so by us. The author quoted expressed the view that, for example, it is useful to give students "impossible" problems, such as computing the area of a right triangle, given one of its legs and the projections of both of its legs onto the hypotenuse (it is evident that there are too many givens here). This, in his view, develops their capacity for observation and for working in the real world, where problems are not necessarily correctly posed. The type "Looking" lends itself especially to the study of three-dimensional geometry, in which it is important to be able to reach a conclusion by looking at a drawing.

Note that in addition to the types listed above, there are also others, and combinations of the types represented above are possible as well. As an example, consider the following problem (Werner & Karp, 2019, p. 336):

Consider four points in space, A, B, C, D. We would like to construct all planes that contain at least three of these points. It is not true that these four points could be chosen such that it would be possible to construct:

a) only one plane;

b) only two planes;

c) only four planes;

d) infinitely many planes.

This is a kind of multiple-choice problem, but what it requires of the student is not only the answer, but also the reasoning behind it. In other words, the student must imagine (draw!) such positions for the four points for which it will be possible to construct one, four, or infinitely many planes, and then to prove that it will be impossible to construct only two planes. This is a kind of investigation problem, involving various kinds of activity.

Working teachers, in our view, must be familiar with different types of activities in which students are involved in the process of studying geometry, without limiting themselves to the classical types. Our conception of geometry today is broader than it was two thousand years ago. Without giving up classical problems involving proofs and calculations, we should broaden the palette of possibilities – which poses new problems both for textbook authors and for teacher educators.

## References

* Anon. (1849). [Review]. *Otechestvennye zapiski, LXIV*, Part VI, p. 45.

* Atanasyan, L.S., Butuzov, V.F., Kadomtsev, S.B., Shestakov, S.A., & Yudina, I.I. (1996). *Geometriya 8. Dopolnitelnye glavy k shkol'nomu uchebniku* [Geometry 8. Additional Chapters for the school textbook]. Prosveschenie.

* Atanasyan, L.S., Butuzov, V.F., Kadomtsev, S.B., Kiseleva, L.S., & Poznyak, E. G. (2000). *Geometriya 10-11*. [Geometry 10-11]. Prosveschenie.

* Atanasyan, L.S., Butuzov, V.F., Kadomtsev, S.B., Poznyak, E. G., & Yudina, I.I. (2004). *Geometriya 7-9* [Geometry 7-9]. Prosveschenie.

* Høyrup, J. (2014). *Mathematics Education in the European Middle Ages*. In A. Karp & G. Schubring (Eds.), *Handbook on the history of mathematics education* (pp. 109-127). Springer.

* Jäder, J., Lithner, J., & Sidenvall, J. (2020). Mathematical problem solving in textbooks from twelve countries. *International Journal of Mathematical Education in Science & Technology, 51*(7), 1120-1136.
Karp, A. (2023). Problem sets in school textbooks: Examples from the United States. In R. Leikin (Ed.), *Mathematical challenges for all* (pp. 459-479). Springer.

Karp, A. (2020). Highest mathematics: How mathematics was taught to future Russian tsars. In E. Barbin et al. (Eds.). "*Dig where you stand*" 6 (pp. 115-128). WTM.

Karp, A. (2015). Problems in old Russian textbooks: How they were selected In K. Bjarnadottir et al. (Eds.), "*Dig where you stand*" 3 (pp. 203-218). Uppsala University.

Karp, A. (2012). Soviet mathematics education between 1918 and 1931: A time of radical reforms. *ZDM/International Mathematics Education,44*(4), 551-561.

Karp, A., & Werner, A. (2011). On the teaching of geometry in Russia. In A. Karp & B. Vogeli (Eds.), *Russian mathematics education: Programs and practices* (pp. 81-128). World Scientific Publishing.

Krutetskii, V. A. (1976). *The psychology of mathematical abilities in schoolchildren*. The University of Chicago Press.

Leikin, R. (Ed.). (2023). *Mathematical challenges for all*. Springer.

Pogorelov, A. (1985). *Geometriya 6-10* [Geometry 6-10]. Prosveschenie.

Rezat, S., Fan, L., Hattermann, M., Schumacher, J., & Wuschke, H. (Eds.). (2019). *Proceedings of the Third International Conference on Mathematics Textbook Research and Development*. Universitätsbibliothek Paderborn.

Schubring, G. (2023). *Analysing historical mathematics textbooks*. Springer.

Sharygin, I.F. (1999). *Geometriya 10-11*. [Geometry 10-11]. Drofa.

Smirnova I.M., & Smirnov, V.A. (2001). *Geometriya 7-9* [Geometry 7-9]. Prosveschenie.

Stylianides, G. (2009). Reasoning-and-Proving in School Mathematics Textbooks. *Mathematical Thinking and Learning, 11*(4), 258-288.

Usiskin, Z. (2014). Transformations in U.S. commercial high school geometry textbooks since 1960: A brief report. In K. Jones, C. Bokhove, G. Howson & L. Fan (Eds.), *Proceedings of the International Conference on Mathematics Textbook Research and Development* (pp. 471-476). Southampton Education School, University of Southampton.

Werner, A., & Karp, A. (2019). *Matematika 10*. [Mathematics for Grade 10]. Prosveschenie.
Karp, A. (2023). Problem sets in school textbooks: Examples from the United States. In R. Leikin (Ed.), *Mathematical challenges for all* (pp. 459-479). Springer.

Karp, A. (2020). Highest mathematics: How mathematics was taught to future Russian tsars. In E. Barbin et al. (Eds.). "Dig where you stand" 6 (pp. 115-128). WTM.

Karp, A. (2015). Problems in old Russian textbooks: How they were selected In K. Bjarnadottir et al. (Eds.), "Dig where you stand" 3 (pp. 203-218). Uppsala University.

Karp, A. (2012). Soviet mathematics education between 1918 and 1931: A time of radical reforms. *ZDM/International Mathematics Education,44*(4), 551-561.

Karp, A., & Werner, A. (2011). On the teaching of geometry in Russia. In A. Karp & B. Vogeli (Eds.), *Russian mathematics education: Programs and practices* (pp. 81-128). World Scientific Publishing.

Krutetskii, V. A. (1976). *The psychology of mathematical abilities in schoolchildren*. The University of Chicago Press.

Leikin, R. (Ed.). (2023). *Mathematical challenges for all*. Springer.

Pogorelov, A. (1985). *Geometriya 6-10* [Geometry 6-10]. Prosveschenie.

Rezat, S., Fan, L., Hattermann, M., Schumacher, J., & Wuschke, H. (Eds.). (2019). *Proceedings of the Third International Conference on Mathematics Textbook Research and Development*. Universitätsbibliothek Paderborn.

Schubring, G. (2023). *Analysing historical mathematics textbooks*. Springer.

Sharygin, I.F. (1999). *Geometriya 10-11*. [Geometry 10-11]. Drofa.

Smirnova I.M., & Smirnov, V.A. (2001). *Geometriya 7-9* [Geometry 7-9]. Prosveschenie.

Stylianides, G. (2009). Reasoning-and-Proving in School Mathematics Textbooks. *Mathematical Thinking and Learning, 11*(4), 258-288.

Usiskin, Z. (2014). Transformations in U.S. commercial high school geometry textbooks since 1960: A brief report. In K. Jones, C. Bokhove, G. Howson & L. Fan (Eds.), *Proceedings of the International Conference on Mathematics Textbook Research and Development* (pp. 471-476). Southampton Education School, University of Southampton.

Werner, A., & Karp, A. (2019). *Matematika 10*. [Mathematics for Grade 10]. Prosveschenie.
# CASUAL AND GEOMETRIC PRAXEOLOGIES IN A STUDY OF SYMMETRIC FIGURES

Margo Kondratieva

Memorial University, Canada

In this paper we collect various solutions produced by in-service teachers in response to problems involving regular polygons. These solutions allow us to identify the most popular ideas found in the contemporary K-12 geometry curriculum. In line with previous research, we found that while the figures in question possess axial symmetry, this property is very rarely used by teachers. Despite the fact that the concept of symmetry is familiar to teachers at the intuitive level, comprehension of this concept is lacking the mathematical precision. Thus, this visual characteristic of the objects of study fails to be employed in teachers' geometric reasoning.

## INTRODUCTION

Symmetric figures do not change when reflected, rotated or translated in a certain way. This property could be both intriguing for human visual perception and useful for our practical and theoretical needs. Symmetry “creates patterns that help us organize our world conceptually” (Knuchel, 2004). The idea of symmetry connects between various branches of mathematics and plays an important role in problem solving (Leikin et al, 2000). In addition, “symmetry is a paramount tool for bridging the gap between science and art” (Livio, 2005). Since ancient times of Aristotle and Plato symmetry has been associated with beauty as well as played a significant role in architectural design.

Psychologists investigated the relationship between symmetry and aesthetic preference by asking people to make a “visually pleasing” arrangement of objects given certain constrains. It was found that the majority of arrangements were symmetrical in nature (see e.g. Szilagyi & Baird, 1977). Birkhoff (1933) attempted to develop a mathematical theory of aesthetic value. He proposed that at a given level of complexity, the aesthetic value is higher for more symmetric objects. The role of symmetry in perception was stressed by Gestalt psychology. The principle of Prägnanz (or “good figure”) states: “of several possible geometric organizations, the one that is seen is the one that possesses the best simplest and most stable shape” (Livio, 2005), which in practice oftentimes is symmetric. For example, a picture showing four dots which are vertices of a square is seen as a square - a symmetric, stable and closed shape. A perceived stability may be related to the fact that frequently physical systems possessing symmetry are stable because they furnish minimum potential energy.

Mirror-reflection symmetry is a specific form of symmetry which characterizes the animal kingdom. The prevalence of the bilateral symmetry in animal appearance for many million years (which is evidenced by both fossils and contemporary leaving creatures) could be explained by evolutionary biology and laws of physics. Gravity and locomotion produce distinctions between top and bottom or back and front. However, the animals' environment is much more homogeneous when it comes to distinguishing between left and right. Therefore, it is not surprising that the most popular interpretation of the word “symmetry” actually is reflection symmetry. According to Webster's Third
# CASUAL AND GEOMETRIC PRAXEOLOGIES IN A STUDY OF SYMMETRIC FIGURES

Margo Kondratieva

Memorial University, Canada

In this paper we collect various solutions produced by in-service teachers in response to problems involving regular polygons. These solutions allow us to identify the most popular ideas found in the contemporary K-12 geometry curriculum. In line with previous research, we found that while the figures in question possess axial symmetry, this property is very rarely used by teachers. Despite the fact that the concept of symmetry is familiar to teachers at the intuitive level, comprehension of this concept is lacking the mathematical precision. Thus, this visual characteristic of the objects of study fails to be employed in teachers' geometric reasoning.

## INTRODUCTION

Symmetric figures do not change when reflected, rotated or translated in a certain way. This property could be both intriguing for human visual perception and useful for our practical and theoretical needs. Symmetry “creates patterns that help us organize our world conceptually” (Knuchel, 2004). The idea of symmetry connects between various branches of mathematics and plays an important role in problem solving (Leikin et al, 2000). In addition, “symmetry is a paramount tool for bridging the gap between science and art” (Livio, 2005). Since ancient times of Aristotle and Plato symmetry has been associated with beauty as well as played a significant role in architectural design.

Psychologists investigated the relationship between symmetry and aesthetic preference by asking people to make a “visually pleasing” arrangement of objects given certain constrains. It was found that the majority of arrangements were symmetrical in nature (see e.g. Szilagyi & Baird, 1977). Birkhoff (1933) attempted to develop a mathematical theory of aesthetic value. He proposed that at a given level of complexity, the aesthetic value is higher for more symmetric objects. The role of symmetry in perception was stressed by Gestalt psychology. The principle of Prägnanz (or “good figure”) states: “of several possible geometric organizations, the one that is seen is the one that possesses the best simplest and most stable shape” (Livio, 2005), which in practice oftentimes is symmetric. For example, a picture showing four dots which are vertices of a square is seen as a square - a symmetric, stable and closed shape. A perceived stability may be related to the fact that frequently physical systems possessing symmetry are stable because they furnish minimum potential energy.

Mirror-reflection symmetry is a specific form of symmetry which characterizes the animal kingdom. The prevalence of the bilateral symmetry in animal appearance for many million years (which is evidenced by both fossils and contemporary leaving creatures) could be explained by evolutionary biology and laws of physics. Gravity and locomotion produce distinctions between top and bottom or back and front. However, the animals' environment is much more homogeneous when it comes to distinguishing between left and right. Therefore, it is not surprising that the most popular interpretation of the word “symmetry” actually is reflection symmetry. According to Webster's Third
New International Dictionary, symmetry is “correspondence in size, shape and relative position of parts that are on opposite sides of a dividing line or median plane”. Mach (1914) noted that figures are perceived as symmetrical primarily as a result of reflection symmetry about a vertical axis. In particular, perception of reflection symmetry may depend on the orientation of the object; the symmetry may not be detected by an observer unless one turns the object in such a way that the axis of symmetry becomes vertical.

Based on the short summary presented above, it is reasonable to assume that humans have a natural ability to recognize and even are drawn to symmetrical objects, in particular to those possessing the vertical axis of symmetry. In this article we aim to investigate the practicality of this ability for a group of in-service teachers confronted with a geometrical question involving symmetric shapes.

## THEORETICAL FRAMEWORK

We draw our theoretical framework from the Anthropological Theory of the Didactic (ATD), in particular the notions of praxeology and institutionalization (Bosch et al, 2019). Praxeology is a basic notion to describe human acts in particular situations. Praxeology consists of praxis and logos blocks. Further, the praxis block consists of types of tasks and techniques to solve them, while logos consists of rationales explaining and justifying the techniques used in praxis and theories within which these rationales exist. Possession of certain praxeologies by an individual constitute their praxeological equipment for working with a respective class of objects known to them.

ATD proposes that praxeologies undergo a transposition as they occur in different social institutions such as research communities, practice communities, or learning communities e.g. schools. Within different institutions different conditions are present that support or restrict individuals in the process of familiarization with objects of their study.

By observing representatives of a given institution one can judge about their prevalent praxeologies for dealing with certain tasks. Thus, we can talk about geometric praxeologies that could be learned in a course of study of this subject. These could be contrasted with casual praxeologies based on intuitive or visual rationales lacking a formal theory. Casual praxeologies can be present in performance of casual tasks for which a person had not been trained, such as making a visually pleasing arrangement of objects in the experiments of Szilagyi & Baird (1977). In particular, the idea of symmetry could be developed by osmosis, a process of subconscious assimilation of this idea from visual information, and be “based on experiential meanings of reflective symmetry as yet not connected with any accepted mathematical definition” (Hoyles, 1997). Alternatively, the notion of symmetry based on “socially-shared understanding of reflection from [...] everyday experiences” could be “expanded and connected to a mathematical meaning for the transformation,” (Hoyles, 1997) providing more precision. The two scenarios lead to developing different types of praxeologies for working with symmetric figures.

## DATA COLLECTION

The data had been collected in a graduate course offered at Memorial University (Canada) online for in-service K-12 teachers. The course focuses on mathematical thinking (Mason et al, 2010) and includes such activities as individual problem solving, group discussions, and journal writing. One major assessment is based on the interconnecting problems, defined in (Kondratieva, 2011) as problems that (1) allow a simple formulation; (2) allow various solutions at both elementary and
New International Dictionary, symmetry is “correspondence in size, shape and relative position of parts that are on opposite sides of a dividing line or median plane”. Mach (1914) noted that figures are perceived as symmetrical primarily as a result of reflection symmetry about a vertical axis. In particular, perception of reflection symmetry may depend on the orientation of the object; the symmetry may not be detected by an observer unless one turns the object in such a way that the axis of symmetry becomes vertical.

Based on the short summary presented above, it is reasonable to assume that humans have a natural ability to recognize and even are drawn to symmetrical objects, in particular to those possessing the vertical axis of symmetry. In this article we aim to investigate the practicality of this ability for a group of in-service teachers confronted with a geometrical question involving symmetric shapes.

## THEORETICAL FRAMEWORK

We draw our theoretical framework from the Anthropological Theory of the Didactic (ATD), in particular the notions of praxeology and institutionalization (Bosch et al, 2019). Praxeology is a basic notion to describe human acts in particular situations. Praxeology consists of praxis and logos blocks. Further, the praxis block consists of types of tasks and techniques to solve them, while logos consists of rationales explaining and justifying the techniques used in praxis and theories within which these rationales exist. Possession of certain praxeologies by an individual constitute their praxeological equipment for working with a respective class of objects known to them.

ATD proposes that praxeologies undergo a transposition as they occur in different social institutions such as research communities, practice communities, or learning communities e.g. schools. Within different institutions different conditions are present that support or restrict individuals in the process of familiarization with objects of their study.

By observing representatives of a given institution one can judge about their prevalent praxeologies for dealing with certain tasks. Thus, we can talk about geometric praxeologies that could be learned in a course of study of this subject. These could be contrasted with casual praxeologies based on intuitive or visual rationales lacking a formal theory. Casual praxeologies can be present in performance of casual tasks for which a person had not been trained, such as making a visually pleasing arrangement of objects in the experiments of Szilagyi & Baird (1977). In particular, the idea of symmetry could be developed by osmosis, a process of subconscious assimilation of this idea from visual information, and be “based on experiential meanings of reflective symmetry as yet not connected with any accepted mathematical definition” (Hoyles, 1997). Alternatively, the notion of symmetry based on “socially-shared understanding of reflection from [...] everyday experiences” could be “expanded and connected to a mathematical meaning for the transformation,” (Hoyles, 1997) providing more precision. The two scenarios lead to developing different types of praxeologies for working with symmetric figures.

## DATA COLLECTION

The data had been collected in a graduate course offered at Memorial University (Canada) online for in-service K-12 teachers. The course focuses on mathematical thinking (Mason et al, 2010) and includes such activities as individual problem solving, group discussions, and journal writing. One major assessment is based on the interconnecting problems, defined in (Kondratieva, 2011) as problems that (1) allow a simple formulation; (2) allow various solutions at both elementary and
advanced levels; (3) may be solved by various mathematical tools from different mathematical branches, which leads to finding multiple solutions, and (4) can be used in different grades and courses and understood in various contexts. The Hexagon Problem is an example of an interconnecting problem used in the course and served as a generating question for this study: *Show that in a regular hexagon there is a diagonal that is perpendicular to one of the sides.*

Students discussed the given interconnecting problem in randomly formed groups (normally composed of 2 elementary and 2 secondary school teachers. There were no restrictions on using any resources or technology. Each group was required to construct at least 3 different solutions at various levels of sophistication. Each problem was discussed by students online during two weeks. Students submitted their groups' reports in both written format and as a short (20-30 minutes) video where they presented and discussed these solutions within their group.

In addition, students were asked to report in their individual journals regarding an extension problem: *Describe all regular polygons that have a diagonal perpendicular to one of the sides.*

The data collection took place in 2022 and 2023 in the school terms when the author was teaching the course. There was a total of 29 students enrolled in the course during this period. Participants' video, written group reports, and individual journals reflecting student's experiences with problem solving in the course had been analyzed. The most popular solutions are discussed in the next section.

## The research questions

(1) What are the common techniques proposed by K-12 teachers to solve the Hexagon Problem?

(2) What is the role of symmetry in teachers' reasoning about this problem and its extension?

## RESULTS AND DISCUSSION

As instructed in the textbook for the course, students started their investigations from listing the facts they know about regular hexagons such as "all six sides are equal", "all six interior angles are equal" and the sum of interior angles is $(6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ}$. They also clarified the terminology, for example, "perpendicular means intersecting at a right (90^{\circ}) angle" or "a diagonal is a line joining two non-adjacent corners of a straight-sided shape." Then they aimed to connect these facts to the statement to be confirmed.

I began solving this problem by identifying what I knew about the question and what I wanted to know. From prior knowledge, I knew that the sum of a hexagon's interior angles is $720^{\circ}$, which means that each interior angle must be $120^{\circ}$. Initially, I visualized the perpendicular diagonal I wanted to attack, but I wasn't quite sure how to get there.

Since it was required to find three solutions, students were searching for several useful ideas and possibilities.

I drew a hexagon and quickly jotted down previous knowledge about this shape that I believed could aid in finding a solution, which included its sum of angles, the degree (120^{\circ}) of each interior angle, and the possible division of this shape into six equilateral triangles. [...] Yet, I decided that using right-angle triangles would be better suited to aid in this goal since I knew that applying the Pythagorean theorem and the slope formula could easily be applied to this specific triangle.

The most popular solution of the Hexagon problem was as follows (see also Figure 1).
advanced levels; (3) may be solved by various mathematical tools from different mathematical branches, which leads to finding multiple solutions, and (4) can be used in different grades and courses and understood in various contexts. The Hexagon Problem is an example of an interconnecting problem used in the course and served as a generating question for this study: *Show that in a regular hexagon there is a diagonal that is perpendicular to one of the sides.*

Students discussed the given interconnecting problem in randomly formed groups (normally composed of 2 elementary and 2 secondary school teachers. There were no restrictions on using any resources or technology. Each group was required to construct at least 3 different solutions at various levels of sophistication. Each problem was discussed by students online during two weeks. Students submitted their groups' reports in both written format and as a short (20-30 minutes) video where they presented and discussed these solutions within their group.

In addition, students were asked to report in their individual journals regarding an extension problem: *Describe all regular polygons that have a diagonal perpendicular to one of the sides.*

The data collection took place in 2022 and 2023 in the school terms when the author was teaching the course. There was a total of 29 students enrolled in the course during this period. Participants' video, written group reports, and individual journals reflecting student's experiences with problem solving in the course had been analyzed. The most popular solutions are discussed in the next section.

## The research questions

(1) What are the common techniques proposed by K-12 teachers to solve the Hexagon Problem?

(2) What is the role of symmetry in teachers' reasoning about this problem and its extension?

## RESULTS AND DISCUSSION

As instructed in the textbook for the course, students started their investigations from listing the facts they know about regular hexagons such as "all six sides are equal", "all six interior angles are equal" and the sum of interior angles is $(6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ}$. They also clarified the terminology, for example, "perpendicular means intersecting at a right (90^{\circ}) angle" or "a diagonal is a line joining two non-adjacent corners of a straight-sided shape." Then they aimed to connect these facts to the statement to be confirmed.

I began solving this problem by identifying what I knew about the question and what I wanted to know. From prior knowledge, I knew that the sum of a hexagon's interior angles is $720^{\circ}$, which means that each interior angle must be $120^{\circ}$. Initially, I visualized the perpendicular diagonal I wanted to attack, but I wasn't quite sure how to get there.

Since it was required to find three solutions, students were searching for several useful ideas and possibilities.

I drew a hexagon and quickly jotted down previous knowledge about this shape that I believed could aid in finding a solution, which included its sum of angles, the degree (120^{\circ}) of each interior angle, and the possible division of this shape into six equilateral triangles. [...] Yet, I decided that using right-angle triangles would be better suited to aid in this goal since I knew that applying the Pythagorean theorem and the slope formula could easily be applied to this specific triangle.

The most popular solution of the Hexagon problem was as follows (see also Figure 1).
I drew my suspected perpendicular diagonal (the green line) from here. Further specialization created an isosceles triangle with two equal sides and, thus, two equal angles. [...] I determined that these two angles must be 30° each. Knowing that the entire black dotted interior angle (as shown in the second hexagon) was 120° and that ∠b was 30°, I determined that my green line was perpendicular due to a 90° angle. Generalizing this solution to the rest of the hexagon, I could now clearly state that any diagonal connecting two vertices one vertex apart would be perpendicular to a side of the hexagon.

<figure><img src="image_58.png" /><figcaption>Figure 1. The most popular solution of the Hexagon Problem</figcaption></figure>

Another solution based on angles consideration used triangular decomposition of a regular hexagon:

When the diagonals are drawn, six equilateral triangles are formed (Figure 2). They are equilateral because in a regular hexagon, the radius of the hexagon (distance from the center to a vertex) is equal to the side length. In an equilateral triangle, all angles are 60°.

It follows that <DAF = 60° and <DAB = 60°. The diagonals bisect the angle, making <EAF = 30°, <DAE = 30°, and <CAD = 30°. Therefore, <CAF = 30 + 30 + 30 = 90°. Since <CAF = 90°, diagonal AC is perpendicular to side AF. In the same way, diagonal AC is also perpendicular to side CD.

<figure><img src="image_59.png" /><figcaption>Figure 2. All diagonals and equilateral triangles in a regular hexagon</figcaption></figure>

The idea that “the diagonal bisects the angle” was further explained as follows.

We also know that when we divide an equilateral triangle in half, starting at one of the three vertices, 2 right triangles are created, resulting in angles of 90°, 60°, and 30°.

This property follows from the axial symmetry of an equilateral triangle, however students did not explicitly refer to the idea of symmetry. The idea of this approach can be developed into a rigorous
I drew my suspected perpendicular diagonal (the green line) from here. Further specialization created an isosceles triangle with two equal sides and, thus, two equal angles. [...] I determined that these two angles must be 30° each. Knowing that the entire black dotted interior angle (as shown in the second hexagon) was 120° and that $\angle b$ was 30°, I determined that my green line was perpendicular due to a 90° angle. Generalizing this solution to the rest of the hexagon, I could now clearly state that any diagonal connecting two vertices one vertex apart would be perpendicular to a side of the hexagon.

<figure><img src="image_64.png" /><figcaption>Figure 1. The most popular solution of the Hexagon Problem</figcaption></figure>

Another solution based on angles consideration used triangular decomposition of a regular hexagon:

When the diagonals are drawn, six equilateral triangles are formed (Figure 2). They are equilateral because in a regular hexagon, the radius of the hexagon (distance from the center to a vertex) is equal to the side length. In an equilateral triangle, all angles are 60°.

It follows that $<DAF = 60°$ and $<DAB = 60°$. The diagonals bisect the angle, making $<EAF = 30°$, $<DAE = 30°$, and $<CAD = 30°$. Therefore, $<CAF = 30° + 30° + 30° = 90°$. Since $<CAF = 90°$, diagonal AC is perpendicular to side AF. In the same way, diagonal AC is also perpendicular to side CD.

<figure><img src="image_65.png" /><figcaption>Figure 2. All diagonals and equilateral triangles in a regular hexagon</figcaption></figure>

The idea that “the diagonal bisects the angle” was further explained as follows.

We also know that when we divide an equilateral triangle in half, starting at one of the three vertices, 2 right triangles are created, resulting in angles of 90°, 60°, and 30°.

This property follows from the axial symmetry of an equilateral triangle, however students did not explicitly refer to the idea of symmetry. The idea of this approach can be developed into a rigorous
proof by noticing that the angles EAF, DAE, and CAD support equal chords of the circumcircle.

The third approach employed analytic geometry, introduction of coordinates and the notion of slope. The following solution started from specializing by taking side length equal 3 units and calculating slopes of lines AB and BG (Figure 3).

<figure><img src="image_60.png" /><figcaption>Figure 3. A solution based on introduction of coordinates and calculation of slopes of AB and BG</figcaption></figure>

Note that the student made approximate calculations before moving to the general case:

My goal was a generalization, so I let $x$ represent any possible side length of the hexagon and set vertex G of the hexagon as the origin, (0, 0), of my graph since it is where I initially drew my diagonals. With the use of my previous specialization and Pythagoras' theorem, I was able to determine the coordinates of vertex A, $(-2x, 0)$, and vertex B, $(\frac{-3x}{2}, \frac{x\sqrt{3}}{2})$ as shown. Next, using the slope formula, I found the slope of $\overline{AB} = \sqrt{3}$, and the slope of $\overline{BG} = -\frac{\sqrt{3}}{3}$. This was my AHA! moment. First, upon completing a quick calculation, it became evident that these slopes were negative reciprocals of one another and thus perpendicular. Furthermore, after reviewing my newfound solution, I recognized that the side length is irrelevant throughout my calculations since it is ultimately eliminated. This reveals that the slopes of these lines will always be the same regardless of the size of the hexagon.

In the fourth solution students assumed the side length of the hexagon to be $x$. Therefore, the long diagonal was measured as $2x$. The goal was to find the length $y$ of the short diagonal. This can be done using the Cosine theorem as shown in Figure 4. Then, since the sides of the triangle formed by the long diagonal, the short diagonal and the side of the hexagon satisfy the Pythagorean formula, it follows that these segments form a right triangle, and so the short diagonal is perpendicular to the side of the hexagon.

<figure><img src="image_61.png" /><figcaption>Figure 4. A solution based on the Cosine and inverse Pythagorean theorems</figcaption></figure>
proof by noticing that the angles EAF, DAE, and CAD support equal chords of the circumcircle.

The third approach employed analytic geometry, introduction of coordinates and the notion of slope. The following solution started from specializing by taking side length equal 3 units and calculating slopes of lines AB and BG (Figure 3).

<figure><img src="image_66.png" /><figcaption>Figure 3. A solution based on introduction of coordinates and calculation of slopes of AB and BG
Note that the student made approximate calculations before moving to the general case:</figcaption></figure>

My goal was a generalization, so I let $x$ represent any possible side length of the hexagon and set vertex G of the hexagon as the origin, (0, 0), of my graph since it is where I initially drew my diagonals. With the use of my previous specialization and Pythagoras' theorem, I was able to determine the coordinates of vertex A, $(-2x, 0)$, and vertex B, $(\frac{-3x}{2}, \frac{x\sqrt{3}}{2})$ as shown. Next, using the slope formula, I found the slope of $\overline{AB} = \sqrt{3}$, and the slope of $\overline{BG} = -\frac{\sqrt{3}}{3}$. This was my AHA! moment. First, upon completing a quick calculation, it became evident that these slopes were negative reciprocals of one another and thus perpendicular. Furthermore, after reviewing my newfound solution, I recognized that the side length is irrelevant throughout my calculations since it is ultimately eliminated. This reveals that the slopes of these lines will always be the same regardless of the size of the hexagon.

In the fourth solution students assumed the side length of the hexagon to be $x$. Therefore, the long diagonal was measured as $2x$. The goal was to find the length $y$ of the short diagonal. This can be done using the Cosine theorem as shown in Figure 4. Then, since the sides of the triangle formed by the long diagonal, the short diagonal and the side of the hexagon satisfy the Pythagorean formula, it follows that these segments form a right triangle, and so the short diagonal is perpendicular to the side of the hexagon.

<figure><img src="image_67.png" /><figcaption>Figure 4. A solution based on the Cosine and inverse Pythagorean theorems</figcaption></figure>
We note that most students availed of various technological tools helping to draw an accurate picture of a regular hexagon with its diagonals. Some of them even imitated a construction of this shape using a compass. Figure 5 gives one such example and also shows the right angle between a diagonal and a side. This conclusion could alternatively be made with reference to the Thales theorem (about three distinct points on a circle forming a right-angled triangle provided that two of the points lie on the opposite ends of a diameter).

<figure><img src="image_62.png" /><figcaption>Figure 5. Construction of a regular hexagon using a compass and the Thales theorem</figcaption></figure>

For some teachers using measurement tools was the first step and actually the only way of verifying the statement. One of them explained in their journal:

> Geometry is something that I am not super familiar with and is definitely something I still struggle with. When I originally solved the hexagon problem, I used measurements to help me solve the problem. The first way I did it was involving using a protractor. The second way I solve it was using GeoGebra to find the slopes of the two lines and prove that they were perpendicular because they were negative reciprocals. Both of these involved measurements and were kind of a way around using geometry.

A couple of teachers “confused the meaning of perpendicular with parallel” and many of them found this problem challenging because “these are definitely topics we do not touch on often within the current mathematics curriculum.” In their journal a teacher confessed:

> Geometry is an essential area of mathematics, but its placement in the curriculum and the required activities would seem as if it's disappearing from the curriculum. In the junior high math curriculum suggested sequence, geometry is the last unit. Teachers are rushing to finish the course. The quality of work done in this unit is frequently limited, with a lower level of critical thinking.

Another teacher echoed the idea:

> It is a shame, honestly, that geometry is often pushed to the back burner because teachers find other components of math to be more important.

Nevertheless, certain geometrical ideas were well known to the teachers and allowed them to successfully approach the Hexagon Problem as is evident from the solutions presented above.

## CONCLUDING REMARKS

The notion of symmetry is present in the K-12 mathematics curriculum. “Paper-folding or grids are used in early activities to assist in “the doing” of a reflection” (Hoyles & Healy, 2002). Primary and elementary students also use manipulatives such as Mira Geomirror to obtain reflections and thus creating a symmetric figure. More broadly, reflections and symmetric shapes can be produced using various technologies such as dynamic geometry software (Mackrell, 2002). Secondary school students observe reflection symmetry when they graph even functions such as parabolas or a composition with the absolute value $y=f(|x|)$. An instrumental idea that gives precise meaning to the
We note that most students availed of various technological tools helping to draw an accurate picture of a regular hexagon with its diagonals. Some of them even imitated a construction of this shape using a compass. Figure 5 gives one such example and also shows the right angle between a diagonal and a side. This conclusion could alternatively be made with reference to the Thales theorem (about three distinct points on a circle forming a right-angled triangle provided that two of the points lie on the opposite ends of a diameter).

<figure><img src="image_68.png" /><figcaption>Figure 5. Construction of a regular hexagon using a compass and the Thales theorem</figcaption></figure>

For some teachers using measurement tools was the first step and actually the only way of verifying the statement. One of them explained in their journal:

> Geometry is something that I am not super familiar with and is definitely something I still struggle with. When I originally solved the hexagon problem, I used measurements to help me solve the problem. The first way I did it was involving using a protractor. The second way I solve it was using GeoGebra to find the slopes of the two lines and prove that they were perpendicular because they were negative reciprocals. Both of these involved measurements and were kind of a way around using geometry.

A couple of teachers “confused the meaning of perpendicular with parallel” and many of them found this problem challenging because “these are definitely topics we do not touch on often within the current mathematics curriculum.” In their journal a teacher confessed:

> Geometry is an essential area of mathematics, but its placement in the curriculum and the required activities would seem as if it's disappearing from the curriculum. In the junior high math curriculum suggested sequence, geometry is the last unit. Teachers are rushing to finish the course. The quality of work done in this unit is frequently limited, with a lower level of critical thinking.

Another teacher echoed the idea:

> It is a shame, honestly, that geometry is often pushed to the back burner because teachers find other components of math to be more important.

Nevertheless, certain geometrical ideas were well known to the teachers and allowed them to successfully approach the Hexagon Problem as is evident from the solutions presented above.

## CONCLUDING REMARKS

The notion of symmetry is present in the K-12 mathematics curriculum. “Paper-folding or grids are used in early activities to assist in “the doing” of a reflection” (Hoyles & Healy, 2002). Primary and elementary students also use manipulatives such as Mira Geomirror to obtain reflections and thus creating a symmetric figure. More broadly, reflections and symmetric shapes can be produced using various technologies such as dynamic geometry software (Mackrell, 2002). Secondary school students observe reflection symmetry when they graph even functions such as parabolas or a composition with the absolute value $y=f(|x|)$. An instrumental idea that gives precise meaning to the
reflection symmetry is that the segment connecting any point with its image is perpendicular to the axis of reflection. Thus, a point and its image are at the same *perpendicular* distance from the axis.

A regular hexagon has six lines of symmetry: three lines passing through its opposite vertices and three lines passing through the midpoints of its opposite sides. In particular, a regular hexagon ABCDEF shown in Figure 6 is symmetric with respect to the line passing through point G and H, where G is the midpoint of the side BC and H is the midpoint of the side FE. The pairs of points B and C, A and D, F and E are reflections of each other with respect to the line GH. Therefore, the segments BC, AD and FE are perpendicular to GH, and thus parallel to each other. The hexagon ABCDEF is also symmetric with respect to the line passing through vertices A and D. The sides BC and EF are parallel to the line AD, where the vertex F is a reflection of B and the vertex E is a reflection of C. Therefore, the diagonals BF and CE are perpendicular AD and to the sides BC and EF.

<figure><img src="image_63.png" /><figcaption>Figure 6. Two lines of symmetry in a regular hexagon and a symmetry-based solution</figcaption></figure>

In my experience, the reasoning along these lines was a popular answer to the Hexagon Problem amongst my colleagues-mathematicians. At the same time, none on the 29 in-service mathematics teachers, whose solutions were collected, proposed that the symmetry could be useful. Two students stated the facts that a regular hexagon has reflection and rotational symmetries, but they did not develop any approaches based on this property.

One group of teachers noted that BCEF appears to be a rectangle (Figure 7), but during the videotaped group discussion they found that they did not know how to prove that all angles in this parallelogram are 90°.

<figure><img src="image_64.png" /><figcaption>Figure 7. A group of students conjectured that BCEF was a rectangle</figcaption></figure>

The thoughts about symmetry did not cross students’ minds at that time as is evident from the following journal entry.

I could clearly see a rectangle mixed into the crossings of the diagonals and sides of the hexagon. From here, I used the properties of a rectangle to prove that one of the diagonals was perpendicular to one of the sides. It was not until I read our group feedback that I thought of/considered using lines of symmetry to solve this problem and prove my solution was valid. With symmetrical lines in mind, I know that an octagon
reflection symmetry is that the segment connecting any point with its image is perpendicular to the axis of reflection. Thus, a point and its image are at the same *perpendicular* distance from the axis.

A regular hexagon has six lines of symmetry: three lines passing through its opposite vertices and three lines passing through the midpoints of its opposite sides. In particular, a regular hexagon ABCDEF shown in Figure 6 is symmetric with respect to the line passing through point G and H, where G is the midpoint of the side BC and H is the midpoint of the side FE. The pairs of points B and C, A and D, F and E are reflections of each other with respect to the line GH. Therefore, the segments BC, AD and FE are perpendicular to GH, and thus parallel to each other. The hexagon ABCDEF is also symmetric with respect to the line passing through vertices A and D. The sides BC and EF are parallel to the line AD, where the vertex F is a reflection of B and the vertex E is a reflection of C. Therefore, the diagonals BF and CE are perpendicular AD and to the sides BC and EF.

<figure><img src="image_69.png" /><figcaption>Figure 6. Two lines of symmetry in a regular hexagon and a symmetry-based solution</figcaption></figure>

In my experience, the reasoning along these lines was a popular answer to the Hexagon Problem amongst my colleagues-mathematicians. At the same time, none on the 29 in-service mathematics teachers, whose solutions were collected, proposed that the symmetry could be useful. Two students stated the facts that a regular hexagon has reflection and rotational symmetries, but they did not develop any approaches based on this property.

One group of teachers noted that BCEF appears to be a rectangle (Figure 7), but during the videotaped group discussion they found that they did not know how to prove that all angles in this parallelogram are 90°.

<figure><img src="image_70.png" /><figcaption>Figure 7. A group of students conjectured that BCEF was a rectangle</figcaption></figure>

The thoughts about symmetry did not cross students' minds at that time as is evident from the following journal entry.

I could clearly see a rectangle mixed into the crossings of the diagonals and sides of the hexagon. From here, I used the properties of a rectangle to prove that one of the diagonals was perpendicular to one of the sides. It was not until I read our group feedback that I thought of/considered using lines of symmetry to solve this problem and prove my solution was valid. With symmetrical lines in mind, I know that an octagon
would also be another example of a regular polygon that has a diagonal that is perpendicular to one of its sides.

It was reassuring to see that the idea of symmetry (suggested in the instructor's feedback to the group) was used for the extension problem by the above-mentioned student. However, many other students did not talk explicitly about lines of symmetry but rather noted a property that uses this fact implicitly.

I concluded that in polygons with an even number of sides, the opposing sides will always be parallel, allowing for a 90° angle formation when a diagonal is drawn between them.

We conclude that the teachers participating in this study were familiar with several praxeologies and ideas related to geometry, such as the sum of angles in a regular n-gon, angles in an isosceles triangle, Pythagorean theorem and slopes of perpendicular lines. These ideas are clearly present in current geometrical curriculum and are well understood and employed by the teachers.

However, our results echo the observation that mathematics teachers usually do not use symmetry in problem solving unless they are trained in this direction (Leikin et al, 2000). Teachers seem to be lacking precise mathematical understanding of symmetric figures and thus perceive symmetry-based reasoning as unreliable. There was an apparent gap between teachers' intuitive sense of symmetry of an entire hexagon and a point-wise symmetry of its vertices that would allow them to develop a geometric argument resolving the Hexagon Problem. In other words, teachers' geometric praxeological equipment involving symmetry did not seem to contain a praxis developed enough for using symmetry in a functional and technical way to solve problems and not only to describe figures' properties. Consequently, teachers' geometric reasoning was not based on this visual characteristic of the figures.

## References

* Birkhoff, G.D. (1933). *Aesthetic measure*. Harvard University Press.

* Bosch, M., Chevallard, Y., García, F.J., & Monaghan, J. (Eds) (2019). *Working with the anthropological theory of the didactic in mathematics education. A comprehensive casebook*. Routledge.

* Hoyles, C., Healy, L. (1997). Unfolding meanings for reflective symmetry. *International Journal of Computers in Mathematical Learning, 2*, 27-59.

* Knuchel, C. (2004). Teaching Symmetry in the Elementary Curriculum. *The Montana Mathematics Enthusiast, 1*(1), 3-8. https://scholarworks.umt.edu/tme/vol1/iss1/2

* Kondratieva, M. (2011). The promise of interconnecting problems for enriching students' experiences in mathematics. *The Montana Mathematics Enthusiast, 8*(1-2), 355-382.

* Leikin, R., Berman, A. & Zaslavsky, O. (2000). Applications of symmetry to problem solving. *International Journal of Mathematical Education in Science and Technology, 31*, 799-809.

* Livio, M. (2005). *The equation that couldn't be solved*. Simon & Schuster.

* Mach, E. (1914). *The Analysis of sensation*. Open Court; republished in 1959 by Dover.

* Mackrell, K. (2002). Polygons with primary students. *Micromath, 18*(3), 26-28

* Mason, J., Burton, L., & Stacey, K. (2010). *Thinking Mathematically*. Addison Wesley.

* Szilagyi, P.G., & Baird, J.C. (1977). A quantitative approach to the study of visual symmetry. *Perception and Psychophysics, 22*(3), 287.
would also be another example of a regular polygon that has a diagonal that is perpendicular to one of its sides.

It was reassuring to see that the idea of symmetry (suggested in the instructor's feedback to the group) was used for the extension problem by the above-mentioned student. However, many other students did not talk explicitly about lines of symmetry but rather noted a property that uses this fact implicitly.

I concluded that in polygons with an even number of sides, the opposing sides will always be parallel, allowing for a 90° angle formation when a diagonal is drawn between them.

We conclude that the teachers participating in this study were familiar with several praxeologies and ideas related to geometry, such as the sum of angles in a regular n-gon, angles in an isosceles triangle, Pythagorean theorem and slopes of perpendicular lines. These ideas are clearly present in current geometrical curriculum and are well understood and employed by the teachers.

However, our results echo the observation that mathematics teachers usually do not use symmetry in problem solving unless they are trained in this direction (Leikin et al, 2000). Teachers seem to be lacking precise mathematical understanding of symmetric figures and thus perceive symmetry-based reasoning as unreliable. There was an apparent gap between teachers' intuitive sense of symmetry of an entire hexagon and a point-wise symmetry of its vertices that would allow them to develop a geometric argument resolving the Hexagon Problem. In other words, teachers' geometric praxeological equipment involving symmetry did not seem to contain a praxis developed enough for using symmetry in a functional and technical way to solve problems and not only to describe figures' properties. Consequently, teachers' geometric reasoning was not based on this visual characteristic of the figures.

## References

* Birkhoff, G.D. (1933). *Aesthetic measure*. Harvard University Press.

* Bosch, M., Chevallard, Y., García, F.J., & Monaghan, J. (Eds) (2019). *Working with the anthropological theory of the didactic in mathematics education. A comprehensive casebook*. Routledge.

* Hoyles, C., Healy, L. (1997). Unfolding meanings for reflective symmetry. *International Journal of Computers in Mathematical Learning, 2*, 27-59.

* Knuchel, C. (2004). Teaching Symmetry in the Elementary Curriculum. *The Montana Mathematics Enthusiast, 1*(1), 3-8. https://scholarworks.umt.edu/tme/vol1/iss1/2

* Kondratieva, M. (2011). The promise of interconnecting problems for enriching students' experiences in mathematics. *The Montana Mathematics Enthusiast, 8*(1-2), 355-382.

* Leikin, R., Berman, A. & Zaslavsky, O. (2000). Applications of symmetry to problem solving. *International Journal of Mathematical Education in Science and Technology, 31*, 799-809.

* Livio, M. (2005). *The equation that couldn't be solved*. Simon & Schuster.

* Mach, E. (1914). *The Analysis of sensation*. Open Court; republished in 1959 by Dover.

* Mackrell, K. (2002). Polygons with primary students. *Micromath, 18*(3), 26-28

* Mason, J., Burton, L., & Stacey, K. (2010). *Thinking Mathematically*. Addison Wesley.

* Szilagyi, P.G., & Baird, J.C. (1977). A quantitative approach to the study of visual symmetry. *Perception and Psychophysics, 22*(3), 287.
# LEARNING STAKES TARGETED BY TEACHERS IN A FIGURE RESTORATION ACTIVITY

Christine Mangiante-Orsola¹, Claire Guille-Biel Winder²

¹ LML, University of Lille, ² ADEF, Aix-Marseille University

Our research focuses on teachers' practices when teaching geometry. In this text, we report on the way in which some teachers (whom we have had the opportunity to follow) take into account the stakes of geometry teaching in the 'heat of the moment'. More precisely, we use the notion of didactic vigilance, to report on the manner in which these teachers prioritize (or not) the finality of geometric learning when implementing a figure restoration situation in elementary school.

## INTRODUCTION

Numerous studies conducted in the field of mathematics didactics show that geometry teaching presents specific difficulties for teachers (Mathé, Barrier & Perrin-Glorian, 2020), and that it is often focused on vocabulary acquisition, knowledge of some properties and mastery of the usual instruments (Mangiante-Orsola, 2014). More generally, our work shows that knowledge at stake are difficult for teachers to perceive and to take into account when implementing teaching-learning situations in geometry (Guille-Biel Winder, 2021; Guille-Biel Winder & Mangiante-Orsola, 2023): for example, teachers often give priority to the accuracy of plotting, to the detriment of the validity of figure construction procedures. This text is based on a larger clinical study (Mangiante-Orsola, 2023) which reports on the follow-up of nine teachers carried out as part of an in-service training program. In this program, training sessions alternate with in-class experimentations. We provided teachers with a document describing the geometry sequence *Triangles on quadrilaterals* (ibid.), and then observed their practices as they implemented the sequence. In this text, we present an analysis of the two teachers' practices, Vic (who teaches pupils aged 9 to 11) and Lea (who teaches pupils aged 9 to 10), based on the data collected: for each teacher, video recordings of the four sessions implemented and audio recording of the interview conducted after each session. Our intention is to report on the professional gestures of these teachers to better understand how, 'in the heat of the moment', they adjust their responses according to the students' proposals and consider as much as possible the stakes of geometry teaching. After introducing our theoretical framework and the methodology, we present the teaching-learning situation and analysis. This reveals three aspects of the situation that a teacher should be vigilant about implementing. We then present an analysis of the two teachers' practices regarding these three aspects. This analysis enables us to identify the teaching priorities of each teacher.

## THEORETICAL FRAMEWORK

We believe that it is important to draw on the observation of actual practices to analyze the practices of schoolteachers when teaching mathematics (Butlen, Mangiante-Orsola & Masselot, 2017). To study these practices, we rely on the theoretical tools developed by Butlen, Charles-Pézard and Masselot (op. cit., 2012; Charles-Pézard, 2010; Butlen & Masselot, 2019). We take up the notion of
# LEARNING STAKES TARGETED BY TEACHERS IN A FIGURE RESTORATION ACTIVITY

Christine Mangiante-Orsola¹, Claire Guille-Biel Winder²

¹ LML, University of Lille, ² ADEF, Aix-Marseille University

Our research focuses on teachers' practices when teaching geometry. In this text, we report on the way in which some teachers (whom we have had the opportunity to follow) take into account the stakes of geometry teaching in the 'heat of the moment'. More precisely, we use the notion of didactic vigilance, to report on the manner in which these teachers prioritize (or not) the finality of geometric learning when implementing a figure restoration situation in elementary school.

## INTRODUCTION

Numerous studies conducted in the field of mathematics didactics show that geometry teaching presents specific difficulties for teachers (Mathé, Barrier & Perrin-Glorian, 2020), and that it is often focused on vocabulary acquisition, knowledge of some properties and mastery of the usual instruments (Mangiante-Orsola, 2014). More generally, our work shows that knowledge at stake are difficult for teachers to perceive and to take into account when implementing teaching-learning situations in geometry (Guille-Biel Winder, 2021; Guille-Biel Winder & Mangiante-Orsola, 2023): for example, teachers often give priority to the accuracy of plotting, to the detriment of the validity of figure construction procedures. This text is based on a larger clinical study (Mangiante-Orsola, 2023) which reports on the follow-up of nine teachers carried out as part of an in-service training program. In this program, training sessions alternate with in-class experimentations. We provided teachers with a document describing the geometry sequence *Triangles on quadrilaterals* (ibid.), and then observed their practices as they implemented the sequence. In this text, we present an analysis of the two teachers' practices, Vic (who teaches pupils aged 9 to 11) and Lea (who teaches pupils aged 9 to 10), based on the data collected: for each teacher, video recordings of the four sessions implemented and audio recording of the interview conducted after each session. Our intention is to report on the professional gestures of these teachers to better understand how, 'in the heat of the moment', they adjust their responses according to the students' proposals and consider as much as possible the stakes of geometry teaching. After introducing our theoretical framework and the methodology, we present the teaching-learning situation and analysis. This reveals three aspects of the situation that a teacher should be vigilant about implementing. We then present an analysis of the two teachers' practices regarding these three aspects. This analysis enables us to identify the teaching priorities of each teacher.

## THEORETICAL FRAMEWORK

We believe that it is important to draw on the observation of actual practices to analyze the practices of schoolteachers when teaching mathematics (Butlen, Mangiante-Orsola & Masselot, 2017). To study these practices, we rely on the theoretical tools developed by Butlen, Charles-Pézard and Masselot (op. cit., 2012; Charles-Pézard, 2010; Butlen & Masselot, 2019). We take up the notion of
professional gestures that Butlen et al. (2011) interpret in terms of schemes<fnref n="11" /> (Vergnaud, 1997) and define as elementary activities built up through professional experience (for example, giving instructions, or managing the layout of what is written on the blackboard...). We use the expression ‘gestures compositions’ to designate sets of professional gestures with the same aim (for example, managing a clarification phase, support students during the research phase...). We also consider the notion of ‘didactic vigilance’ (denoted DV in the following) defined by Charles-Pézard (2010) as a permanent adjustment by the teacher to prioritize the session ‘as close as possible’ to the targeted mathematical learning. This concept allows us to study how teachers take teaching stakes into account (Butlen et al., 2011). Exercising some didactic vigilance requires not only the teacher's mastery of mathematical and didactic knowledge, and the implementation of gestures and gestures compositions regarding the targeted learning, but also the preponderance of mathematical learning purposes. This leads us to model these conditions as a triplet (Mathematical and Didactic Knowledge, Gestures and Gestures compositions, Mathematical Learning Purposes) and to consider the three elements of this triplet together (Guille-Biel Winder & Mangiante-Orsola, 2023). The notion of DV can be related to other notions developed in the field of mathematics didactics. For example, the concept of ‘knowing- to act in the moment’ (Mason & Spence, 1999) highlights the teacher's ability to draw on various types of knowledge to react to the situation, and involves a state of awareness, of readiness to ‘see in the moment’. Like DV, this concept emphasizes the need for the teacher to react in the heat of the moment. The notion of ‘cognitive clarity’ (Downing & Fijalkow, 1984) stresses the need to help pupils recognize the object of learning and, like DV, leads us to question the teacher's ability to stand back and identify “what he or she is supposed to be leading up to”. However, the dynamic nature of the concept and the need to simultaneously take into account the three conditions modeled as a triplet distinguishes the notion of DV from the notions previously mentioned. Reacting in the moment in relation to the learning targeted must be done throughout the session and supported by mathematical and didactic knowledge through the implementation of gestures and compositions of gestures.

## GENERAL ANALYSIS METHOD

In the following, our intention is to describe how each teacher considers the geometric learning issues at stake. To this end, we have highlighted triplets for each of them, which model their exercise of DV and are contextualized to the selected teaching-learning situation. According to our previous work (Guille-Biel Winder & Mangiante-Orsola, 2023), our analytical approach is divided into several steps (Figure 1). We carry out an *a priori* analysis of the tasks proposed by the teaching-learning situation, and we pay particular attention to the procedures that pupils need to implement in order to achieve the expected layouts using the instruments provided for them (step 1). We then do an anticipatory analysis (step 2): that consists in questioning the task expected of the teacher, so as to identify the main difficulties he or she might encounter, as well as the aspects about which he or she should be vigilant with regard to the targeted learning – what we have termed ‘vigilance points’. We analyze the mathematical knowledge at stake as well as the instrumented procedures enabling these tasks to be performed. Based on an initial *a posteriori* analysis of sessions of the same teaching-learning situation carried out in several classes (step 2bis), we design tools adapted to the leads that we have identified in the anticipatory analysis (step 3). We use these tools in specific *a posteriori* analyses to identify traces of the didactic vigilance that is exercised (step 4). Although we follow the different

<footnote n="11">¹¹ A scheme is an invariant form of activity organization for a specific class of situations (Vergnaud, 1997).</footnote>
professional gestures that Butlen et al. (2011) interpret in terms of schemes<fnref n="11" /> (Vergnaud, 1997) and define as elementary activities built up through professional experience (for example, giving instructions, or managing the layout of what is written on the blackboard...). We use the expression ‘gestures compositions’ to designate sets of professional gestures with the same aim (for example, managing a clarification phase, support students during the research phase...). We also consider the notion of ‘didactic vigilance’ (denoted DV in the following) defined by Charles-Pézard (2010) as a permanent adjustment by the teacher to prioritize the session ‘as close as possible’ to the targeted mathematical learning. This concept allows us to study how teachers take teaching stakes into account (Butlen et al., 2011). Exercising some didactic vigilance requires not only the teacher's mastery of mathematical and didactic knowledge, and the implementation of gestures and gestures compositions regarding the targeted learning, but also the preponderance of mathematical learning purposes. This leads us to model these conditions as a triplet (Mathematical and Didactic Knowledge, Gestures and Gestures compositions, Mathematical Learning Purposes) and to consider the three elements of this triplet together (Guille-Biel Winder & Mangiante-Orsola, 2023). The notion of DV can be related to other notions developed in the field of mathematics didactics. For example, the concept of ‘knowing- to act in the moment’ (Mason & Spence, 1999) highlights the teacher's ability to draw on various types of knowledge to react to the situation, and involves a state of awareness, of readiness to ‘see in the moment’. Like DV, this concept emphasizes the need for the teacher to react in the heat of the moment. The notion of ‘cognitive clarity’ (Downing & Fijalkow, 1984) stresses the need to help pupils recognize the object of learning and, like DV, leads us to question the teacher's ability to stand back and identify “what he or she is supposed to be leading up to”. However, the dynamic nature of the concept and the need to simultaneously take into account the three conditions modeled as a triplet distinguishes the notion of DV from the notions previously mentioned. Reacting in the moment in relation to the learning targeted must be done throughout the session and supported by mathematical and didactic knowledge through the implementation of gestures and compositions of gestures.

## GENERAL ANALYSIS METHOD

In the following, our intention is to describe how each teacher considers the geometric learning issues at stake. To this end, we have highlighted triplets for each of them, which model their exercise of DV and are contextualized to the selected teaching-learning situation. According to our previous work (Guille-Biel Winder & Mangiante-Orsola, 2023), our analytical approach is divided into several steps (Figure 1). We carry out an *a priori* analysis of the tasks proposed by the teaching-learning situation, and we pay particular attention to the procedures that pupils need to implement in order to achieve the expected layouts using the instruments provided for them (step 1). We then do an anticipatory analysis (step 2): that consists in questioning the task expected of the teacher, so as to identify the main difficulties he or she might encounter, as well as the aspects about which he or she should be vigilant with regard to the targeted learning – what we have termed ‘vigilance points’. We analyze the mathematical knowledge at stake as well as the instrumented procedures enabling these tasks to be performed. Based on an initial *a posteriori* analysis of sessions of the same teaching-learning situation carried out in several classes (step 2bis), we design tools adapted to the leads that we have identified in the anticipatory analysis (step 3). We use these tools in specific *a posteriori* analyses to identify traces of the didactic vigilance that is exercised (step 4). Although we follow the different

<footnote n="11">¹¹ A scheme is an invariant form of activity organization for a specific class of situations (Vergnaud, 1997).</footnote>
steps of our methodology, we do not detail all the analyses carried out here, but only highlight the
most striking elements regarding the presented data.

<figure><img src="image_65.png" /><figcaption>Figure 1: General analysis method</figcaption></figure>

PRESENTATION OF THE TEACHING-LEARNING SITUATION

In the early 2000s, research was carried out with the aim of proposing a coherent progression of geometry throughout compulsory education, to better manage the transition from instrument-based tracing geometry to deductive geometry (Mathé, Barrier & Perrin-Glorian, 2020). Duval's work shows that the way a pupil perceives a geometric figure is decisive (Duval, 2008). Indeed, for them, the natural apprehension of figures is a vision of juxtaposed surfaces; in mathematics, however, geometrical figures are defined by relations linking lines and points. To help students enrich their geometric vision of figures, i.e., to help them see the relationships between the smaller elements (lines, points, circles) of a figure, a fundamental situation and its didactic variables have been identified: the figure restoration. A figure restoration is an action situation (Brousseau, 2002) that consists of reproducing a figure (full-scale or not) from a 'started figure' (part of the figure to obtain), with identified instruments (see examples figure 2). We note that "the reproduction of figures corresponds to the case where there is no started figure" (Mathé & Perrin-Glorian, 2023, p. 9). The 'started figure' or some of the tools can be used to transfer information from the initial figure (the full figure to be reproduced, called 'model figure' in the following), but without giving all the information. The pupil can then start restoration more easily with figures in terms of surfaces. The choice of 'started figure' and tools (templates, geometric tools, etc.) has a significant impact on the expected procedures: it can lead the pupils to apprehend the figure as being made up of smaller elements (lines or points). This also enables the teacher to support a change in the pupils' apprehension of figures (Mangiante-Orsola, 2023).

The *Triangles on Quadrilateral* sequence (ibid.) is organized in four successive phases. In each phase, pupils are asked to restore a figure, always the same one (an example is given in figure 2a) but using a different 'started figure' and/or various instruments. The four-phase organization should enable pupils to enrich their analysis of the model figure: identify simple figures (quadrilaterals, triangles, etc.) that make it up and the links that exist between them (apprehend the figure as combination of surfaces); identify locate alignments, extended lines, get a point by intersection of lines...(analyze the figure by relations linking lines and point). We clarify these elements in the following. In the first phase, the 'started figure' is the quadrilateral frame, and the instruments are the templates of the two large triangles, which we call T1 and T2 (figure 2b). To restore the figure, pupils need to distinguish the various simple figures that make it up (a quadrilateral, different triangles) and to use objects representing surfaces (templates). In the second phase, students are provided with another 'started figure' (figure 2c) and a template of the 'nibbled' T'2 triangle. To place the template and thus trace the missing triangle, they must extend the sides of the small triangle already traced (T'1) and use the alignment of the sides of the small triangles (segments: parts of lines objects). Two further phases are
steps of our methodology, we do not detail all the analyses carried out here, but only highlight the
most striking elements regarding the presented data.

<figure><img src="image_71.png" /><figcaption>Figure 1: General analysis method</figcaption></figure>

PRESENTATION OF THE TEACHING-LEARNING SITUATION

In the early 2000s, research was carried out with the aim of proposing a coherent progression of geometry throughout compulsory education, to better manage the transition from instrument-based tracing geometry to deductive geometry (Mathé, Barrier & Perrin-Glorian, 2020). Duval's work shows that the way a pupil perceives a geometric figure is decisive (Duval, 2008). Indeed, for them, the natural apprehension of figures is a vision of juxtaposed surfaces; in mathematics, however, geometrical figures are defined by relations linking lines and points. To help students enrich their geometric vision of figures, i.e., to help them see the relationships between the smaller elements (lines, points, circles) of a figure, a fundamental situation and its didactic variables have been identified: the figure restoration. A figure restoration is an action situation (Brousseau, 2002) that consists of reproducing a figure (full-scale or not) from a 'started figure' (part of the figure to obtain), with identified instruments (see examples figure 2). We note that "the reproduction of figures corresponds to the case where there is no started figure" (Mathé & Perrin-Glorian, 2023, p. 9). The 'started figure' or some of the tools can be used to transfer information from the initial figure (the full figure to be reproduced, called 'model figure' in the following), but without giving all the information. The pupil can then start restoration more easily with figures in terms of surfaces. The choice of 'started figure' and tools (templates, geometric tools, etc.) has a significant impact on the expected procedures: it can lead the pupils to apprehend the figure as being made up of smaller elements (lines or points). This also enables the teacher to support a change in the pupils' apprehension of figures (Mangiante-Orsola, 2023).

The *Triangles on Quadrilateral* sequence (ibid.) is organized in four successive phases. In each phase, pupils are asked to restore a figure, always the same one (an example is given in figure 2a) but using a different 'started figure' and/or various instruments. The four-phase organization should enable pupils to enrich their analysis of the model figure: identify simple figures (quadrilaterals, triangles, etc.) that make it up and the links that exist between them (apprehend the figure as combination of surfaces); identify locate alignments, extended lines, get a point by intersection of lines...(analyze the figure by relations linking lines and point). We clarify these elements in the following. In the first phase, the 'started figure' is the quadrilateral frame, and the instruments are the templates of the two large triangles, which we call T1 and T2 (figure 2b). To restore the figure, pupils need to distinguish the various simple figures that make it up (a quadrilateral, different triangles) and to use objects representing surfaces (templates). In the second phase, students are provided with another 'started figure' (figure 2c) and a template of the 'nibbled' T'2 triangle. To place the template and thus trace the missing triangle, they must extend the sides of the small triangle already traced (T'1) and use the alignment of the sides of the small triangles (segments: parts of lines objects). Two further phases are
planned. Phase 3 is designed to help pupils to go further in analyzing the model figure. Indeed, they must work out how to obtain one of the sides of the quadrilateral from points not already constructed, and to identify the links between the simple figures that make up the model figure (some of the sides of triangles T1 and T2 are carried by the diagonals of the quadrilateral). This involves identifying alignments. Phase 4 leads students to reinvest this knowledge. In the following section, we present the elements of the *a priori* analysis of phase 2 on which we are focusing.

<figure><img src="image_66.png" /><figcaption>Figure 2: Model figure (a), starting figure and instruments for phases 1 (b) and 2 (c)</figcaption></figure>

## FIRST ANALYSIS (STEP 1 AND STEP 2 OF OUR ANALYTICAL APPROACH)

To complete the started figure of this phase (quadrilateral and triangle T'1) using a partially ‘nibbled’ template of triangle T'2 and a non-scale ruler (figure 2c), pupils must extend two sides of triangle T'1 to be able to place the ‘nibbled’ template of triangle T'2, and then use this template to trace the missing segments. This procedure can be analyzed in three different stages<fnref n="12" /> (figure 3). It involves several difficulties in apprehending figures using instruments. For each stage, pupils must identify elements of the figure (graphic objects representing geometric objects) enabling them to position the instruments.

<figure><img src="image_67.png" /><figcaption>Figure 3: A three stages procedure</figcaption></figure>

On Stage 1, pupils must identify the element of the model figure that corresponds to the ‘nibbled’ template provided (the T'2 triangle). To do this, they must match some of the edges of the ‘nibbled’ template (surface) to the outline of a simple figure (a triangle) on the model figure (surface outline). On Stage 2, pupils must consider that they need information to place the template on the started figure: at least two directions of straight lines are required (without using visual perception). They must then extend two sides of the T'1 triangle on the started figure. Note that this presupposes that they can use the outline of the triangle (outline of a surface) to draw straight lines with the non-scale ruler (network of straight lines). To extend the sides of T'1 triangle, pupils implicitly use the fact that some sides of T'1 and T'2 triangles are aligned. When placing the ‘nibbled’ template, they must then pay attention to both sides at the same time (as when positioning the square and paying attention to both sides to draw the perpendicular to a straight line passing through a specific point not belonging to this straight line): this cannot be done in a single movement. On Stage 3, pupils must finally use the template to draw the missing elements. They can either draw the outline or mark the direction of lines with graphic elements (lines of a network), that can then be extended. The alignment of the

<footnote n="12">¹² As in any research activity, pupils can grope, make mistakes and start again. For this reason, these three phases are not necessarily carried out in chronological order.</footnote>
planned. Phase 3 is designed to help pupils to go further in analyzing the model figure. Indeed, they
must work out how to obtain one of the sides of the quadrilateral from points not already constructed,
and to identify the links between the simple figures that make up the model figure (some of the sides
of triangles T1 and T2 are carried by the diagonals of the quadrilateral). This involves identifying
alignments. Phase 4 leads students to reinvest this knowledge. In the following section, we present
the elements of the *a priori* analysis of phase 2 on which we are focusing.

<figure><img src="image_72.png" /><figcaption>Figure 2: Model figure (a), starting figure and instruments for phases 1 (b) and 2 (c)</figcaption></figure>

FIRST ANALYSIS (STEP 1 AND STEP 2 OF OUR ANALYTICAL APPROACH)

To complete the started figure of this phase (quadrilateral and triangle T'1) using a partially ‘nibbled’
template of triangle T'2 and a non-scale ruler (figure 2c), pupils must extend two sides of triangle T'1
to be able to place the ‘nibbled’ template of triangle T'2, and then use this template to trace the missing
segments. This procedure can be analyzed in three different stages<fnref n="12" /> (figure 3). It involves several
difficulties in apprehending figures using instruments. For each stage, pupils must identify elements
of the figure (graphic objects representing geometric objects) enabling them to position the
instruments.

<figure><img src="image_73.png" /><figcaption>Figure 3: A three stages procedure</figcaption></figure>

On Stage 1, pupils must identify the element of the model figure that corresponds to the ‘nibbled’
template provided (the T'2 triangle). To do this, they must match some of the edges of the ‘nibbled’
template (surface) to the outline of a simple figure (a triangle) on the model figure (surface outline).
On Stage 2, pupils must consider that they need information to place the template on the started
figure: at least two directions of straight lines are required (without using visual perception). They
must then extend two sides of the T'1 triangle on the started figure. Note that this presupposes that
they can use the outline of the triangle (outline of a surface) to draw straight lines with the non-scale
ruler (network of straight lines). To extend the sides of T'1 triangle, pupils implicitly use the fact that
some sides of T'1 and T'2 triangles are aligned. When placing the ‘nibbled’ template, they must then
pay attention to both sides at the same time (as when positioning the square and paying attention to
both sides to draw the perpendicular to a straight line passing through a specific point not belonging
to this straight line): this cannot be done in a single movement. On Stage 3, pupils must finally use
the template to draw the missing elements. They can either draw the outline or mark the direction of
lines with graphic elements (lines of a network), that can then be extended. The alignment of the

<footnote n="12">¹² As in any research activity, pupils can grope, make mistakes and start again. For this reason, these three phases are not necessarily carried out in chronological order.</footnote>
segments to be traced with one of the vertices of the quadrilateral constitutes an element of control
of the directions of the obtained lines.

Anticipatory analysis of the didactic vigilance exercising. Like any figure restoration situation, the *Triangles on Quadrilateral* sequence involves tracing and consequently requires managing the transition between actions on material objects and/or various graphic representations and the geometric knowledge involved. This is a major challenge for teachers (Mangiante-Orsola, 2014; Guille-Biel Winder, 2021). With this in mind, we use the *a priori* analysis of phase 2 to identify ‘points of vigilance’ for the teacher in each of the three phases identified above. More specifically, we highlight the importance of getting pupils to identify the simple figure on the model that coincides with the template (phase 1), to identify and draw the geometric objects required to position the template (phase 2), and to identify and draw the missing geometric objects using the template (phase 3).

## ANALYSIS OF THE DIDACTIC VIGILANCE EXERCISING

According to our methodology, we rely on a model of DV exercising in the form of a triplet. We first identify the gesture or gestures composition relating to the identification by pupils of the elements of the figure that enable them to position the instruments, and to the linking graphic objects with the corresponding geometric objects. Finally, to identify the learning purpose and evaluate the extent to which geometric learning is a priority, we relate the interventions of each teacher (her actions, her words) to the two possible priorities, situating them between justifying the actions to be performed by the precision of the tracings (graphic priority) and emphasizing to pupils the geometric knowledge involved (geometric priority). Based on *a posteriori* analyses of class sessions (step 2bis, not presented here), we have identified four points of analysis: does the teacher highlight the need to identify the elements required to place an instrument (template or non-scale ruler here)? Does she specify how many are necessary and which ones? What lexicon does she use to describe them? What type of argument does she use to convince the pupil? To find answers to these questions, we rely on our analysis of verbal interactions and actual gestures. In this text, we selected three ‘remarkable events’ (Leutenegger, 2000) in terms of considering the stakes of geometry teaching in the ‘heat of the moment’ referring to each stage of phase 2. But our analyses are also based on other data collected as part of our monitoring of these teachers (Mangiante-Orsola, 2023).

### Stage 1 - Place the template on the model figure

The first remarkable event takes place during the group discussion of phase 2 in Vic's class. The teacher asked a pupil about the way he had placed the template on the model figure. She asks: 'What else did you have to do? Did you have to...? Put the triangle in the triangle's house, because it's almost his house, and it fit really well'. We identify a first element of the triplet modeling the DV exercising: the gestures composition performed by Vic is linked to the institutionalization process. It consists in explaining how to place the template on the model figure, to identify the corresponding elements of the model figure (here a triangle). The mathematical and didactic knowledge to be mobilized by the teacher concerns the fact that a triangle is defined by the contour of a surface. To associate a simple figure seen as a surface with a template, the template must be placed in such a way that its edges coincide with the contour of the surface. In order to characterize Vic's learning objective, we identify few possibilities. In this situation, Vic could have simply given a manipulative tip (e.g., explain that
segments to be traced with one of the vertices of the quadrilateral constitutes an element of control
of the directions of the obtained lines.

Anticipatory analysis of the didactic vigilance exercising. Like any figure restoration situation, the *Triangles on Quadrilateral* sequence involves tracing and consequently requires managing the transition between actions on material objects and/or various graphic representations and the geometric knowledge involved. This is a major challenge for teachers (Mangiante-Orsola, 2014; Guille-Biel Winder, 2021). With this in mind, we use the *a priori* analysis of phase 2 to identify ‘points of vigilance’ for the teacher in each of the three phases identified above. More specifically, we highlight the importance of getting pupils to identify the simple figure on the model that coincides with the template (phase 1), to identify and draw the geometric objects required to position the template (phase 2), and to identify and draw the missing geometric objects using the template (phase 3).

## ANALYSIS OF THE DIDACTIC VIGILANCE EXERCISING

According to our methodology, we rely on a model of DV exercising in the form of a triplet. We first identify the gesture or gestures composition relating to the identification by pupils of the elements of the figure that enable them to position the instruments, and to the linking graphic objects with the corresponding geometric objects. Finally, to identify the learning purpose and evaluate the extent to which geometric learning is a priority, we relate the interventions of each teacher (her actions, her words) to the two possible priorities, situating them between justifying the actions to be performed by the precision of the tracings (graphic priority) and emphasizing to pupils the geometric knowledge involved (geometric priority). Based on *a posteriori* analyses of class sessions (step 2bis, not presented here), we have identified four points of analysis: does the teacher highlight the need to identify the elements required to place an instrument (template or non-scale ruler here)? Does she specify how many are necessary and which ones? What lexicon does she use to describe them? What type of argument does she use to convince the pupil? To find answers to these questions, we rely on our analysis of verbal interactions and actual gestures. In this text, we selected three ‘remarkable events’ (Leutenegger, 2000) in terms of considering the stakes of geometry teaching in the ‘heat of the moment’ referring to each stage of phase 2. But our analyses are also based on other data collected as part of our monitoring of these teachers (Mangiante-Orsola, 2023).

### Stage 1 - Place the template on the model figure

The first remarkable event takes place during the group discussion of phase 2 in Vic's class. The teacher asked a pupil about the way he had placed the template on the model figure. She asks: 'What else did you have to do? Did you have to...? Put the triangle in the triangle's house, because it's almost his house, and it fit really well'. We identify a first element of the triplet modeling the DV exercising: the gestures composition performed by Vic is linked to the institutionalization process. It consists in explaining how to place the template on the model figure, to identify the corresponding elements of the model figure (here a triangle). The mathematical and didactic knowledge to be mobilized by the teacher concerns the fact that a triangle is defined by the contour of a surface. To associate a simple figure seen as a surface with a template, the template must be placed in such a way that its edges coincide with the contour of the surface. In order to characterize Vic's learning objective, we identify few possibilities. In this situation, Vic could have simply given a manipulative tip (e.g., explain that
the template should be held firmly with the fingers, turned over if necessary or rotated). Vic does,
however, make explicit the element to be considered when positioning a template on the model figure,
namely an outline, but she designates it as a graphic object and even iconically (a ‘little house’). Other
data lead us to conclude that Vic is demanding when it comes to the precision of tracings. In fact,
right from the first session of the sequence, she explains to students that, in geometry, they must be
‘precise’, ‘clean’, careful’, they must not ‘press too hard on the pencil’, ‘extend’, ‘go beyond’, they
are ‘allowed to erase’. Hence, she is simply proposing an explanation of the procedure for placing
the template on the model figure to obtain more precise plotting (graphic priority), without naming
the involved geometric objects.

Stage 2 - Place the template on the started figure

The second remarkable event occurred in Lea's class. She intervenes with a pupil who has only extended one side of the triangle and is therefore unable to place the nibbled template correctly. Lea moves the template along the straight line drawn to show that its position cannot be determined precisely (figure 3).

Lea: Where do you place the template on the model figure? (...) You're missing something to be more precise! (...) You need at least two directions, in other words, two straight lines to be able to place the template! Extending just one side of the triangle isn't enough!

The gestures compositions highlighted by this extract helps to regulate the student's activity. It consists of helping him to place the template on the starting figure to be able to draw a triangle. Lea's useful mathematical and didactical knowledge concerns the fact that a triangle is defined by a surface/circumference (angles, side lengths), three straight line directions or three points. Her comments suggest that she is pursuing a geometric stake. In fact, she indicates to the pupil the elements required to place the template and designates them not as graphic objects (lines) but as geometric objects ('direction'). What's more, she's able to explain the number of directions required to place the template ('you need two directions'). We can find other examples of Lea's such propositions during the sequence, as for example: 'you need two points to draw a straight line with the ruler'. In this way, Lea makes explicit the geometric proposition underpinning the appropriate use of the instrument. This relates to practical axiomatic theory (Mathé and Perrin-Glorian, 2023).

Stage 3 - Complete the tracing required to complete the started figure

The third remarkable event takes place in Vic's class. During the group discussion, Vic asked a student (Jim) who extends only one side of the triangle, about how to use the template to draw the missing elements:

Vic: How did you manage to reproduce the same thing here? (...) Here's a first tracing. Next? [Jim misplaces the set square, which he uses as a ruler] Position your set square correctly. Which side of the triangle have you extended? (...) [Jim points to the side of the triangle to be extended]. What do you have to do to position it correctly? What are you going to do with the template because you can't go on like this? [Vic waits for Jim to draw little lines to mark the directions then speaks to the whole class] (...) Take a benchmark, do you remember? [Jim draws a small line along one edge of the template. Vic takes the setsquare in her hands, gives it to Jim and invites him to extend the line].

The gestures composition featured in this extract is part of the institutionalization process. The
mathematical and didactical knowledge at stake relates to the fact that a triangle is defined by three
the template should be held firmly with the fingers, turned over if necessary or rotated). Vic does,
however, make explicit the element to be considered when positioning a template on the model figure,
namely an outline, but she designates it as a graphic object and even iconically (a ‘little house’). Other
data lead us to conclude that Vic is demanding when it comes to the precision of tracings. In fact,
right from the first session of the sequence, she explains to students that, in geometry, they must be
‘precise’, ‘clean’, careful’, they must not ‘press too hard on the pencil’, ‘extend’, ‘go beyond’, they
are ‘allowed to erase’. Hence, she is simply proposing an explanation of the procedure for placing
the template on the model figure to obtain more precise plotting (graphic priority), without naming
the involved geometric objects.

Stage 2 - Place the template on the started figure

The second remarkable event occurred in Lea's class. She intervenes with a pupil who has only extended one side of the triangle and is therefore unable to place the nibbled template correctly. Lea moves the template along the straight line drawn to show that its position cannot be determined precisely (figure 3).

Lea: Where do you place the template on the model figure? (...) You're missing something to be more precise! (...) You need at least two directions, in other words, two straight lines to be able to place the template! Extending just one side of the triangle isn't enough!

The gestures compositions highlighted by this extract helps to regulate the student's activity. It consists of helping him to place the template on the starting figure to be able to draw a triangle. Lea's useful mathematical and didactical knowledge concerns the fact that a triangle is defined by a surface/circumference (angles, side lengths), three straight line directions or three points. Her comments suggest that she is pursuing a geometric stake. In fact, she indicates to the pupil the elements required to place the template and designates them not as graphic objects (lines) but as geometric objects ('direction'). What's more, she's able to explain the number of directions required to place the template ('you need two directions'). We can find other examples of Lea's such propositions during the sequence, as for example: 'you need two points to draw a straight line with the ruler'. In this way, Lea makes explicit the geometric proposition underpinning the appropriate use of the instrument. This relates to practical axiomatic theory (Mathé and Perrin-Glorian, 2023).

Stage 3 - Complete the tracing required to complete the started figure

The third remarkable event takes place in Vic's class. During the group discussion, Vic asked a student (Jim) who extends only one side of the triangle, about how to use the template to draw the missing elements:

Vic: How did you manage to reproduce the same thing here? (...) Here's a first tracing. Next? [Jim misplaces the set square, which he uses as a ruler] Position your set square correctly. Which side of the triangle have you extended? (...) [Jim points to the side of the triangle to be extended]. What do you have to do to position it correctly? What are you going to do with the template because you can't go on like this? [Vic waits for Jim to draw little lines to mark the directions then speaks to the whole class] (...) Take a benchmark, do you remember? [Jim draws a small line along one edge of the template. Vic takes the setsquare in her hands, gives it to Jim and invites him to extend the line].

The gestures composition featured in this extract is part of the institutionalization process. The
mathematical and didactical knowledge at stake relates to the fact that a triangle is defined by three
straight line directions, and that identifying straight line directions enables pupils to enrich their analysis of the figure by considering the triangle not as defined by a contour, but by a network of straight lines. Vic, although always very attentive to the precision of the drawings, seems here to go beyond simple manipulative considerations for graphic purposes. She guides Jim more in the analysis of the figure through the clarification of elements to be identified to place the instruments (the non-graduated ruler). She doesn't use a geometric lexicon (there's no mention here of straight lines to be extended, points of intersection, aligned vertices), but it's interesting to note here her emphasis on the word 'benchmark', which enables her to designate the graphic element to be used to place the ruler correctly (the direction of a straight line). We also note that Vic specifies some geometric objects (the side of the triangle) and, in so doing, leads Jim to consider and draw each side of the triangle in turn, thereby segmenting its outline.

## CONCLUSION

By analyzing these three remarkable events in terms of triplets, we can give an account of the way in which these two teachers exercise their didactic vigilance (these analyses are confirmed by other data not presented here). During Stage 1, Vic's DV exercising is characterized by learning stakes limited to graphic aspects, even at the knowledge institutionalization. On the other hand, Lea exercises during Stage 2 a significant DV on geometrical learning stakes: she intervenes regularly with pupils to identify the geometric objects required to position the template and designates them using geometric vocabulary. Our analysis of Stage 3 highlights an 'in-between' situation: it confirms Vic's particular attention to the identification of elements for positioning the instruments, since she insists on the need to 'take reference points'. But even if the geometric objects are not explicitly described using a geometric lexicon, it's clear that Vic's often-expressed concern for precision leads her to encourage pupils to use graphic elements, and through them, the geometric objects they represent. We can conclude from this that these two teachers' DV exercising helps to develop in the pupils a use of instruments based on a set of rules that Mathé and Perrin-Glorian (2023) refer to as practical axiomatic theory. However, our analyses reveal a gradation between two possible outcomes: justifying the actions to be performed by the precision of the lines (graphic priority) and highlighting to pupils the geometric knowledge involved (geometric priority). From this observation, we have highlighted specific difficulties for teachers who are often situated in this 'in-between' situation, between graphic and geometric learning goals. In training, making teachers aware of the existence of geometric propositions that underpin the proper use of instruments could, in our view, help them take better account of the learning stakes involved in geometry teaching.

## Acknowledgements

This research is financially supported by the Institut français de l'éducation (réseau des LéA).

## References

* Brousseau, G. (2002). *Theory of didactical situations in mathematics*. Mathematics Education Library, Kluwer Academic Publishers.

* Duval, R. (2008). Eight problems for a semiotic approach in mathematics education. In L.Radford, G. Schubring, & F. Seeger (Eds.), *Semiotics in mathematics education; epistemology, history, classroom and culture* (pp. 39-61). Sense Publishers.

* Butlen, D., Charles-Pezard, M., & Masselot, P. (2011). Deux dimensions de l'activité du professeur des écoles exerçant dans des classes de milieux défavorisés: installer la paix scolaire, exercer une vigilance didactique.
straight line directions, and that identifying straight line directions enables pupils to enrich their analysis of the figure by considering the triangle not as defined by a contour, but by a network of straight lines. Vic, although always very attentive to the precision of the drawings, seems here to go beyond simple manipulative considerations for graphic purposes. She guides Jim more in the analysis of the figure through the clarification of elements to be identified to place the instruments (the non-graduated ruler). She doesn't use a geometric lexicon (there's no mention here of straight lines to be extended, points of intersection, aligned vertices), but it's interesting to note here her emphasis on the word 'benchmark', which enables her to designate the graphic element to be used to place the ruler correctly (the direction of a straight line). We also note that Vic specifies some geometric objects (the side of the triangle) and, in so doing, leads Jim to consider and draw each side of the triangle in turn, thereby segmenting its outline.

## CONCLUSION

By analyzing these three remarkable events in terms of triplets, we can give an account of the way in which these two teachers exercise their didactic vigilance (these analyses are confirmed by other data not presented here). During Stage 1, Vic's DV exercising is characterized by learning stakes limited to graphic aspects, even at the knowledge institutionalization. On the other hand, Lea exercises during Stage 2 a significant DV on geometrical learning stakes: she intervenes regularly with pupils to identify the geometric objects required to position the template and designates them using geometric vocabulary. Our analysis of Stage 3 highlights an 'in-between' situation: it confirms Vic's particular attention to the identification of elements for positioning the instruments, since she insists on the need to 'take reference points'. But even if the geometric objects are not explicitly described using a geometric lexicon, it's clear that Vic's often-expressed concern for precision leads her to encourage pupils to use graphic elements, and through them, the geometric objects they represent. We can conclude from this that these two teachers' DV exercising helps to develop in the pupils a use of instruments based on a set of rules that Mathé and Perrin-Glorian (2023) refer to as practical axiomatic theory. However, our analyses reveal a gradation between two possible outcomes: justifying the actions to be performed by the precision of the lines (graphic priority) and highlighting to pupils the geometric knowledge involved (geometric priority). From this observation, we have highlighted specific difficulties for teachers who are often situated in this 'in-between' situation, between graphic and geometric learning goals. In training, making teachers aware of the existence of geometric propositions that underpin the proper use of instruments could, in our view, help them take better account of the learning stakes involved in geometry teaching.

## Acknowledgements

This research is financially supported by the *Institut français de l'éducation* (réseau des LéA).

## References

* Brousseau, G. (2002). *Theory of didactical situations in mathematics*. Mathematics Education Library, Kluwer Academic Publishers.

* Duval, R. (2008). Eight problems for a semiotic approach in mathematics education. In L.Radford, G. Schubring, & F. Seeger (Eds.), *Semiotics in mathematics education; epistemology, history, classroom and culture* (pp. 39-61). Sense Publishers.

* Butlen, D., Charles-Pezard, M., & Masselot, P. (2011). Deux dimensions de l'activité du professeur des écoles exerçant dans des classes de milieux défavorisés: installer la paix scolaire, exercer une vigilance didactique.
*Colloque Le travail enseignant au XXIe siècle. Perspectives croisées: didactiques et didactique professionnelle*. http://www.inrp.fr/archives/colloques/travail-enseignant/contrib/70.pdf

Butlen, D., Mangiante-Orsola, C., & Masselot, P. (2017). Routines et gestes professionnels, un outil pour l'analyse des pratiques effectives et pour la formation des pratiques des professeurs des écoles en mathématiques [Routines and professional gestures, a tool for analyzing actual practices and training schoolteachers in mathematics]. *Recherches en Didactique des Mathématiques, 2017*(2), 25-40.

Butlen, D., & Masselot, P. (2019). Enjeux et modalités de formation pour les professeurs des écoles en didactique des mathématiques [Challenges and training methods for schoolteachers in mathematics didactics]. *Canadian Journal of Science, Mathematics and Technology Education, 19*(2), 91-106.

Charles-Pezard, M. (2010). Installer la paix scolaire, exercer une vigilance didactique [To install the school peace, to exercise some didactic vigilance]. *Recherches en Didactique des Mathématiques, 30*(2), 197-261.

Charles-Pezard, M., Butlen, D., & Masselot, P. (2012). *Professeurs des écoles débutants enseignant en ZEP, quelles pratiques ? Quelle formation? La Pensée Sauvage*.

Downing, J., & Fijalkow, J. (1984). *Lire et raisonner*. Privat.

Guille-Biel Winder, C. (2021). Impact du langage de l'enseignant sur les relations entre les élèves et le milieu dans une situation d'action en géométrie [Impact of the teacher's language on relations between students and the milieu during an action situation in geometry]. *Recherches en Didactique des Mathématiques, 41*(1), 55-96.

Guille-Biel Winder, C., & Mangiante-Orsola, C. (2023). Contribution à l'étude de l'exercice de la vigilance didactique. [Contribution to the study of the didactic vigilance]. *Recherches en Didactique des Mathématiques, 43*(2), 199-241.

Leutenegger, F. (2000). Construction d'une « clinique » pour le didactique: une étude des phénomènes temporels de l'enseignement [Building a 'clinic' for didactics: a study of temporal phenonema in teaching]. *Recherches en Didactique des Mathématiques, 20*(2), 209-250.

Mangiante-Orsola, C. (2014). Une étude du processus d'appropriation par des enseignants de situations produites par la recherche pour l'enseignement de la géométrie. In S. Coppé, & M. Haspekian (Eds.), *Actes du Séminaire National de Didactique des Mathématiques 2013* (pp. 137-158). IREM de Paris.

Mangiante-Orsola, C. (2023). Study on teacher appropriation of a geometry education resource. In C. Guille-Biel Winder, & T. Assude (Eds.), *Articulations between tangible space, graphical space and geometrical space, resources practices and training* (pp. 199-219). Iste Wiley.

Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing to act in the moment. *Educational Studies in Mathematics, 38*, 135-161.

Mathé, A.C., Barrier, T., & Perrin-Glorian, M.- J. (2020). *Enseigner la géométrie à l'école élémentaire: enjeux, ruptures et continuités*. Academia-L'Harmattan.

Mathé, A.C., & Perrin-Glorian, M.- J. (2023). The geometry of tracing, a possible link between geometric drawing and Euclid's geometry. In C. Guille-Biel Winder, & T. Assude (Eds.), *Articulations between tangible space, graphical space and geometrical space, resources practices and training* (pp. 3-33). Iste Wiley.

Vergnaud, G. (1997). The nature of mathematical concepts. In T. Nunes, & T. Bryant (Eds), *Learning and teaching mathematics: An international perspective*. Psychology Press.
*Colloque Le travail enseignant au XXIe siècle. Perspectives croisées: didactiques et didactique professionnelle*. http://www.inrp.fr/archives/colloques/travail-enseignant/contrib/70.pdf

Butlen, D., Mangiante-Orsola, C., & Masselot, P. (2017). Routines et gestes professionnels, un outil pour l'analyse des pratiques effectives et pour la formation des pratiques des professeurs des écoles en mathématiques [Routines and professional gestures, a tool for analyzing actual practices and training schoolteachers in mathematics]. *Recherches en Didactique des Mathématiques, 2017*(2), 25-40.

Butlen, D., & Masselot, P. (2019). Enjeux et modalités de formation pour les professeurs des écoles en didactique des mathématiques [Challenges and training methods for schoolteachers in mathematics didactics]. *Canadian Journal of Science, Mathematics and Technology Education, 19*(2), 91-106.

Charles-Pezard, M. (2010). Installer la paix scolaire, exercer une vigilance didactique [To install the school peace, to exercise some didactic vigilance]. *Recherches en Didactique des Mathématiques, 30*(2), 197-261.

Charles-Pezard, M., Butlen, D., & Masselot, P. (2012). *Professeurs des écoles débutants enseignant en ZEP, quelles pratiques ? Quelle formation? La Pensée Sauvage*.

Downing, J., & Fijalkow, J. (1984). *Lire et raisonner*. Privat.

Guille-Biel Winder, C. (2021). Impact du langage de l'enseignant sur les relations entre les élèves et le milieu dans une situation d'action en géométrie [Impact of the teacher's language on relations between students and the milieu during an action situation in geometry]. *Recherches en Didactique des Mathématiques, 41*(1), 55-96.

Guille-Biel Winder, C., & Mangiante-Orsola, C. (2023). Contribution à l'étude de l'exercice de la vigilance didactique. [Contribution to the study of the didactic vigilance]. *Recherches en Didactique des Mathématiques, 43*(2), 199-241.

Leutenegger, F. (2000). Construction d'une « clinique » pour le didactique: une étude des phénomènes temporels de l'enseignement [Building a 'clinic' for didactics: a study of temporal phenonema in teaching]. *Recherches en Didactique des Mathématiques, 20*(2), 209-250.

Mangiante-Orsola, C. (2014). Une étude du processus d'appropriation par des enseignants de situations produites par la recherche pour l'enseignement de la géométrie. In S. Coppé, & M. Haspekian (Eds.), *Actes du Séminaire National de Didactique des Mathématiques 2013* (pp. 137-158). IREM de Paris.

Mangiante-Orsola, C. (2023). Study on teacher appropriation of a geometry education resource. In C. Guille-Biel Winder, & T. Assude (Eds.), *Articulations between tangible space, graphical space and geometrical space, resources practices and training* (pp. 199-219). Iste Wiley.

Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing to act in the moment. *Educational Studies in Mathematics, 38*, 135-161.

Mathé, A.C., Barrier, T., & Perrin-Glorian, M.-J. (2020). *Enseigner la géométrie à l'école élémentaire: enjeux, ruptures et continuités*. Academia-L'Harmattan.

Mathé, A.C., & Perrin-Glorian, M.-J. (2023). The geometry of tracing, a possible link between geometric drawing and Euclid's geometry. In C. Guille-Biel Winder, & T. Assude (Eds.), *Articulations between tangible space, graphical space and geometrical space, resources practices and training* (pp. 3-33). Iste Wiley.

Vergnaud, G. (1997). The nature of mathematical concepts. In T. Nunes, & T. Bryant (Eds), *Learning and teaching mathematics: An international perspective*. Psychology Press.
# WHAT CAN THE STRUCTURE OF A GEOMETRY TASK STATEMENT BE TO PROMOTE A CERTAIN TYPE OF ARGUMENT?

Oscar Molina, Carmen Samper, Claudia Vargas, Leonor Camargo, Patricia Perry

Universidad Pedagógica Nacional (Colombia)

> We propose the structure that we think a task statement should have to promote the production of a specific type of argument (inductive, abductive, or deductive) during the task solution process in which, preferably, a Dynamic Geometry System (DGS) must be used. To support the idea, solution strategies produced by students of a plane geometry course of the preservice program for mathematics teachers of our University, when they address tasks that involve the perpendicular bisector of a segment, are presented. Using the argument model proposed by Toulmin to analyze these strategies, the type of main argument enounced according to the type of statement was identified. We point out how the type of task can contribute to the didactic-mathematical knowledge of the geometry teacher.

## PRESENTATION

In the last three decades there has been a growing interest in including argumentation in the school curriculum as a means to develop the students' mathematical proficiency (Boero et al., 2010; McNeill & Knight, 2013; Stylianides et al., 2016). It is assumed that it favors proper mathematical actions such as inducing or abducing properties from empirical evidence to formulate conjectures, providing formal deductive proofs to validate their conjectures, and communicating new ideas. Favoring argumentation in the classroom requires that the teacher use not only his knowledge (or his beliefs) about different types of arguments, but also that relative to didactical aspects since it is useful when selecting learning tasks that promote argumentation and helps to foresee (and later evaluate) students' activity (Lin et al., 2012). Therefore, knowing about different types of arguments and recognizing the difference between them should be part of elementary and secondary school mathematics teachers' knowledge. Even more, knowing about different types of tasks related to argumentation, how to manage them, and recognizing the relation between these and the mathematical activity they spur should be another element of a teacher's knowledge (Stylianides & Ball, 2008).

To promote preservice mathematics teachers' involvement in argumentative practices in geometry, for more than a decade we have been immersed in a curriculum innovation process with the purpose of generating a favorable environment in which it can be achieved. The didactic effort of such an innovation includes the following elements: (i) mathematics tasks which require diverse mathematical activity and the discovery of interesting geometry relations, around which the corresponding theoretic system is conformed; (ii) social interaction in the classroom, among the students while jointly solving the problems, or between teacher and students, when they publicly expose their solution and argument their ideas, and (iii) the use of a Dynamic Geometry System (DGS) as an artifact that makes possible the exploration of the proposed tasks to find a solution.
# WHAT CAN THE STRUCTURE OF A GEOMETRY TASK STATEMENT BE TO PROMOTE A CERTAIN TYPE OF ARGUMENT?

Oscar Molina, Carmen Samper, Claudia Vargas, Leonor Camargo, Patricia Perry

Universidad Pedagógica Nacional (Colombia)

> We propose the structure that we think a task statement should have to promote the production of a specific type of argument (inductive, abductive, or deductive) during the task solution process in which, preferably, a Dynamic Geometry System (DGS) must be used. To support the idea, solution strategies produced by students of a plane geometry course of the preservice program for mathematics teachers of our University, when they address tasks that involve the perpendicular bisector of a segment, are presented. Using the argument model proposed by Toulmin to analyze these strategies, the type of main argument enounced according to the type of statement was identified. We point out how the type of task can contribute to the didactic-mathematical knowledge of the geometry teacher.

## PRESENTATION

In the last three decades there has been a growing interest in including argumentation in the school curriculum as a means to develop the students' mathematical proficiency (Boero et al., 2010; McNeill & Knight, 2013; Stylianides et al., 2016). It is assumed that it favors proper mathematical actions such as inducing or abducing properties from empirical evidence to formulate conjectures, providing formal deductive proofs to validate their conjectures, and communicating new ideas. Favoring argumentation in the classroom requires that the teacher use not only his knowledge (or his beliefs) about different types of arguments, but also that relative to didactical aspects since it is useful when selecting learning tasks that promote argumentation and helps to foresee (and later evaluate) students' activity (Lin et al., 2012). Therefore, knowing about different types of arguments and recognizing the difference between them should be part of elementary and secondary school mathematics teachers' knowledge. Even more, knowing about different types of tasks related to argumentation, how to manage them, and recognizing the relation between these and the mathematical activity they spur should be another element of a teacher's knowledge (Stylianides & Ball, 2008).

To promote preservice mathematics teachers' involvement in argumentative practices in geometry, for more than a decade we have been immersed in a curriculum innovation process with the purpose of generating a favorable environment in which it can be achieved. The didactic effort of such an innovation includes the following elements: (i) mathematics tasks which require diverse mathematical activity and the discovery of interesting geometry relations, around which the corresponding theoretic system is conformed; (ii) social interaction in the classroom, among the students while jointly solving the problems, or between teacher and students, when they publicly expose their solution and argument their ideas, and (iii) the use of a Dynamic Geometry System (DGS) as an artifact that makes possible the exploration of the proposed tasks to find a solution.
In this paper, we expose the relation between task statements and types of arguments that each statement favors. To illustrate, we use examples of class happenings in the plane geometry course of the preservice teacher program offered by our University. Concretely, we present how each type of task, characterized by the structure of its statement, principally favors a specific type of argument. Although various researchers have shown how tasks which must be solved with a DGS promote argumentation (e.g., Arzarello et al., 2012; Baccaglini-Frank & Mariotti, 2010), they do not expose how the structure of the task statement can contribute to the production of different types of arguments. Following the idea exposed by Stylianides and Stylianides (2006), the structure of the task statements that we propose, besides promoting preservice teachers’ argumentation, also becomes a concrete tool when they design tasks to favor this mathematical process among their future students.

In what follows, we initially present conceptual references with respect to argumentation and argument; then we expose the methodological aspects used for the analysis. Later, we describe the type of task statements, their relation with the types of arguments and illustrate each case with an example. To conclude, we communicate some reflections.

## CONCEPTUAL REFERENCES

In this section, we initially present what we conceive as argumentation, argument and types of arguments, using Toulmin's proposal (2003) as our principal reference. Then we expose our conceptualization of learning task and argumentation task; we describe the aspects that compose the task statement and elements that can be used to design statements of argumentation tasks.

### Argument and argumentation

Our conceptualization of argument and argumentation comes from a sociocultural and discursive posture; for it we used the definition of argumentation proposed by Durand-Guerrier, Boero, Douek, Epp and Tanguay (2012) and the commented narration that Knipping and Reid (2019) present with respect to the functional structure of an argument proposed by Toulmin (2003). We consider an argument to be an expositive discursive expression, according to shared norms, that presents an assertion and reasons that sustain it. The assertion can be presented as one of the following: a proposition (i.e. a sentence of which it can be said it is true or false) that affirms or negates an idea; a sentence in which a posture is stated; or a physical action with which an idea or posture is expressed. Of the idea exposed it is of interest to sustain its veracity; of the posture presented, its acceptability. The reasons can be presented as sentences (propositions or not) or as actions. The set of reasons that sustain the veracity or acceptability of an assertion conform its justification. We understand argumentation as a discursive and social cultural process destined to increase the acceptability of a point of view or to determine the veracity of an exposed idea through the presentation of propositions or actions destined to justify (or refute) them. In such process arguments arise.

In its most simple version, an argument consists of three elements –data, assertion, warrant– and has a functional structure that displays the relations among these elements, according to the role they each have (Toulmin, 2003). The data sustains the assertion; it is evidence that supports the assertion. The warrant expresses the relation between the data and the assertion in a general statement that indicates why the data is evidence to sustain the assertion. The data and the warrant (which can be explicit or implicit) are the reasons with which the assertion is justified.
In this paper, we expose the relation between task statements and types of arguments that each statement favors. To illustrate, we use examples of class happenings in the plane geometry course of the preservice teacher program offered by our University. Concretely, we present how each type of task, characterized by the structure of its statement, principally favors a specific type of argument. Although various researchers have shown how tasks which must be solved with a DGS promote argumentation (e.g., Arzarello et al., 2012; Baccaglini-Frank & Mariotti, 2010), they do not expose how the structure of the task statement can contribute to the production of different types of arguments. Following the idea exposed by Stylianides and Stylianides (2006), the structure of the task statements that we propose, besides promoting preservice teachers’ argumentation, also becomes a concrete tool when they design tasks to favor this mathematical process among their future students.

In what follows, we initially present conceptual references with respect to argumentation and argument; then we expose the methodological aspects used for the analysis. Later, we describe the type of task statements, their relation with the types of arguments and illustrate each case with an example. To conclude, we communicate some reflections.

## CONCEPTUAL REFERENCES

In this section, we initially present what we conceive as argumentation, argument and types of arguments, using Toulmin's proposal (2003) as our principal reference. Then we expose our conceptualization of learning task and argumentation task; we describe the aspects that compose the task statement and elements that can be used to design statements of argumentation tasks.

### Argument and argumentation

Our conceptualization of argument and argumentation comes from a sociocultural and discursive posture; for it we used the definition of argumentation proposed by Durand-Guerrier, Boero, Douek, Epp and Tanguay (2012) and the commented narration that Knipping and Reid (2019) present with respect to the functional structure of an argument proposed by Toulmin (2003). We consider an argument to be an expositive discursive expression, according to shared norms, that presents an assertion and reasons that sustain it. The assertion can be presented as one of the following: a proposition (i.e. a sentence of which it can be said it is true or false) that affirms or negates an idea; a sentence in which a posture is stated; or a physical action with which an idea or posture is expressed. Of the idea exposed it is of interest to sustain its veracity; of the posture presented, its acceptability. The reasons can be presented as sentences (propositions or not) or as actions. The set of reasons that sustain the veracity or acceptability of an assertion conform its justification. We understand argumentation as a discursive and social cultural process destined to increase the acceptability of a point of view or to determine the veracity of an exposed idea through the presentation of propositions or actions destined to justify (or refute) them. In such process arguments arise.

In its most simple version, an argument consists of three elements –data, assertion, warrant– and has a functional structure that displays the relations among these elements, according to the role they each have (Toulmin, 2003). The data sustains the assertion; it is evidence that supports the assertion. The warrant expresses the relation between the data and the assertion in a general statement that indicates why the data is evidence to sustain the assertion. The data and the warrant (which can be explicit or implicit) are the reasons with which the assertion is justified.
## Types of arguments

We present our proposal of types of arguments, following an analogous idea exposed by Pedemonte (2007). As mentioned in the definition, the three basic elements of a simple argument have a functional relation that is always the same. Considering the course of a specific argumentation, more precisely, which of the elements of a simple argument is inferred and which of the elements is taken as given, it is possible to recognize three different argumentative situations, which we denominate inductive argumentation, abductive argumentation and deductive argumentation. Abusing the language, we classify arguments according to the type of argumentation in which it appears.

Deductive argumentation begins with two types of information that is accepted as true: a general rule (if *p*, then *q*) and *data* that consists of a particular case of what the antecedent of the general rule verses. Tacitly or explicitly, the general rule is used to eventually infer an *assertion* as a necessary result, that corresponds to the respective particular case of the consequent of the general rule. In short, a simple deductive argument can be enounced as:

*p*<sub>i</sub> (*data*) is given; therefore *q*<sub>i</sub> (*assertion*) is concluded, due to if *p*, then *q* (*warrant*).

Inductive argumentation occurs within the framework of a reference set (i.e., a set *A* of objects that have a common attribute *p*) that constitutes information as *data*, that is, information accepted as true. It is recognized that some cases (subsets of *A*) share another attribute *q*. By considering one or several particular unexamined cases of the reference set, inductive argumentation leads to inferring a possible pattern of generality (*warrant*) and also the possible *assertion* which states that at least one of these other cases (or all) also has attribute *q*. In summary, a simple inductive argument can be stated as:

The elements of *A* have attribute *p*, some elements of *A* have attribute *q*, about *other* elements of *A* it is not known if they have attribute *q* (*data*). Therefore, at least *another* element of *A* also has attribute *q* (*assertion*), because *the* elements of *A* have attribute *q* (*warrant*).

In abductive argumentation, there is an assertion that, from the beginning, is assumed to be true or acceptable; the *data* is the element that is inferred with the intention of providing a reason that supports the veracity of the assertion, based on a general rule that is believed or known to be true. The data is obtained as a plausible or probable inference. A key feature of this type of argument is the establishment of a fact or, a fact and a warrant, that could support the assertion. In summary, a simple abductive argument can be stated like this:

Since we have *q*<sub>i</sub> (*assertion*), given that if *p* then *q* (*known or inferred warrant*), we could have *p*<sub>i</sub> (*data*).

## Learning tasks and argumentation tasks

We conceive a learning task as an action (or actions) that the teacher asks students to perform with the intention of providing them with an opportunity to achieve the learning expectation that he has established (Gómez et al., 2018). The statement of a learning task (task statement for short) states information about what is being asked to do (*request*). It also exposes information that places the action requested (*situation*) within a specific knowledge and practice, in this case mathematics; Eventually, it exposes suggestions to support or conditions to limit the execution of the action (*indications*). The *learning expectation* is not explicit for the students; an expert should be able to unravel it from the statement. If the learning expectation of a task is the production and exposition of arguments, it is called an *argumentation task*. In this case, the statement must contribute to the generation of a scenario in which the need to take a position or expose ideas that must be supported
## Types of arguments

We present our proposal of types of arguments, following an analogous idea exposed by Pedemonte (2007). As mentioned in the definition, the three basic elements of a simple argument have a functional relation that is always the same. Considering the course of a specific argumentation, more precisely, which of the elements of a simple argument is inferred and which of the elements is taken as given, it is possible to recognize three different argumentative situations, which we denominate inductive argumentation, abductive argumentation and deductive argumentation. Abusing the language, we classify arguments according to the type of argumentation in which it appears.

Deductive argumentation begins with two types of information that is accepted as true: a general rule (if *p*, then *q*) and *data* that consists of a particular case of what the antecedent of the general rule verses. Tacitly or explicitly, the general rule is used to eventually infer an *assertion* as a necessary result, that corresponds to the respective particular case of the consequent of the general rule. In short, a simple deductive argument can be enounced as:

*p*<sub>i</sub> (*data*) is given; therefore *q*<sub>i</sub> (*assertion*) is concluded, due to if *p*, then *q* (*warrant*).

Inductive argumentation occurs within the framework of a reference set (i.e., a set *A* of objects that have a common attribute *p*) that constitutes information as *data*, that is, information accepted as true. It is recognized that some cases (subsets of *A*) share another attribute *q*. By considering one or several particular unexamined cases of the reference set, inductive argumentation leads to inferring a possible pattern of generality (*warrant*) and also the possible *assertion* which states that at least one of these other cases (or all) also has attribute *q*. In summary, a simple inductive argument can be stated as:

The elements of *A* have attribute *p*, some elements of *A* have attribute *q*, about *other* elements of *A* it is not known if they have attribute *q* (*data*). Therefore, at least *another* element of *A* also has attribute *q* (*assertion*), because *the* elements of *A* have attribute *q* (*warrant*).

In abductive argumentation, there is an assertion that, from the beginning, is assumed to be true or acceptable; the *data* is the element that is inferred with the intention of providing a reason that supports the veracity of the assertion, based on a general rule that is believed or known to be true. The data is obtained as a plausible or probable inference. A key feature of this type of argument is the establishment of a fact or, a fact and a warrant, that could support the assertion. In summary, a simple abductive argument can be stated like this:

Since we have *q*<sub>i</sub> (*assertion*), given that if *p* then *q* (*known or inferred warrant*), we could have *p*<sub>i</sub> (*data*).

## Learning tasks and argumentation tasks

We conceive a learning task as an action (or actions) that the teacher asks students to perform with the intention of providing them with an opportunity to achieve the learning expectation that he has established (Gómez et al., 2018). The statement of a learning task (task statement for short) states information about what is being asked to do (*request*). It also exposes information that places the action requested (*situation*) within a specific knowledge and practice, in this case mathematics; Eventually, it exposes suggestions to support or conditions to limit the execution of the action (*indications*). The *learning expectation* is not explicit for the students; an expert should be able to unravel it from the statement. If the learning expectation of a task is the production and exposition of arguments, it is called an *argumentation task*. In this case, the statement must contribute to the generation of a scenario in which the need to take a position or expose ideas that must be supported
is felt. We now present six elements that we have considered when designing the statements of
argumentation tasks, based on references from the literature that, implicitly or explicitly, have been
proposed for that same purpose (e.g., Ayalon & Nama, 2023; Lin et al, 2012):

E1: What is stated in the situation and/or request has the potential to generate doubt, curiosity, uncertainty, or controversy, which when resolved leads to exposing ideas or positions.

E2: The situation presents information related to a definition, theorem and/or fact that can be used in an argument.

E3: The request asks (implicitly or explicitly) for the presentation of reasons that serve to support or refute the veracity of a proposition, or the acceptability of a position raised or an action taken (exposition of arguments).

E4: The indications are a guide that supports or limits the execution of actions in order to establish positions or formulate new ideas, and to expose arguments.

E5: The short-term learning expectation is that students engage in argumentation and formulate and ennounce arguments, relative to a specific mathematical proposition, in accordance with established classroom norms.

E6: The elements of the main argument (data, assertion, warrant) that may arise, when carrying out the task, are explicitly stated, suggested or requested in the task statement.

## METHODOLOGY

The data in this report is part of a research study that had as a goal designing a sequence of tasks to promote preservice mathematics teachers’ learning about argumentation. The methodological strategy used in the study was "the development of the mathematics curriculum as a scientific effort" (Battista and Clements, 2000). The experimental context was a plane Euclidean geometry course, located in the second semester of the program in our University, one of its objectives being learning to prove. In previous courses, what an argument as an object is was not addressed. The methodological strategy used in the course is characterized by the use of tasks that satisfy the aforementioned elements that engage students (working in groups of two or three) to solve them (with the use of a DGS) and requires the formulation of conjectures that must be validated with elements of the theoretical system that is being conformed. That is, the tasks proposed mainly favor abductive or inductive argumentation because conjectures must be formulated.

Each group of students must present their productions in writing, responding to specific *requests* given in the task statement: i) a narration of how they proceeded to establish the solution to the problem (e.g., what was constructed to obtain the configuration obtained in a DGS, what empirical exploration was done, what type of invariant was determined); ii) the conjecture (solution of the problem) stated as a conditional proposition that reports the relationship between what was constructed and what was discovered; and iii) an explanation of what they use to establish that the conjecture is true. The teacher allows each group to present their productions to the class, discuss which conjectures solve the problem, and adjust them, if necessary, so that they are mathematically correct; they then proceed to collectively build the proof.

The data for this report are the written productions of the students in which they report the solution to the two specific tasks. The statements include the situation and the main request; the other requests included in the original statement, mentioned in the above section, are not reported.
is felt. We now present six elements that we have considered when designing the statements of
argumentation tasks, based on references from the literature that, implicitly or explicitly, have been
proposed for that same purpose (e.g., Ayalon & Nama, 2023; Lin et al, 2012):

E1: What is stated in the situation and/or request has the potential to generate doubt, curiosity, uncertainty, or controversy, which when resolved leads to exposing ideas or positions.

E2: The situation presents information related to a definition, theorem and/or fact that can be used in an argument.

E3: The request asks (implicitly or explicitly) for the presentation of reasons that serve to support or refute the veracity of a proposition, or the acceptability of a position raised or an action taken (exposition of arguments).

E4: The indications are a guide that supports or limits the execution of actions in order to establish positions or formulate new ideas, and to expose arguments.

E5: The short-term learning expectation is that students engage in argumentation and formulate and ennounce arguments, relative to a specific mathematical proposition, in accordance with established classroom norms.

E6: The elements of the main argument (data, assertion, warrant) that may arise, when carrying out the task, are explicitly stated, suggested or requested in the task statement.

## METHODOLOGY

The data in this report is part of a research study that had as a goal designing a sequence of tasks to promote preservice mathematics teachers’ learning about argumentation. The methodological strategy used in the study was "the development of the mathematics curriculum as a scientific effort" (Battista and Clements, 2000). The experimental context was a plane Euclidean geometry course, located in the second semester of the program in our University, one of its objectives being learning to prove. In previous courses, what an argument as an object is was not addressed. The methodological strategy used in the course is characterized by the use of tasks that satisfy the aforementioned elements that engage students (working in groups of two or three) to solve them (with the use of a DGS) and requires the formulation of conjectures that must be validated with elements of the theoretical system that is being conformed. That is, the tasks proposed mainly favor abductive or inductive argumentation because conjectures must be formulated.

Each group of students must present their productions in writing, responding to specific *requests* given in the task statement: i) a narration of how they proceeded to establish the solution to the problem (e.g., what was constructed to obtain the configuration obtained in a DGS, what empirical exploration was done, what type of invariant was determined); ii) the conjecture (solution of the problem) stated as a conditional proposition that reports the relationship between what was constructed and what was discovered; and iii) an explanation of what they use to establish that the conjecture is true. The teacher allows each group to present their productions to the class, discuss which conjectures solve the problem, and adjust them, if necessary, so that they are mathematically correct; they then proceed to collectively build the proof.

The data for this report are the written productions of the students in which they report the solution to the two specific tasks. The statements include the situation and the main request; the other requests included in the original statement, mentioned in the above section, are not reported.
Task 1: Consider three non-collinear points A, B and C. Let m be the perpendicular line to AB through its midpoint and n the perpendicular line to BC through its midpoint. Let T be the intersection point of m and n. ¿What geometric characteristic does point T have as point B is moved?

Task 2: $\overline{AB}$ and $\overline{CD}$ are congruent. ¿Does there exist a point E so that $\Delta ABE$ y $\Delta CDE$ are congruent?

The statements are representative of the typology that will be described; they involve the same geometric object (*perpendicular bisector of a segment*<fnref n="13" />), reason why we chose them for this report. We had two other sources: i) class notes prepared after each class by groups of two or three students and delivered to the teacher, in which the main aspects covered in the class are reconstructed (specifying the solution presented by each group of students, and the teacher’s or students’ comments about them that arise); ii) a *retrospective recall* (Sherin et al, 2011) done by the teacher after each class session, which made it possible to complement the information recorded in the class notes.

## Data analysis

The data analysis consists of two parts. A preliminary analysis (of a theoretical nature), which briefly describes how a student should proceed to establish his solution, according to the type of problem and suggests, a priori, the type of argument that is expected. A retrospective analysis (empirical in nature) contrasted with the theoretical, where both the most common problem-solving strategies and their respective solution-conjectures are identified in the students' productions. From both analyses, we obtained input to identify the types of arguments, using the Toulmin Model, that are necessarily provoked when students solve a task with a particular type of statement. The *retrospective recall* made by the teacher with respect to the students' forms of exploration helped to complement the information. A reading of these same elements in the students' written reports, together with what is evident in the class notes written by the students, led us to identify the types of arguments that emerged from the students’ activity.

## SOME RESULTS

We found that the statement structure of an argumentation task may primarily promote different types of main arguments. This is because in the argumentation process that is favored by the statement, both the information available and the information that is inferred play a certain role in the functional structure of the main argument that emerges. For the purpose of this report, we summarize in Table 1 the expected learning in terms of the type of argument that is favored and the corresponding structure of the task statement that promotes it. Additionally, we include a narration of the production of groups of students for Task 1 and Task 2 that illustrate the results presented in Table 1.

Task 1 statement is an example of the type *Consequent search*. In it a reference set is provided that characterizes the *situation* (non-collinear points A, B and C, and perpendicular bisector of $\overline{AB}$ and $\overline{BC}$); the question *requests* finding an attribute for *T*. As a solution, some students represent the situation in GeoGebra and explore it empirically, moving *B* around the screen; they use the DGS “Trace” tool for *T*. This action leads them to see that, when *B* moves, the path of *T* is a straight line; they expose the attribute of *T*: for any point *B* in that plane the intersection (*T*) of lines *m* and *n* will always belong to the perpendicular bisector of $\overline{AC}$ (Figure 1).

<footnote n="13">¹³ The geometric characterizations of that object are: 1) the locus of points that are equidistant from the endpoints of the segment and 2) the line perpendicular to the segment that contains its midpoint. Either one can be the definition of the perpendicular bisector; the other would be a theorem. Option 1 was the definition used in the course.</footnote>
Task 1: Consider three non-collinear points A, B and C. Let m be the perpendicular line to AB through its midpoint and n the perpendicular line to BC through its midpoint. Let T be the intersection point of m and n. ¿What geometric characteristic does point T have as point B is moved?

Task 2: $\overline{AB}$ and $\overline{CD}$ are congruent. ¿Does there exist a point E so that $\Delta ABE$ y $\Delta CDE$ are congruent?

The statements are representative of the typology that will be described; they involve the same geometric object (*perpendicular bisector of a segment*<fnref n="13" />), reason why we chose them for this report. We had two other sources: i) class notes prepared after each class by groups of two or three students and delivered to the teacher, in which the main aspects covered in the class are reconstructed (specifying the solution presented by each group of students, and the teacher’s or students’ comments about them that arise); ii) a *retrospective recall* (Sherin et al, 2011) done by the teacher after each class session, which made it possible to complement the information recorded in the class notes.

## Data analysis

The data analysis consists of two parts. A preliminary analysis (of a theoretical nature), which briefly describes how a student should proceed to establish his solution, according to the type of problem and suggests, a priori, the type of argument that is expected. A retrospective analysis (empirical in nature) contrasted with the theoretical, where both the most common problem-solving strategies and their respective solution-conjectures are identified in the students' productions. From both analyses, we obtained input to identify the types of arguments, using the Toulmin Model, that are necessarily provoked when students solve a task with a particular type of statement. The *retrospective recall* made by the teacher with respect to the students' forms of exploration helped to complement the information. A reading of these same elements in the students' written reports, together with what is evident in the class notes written by the students, led us to identify the types of arguments that emerged from the students’ activity.

## SOME RESULTS

We found that the statement structure of an argumentation task may primarily promote different types of main arguments. This is because in the argumentation process that is favored by the statement, both the information available and the information that is inferred play a certain role in the functional structure of the main argument that emerges. For the purpose of this report, we summarize in Table 1 the expected learning in terms of the type of argument that is favored and the corresponding structure of the task statement that promotes it. Additionally, we include a narration of the production of groups of students for Task 1 and Task 2 that illustrate the results presented in Table 1.

Task 1 statement is an example of the type *Consequent search*. In it a reference set is provided that characterizes the *situation* (non-collinear points A, B and C, and perpendicular bisector of $\overline{AB}$ and $\overline{BC}$); the question *requests* finding an attribute for *T*. As a solution, some students represent the situation in GeoGebra and explore it empirically, moving *B* around the screen; they use the DGS “Trace” tool for *T*. This action leads them to see that, when *B* moves, the path of *T* is a straight line; they expose the attribute of *T*: for any point *B* in that plane the intersection (*T*) of lines *m* and *n* will always belong to the perpendicular bisector of $\overline{AC}$ (Figure 1).

<footnote n="13">¹³ The geometric characterizations of that object are: 1) the locus of points that are equidistant from the endpoints of the segment and 2) the line perpendicular to the segment that contains its midpoint. Either one can be the definition of the perpendicular bisector; the other would be a theorem. Option 1 was the definition used in the course.</footnote>
<table><caption>Table 1. Type of statement according to the type of argument that is mainly promoted</caption><thead><tr><th>Type of argument</th><th>Type of statement</th><th>Statement structure</th><th>Solving process</th><th>Argument structure</th></tr></thead><tbody><tr><td>Inductive argument</td><td>Consequent search</td><td><em>Situation</em>: provides or insinuates the reference set of geometric figures.<br><em>Request</em>: Asks about another attribute that the objects in the set might have.</td><td>Representation of the objects of the reference set. Exploration, using wandering dragging (Arzarello et al., 2002), of cases to determine another attribute of those objects</td><td>For some objects in the set another attribute (data) is satisfied. Therefore, other object(s) in the set also have that attribute (assertion). The established conjecture would be the warrant.</td></tr><tr><td>Abductive argument</td><td>Antecedent search</td><td><em>Situation</em>: provides or insinuates a fact.<br><em>Request</em>: Asks about conditions that could support the fact.</td><td>Exploration, using maintaining dragging (Baccaglini-Frank & Mariotti, 2010), of properties that can lead to the fact.</td><td>The proposition that is provided or implied is a fact (assertion), since the conditions found (data) exist. The established conjecture would be the warrant.</td></tr></tbody></table>

<figure><img src="image_68.png" /><figcaption>Figure 1. Graphic representation that illustrates the exploration that leads to an inductive argument</figcaption></figure>

Finally, they report the following inductive argument in response to the request about what they use to be certain of the established invariant: for some points *B* of the plane, the corresponding points *T* are part of a certain subset of the line that appears to be the perpendicular bisector (data). Therefore, we can say that for the other points *B* [of the plane], the point *T* is always on that line, the perpendicular bisector [of *AC*] (assertion). Implicitly, the warrant of the argument is the conjecture that they report: Let *A*, *B* and *C* be non-collinear points, *m* the perpendicular bisector of *AB* and *n* the perpendicular bisector of *BC*. Let *T* be the intersection point of *m* and *n*. Then *T* belongs to the perpendicular bisector of *AC*.

Task 2 statement is an example of the type *Antecedent search*. The *situation* corresponds to the two triangles with the aforementioned property; the *request* is to find data so that the triangles are congruent. As a solution, a group of students reported that they needed congruent corresponding sides to obtain congruent triangles. Therefore, they proceeded to determine the condition *E* must have so that $\overline{AE} \cong \overline{CE}$, as well as $\overline{BE} \cong \overline{DE}$. They report that their strategy consisted of taking any point *E*, activating its trace and moving it with the purpose of maintaining the first congruency; this trace allowed them to recognize that *E* should belong to the perpendicular bisector of $\overline{AC}$ (Figure 2); likewise, they determined that *E* should also be in the perpendicular bisector of $\overline{BD}$. They concluded
<table><caption>Table 1. Type of statement according to the type of argument that is mainly promoted</caption><thead><tr><th>Type of argument</th><th>Type of statement</th><th>Statement structure</th><th>Solving process</th><th>Argument structure</th></tr></thead><tbody><tr><td>Inductive argument</td><td>Consequent search</td><td><i>Situation</i>: provides or insinuates the reference set of geometric figures.<br><i>Request</i>: Asks about another attribute that the objects in the set might have.</td><td>Representation of the objects of the reference set. Exploration, using wandering dragging (Arzarello et al., 2002), of cases to determine another attribute of those objects</td><td>For some objects in the set another attribute (data) is satisfied. Therefore, other object(s) in the set also have that attribute (assertion). The established conjecture would be the warrant.</td></tr><tr><td>Abductive argument</td><td>Antecedent search</td><td><i>Situation</i>: provides or insinuates a fact.<br><i>Request</i>: Asks about conditions that could support the fact.</td><td>Exploration, using maintaining dragging (Baccaglini-Frank & Mariotti, 2010), of properties that can lead to the fact.</td><td>The proposition that is provided or implied is a fact (assertion), since the conditions found (data) exist. The established conjecture would be the warrant.</td></tr></tbody></table>

<figure><img src="image_74.png" /></figure>

Figure 1. Graphic representation that illustrates the exploration that leads to an inductive argument. Finally, they report the following inductive argument in response to the request about what they use to be certain of the established invariant: for some points *B* of the plane, the corresponding points *T* are part of a certain subset of the line that appears to be the perpendicular bisector (data). Therefore, we can say that for the other points *B* [of the plane], the point *T* is always on that line, the perpendicular bisector [of *AC*] (assertion). Implicitly, the warrant of the argument is the conjecture that they report: Let *A*, *B* and *C* be non-collinear points, *m* the perpendicular bisector of *AB* and *n* the perpendicular bisector of *BC*. Let *T* be the intersection point of *m* and *n*. Then *T* belongs to the perpendicular bisector of *AC*.

Task 2 statement is an example of the type *Antecedent search*. The *situation* corresponds to the two triangles with the aforementioned property; the *request* is to find data so that the triangles are congruent. As a solution, a group of students reported that they needed congruent corresponding sides to obtain congruent triangles. Therefore, they proceeded to determine the condition *E* must have so that $\overline{AE} \cong \overline{CE}$, as well as $\overline{BE} \cong \overline{DE}$. They report that their strategy consisted of taking any point *E*, activating its trace and moving it with the purpose of maintaining the first congruency; this trace allowed them to recognize that *E* should belong to the perpendicular bisector of $\overline{AC}$ (Figure 2); likewise, they determined that *E* should also be in the perpendicular bisector of $\overline{BD}$. They concluded
that the congruency of the triangles is achieved when *E* is the intersection of the perpendicular bisectors (Figure 3). As an argument they reported the following: ΔABE and ΔCDE are congruent (assertion) because *E* is the intersection of the perpendicular bisectors of $\overline{AC}$ and $\overline{BD}$ (inferred data).

<figure><img src="image_69.png" /><figcaption>Figure 2. Exploration to determine the position of *E* so that $\overline{AE}$ and $\overline{CE}$ are congruent</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>Figure 3. Representation of the solution for the second problem</figcaption></figure>

FINAL REMARKS

As a result of the implementation of these tasks, our students recognize differences between types of task statements, identify the type of argument that each favor, and distinguish their role in the mathematical activity (inductive and abductive to conjecture, deductive to validate). This leads us to state that statements such as those illustrated contribute to a classroom environment in which the future teacher has the opportunity, not only to do mathematical activity close to that of a professional mathematician (in terms of exploring a situation, conjecturing, and arguing), but to experiment the potential of those types of task statements to carry out that activity. Yet we think that having experiences that lead them to argue is not enough for preservice teachers to gain specialized knowledge about argumentation. As several authors advocate (Stylianides and Ball; 2008; McNeill & Knight, 2013) to promote this process in their classes, it is necessary for this construct to become an object of study. Therefore, besides being able to establish the relationship between types of task statements and types of arguments that they favor, the students must construct their own perspective about argumentation, develop the ability to design task statements, identify and classify the arguments that are favored, identify their role in the mathematical activity, and recognize the arguments in what their students say. We would like to believe that, with what we present here, we partially respond to the concern that Stylianides et al. (2016) state about the necessity to continue thinking about features of tasks or classroom environments that promote argumentation in the mathematics classroom.

References

Arzarello, F., Bartolini-Bussi, M., Leung, A., Mariotti, M., & Stevenson, I. (2012). Experimental approaches to theoretical thinking: artefacts and proofs. In G. Hanna, & M. de Villiers (Eds.), *Proof and proving in mathematics education* (pp. 97-146). Springer.

Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. *ZDM, 34*(3), 66-72.

Ayalon, M. & Nama, S. (2023). Secondary school mathematics teacher-perceived factors involved in argumentation: an emerging framework. *Research in Mathematics Education*. https://doi.org/10.1080/14794802.2022.2156585

Baccaglini-Frank, A., & Mariotti, M. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. *International Journal of Computers for Mathematical Learning, 15*, 225-253.
that the congruency of the triangles is achieved when *E* is the intersection of the perpendicular bisectors (Figure 3). As an argument they reported the following: ΔABE and ΔCDE are congruent (assertion) because *E* is the intersection of the perpendicular bisectors of $\overline{AC}$ and $\overline{BD}$ (inferred data).

<figure><img src="image_75.png" /><figcaption>Figure 2. Exploration to determine the position of *E* so that $\overline{AE}$ and $\overline{CE}$ are congruent</figcaption></figure>

<figure><img src="image_76.png" /><figcaption>Figure 3. Representation of the solution for the second problem</figcaption></figure>

## FINAL REMARKS

As a result of the implementation of these tasks, our students recognize differences between types of task statements, identify the type of argument that each favor, and distinguish their role in the mathematical activity (inductive and abductive to conjecture, deductive to validate). This leads us to state that statements such as those illustrated contribute to a classroom environment in which the future teacher has the opportunity, not only to do mathematical activity close to that of a professional mathematician (in terms of exploring a situation, conjecturing, and arguing), but to experiment the potential of those types of task statements to carry out that activity. Yet we think that having experiences that lead them to argue is not enough for preservice teachers to gain specialized knowledge about argumentation. As several authors advocate (Stylianides and Ball; 2008; McNeill & Knight, 2013) to promote this process in their classes, it is necessary for this construct to become an object of study. Therefore, besides being able to establish the relationship between types of task statements and types of arguments that they favor, the students must construct their own perspective about argumentation, develop the ability to design task statements, identify and classify the arguments that are favored, identify their role in the mathematical activity, and recognize the arguments in what their students say. We would like to believe that, with what we present here, we partially respond to the concern that Stylianides et al. (2016) state about the necessity to continue thinking about features of tasks or classroom environments that promote argumentation in the mathematics classroom.

## References

* Arzarello, F., Bartolini-Bussi, M., Leung, A., Mariotti, M., & Stevenson, I. (2012). Experimental approaches to theoretical thinking: artefacts and proofs. In G. Hanna, & M. de Villiers (Eds.), *Proof and proving in mathematics education* (pp. 97-146). Springer.

* Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. *ZDM, 34*(3), 66-72.

* Ayalon, M. & Nama, S. (2023). Secondary school mathematics teacher-perceived factors involved in argumentation: an emerging framework. *Research in Mathematics Education*. https://doi.org/10.1080/14794802.2022.2156585

* Baccaglini-Frank, A., & Mariotti, M. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. *International Journal of Computers for Mathematical Learning, 15*, 225-253.
Battista, M., & Clements, D. (2000). Mathematics curriculum development as a scientific endeavor. In A. Kelly, & R. Lesh (Eds.), *Education, handbook of research design in mathematics and science* (pp. 737-760). Lawrence Erlbaum Associates.

Boero, P., Douek, N., Morselli, F., & Pedemonte, B. (2010). Argumentation and proof: A contribution to theoretical perspectives and their classroom implementation. In M. M. Pinto, & T. F. Kawasaki (Eds.), *Proc. 34th Conf. of the Int. Group for the Psychology of Mathematics Education*. (Vol 1, pp. 179-204). PME.

Durand-Guerrier, V., Boero, P., Douek, N., Epp, S., & Tanguay, D. (2012). Argumentation and proof in the mathematics classroom. In G. Hanna, & M. de Villiers (Eds.), *Proof and proving in mathematics education* (Vol. 15, pp. 349-368). Springer. https://doi.org/10.1007/978-94-007-2129-6_15

Gómez, P., Mora, M., & Velasco, C. (2018). Análisis de instrucción. En P. Gómez, *Formación de profesores de matemáticas y práctica de aula: conceptos técnicas curriculares* (pp. 197-268)). Universidad de los Andes.

Knipping, C., & Reid, D. (2019). Argumentation analysis for early career researchers. In G. Kaiser & N. Presmeg (Eds.), *Compendium for Early Career Researchers in Mathematics Education. ICME-13 Monographs*. (pp. 3-31). Springer. https://doi.org/10.1007/978-3-030-15636-7_1

Lin, F., Yang, K., Lo, J., Tsamir, P., Tirosh, D., & Stylianides, G. (2012). Teachers’ professional learning of teaching proof and proving. In G. Hanna, & M. de Villiers (Eds), *Proof and proving in mathematics education*. (Vol. 15, pp. 327–346). Springer. doi:https://doi.org/10.1007/978-94-007-2129-6_14

McNeill, K. L., & Knight, A. M. (2013). Teachers’ pedagogical content knowledge of scientific argumentation: The impact of professional development on k-12 teachers. *Science Education, 97*(6), 936–972.

Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? *Educational Studies in Mathematics*, **66**, 23–41. doi:https://doi.org/10.1007/s10649-006-9057-x

Stylianides, A., & Stylianides, G. (2006). Content knowledge for mathematics teaching: The case of reasoning and proving. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), *Proc. 30th Conf. of the Int. Group for the Psychology of Mathematics Education. 1* (Vol. 5, pp. 201-208). PME.

Stylianides, A., & Ball, D. (2008). Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving. *Journal of Mathematics Teacher Education, 11*(4), 307–332.

Stylianides, A., Bieda, K., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Guitérrez, G. Leder, & P. Boero (Eds.), *The Second Handbook of Research on the Psychology of Mathematics Education* (pp. 315-352). Sense Publishers.

Toulmin, S. (2003). *The Uses of Arguments*. Cambridge University Press.
Battista, M., & Clements, D. (2000). Mathematics curriculum development as a scientific endeavor. In A. Kelly, & R. Lesh (Eds.), *Education, handbook of research design in mathematics and science* (pp. 737-760). Lawrence Erlbaum Associates.

Boero, P., Douek, N., Morselli, F., & Pedemonte, B. (2010). Argumentation and proof: A contribution to theoretical perspectives and their classroom implementation. In M. M. Pinto, & T. F. Kawasaki (Eds.), *Proc. 34th Conf. of the Int. Group for the Psychology of Mathematics Education*. (Vol 1, pp. 179-204). PME.

Durand-Guerrier, V., Boero, P., Douek, N., Epp, S., & Tanguay, D. (2012). Argumentation and proof in the mathematics classroom. In G. Hanna, & M. de Villiers (Eds.), *Proof and proving in mathematics education* (Vol. 15, pp. 349-368). Springer. https://doi.org/10.1007/978-94-007-2129-6_15

Gómez, P., Mora, M., & Velasco, C. (2018). Análisis de instrucción. En P. Gómez, *Formación de profesores de matemáticas y práctica de aula: conceptos técnicas curriculares* (pp. 197-268)). Universidad de los Andes.

Knipping, C., & Reid, D. (2019). Argumentation analysis for early career researchers. In G. Kaiser & N. Presmeg (Eds.), *Compendium for Early Career Researchers in Mathematics Education. ICME-13 Monographs*. (pp. 3-31). Springer. https://doi.org/10.1007/978-3-030-15636-7_1

Lin, F., Yang, K., Lo, J., Tsamir, P., Tirosh, D., & Stylianides, G. (2012). Teachers’ professional learning of teaching proof and proving. In G. Hanna, & M. de Villiers (Eds), *Proof and proving in mathematics education*. (Vol. 15, pp. 327–346). Springer. doi:https://doi.org/10.1007/978-94-007-2129-6_14

McNeill, K. L., & Knight, A. M. (2013). Teachers’ pedagogical content knowledge of scientific argumentation: The impact of professional development on k-12 teachers. *Science Education, 97*(6), 936–972.

Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? *Educational Studies in Mathematics, 66*, 23–41. doi:https://doi.org/10.1007/s10649-006-9057-x

Stylianides, A., & Stylianides, G. (2006). Content knowledge for mathematics teaching: The case of reasoning and proving. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), *Proc. 30th Conf. of the Int. Group for the Psychology of Mathematics Education. 1* (Vol. 5, pp. 201-208). PME.

Stylianides, A., & Ball, D. (2008). Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving. *Journal of Mathematics Teacher Education, 11*(4), 307–332.

Stylianides, A., Bieda, K., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Guitérrez, G. Leder, & P. Boero (Eds.), *The Second Handbook of Research on the Psychology of Mathematics Education* (pp. 315-352). Sense Publishers.

Toulmin, S. (2003). *The Uses of Arguments*. Cambridge University Press.
# ANALYSIS OF VISUALIZATION AS AN INDICATOR OF MATHEMATICAL GIFTEDNESS

Mónica Mora¹, Angel Gutiérrez², Adela Jaime²

¹ Universidad Estatal a Distancia (Costa Rica), ² Universidad de Valencia (Spain)

We present results from a research project we have conducted with primary school students who participated in an olympiad. We focus on the use of visualization as a possible characteristic of mathematically gifted students. We designed several types of problems requiring the use of visualization, to be solved in the olympiad, and observed which problems were useful for such identification and also which components of visualization, evidenced in students' solutions, were good discriminating the students with best scores, who can be considered potential mathematically gifted students. We have identified and defined seven descriptors associated with the problem-solving process, and identified the descriptors used by students in each problem. Here we show examples of the different types of problems and evidence of the descriptors of visualization evidenced in students' answers.

## INTRODUCTION

There is broad agreement among researchers in mathematics education that mathematically gifted students (MG students hereafter) develop faster and use more effectively than their peers certain central abilities or skills in mathematics, such as generalization, abstraction, transfer, deductive reasoning, creativity, or flexibility (Krutetskii, 1976; Leikin, 2021; Miller, 1990). However, when analyzing the relationship between MG and visualization in mathematics, there are discrepant research results. Clements (1999) points out that “many studies have shown that children with specific spatial abilities are more mathematically competent. However, other research indicates that students who process mathematical information by verbal-logical means outperform students who process information visually” (p. 72). This diversity of findings is still present in more recent literature. It is therefore relevant for mathematics education to study this apparent contradiction and provide information to help decide whether mathematical visualization is a trait of MG students.

We present part of a research in which we have analyzed the presence of mathematical generalization, visualization, and flexibility in the solutions of students who participated in the mathematical olympiad for primary school of Costa Rica (OLCOMEП). We will focus on the ability of visualization and the problems we posed in the olympiad that required its use. One objective of our research is to identify which of these problems were more suitable for discriminating students with higher scores and, therefore, potential MG students. Another research objective, related to the previous one, is to identify which visualization features evidenced in students' solutions showed adequate power to discriminate the students with higher scores. The objective of this paper is to identify the descriptors of visualization evidenced in students' solutions and inform on their power to discriminate potential MG students. To this end, we present the different types of visualization problems and an analysis of the descriptors of visualization that some examples of students' solutions used to solve them.
# ANALYSIS OF VISUALIZATION AS AN INDICATOR OF MATHEMATICAL GIFTEDNESS

Mónica Mora¹, Angel Gutiérrez², Adela Jaime²

¹ Universidad Estatal a Distancia (Costa Rica), ² Universidad de Valencia (Spain)

We present results from a research project we have conducted with primary school students who participated in an olympiad. We focus on the use of visualization as a possible characteristic of mathematically gifted students. We designed several types of problems requiring the use of visualization, to be solved in the olympiad, and observed which problems were useful for such identification and also which components of visualization, evidenced in students' solutions, were good discriminating the students with best scores, who can be considered potential mathematically gifted students. We have identified and defined seven descriptors associated with the problem-solving process, and identified the descriptors used by students in each problem. Here we show examples of the different types of problems and evidence of the descriptors of visualization evidenced in students' answers.

## INTRODUCTION

There is broad agreement among researchers in mathematics education that mathematically gifted students (MG students hereafter) develop faster and use more effectively than their peers certain central abilities or skills in mathematics, such as generalization, abstraction, transfer, deductive reasoning, creativity, or flexibility (Krutetskii, 1976; Leikin, 2021; Miller, 1990). However, when analyzing the relationship between MG and visualization in mathematics, there are discrepant research results. Clements (1999) points out that “many studies have shown that children with specific spatial abilities are more mathematically competent. However, other research indicates that students who process mathematical information by verbal-logical means outperform students who process information visually” (p. 72). This diversity of findings is still present in more recent literature. It is therefore relevant for mathematics education to study this apparent contradiction and provide information to help decide whether mathematical visualization is a trait of MG students.

We present part of a research in which we have analyzed the presence of mathematical generalization, visualization, and flexibility in the solutions of students who participated in the mathematical olympiad for primary school of Costa Rica (OLCOMEП). We will focus on the ability of visualization and the problems we posed in the olympiad that required its use. One objective of our research is to identify which of these problems were more suitable for discriminating students with higher scores and, therefore, potential MG students. Another research objective, related to the previous one, is to identify which visualization features evidenced in students’ solutions showed adequate power to discriminate the students with higher scores. The objective of this paper is to identify the descriptors of visualization evidenced in students’ solutions and inform on their power to discriminate potential MG students. To this end, we present the different types of visualization problems and an analysis of the descriptors of visualization that some examples of students’ solutions used to solve them.
# REVIEW OF LITERATURE

Several authors (Clements & Batista, 1992; Giaquinto, 2007; Gutiérrez, 1996) point out the relevance of diagrams, figures, real objects, etc., for both mathematics teaching and mathematical research. Visualization is “being recognized as a key component of reasoning (deeply engaging with the conceptual and not the merely perceptual), problem solving, and even proving” (Arcavi, 2003, p. 235) in school contexts and, on the other hand, “research mathematicians can think through and communicate their proofs to one another in a quite casual way, ... often indicating their intentions with diagrams” (Giaquinto, 2007, p. 81). From the cognitive point of view, Krutetskii (1976) identifies analytic and geometric (visual) types of mathematical thinking and Cain (2019) recalls that some great mathematicians showed preference for analytic thinking (e.g., Hardy or Russell) while others used geometric thinking in their research (e.g., Hadamard).

To evaluate the use of visualization by mathematics students, both psychometric tests and mathematics problem sets are used. Several researchers conclude that it is more effective and reliable to analyze solutions to problems than answers to tests (Ramírez & Flores, 2017).

## Interpretations of visualization from mathematics education

Several authors have proposed different conceptualizations of visualization and its use in mathematics: Gutiérrez (1996) states that mathematical visualization is composed of three main elements: mental images (created and manipulated to solve problems or any other mathematical activity; Presmeg, 1986), processes (of converting available information into mental images and interpreting these to obtain new information; Bishop, 1983) and abilities (the skills necessary to carry out the processes and create or manipulate images; Del Grande, 1990). Clements (1999) considers that spatial thinking, or spatial sense, is composed of spatial orientation, spatial visualization, and imagery, along with skills such as creating and transforming mental images and representing them graphically. More recently, Lowrie et al. (2020) consider spatial reasoning as integrated by spatial visualization, mental rotation, spatial orientation, and spatial structuring. Miragliotta and Baccaglini-Frank (2017) call spatial reasoning “the set of cognitive processes by which mental representation for spatial objects, relationships, and transformations are constructed and manipulated” (p. 3953) and characterize it by various skills, in which the use of memory plays an important role. These interpretations of visualization are different but have as common characteristics the creation and transformation of mental images and the need to develop skills to properly use visualization.

## Visualization and mathematical giftedness

There is a disparity of conclusions in the research on the use of visualization by MG students. Krutetskii (1976) recognizes the importance of visualization in mathematics by including, among “the component mathematical abilities that arise from the basic characteristics of mathematical thought ... the ability for spatial concepts” (pp. 87-88). But, when he analyzes the presence of such components in MG students’ solutions, he concludes that some of them “are not obligatory in the structure of mathematical giftedness” (p. 351), among which he mentions “an ability for spatial concepts” and “an ability to visualize abstract mathematical relationships and dependencies” (p. 351). Presmeg (1986), based on Krutetskii’s findings, identifies several reasons for MG students’ low use of visualization, internal (natural predisposition and need to use analytical reasoning) and external (textbooks, teaching methodologies, and examination procedures). Van Garderen et al. (2014), after
# REVIEW OF LITERATURE

Several authors (Clements & Batista, 1992; Giaquinto, 2007; Gutiérrez, 1996) point out the relevance of diagrams, figures, real objects, etc., for both mathematics teaching and mathematical research. Visualization is “being recognized as a key component of reasoning (deeply engaging with the conceptual and not the merely perceptual), problem solving, and even proving” (Arcavi, 2003, p. 235) in school contexts and, on the other hand, “research mathematicians can think through and communicate their proofs to one another in a quite casual way, ... often indicating their intentions with diagrams” (Giaquinto, 2007, p. 81). From the cognitive point of view, Krutetskii (1976) identifies analytic and geometric (visual) types of mathematical thinking and Cain (2019) recalls that some great mathematicians showed preference for analytic thinking (e.g., Hardy or Russell) while others used geometric thinking in their research (e.g., Hadamard).

To evaluate the use of visualization by mathematics students, both psychometric tests and mathematics problem sets are used. Several researchers conclude that it is more effective and reliable to analyze solutions to problems than answers to tests (Ramírez & Flores, 2017).

## Interpretations of visualization from mathematics education

Several authors have proposed different conceptualizations of visualization and its use in mathematics: Gutiérrez (1996) states that mathematical visualization is composed of three main elements: mental images (created and manipulated to solve problems or any other mathematical activity; Presmeg, 1986), processes (of converting available information into mental images and interpreting these to obtain new information; Bishop, 1983) and abilities (the skills necessary to carry out the processes and create or manipulate images; Del Grande, 1990). Clements (1999) considers that spatial thinking, or spatial sense, is composed of spatial orientation, spatial visualization, and imagery, along with skills such as creating and transforming mental images and representing them graphically. More recently, Lowrie et al. (2020) consider spatial reasoning as integrated by spatial visualization, mental rotation, spatial orientation, and spatial structuring. Miragliotta and Baccaglini-Frank (2017) call spatial reasoning “the set of cognitive processes by which mental representation for spatial objects, relationships, and transformations are constructed and manipulated” (p. 3953) and characterize it by various skills, in which the use of memory plays an important role. These interpretations of visualization are different but have as common characteristics the creation and transformation of mental images and the need to develop skills to properly use visualization.

## Visualization and mathematical giftedness

There is a disparity of conclusions in the research on the use of visualization by MG students. Krutetskii (1976) recognizes the importance of visualization in mathematics by including, among “the component mathematical abilities that arise from the basic characteristics of mathematical thought ... the ability for spatial concepts” (pp. 87-88). But, when he analyzes the presence of such components in MG students’ solutions, he concludes that some of them “are not obligatory in the structure of mathematical giftedness” (p. 351), among which he mentions “an ability for spatial concepts” and “an ability to visualize abstract mathematical relationships and dependencies” (p. 351). Presmeg (1986), based on Krutetskii’s findings, identifies several reasons for MG students’ low use of visualization, internal (natural predisposition and need to use analytical reasoning) and external (textbooks, teaching methodologies, and examination procedures). Van Garderen et al. (2014), after
observing students with learning disabilities (LD) and high ability (HA), obtain similar conclusions: "a higher percentage of HA students than students with LD did not consistently utilize a diagram as a strategy to solve a problem. ... [and] a higher percentage of students identified as HA than students with LD had difficulty using a diagram to reason with as they solved the problem" (p. 147).

On the other hand, Ramírez (2012), analyzing the strategies of secondary school students, concludes that "mathematically talented students have shown significantly higher intelligence and visual ability in tests than students in the control group" (p. 336). Other research shows that, when solving math problems in which visualization facilitates the solution, (potentially) MG students use it more consistently and effectively than average students (Escrivá et al., 2017; Ramírez & Flores, 2017). Therefore, we agree with Webb, Lubinski and Benbow (2007) that observing visualization ability, through problem solving, should be included in the processes of identifying MG students, in order to identify a type of MG students that is sometimes uncovered.

## THEORETICAL FRAMEWORK

Based on Gutiérrez (1996), we define the *capacity of visualization* as the capacity to reason using spatial or visual elements, both physical (photos, diagrams, solid objects, etc.) and mental (mental images), to solve problems, perform proofs, or understand mathematical concepts and properties. Furthermore, based on this author and Del Grande (1990), we define *abilities of visualization* as skills that must be acquired and improved by students to effectively execute the necessary visualization processes with the appropriate mental images when solving problems. For our research, we have taken the definitions proposed by these authors for visual identification, mental rotation, conservation of perception, recognition of positions-in-space, recognition of spatial relationships, and visual discrimination. From them, we have defined a set of operational *specific descriptors* of the capacity of visualization that allowed us to observe evidence of the capacity and skills of visualization in students' solutions to the problems in our experiment:

* DV1. *Visual identification:* to identify a figure that is part of a complex context by isolating it from the context.

* DV2. *Mental rotation:* to produce or transform a mental image by rotating it in space. It can be based on a dynamic image or on a pair of pictorial images.

* DV3.1. *Conservation of perception of partially hidden figures:* to identify a regularity of a partially hidden figure that allows to imagine how the figure would look like if it could be fully seen.

* DV3.2. *Conservation of perception of completely hidden figures:* to identify a regularity of a structure that allows to assume that it includes hidden figures, and how the figures would look like if they could be fully seen.

* DV4.1. *Recognition of positions-in-space with respect to oneself as observer:* to recognize the positions of objects located in space in relation to oneself as observer and as the reference center.

* DV4.2. *Recognition of positions-in-space with respect to a reference object:* to recognize the positions of objects located in space in relation to another fixed object acting as a reference center.

* DV5.1. *Recognition of spatial relationships between objects:* to recognize the positions of objects located in space in relation to each other.
observing students with learning disabilities (LD) and high ability (HA), obtain similar conclusions: "a higher percentage of HA students than students with LD did not consistently utilize a diagram as a strategy to solve a problem. ... [and] a higher percentage of students identified as HA than students with LD had difficulty using a diagram to reason with as they solved the problem" (p. 147).

On the other hand, Ramírez (2012), analyzing the strategies of secondary school students, concludes that "mathematically talented students have shown significantly higher intelligence and visual ability in tests than students in the control group" (p. 336). Other research shows that, when solving math problems in which visualization facilitates the solution, (potentially) MG students use it more consistently and effectively than average students (Escrivá et al., 2017; Ramírez & Flores, 2017). Therefore, we agree with Webb, Lubinski and Benbow (2007) that observing visualization ability, through problem solving, should be included in the processes of identifying MG students, in order to identify a type of MG students that is sometimes uncovered.

## THEORETICAL FRAMEWORK

Based on Gutiérrez (1996), we define the *capacity of visualization* as the capacity to reason using spatial or visual elements, both physical (photos, diagrams, solid objects, etc.) and mental (mental images), to solve problems, perform proofs, or understand mathematical concepts and properties. Furthermore, based on this author and Del Grande (1990), we define *abilities of visualization* as skills that must be acquired and improved by students to effectively execute the necessary visualization processes with the appropriate mental images when solving problems. For our research, we have taken the definitions proposed by these authors for visual identification, mental rotation, conservation of perception, recognition of positions-in-space, recognition of spatial relationships, and visual discrimination. From them, we have defined a set of operational *specific descriptors* of the capacity of visualization that allowed us to observe evidence of the capacity and skills of visualization in students' solutions to the problems in our experiment:

* DV1. *Visual identification:* to identify a figure that is part of a complex context by isolating it from the context.

* DV2. *Mental rotation:* to produce or transform a mental image by rotating it in space. It can be based on a dynamic image or on a pair of pictorial images.

* DV3.1. *Conservation of perception of partially hidden figures:* to identify a regularity of a partially hidden figure that allows to imagine how the figure would look like if it could be fully seen.

* DV3.2. *Conservation of perception of completely hidden figures:* to identify a regularity of a structure that allows to assume that it includes hidden figures, and how the figures would look like if they could be fully seen.

* DV4.1. *Recognition of positions-in-space with respect to oneself as observer:* to recognize the positions of objects located in space in relation to oneself as observer and as the reference center.

* DV4.2. *Recognition of positions-in-space with respect to a reference object:* to recognize the positions of objects located in space in relation to another fixed object acting as a reference center.

* DV5.1. *Recognition of spatial relationships between objects:* to recognize the positions of objects located in space in relation to each other.
* DV5.2. *Recognition of spatial relationships between elements of objects:* to recognize the positions of elements of one or more objects located in space in relation to each other.

* DV6. *Visual discrimination:* to compare objects, identifying their visual similarities and differences.

As we have shown in the previous section, some authors argue that being able to choose and effectively use the most appropriate visualization skills to solve a problem is a skill that differentiates MG students.

## METHODOLOGY

We gathered the solutions by a sample of 300 students in grades 2 and 4 to 6 (7 to 12 years old) who participated in the phases 2, 3, and 4 (final) of the OLCOMEP olympiad and analyzed the ways in which the students used their capacity of visualization, by identifying which descriptors were evidenced in their solutions. We created 13 problems of visualization, posed in the successive phases of the olympiad of the mentioned grades, and classified them into three types: A) given a complex structure made of simple objects, analyze its different side-views to get some information; B) given a complex structure made of simple objects, identify the quantity or types of objects that form it; C) given a structure, mentally disassemble or rearrange its elements, to fit the instructions stated in the problem. Several problems are presented in Figures 1, 3, and 5 and in Mora and Gutiérrez (2021).

We have made a mixed analysis of the data provided by the students' solutions. The qualitative analysis has allowed us to classify the solutions according to the descriptors evidenced by each one, as shown below. The quantitative analysis has been oriented to identify the problems and descriptors that have shown the highest power of discrimination of students with better visualization skills (Mora et al., 2023).

## RESULTS

In this section we analyze several problems, as examples of the types described above, and present a synthesis of the responses to each problem, analyzed in terms of the visualization descriptors evidenced by the students in each problem.

### Type A problems: analyze side-views of a complex structure made of simple elements

Luis builds a city with Lego, that has buildings of three colors and the buildings of the same color have the same height. He places them in a grid of 3x3 buildings, so that in the same row and column there are no buildings of the same height. Ana and Juan are two habitants of the city that are placed as shown in the figure.

<figure><img src="image_71.png" /></figure>

a) If Juan looks towards the city, how many buildings can he see?
Justify your answer.

b) If Anna looks towards the city, how many buildings can she see? Justify your answer.

c) In what position, different from Juan's, could Ana stand to see the same number of buildings that Juan sees?

<caption>Figure 1: Problem of phase 3, grade 4</caption>

Figure 2 shows a student's solution to the problem in Figure 1. To solve it, students needed to use *visual identification* (DV1), to correctly interpret the information, to identify the polygonal shapes that form the visible faces of the buildings, the complete buildings (with their walls and roofs), and
* DV5.2. *Recognition of spatial relationships between elements of objects:* to recognize the positions of elements of one or more objects located in space in relation to each other.

* DV6. *Visual discrimination:* to compare objects, identifying their visual similarities and differences.

As we have shown in the previous section, some authors argue that being able to choose and effectively use the most appropriate visualization skills to solve a problem is a skill that differentiates MG students.

## METHODOLOGY

We gathered the solutions by a sample of 300 students in grades 2 and 4 to 6 (7 to 12 years old) who participated in the phases 2, 3, and 4 (final) of the OLCOMEP olympiad and analyzed the ways in which the students used their capacity of visualization, by identifying which descriptors were evidenced in their solutions. We created 13 problems of visualization, posed in the successive phases of the olympiad of the mentioned grades, and classified them into three types: A) given a complex structure made of simple objects, analyze its different side-views to get some information; B) given a complex structure made of simple objects, identify the quantity or types of objects that form it; C) given a structure, mentally disassemble or rearrange its elements, to fit the instructions stated in the problem. Several problems are presented in Figures 1, 3, and 5 and in Mora and Gutiérrez (2021).

We have made a mixed analysis of the data provided by the students' solutions. The qualitative analysis has allowed us to classify the solutions according to the descriptors evidenced by each one, as shown below. The quantitative analysis has been oriented to identify the problems and descriptors that have shown the highest power of discrimination of students with better visualization skills (Mora et al., 2023).

## RESULTS

In this section we analyze several problems, as examples of the types described above, and present a synthesis of the responses to each problem, analyzed in terms of the visualization descriptors evidenced by the students in each problem.

### Type A problems: analyze side-views of a complex structure made of simple elements

Luis builds a city with Lego, that has buildings of three colors and the buildings of the same color have the same height. He places them in a grid of 3x3 buildings, so that in the same row and column there are no buildings of the same height. Ana and Juan are two habitants of the city that are placed as shown in the figure.

<figure><img src="image_77.png" /></figure>

a) If Juan looks towards the city, how many buildings can he see? Justify your answer.

b) If Anna looks towards the city, how many buildings can she see? Justify your answer.

c) In what position, different from Juan's, could Ana stand to see the same number of buildings that Juan sees?

<caption>Figure 1: Problem of phase 3, grade 4</caption>

Figure 2 shows a student's solution to the problem in Figure 1. To solve it, students needed to use *visual identification* (DV1), to correctly interpret the information, to identify the polygonal shapes that form the visible faces of the buildings, the complete buildings (with their walls and roofs), and
the complete structure of the city. All students scoring high used DV1 to solve the problem, while only 20% of the students scoring low did it. Therefore, this problem allowed to discriminate very well the use of DV1 by the best visualizers, who are thus potential MG students.

To answer questions a) and b), students had to use the *conservation of perception* to, considering the regularity of the structure, identify the buildings that, from John or Anne's position, are *partially hidden* (DV3.1) and those that are *completely hidden* (DV3.2). In this problem, DV3.1 was observed in 75% of the students and DV3.2 in only 62,5% of them, so using DV3.2 was more difficult for students than DV3.1. In the solutions to this problem, all students scoring high used both descriptors to solve the questions, but only 20% of students scoring low did the same. Then both descriptors may well differentiate students with high visualization ability and thus potential MG students.

<figure><img src="image_72.png" /><figcaption>Figure 2: Student E.E63031's solution to the problem of phase 3, grade 4</figcaption></figure>

To answer question c), students had to do recognition of *positions-in-space*, to choose a position taking as the reference either *themselves as an observer* (DV4.1) or *another fixed object* in the figure (DV4.2). The first option was observed in 18,7% of the students and the second one in 37,5% of them. In the solutions to this problem, all students scoring high used some of these descriptors to solve the questions, but none of the students scoring low did it. Then, this pair of descriptors may well differentiate students with high visualization ability and thus potential MG students. The student in Figure 2 used the descriptors DV1, DV3.1, DV3.2, and DV4.2.

**Type B problems: identify the quantity and/or type of objects that form a complex structure**

In the park there is a sculpture made of columns of cubes, the columns of the same height have the same color. If someone stands at the top of the light blue column, which is the center of the sculpture, they will see the sculpture symmetrical both comparing front and behind and comparing right and left.

a) How many cubes in total were used to build the sculpture?

b) What is the difference between the number of pink cubes and blue cubes?

c) What is the difference between the number of red cubes and brown cubes?

<figure><img src="image_73.png" /><figcaption>Figure 3: Problem of phase 2, grade 5</figcaption></figure>