has connections with Activity 4. This task can be made more challenging for participants, if the altitude AD is not given in the task. The following transcript demonstrates participants' views in relation to Pythagoras theorem and the use of drawing instruments in the task above.
Participant B: It makes sense to me. I mean that the use of compass and straightedge in tasks like this makes Pythagoras theorem alive and usable in classroom activities.

Participant J: Looks to me that Pythagoras theorem could be somehow used in the task where we constructed the centre of the circle [the task from Activity 1]. Need to think more.

## Activity 3: Symmetry of a rectangle

The third activity is based on the central symmetry of a rectangle. An instrument for the third activity is a straightedge only. This activity helps participants develop more understanding of how some sophisticated constructions can be carried out with a straightedge only.

### Construction

Let ABCD be a rectangle and O be a point where the diagonals AC and BD intersect. Then, O is the centre of symmetry of ABCD and any straight line through O divides the rectangle ABCD into two congruent parts.

### Task

Let ABCDEF be a hexagon with five angles of size 90° and one of 270° (Figure 6).

<figure><img src="image_98.png" /><figcaption>Figure 6: Task for Activity 3</figcaption></figure>

Divide the hexagon ABCDEF into two parts of equal area using a straightedge only.

### Analysis

To see the benefits and connections of the construction above with the task, participants are expected to construct a larger rectangle ABKF (Figure 7).

<figure><img src="image_99.png" /><figcaption>Figure 7: Solution</figcaption></figure>

Then, the diagonals of the rectangles ABKF and CKED can be constructed, where J and L are the points of their intersection, respectively. According to the construction above, J and L are the centres of symmetry of the rectangles ABKF and CKED respectively, so a straight line through J and L divides the hexagon ABCDEF into two parts of equal area. The next transcript summarizes the difficulties experienced by many participants.
Participant B: It makes sense to me. I mean that the use of compass and straightedge in tasks like this makes Pythagoras theorem alive and usable in classroom activities.

Participant J: Looks to me that Pythagoras theorem could be somehow used in the task where we constructed the centre of the circle [the task from Activity 1]. Need to think more.

## Activity 3: Symmetry of a rectangle

The third activity is based on the central symmetry of a rectangle. An instrument for the third activity is a straightedge only. This activity helps participants develop more understanding of how some sophisticated constructions can be carried out with a straightedge only.

### Construction

Let ABCD be a rectangle and O be a point where the diagonals AC and BD intersect. Then, O is the centre of symmetry of ABCD and any straight line through O divides the rectangle ABCD into two congruent parts.

### Task

Let ABCDEF be a hexagon with five angles of size 90° and one of 270° (Figure 6).

<figure><img src="image_104.png" /><figcaption>Figure 6: Task for Activity 3</figcaption></figure>

Divide the hexagon ABCDEF into two parts of equal area using a straightedge only.

### Analysis

To see the benefits and connections of the construction above with the task, participants are expected to construct a larger rectangle ABKF (Figure 7).

<figure><img src="image_105.png" /><figcaption>Figure 7: Solution</figcaption></figure>

Then, the diagonals of the rectangles ABKF and CKED can be constructed, where J and L are the points of their intersection, respectively. According to the construction above, J and L are the centres of symmetry of the rectangles ABKF and CKED respectively, so a straight line through J and L divides the hexagon ABCDEF into two parts of equal area. The next transcript summarizes the difficulties experienced by many participants.
Participant M: I see why I was stuck with this task... All three points that can be constructed with a straightedge are hidden here, and the conclusion is still not obvious after that... A very convincing example to demonstrate how powerful a straightedge can be in good hands.

## Activity 4: A cyclic quadrilateral

The fourth activity includes a special construction related to cyclic quadrilaterals. Instruments for this activity are a straightedge and compass. This activity helps participants develop more understanding for the case where additional constructions need to be completed.

### Construction

A cyclic quadrilateral *ABDC* is such that *AB* > *AC* and *D* is the midpoint of the arc *BC* not containing *A*. Then, there exists a point *K* on the side *AB* such that the quadrilateral *ABDC* can be divided into three isosceles triangles, *ACK*, *DKC* and *DBK* (Figure 8).

<figure><img src="image_100.png" /><figcaption>Figure 8: A property of a cyclic quadrilateral</figcaption></figure>

Similarly to Activity 3, participants need to prove this property first, and then work on the task that follows. At first, using a compass, point *K* is constructed on *AB* such that *AK*=*AC*, so participants need to show that the triangles *DKC* and *DBK* are isosceles. Using a straightedge to connect *A* and *D*, and considering congruent triangles *AKD* and *ACD* (SAS) one can conclude that *CD*=*DK* which implies that triangles *DKC* and *DBK* are isosceles.

### Task

Given triangle ABC, where *AB* > *AC*, and its circumcircle. Using a straightedge and compass, construct the midpoint of the arc *BC* not containing *A*. Both instruments can be used just once.

### Analysis

The main challenge for participants remains the same as in Activity 3 – to find out where the construction above is hidden in the task. Extending *DK* beyond *K*, let *DK* intersect the circumcircle of *ABC* at point *L*. Taking into account that angles subtended by the same arc are equal, and vertically opposite angles are equal, $\angle ALK = \angle ALD = \angle ABD = \angle KBD = \angle BKD = \angle AKL$ which implies $AL=AK=AC$. This paves the way to complete the task using a compass to construct a circle of radius *AC* with the centre at *A*, and then a straightedge for a line through the points *K* and *L* (Figure 9).
Participant M: I see why I was stuck with this task... All three points that can be constructed with a straightedge are hidden here, and the conclusion is still not obvious after that... A very convincing example to demonstrate how powerful a straightedge can be in good hands.

## Activity 4: A cyclic quadrilateral

The fourth activity includes a special construction related to cyclic quadrilaterals. Instruments for this activity are a straightedge and compass. This activity helps participants develop more understanding for the case where additional constructions need to be completed.

### Construction

A cyclic quadrilateral *ABDC* is such that *AB* > *AC* and *D* is the midpoint of the arc *BC* not containing *A*. Then, there exists a point *K* on the side *AB* such that the quadrilateral *ABDC* can be divided into three isosceles triangles, *ACK*, *DKC* and *DBK* (Figure 8).

<figure><img src="image_106.png" /><figcaption>Figure 8: A property of a cyclic quadrilateral</figcaption></figure>

Similarly to Activity 3, participants need to prove this property first, and then work on the task that follows. At first, using a compass, point *K* is constructed on *AB* such that *AK=AC*, so participants need to show that the triangles *DKC* and *DBK* are isosceles. Using a straightedge to connect *A* and *D*, and considering congruent triangles *AKD* and *ACD* (SAS) one can conclude that *CD=DK* which implies that triangles *DKC* and *DBK* are isosceles.

### Task

Given triangle ABC, where $AB > AC$, and its circumcircle. Using a straightedge and compass, construct the midpoint of the arc *BC* not containing *A*. Both instruments can be used just once.

### Analysis

The main challenge for participants remains the same as in Activity 3 – to find out where the construction above is hidden in the task. Extending *DK* beyond *K*, let *DK* intersect the circumcircle of *ABC* at point *L*. Taking into account that angles subtended by the same arc are equal, and vertically opposite angles are equal, $\angle ALK = \angle ALD = \angle ABD = \angle KBD = \angle BKD = \angle AKL$ which implies $AL=AK=AC$. This paves the way to complete the task using a compass to construct a circle of radius *AC* with the centre at *A*, and then a straightedge for a line through the points *K* and *L* (Figure 9).
<figure><img src="image_101.png" /><figcaption>Figure 9: Solution</figcaption></figure>

The following short transcript summarizes the most common responses received from participants.

Interviewer: What useful conclusions did you draw from this task?

Participant C: I think I got it. Isosceles triangles represent a good example of how some constructions with a compass can be interpreted. Using one vertex of a triangle as the centre of a circle and the equal sides as radius, we can construct the isosceles triangle that we need.

References

Bartolini Bussi, M.G., Gade, S., Janvier, M., Kahane, J.-P., Matsko, V.J., Maschietto, M., Ouvrier-Buffet, C. & Saul, M. (2009). Mathematics in context: Focusing on students. In P. Taylor & E. Barbeau, (Eds.), *Challenging mathematics in and beyond the classroom*. New ICMI Study Series, (Vol. 12, pp. 171-204). Springer. https://doi.org/10.1007/978-0-387-09603-2_6

Gridos, P., Avgerinos, E., Mamona-Downs, J., & Vlachou, R. (2022). geometrical figure apprehension, construction of auxiliary lines, and multiple solutions in problem solving: Aspects of mathematical creativity in school geometry. *International Journal of Science and Mathematics Education, 20*, 619-636. https://doi.org/10.1007/s10763-021-10155-4

Jacobs, J. K., Koellner, K., Seago, N., Garnier, H., & Wang, C. (2020). Professional Development to Support the Learning and Teaching of Geometry. In P.M. Jenlink, (Ed.), *The Language of Mathematics: How the Teacher's Knowledge of Mathematics Affects Instruction* (pp. 143-173). Rowman & Littlefield.

Seago, N., Jacobs, J., & Driscoll, M. (2010). Transforming middle school geometry: Designing professional development materials that support the teaching and learning of similarity. *Middle Grades Research Journal, 5*(4), 199-211.

Smith, M. S., Silver, E. A., & Stein, M. K. (2005). *Improving instruction in geometry and measurement*, Vol. 3. Teachers College Press.

Vondrova, N., Divisova, B. (2013). Strategies for a certain type of geometric problems solvable without calculations. *Procedia – Social and Behavioral Sciences, 93*, 400-404.

Vygotsky, L.S. (1987). Thinking and speech. In R.W. Rieber & A. S. Carton, (Eds.), *The collected works of L. S. Vygotsky*, (Vol. 1, pp. 39-285), *Problems of general psychology*. Plenum Press.

White, A. L., Jaworski, B., Agudelo-Valderrama, C., & Gooya, Z. (2012). Teachers learning from teachers. In M.A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung, (Eds.), *Third international handbook of mathematics education*, (Vol. B, pp. 393-430). Springer.
<figure><img src="image_107.png" /><figcaption>Figure 9: Solution</figcaption></figure>

The following short transcript summarizes the most common responses received from participants.

Interviewer: What useful conclusions did you draw from this task?

Participant C: I think I got it. Isosceles triangles represent a good example of how some constructions with a compass can be interpreted. Using one vertex of a triangle as the centre of a circle and the equal sides as radius, we can construct the isosceles triangle that we need.

References

Bartolini Bussi, M.G., Gade, S., Janvier, M., Kahane, J.-P., Matsko, V.J., Maschietto, M., Ouvrier-Buffet, C. & Saul, M. (2009). Mathematics in context: Focusing on students. In P. Taylor & E. Barbeau, (Eds.), *Challenging mathematics in and beyond the classroom*. New ICMI Study Series, (Vol. 12, pp. 171-204). Springer. https://doi.org/10.1007/978-0-387-09603-2_6

Gridos, P., Avgerinos, E., Mamona-Downs, J., & Vlachou, R. (2022). geometrical figure apprehension, construction of auxiliary lines, and multiple solutions in problem solving: Aspects of mathematical creativity in school geometry. *International Journal of Science and Mathematics Education, 20*, 619-636. https://doi.org/10.1007/s10763-021-10155-4

Jacobs, J. K., Koellner, K., Seago, N., Garnier, H., & Wang, C. (2020). Professional Development to Support the Learning and Teaching of Geometry. In P.M. Jenlink, (Ed.), *The Language of Mathematics: How the Teacher's Knowledge of Mathematics Affects Instruction* (pp. 143-173). Rowman & Littlefield.

Seago, N., Jacobs, J., & Driscoll, M. (2010). Transforming middle school geometry: Designing professional development materials that support the teaching and learning of similarity. *Middle Grades Research Journal, 5*(4), 199-211.

Smith, M. S., Silver, E. A., & Stein, M. K. (2005). *Improving instruction in geometry and measurement*, Vol. 3. Teachers College Press.

Vondrova, N., Divisova, B. (2013). Strategies for a certain type of geometric problems solvable without calculations. *Procedia – Social and Behavioral Sciences, 93*, 400-404.

Vygotsky, L.S. (1987). Thinking and speech. In R.W. Rieber & A. S. Carton, (Eds.), *The collected works of L. S. Vygotsky*, (Vol. 1, pp. 39-285), *Problems of general psychology*. Plenum Press.

White, A. L., Jaworski, B., Agudelo-Valderrama, C., & Gooya, Z. (2012). Teachers learning from teachers. In M.A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung, (Eds.), *Third international handbook of mathematics education*, (Vol. B, pp. 393-430). Springer.
# TOPIC C

## Resources for teaching and learning geometry
TOPIC C

Resources for teaching and
learning geometry
<figure><img src="image_102.png" /></figure>
<figure><img src="image_108.png" /></figure>
# DESTRUCTURING / RESTRUCTURING OF A VIRTUAL GEOMETRIC ARTIFACT: THE CASE OF THE NUMBER LINE

Tullio Aebischer

Ph.D. School, University of Rome Tor Vergata, Dept. of Mathematics

The use of artifacts in mathematics teaching has become a widely used practice with the introduction of laboratory activities. In this regard, even the advent of information technologies has given impetus to both research and new educational aspects. In particular, the use of artifacts virtually manipulated with the DGS (Dynamic Geometry Software) opens up new scenarios available to both teachers and students. In this work we will present an alternative teaching proposal of manipulating a virtual artifact: its destructuring (reduction into its fundamental elements) and its restructuring (reconstruction of the same artifact) to give students the possibility of also grasping the intrinsic information. We will analyze this didactic proposal of manipulative typology of an artifact in the case of the Number Line.

## ARTIFACT AND ITS MANIPULATION

Mathematical concepts fundamentally face two difficulties in the communication phase. The first is the need for a formal language supported by a natural one that conveys it. From this it can be deduced that mathematics also needs words. The second is that its foundations do not have a counterpart in reality but only a representation of them, if one wants to adhere to Platonic vision (D'Amore, 2003). This situation, which involves the Author from a professional point of view as an Italian secondary school teacher, has prompted throughout history the 'materialization' of the concepts themselves into objects that contained this concept or were effective mediators of it for the students' use. According to what has been said, this materialization is achieved with objects called artifacts.

Artifacts can be of the most diverse shapes, built with the most varied materials and digital. Some are in common use (abacus, ruler, compass, etc.) others are used specifically for educational purposes as we will see in the case of the Number Line (Frykholm, 2010).

The useful theoretical reference regarding the didactic use of artifacts is in the Theory of Semiotic Mediation (Bartolini Bussi & Mariotti, 2008). Here the artifact differs from the tool where the first is a product of Man while the second results from the union of an artifact and a pattern of use (Mariotti & Maffia, 2018). The aforementioned theory analyzes the use of the artifact by the students and the process they follow in solving an assigned task. As far as we are concerned, artifacts can be considered both concrete and digital. In the use of the artifact, its semiotic potential is also highlighted, i.e. its close link between the meaning that is evoked with its use and the mathematical meaning itself evoked. In this context, the teacher has the role of activity planning and guide. The result is that the manipulation of an artifact puts almost the entire body and mind in motion and in relation in the student, involving manual skills, vision, the coordination between the two, cognition and metacognition.
# DESTRUCTURING / RESTRUCTURING OF A VIRTUAL GEOMETRIC ARTIFACT: THE CASE OF THE NUMBER LINE

Tullio Aebischer

Ph.D. School, University of Rome Tor Vergata, Dept. of Mathematics

The use of artifacts in mathematics teaching has become a widely used practice with the introduction of laboratory activities. In this regard, even the advent of information technologies has given impetus to both research and new educational aspects. In particular, the use of artifacts virtually manipulated with the DGS (Dynamic Geometry Software) opens up new scenarios available to both teachers and students. In this work we will present an alternative teaching proposal of manipulating a virtual artifact: its destructuring (reduction into its fundamental elements) and its restructuring (reconstruction of the same artifact) to give students the possibility of also grasping the intrinsic information. We will analyze this didactic proposal of manipulative typology of an artifact in the case of the Number Line.

## ARTIFACT AND ITS MANIPULATION

Mathematical concepts fundamentally face two difficulties in the communication phase. The first is the need for a formal language supported by a natural one that conveys it. From this it can be deduced that mathematics also needs words. The second is that its foundations do not have a counterpart in reality but only a representation of them, if one wants to adhere to Platonic vision (D'Amore, 2003). This situation, which involves the Author from a professional point of view as an Italian secondary school teacher, has prompted throughout history the 'materialization' of the concepts themselves into objects that contained this concept or were effective mediators of it for the students' use. According to what has been said, this materialization is achieved with objects called artifacts.

Artifacts can be of the most diverse shapes, built with the most varied materials and digital. Some are in common use (abacus, ruler, compass, etc.) others are used specifically for educational purposes as we will see in the case of the Number Line (Frykholm, 2010).

The useful theoretical reference regarding the didactic use of artifacts is in the Theory of Semiotic Mediation (Bartolini Bussi & Mariotti, 2008). Here the artifact differs from the tool where the first is a product of Man while the second results from the union of an artifact and a pattern of use (Mariotti & Maffia, 2018). The aforementioned theory analyzes the use of the artifact by the students and the process they follow in solving an assigned task. As far as we are concerned, artifacts can be considered both concrete and digital. In the use of the artifact, its semiotic potential is also highlighted, i.e. its close link between the meaning that is evoked with its use and the mathematical meaning itself evoked. In this context, the teacher has the role of activity planning and guide. The result is that the manipulation of an artifact puts almost the entire body and mind in motion and in relation in the student, involving manual skills, vision, the coordination between the two, cognition and metacognition.
THE GEOMETRICAL NUMBER LINE AND THE NUMBER THEORY

The Number Line is an ideal didactic artifact practically unchanged from the 17th century since it was first used by J. Wallis (1616-1703) in *Treatise of algebra* (1685). In his drawing the negative part was indicated with the dotted line (Figure 1). For this reason it can be defined as a fossil artifact. However, it does not seem that Wallis had designed the segment with the intention of placing the known numbers to order them but only as an example of a 'road' traveled in one direction and/or the other, i.e. more as a reference system for motion on which, then, he operated with the relative integers.

<figure><img src="image_103.png" /><figcaption>Figure 1: The first Number Line published by the English mathematician J. Wallis in 1685</figcaption></figure>

The Number Line must be considered fundamentally abstract since its definition requires geometric points and ideally it has an infinite extension, which cannot occur in material translation. Therefore the artifact that physically finds itself in the hands of the students is a materialisation of those ideal geometric concepts.

The choice to study the Number Line derives from the Author's professional experience regarding the problems of secondary school students about numbers. It was observed that:

* numbers are all considered 'the same', highlighting the predilection for the 'integer' and/or 'decimal' forms as a consequence of the use of the calculator.

* difficulty in managing numbers written in 'mixed' form in which numbers, signs and symbols must be considered at the same time: $-7 \ 2^n \frac{4}{5} \ 2\pi \sqrt{2}$.

* the non-differentiation of numbers leads to a non-deep understanding of their use in certain fields (e.g. Science → measurement).

* the incomplete knowledge of the typology of numbers results in the inaccurate application and understanding of approximation techniques.

* difficulty in ordering the numbers due to their different writing.

Therefore, the geometrical visualization of numbers can be, together with their different way of drawing them, an aid to move with greater awareness in the fundamental notions of mathematics.

Before drawing them, let's see which numerical sets we can place on the Number Line. From number theory we have the following set relationship:

$$ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \equiv \{\mathbb{I}_A \cup \mathbb{I}_{NA} \wedge (\mathbb{I}_A \cap \mathbb{I}_{NA} \equiv \emptyset)\} \subset \mathbb{C} \equiv \{\Re(e(C)) + \Im(C)\} $$

where $\mathbb{I}_A$ are the algebraic irrationals (numbers that cannot be expressed as rationals but solution of polynomial equations with rational coefficients) and $\mathbb{I}_{NA}$ non-algebraic or transcendent irrationals (e.g. $e, \pi$). It should be noted that $\mathfrak{Im}(C)$ cannot be reported on the Number Line since the Argand-Gauss plane is required. Therefore, only the complexes with zero imaginary part, i.e. the real ones, will be considered.
THE GEOMETRICAL NUMBER LINE AND THE NUMBER THEORY

The Number Line is an ideal didactic artifact practically unchanged from the 17th century since it was first used by J. Wallis (1616-1703) in *Treatise of algebra* (1685). In his drawing the negative part was indicated with the dotted line (Figure 1). For this reason it can be defined as a fossil artifact. However, it does not seem that Wallis had designed the segment with the intention of placing the known numbers to order them but only as an example of a 'road' traveled in one direction and/or the other, i.e. more as a reference system for motion on which, then, he operated with the relative integers.

<figure><img src="image_109.png" /><figcaption>Figure 1: The first Number Line published by the English mathematician J. Wallis in 1685</figcaption></figure>

The Number Line must be considered fundamentally abstract since its definition requires geometric points and ideally it has an infinite extension, which cannot occur in material translation. Therefore the artifact that physically finds itself in the hands of the students is a materialisation of those ideal geometric concepts.

The choice to study the Number Line derives from the Author's professional experience regarding the problems of secondary school students about numbers. It was observed that:

* numbers are all considered 'the same', highlighting the predilection for the 'integer' and/or 'decimal' forms as a consequence of the use of the calculator.

* difficulty in managing numbers written in 'mixed' form in which numbers, signs and symbols must be considered at the same time: $-7 \quad 2^n \quad \frac{4}{5} \quad 2\pi \quad \sqrt{2}$.

* the non-differentiation of numbers leads to a non-deep understanding of their use in certain fields (e.g. Science → measurement).

* the incomplete knowledge of the typology of numbers results in the inaccurate application and understanding of approximation techniques.

* difficulty in ordering the numbers due to their different writing.

Therefore, the geometrical visualization of numbers can be, together with their different way of drawing them, an aid to move with greater awareness in the fundamental notions of mathematics.

Before drawing them, let's see which numerical sets we can place on the Number Line. From number theory we have the following set relationship:

$$ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \equiv \{\mathbb{I}_A \cup \mathbb{I}_{NA} \wedge (\mathbb{I}_A \cap \mathbb{I}_{NA} \equiv \emptyset)\} \subset \mathbb{C} \equiv \{\Re(e(\mathbb{C})) + \Im(\mathbb{C})\} $$

where $\mathbb{I}_A$ are the algebraic irrationals (numbers that cannot be expressed as rationals but solution of polynomial equations with rational coefficients) and $\mathbb{I}_{NA}$ non-algebraic or transcendent irrationals (e.g. $e, \pi$). It should be noted that $\mathfrak{Im}(\mathbb{C})$ cannot be reported on the Number Line since the Argand-Gauss plane is required. Therefore, only the complexes with zero imaginary part, i.e. the real ones, will be considered.
The drawing of the Number Line will follow the method described in Euclid's *Elements* (3rd century BC) using only straightedge and compass (Hartshorne, 2000). Ruler and compass for the Greeks were simply tools for tracing straight segments and not for measuring lengths, for tracing circles with a given center and passing through a given point or having a given segment as a radius. These were ideal tools for realizing properties of incidence between points, lines and circles. Their exclusive use started a fruitful analysis of those problems that could not be solved with this method such as the trisection of the angle, the squaring of the circle or the duplication of the cube.

## DESTRUCTURING / RESTRUCTURING OF AN ARTIFACT

We have seen how the artifact must be given to the students with an assignment. The achievement of the task is documented in various ways: written, oral, multimedia. All this allows the teacher to convey the external information *x* of the artifact. It must, however, be taken into consideration that the restructuring of the artifact itself was possible with another knowledge which in the moment of its final realization became intrinsic and not immediately capturable in the teaching activity (Rizzo, 2000). In order to exploit the full potential of the artifact it is necessary to also be able to transmit its intrinsic information *y*.

<figure><img src="image_104.png" /><figcaption>Figure 2: Logical diagram of the destructuring / restructuring of an artifact</figcaption></figure>

The activity of deconstructing and restructuring the artifact begins in the classic way with the teacher delivering the artifact to the student (Figure 2). The student's activity operates on the artifact to conclude with the acquisition of information *x*. At this point, part of the artifact's metacognitive potential is lost. To make the intrinsic information of the artifact usable, the student must deconstruct the artifact itself (not a similar one!; see dotted arrow in Figure 2) to dissecting it into its fundamental parts. The destructuring phase is followed by the restructuring phase during which the student reconstructs the artifact by acquiring the two types of information *x* and *y*.

With this methodological approach, the Number Line is a useful geometric artifact that lends itself very well to the aforementioned logical scheme if transformed into a virtual artifact, i.e. if it is built and manipulated in a DGS environment like GeoGebra®, the environment used for our purposes.

The problem with the geometric Number Line in school textbooks is that this, apart from some variations in the drawing such as the presence of double arrows at the 'extremes' which create disorientation, is date already drawn with the numbers positioned. The delivery of such an artifact already drawn also involves the total disregard of the fact that, as will be highlighted, one does not discriminate between constructible numbers (K set) and those that cannot be built with straightedge and compass. The placement of the various dashes with the numerical equivalents is not defined, remaining an unexplained fact and to be accepted uncritically. Ultimately, the use of the Number Line in school textbooks is weakened. In fact, it is used in primary and the first year of secondary school,
The drawing of the Number Line will follow the method described in Euclid's *Elements* (3rd century BC) using only straightedge and compass (Hartshorne, 2000). Ruler and compass for the Greeks were simply tools for tracing straight segments and not for measuring lengths, for tracing circles with a given center and passing through a given point or having a given segment as a radius. These were ideal tools for realizing properties of incidence between points, lines and circles. Their exclusive use started a fruitful analysis of those problems that could not be solved with this method such as the trisection of the angle, the squaring of the circle or the duplication of the cube.

## DESTRUCTURING / RESTRUCTURING OF AN ARTIFACT

We have seen how the artifact must be given to the students with an assignment. The achievement of the task is documented in various ways: written, oral, multimedia. All this allows the teacher to convey the external information x of the artifact. It must, however, be taken into consideration that the restructuring of the artifact itself was possible with another knowledge which in the moment of its final realization became intrinsic and not immediately capturable in the teaching activity (Rizzo, 2000). In order to exploit the full potential of the artifact it is necessary to also be able to transmit its intrinsic information y.

<figure><img src="image_110.png" /><figcaption>Figure 2: Logical diagram of the destructuring / restructuring of an artifact</figcaption></figure>

The activity of deconstructing and restructuring the artifact begins in the classic way with the teacher delivering the artifact to the student (Figure 2). The student's activity operates on the artifact to conclude with the acquisition of information x. At this point, part of the artifact's metacognitive potential is lost. To make the intrinsic information of the artifact usable, the student must deconstruct the artifact itself (not a similar one!; see dotted arrow in Figure 2) to dissecting it into its fundamental parts. The destructuring phase is followed by the restructuring phase during which the student reconstructs the artifact by acquiring the two types of information x and y.

With this methodological approach, the Number Line is a useful geometric artifact that lends itself very well to the aforementioned logical scheme if transformed into a virtual artifact, i.e. if it is built and manipulated in a DGS environment like GeoGebra®, the environment used for our purposes.

The problem with the geometric Number Line in school textbooks is that this, apart from some variations in the drawing such as the presence of double arrows at the 'extremes' which create disorientation, is date already drawn with the numbers positioned. The delivery of such an artifact already drawn also involves the total disregard of the fact that, as will be highlighted, one does not discriminate between constructible numbers (K set) and those that cannot be built with straightedge and compass. The placement of the various dashes with the numerical equivalents is not defined, remaining an unexplained fact and to be accepted uncritically. Ultimately, the use of the Number Line in school textbooks is weakened. In fact, it is used in primary and the first year of secondary school,