them were prototypical (Figure 2a) and the other 2 built irregular convex quadrilateral. Other 7 students built the polygons almost correctly, but we did not consider them as correct because sometimes the buildings were open and/or one (or more) side was cutting off another (Figure 2b). From these 7 students, 5 students built convex quadrilateral, of which 4 were figures similar to rectangles (prototypical example) (Figure 2b) and the other 2 students built a hexagon and a heptagon. The other 4 students built polygons incorrectly because they put the sticks randomly, without forming any figure similar to a polygon (Figure 2c).

Concerning pentagons, the polygon with the lowest frequency of success, only 4 students were able to build it correctly. The other 13 students were not able to build an example of a pentagon because they did not know to combine the five sticks to make a polygon, where some of them put the sticks randomly, with the sticks crossing each other.

Regarding triangles, 9 students built a triangle correctly, which all were in the prototypical position (the base of the triangle parallel to the border of the table) and most were isosceles. Only 2 students built a triangle almost correctly because one side is cutting off another. The other 6 students did not know to build a triangle with Meccano.

<figure><img src="image_129.png" /><figcaption>Figure 2: Examples of students' buildings</figcaption></figure>

## Evidence of apprehensions when building polygons

In this study, we will only focus on the analysis of the pentagon and triangle buildings. Table 2 shows the frequency of the evidence or no evidence of the apprehensions when students built pentagons and triangles with Meccano. Specifically, whether students show that they anticipated the building through oral discourse or whether they took the necessary sticks to build the requested polygon (sequential apprehension (SA)); whether they built the requested polygon, considering those that were almost polygons because the buildings were open or had one side cutting off another (sequential apprehension (SA)); and whether they justified the properties that the built polygons had (discursive apprehension (DA)). We considered as evidence of discursive apprehension whether students justified the properties of the polygon if the building was correct.

<table><caption>Table 2: Frequency of the evidence of the sequential and discursive apprehensions</caption><thead><tr><th rowspan="2"></th><th colspan="3">Pentagon</th><th colspan="3">Triangle</th></tr><tr><th>SA<br>(anticipate)</th><th>SA<br>(build)</th><th>DA<br>(justify)</th><th>SA<br>(anticipate)</th><th>SA<br>(build)</th><th>DA<br>(justify)</th></tr></thead><tbody><tr><th>Evidence</th><td>7</td><td>4</td><td>4</td><td>11</td><td>11</td><td>10</td></tr><tr><th>No evidence</th><td>10</td><td>13</td><td>13</td><td>6</td><td>6</td><td>7</td></tr></tbody></table>
them were prototypical (Figure 2a) and the other 2 built irregular convex quadrilateral. Other 7 students built the polygons almost correctly, but we did not consider them as correct because sometimes the buildings were open and/or one (or more) side was cutting off another (Figure 2b). From these 7 students, 5 students built convex quadrilateral, of which 4 were figures similar to rectangles (prototypical example) (Figure 2b) and the other 2 students built a hexagon and a heptagon. The other 4 students built polygons incorrectly because they put the sticks randomly, without forming any figure similar to a polygon (Figure 2c).

Concerning pentagons, the polygon with the lowest frequency of success, only 4 students were able to build it correctly. The other 13 students were not able to build an example of a pentagon because they did not know to combine the five sticks to make a polygon, where some of them put the sticks randomly, with the sticks crossing each other.

Regarding triangles, 9 students built a triangle correctly, which all were in the prototypical position (the base of the triangle parallel to the border of the table) and most were isosceles. Only 2 students built a triangle almost correctly because one side is cutting off another. The other 6 students did not know to build a triangle with Meccano.

<figure><img src="image_136.png" /><figcaption>Figure 2: Examples of students' buildings</figcaption></figure>

## Evidence of apprehensions when building polygons

In this study, we will only focus on the analysis of the pentagon and triangle buildings. Table 2 shows the frequency of the evidence or no evidence of the apprehensions when students built pentagons and triangles with Meccano. Specifically, whether students show that they anticipated the building through oral discourse or whether they took the necessary sticks to build the requested polygon (sequential apprehension (SA)); whether they built the requested polygon, considering those that were almost polygons because the buildings were open or had one side cutting off another (sequential apprehension (SA)); and whether they justified the properties that the built polygons had (discursive apprehension (DA)). We considered as evidence of discursive apprehension whether students justified the properties of the polygon if the building was correct.

<table><caption>Table 2: Frequency of the evidence of the sequential and discursive apprehensions</caption><thead><tr><th rowspan="2"></th><th colspan="3">Pentagon</th><th colspan="3">Triangle</th></tr><tr><th>SA<br>(anticipate)</th><th>SA<br>(build)</th><th>DA<br>(justify)</th><th>SA<br>(anticipate)</th><th>SA<br>(build)</th><th>DA<br>(justify)</th></tr></thead><tbody><tr><th>Evidence</th><td>7</td><td>4</td><td>4</td><td>11</td><td>11</td><td>10</td></tr><tr><th>No evidence</th><td>10</td><td>13</td><td>13</td><td>6</td><td>6</td><td>7</td></tr></tbody></table>
Regarding the building of pentagons, in relation to the sequential apprehension, on the one hand, 7 students evidenced the ability to anticipate the needed material to build a pentagon, of which 5 said orally the material they needed and took it and 2 took the necessary material. On the other hand, 4 students evidenced the ability to build a polygon with manipulative material. According to discursive apprehension, 4 students provided a justification of the attributes included in the built pentagons (the same 4 students that built correctly).

Of the 7 students who anticipated what they needed to build the pentagon, 4 were not able to build it. In addition, one student (Student 16), who was able to build and justify the pentagon correctly when she had to anticipate, selected six Meccano sticks to build a pentagon, so she did not anticipate correctly the necessary material. When the student built the pentagon, she realized that the building did not have five sticks and removed one, building correctly the demanded pentagon (Figure 3).

<figure><img src="image_130.png" /><figcaption>Figure 3: Discourse between the teacher-researcher and Student 16 during the building of pentagon. According to the building of triangles, the pieces of evidence of the apprehensions were higher than with the pentagons. In relation to sequential apprehension, on the one hand, 11 students anticipated the needed material to build a triangle, of which 9 said orally the material they needed and took it and 2 only took the necessary material. On the other hand, the same 11 students built the triangles, of which 9 built correctly, and 2 built almost correctly. In relation to discursive apprehension, 10 students evidenced this apprehension because they provided a justification of the attributes included in the built triangles.</figcaption></figure>

Paying attention to combination of both buildings, on the one hand, the 3 students who evidenced the two apprehensions in the building of the pentagon (plus the exception of the case of the Student 16, who removed a stick to correct her building so did not evidence a part of the sequential apprehension (anticipate)), also evidenced these apprehensions in the building of the triangle. In addition, we
Regarding the building of pentagons, in relation to the sequential apprehension, on the one hand, 7 students evidenced the ability to anticipate the needed material to build a pentagon, of which 5 said orally the material they needed and took it and 2 took the necessary material. On the other hand, 4 students evidenced the ability to build a polygon with manipulative material. According to discursive apprehension, 4 students provided a justification of the attributes included in the built pentagons (the same 4 students that built correctly).

Of the 7 students who anticipated what they needed to build the pentagon, 4 were not able to build it. In addition, one student (Student 16), who was able to build and justify the pentagon correctly when she had to anticipate, selected six Meccano sticks to build a pentagon, so she did not anticipate correctly the necessary material. When the student built the pentagon, she realized that the building did not have five sticks and removed one, building correctly the demanded pentagon (Figure 3).

<figure><img src="image_137.png" /><figcaption>Figure 3: Discourse between the teacher-researcher and Student 16 during the building of pentagon</figcaption></figure>

According to the building of triangles, the pieces of evidence of the apprehensions were higher than with the pentagons. In relation to sequential apprehension, on the one hand, 11 students anticipated the needed material to build a triangle, of which 9 said orally the material they needed and took it and 2 only took the necessary material. On the other hand, the same 11 students built the triangles, of which 9 built correctly, and 2 built almost correctly. In relation to discursive apprehension, 10 students evidenced this apprehension because they provided a justification of the attributes included in the built triangles.

Paying attention to combination of both buildings, on the one hand, the 3 students who evidenced the two apprehensions in the building of the pentagon (plus the exception of the case of the Student 16, who removed a stick to correct her building so did not evidence a part of the sequential apprehension (anticipate)), also evidenced these apprehensions in the building of the triangle. In addition, we
consider that Student 16, who removed a stick to correct her building, would also form part of this
group (being a total of 4 students) since, although she did not show evidence of half of the sequential
apprehension (anticipate), she showed evidence of the other half (build) and of the discursive
apprehension during the building of the pentagon. On the other hand, the 4 students who correctly
anticipated the building of the pentagon, but they did not build it correctly and therefore their
justification was not considered correct, showed the two apprehensions in the building of the triangle.

CONCLUSIONS AND DISCUSSION

In this study, we aim to generate information about the manifestation of dimensional deconstruction
through sequential and discursive apprehension when kindergarten students build polygons with
manipulative material. The results obtained in the first section of the results based on the frequencies
of success (correct and almost correct) building polygons and triangle with manipulative material
(13/17 and 11/17, respectively) show that students began to develop the dimensional deconstruction
(Duval, 2017). The students were able to give mathematical meaning to the sticks of Meccano to join
them and build a polygon or triangle, assuming that the sticks were the sides of the polygons.
According to the results obtained in the second section, the pieces of evidence of the sequential and
discursive apprehensions when students build triangles and pentagons, greater in the building of the
triangle than in the pentagon, also manifests a use of dimensional deconstruction to solve the
geometric task. These results show that students, after classroom instruction, began to be able to make
mathematical sense of the parts of the figures to be constructed through sequential apprehension
(anticipating and building) and subsequently, they were able to see the built figures mathematically
to justify the properties of these figures according to the geometric concepts, through discursive
apprehension. Therefore, the building tasks help the development of the figural concept (Kinzer et
al., 2016; Fischbein, 1993), since students can associate the attributes belonging to the concepts with
the representations of geometric figures through the manipulative material and modify a
representation if it does not meet the properties of the concept to be represented (as in the case of
Student 16). On the contrary, the low frequencies obtained with the pentagons lead us to think that
the development of dimensional deconstruction depends on the attribute considered (Bernabeu &
Llinares, 2017), in this case, the number of sides of the polygons.

Also, when building polygons and triangles, most of them were representations of prototypical
examples such as isosceles or similar isosceles triangles (with similarly measured sticks) in
prototypical position and non-polygons similar to rectangles (considered as almost correct because
they had sides that cut off other sides). In this latter case, students forced the sides to cut off to build
a familiar figure for them, the rectangle (see example b in Figure 2). This may be due to the students'
concept image of polygons, which has a limited variety of examples and is still very much rooted in
the examples that students are used to seeing in any school or out-of-school context (Maier & Benz,
2020; Tall & Vinner, 1981; Verdine et al., 2016).

In addition, the results lead us to infer that anticipating is easier than building as it only involves
saying what material is needed or/and taking it but building requires manipulating it so that it fulfils
the attributes presented by the concept (such as being closed and not having crossed sides to be a
polygon). On the other hand, we can infer that if students know how to anticipate and build, they will
possibly be able to justify the attributes of the constructed figures, demonstrating the importance of
combining the perceptual with the conceptual to develop geometric thinking.
consider that Student 16, who removed a stick to correct her building, would also form part of this
group (being a total of 4 students) since, although she did not show evidence of half of the sequential
apprehension (anticipate), she showed evidence of the other half (build) and of the discursive
apprehension during the building of the pentagon. On the other hand, the 4 students who correctly
anticipated the building of the pentagon, but they did not build it correctly and therefore their
justification was not considered correct, showed the two apprehensions in the building of the triangle.

CONCLUSIONS AND DISCUSSION

In this study, we aim to generate information about the manifestation of dimensional deconstruction
through sequential and discursive apprehension when kindergarten students build polygons with
manipulative material. The results obtained in the first section of the results based on the frequencies
of success (correct and almost correct) building polygons and triangle with manipulative material
(13/17 and 11/17, respectively) show that students began to develop the dimensional deconstruction
(Duval, 2017). The students were able to give mathematical meaning to the sticks of Meccano to join
them and build a polygon or triangle, assuming that the sticks were the sides of the polygons.
According to the results obtained in the second section, the pieces of evidence of the sequential and
discursive apprehensions when students build triangles and pentagons, greater in the building of the
triangle than in the pentagon, also manifests a use of dimensional deconstruction to solve the
geometric task. These results show that students, after classroom instruction, began to be able to make
mathematical sense of the parts of the figures to be constructed through sequential apprehension
(anticipating and building) and subsequently, they were able to see the built figures mathematically
to justify the properties of these figures according to the geometric concepts, through discursive
apprehension. Therefore, the building tasks help the development of the figural concept (Kinzer et
al., 2016; Fischbein, 1993), since students can associate the attributes belonging to the concepts with
the representations of geometric figures through the manipulative material and modify a
representation if it does not meet the properties of the concept to be represented (as in the case of
Student 16). On the contrary, the low frequencies obtained with the pentagons lead us to think that
the development of dimensional deconstruction depends on the attribute considered (Bernabeu &
Llinares, 2017), in this case, the number of sides of the polygons.

Also, when building polygons and triangles, most of them were representations of prototypical
examples such as isosceles or similar isosceles triangles (with similarly measured sticks) in
prototypical position and non-polygons similar to rectangles (considered as almost correct because
they had sides that cut off other sides). In this latter case, students forced the sides to cut off to build
a familiar figure for them, the rectangle (see example b in Figure 2). This may be due to the students'
concept image of polygons, which has a limited variety of examples and is still very much rooted in
the examples that students are used to seeing in any school or out-of-school context (Maier & Benz,
2020; Tall & Vinner, 1981; Verdine et al., 2016).

In addition, the results lead us to infer that anticipating is easier than building as it only involves
saying what material is needed or/and taking it but building requires manipulating it so that it fulfils
the attributes presented by the concept (such as being closed and not having crossed sides to be a
polygon). On the other hand, we can infer that if students know how to anticipate and build, they will
possibly be able to justify the attributes of the constructed figures, demonstrating the importance of
combining the perceptual with the conceptual to develop geometric thinking.
## References

Bernabeu, M. (2022). “Tips” para la enseñanza-aprendizaje de las figuras geométricas. *Números. Revista de Didáctica de las Matemáticas*, **110**, 113-128.

Bernabeu, M., & Llinares, S. (2017). Comprensión de las figuras geométricas en niños de 6-9 años. *Educación Matemática*, **29**(2), 9-35. https://doi.org/10.24844/EM2902.01

Bernabeu, M., Moreno, M., & Llinares, S. (2019). El conocimiento geométrico de los/las estudiantes para maestro/a de educación infantil a través de una tarea de clasificación de cuadriláteros. In R. Roig-Vila, (Ed.), *Investigación e innovación en la enseñanza superior: Nuevos contextos, nuevas ideas* (pp. 34-44). Octaedro.

Casey, B. M., Andrews, N., Schindler, H., Kersh, J. E., Samper, A., & Copley, J. (2008). The development of spatial skills through interventions involving block building activities. *Cognition and Instruction*, **26**(3), 269-309.

Clements, D., Swaminathan, S., Hannibal, M., & Sarama, J. (1999). Young children's concepts of shape. *Journal for research in Mathematics Education*, **19**(2), 212-217. https://doi.org/10.2307/749610

Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.), *Exploiting mental imagery with computers in mathematics education* (pp. 142-157). Springer. https://doi.org/10.1007/978-3-642-57771-0_10

Duval, R. (2017). *Understanding the mathematical way of thinking - The registers of semiotic representations*. Springer International Publishing.

Fischbein, E. (1993). The theory of figural concepts. *Educational Studies in Mathematics*, **24**(2), 139-162. https://doi.org/10.1007/BF01273689

Herbst, P., Fujita, T., Halverscheid, S., & Weiss, M. (2017). *The learning and teaching of geometry in secondary schools: A modeling perspective*. Routledge.

Jatisunda, M. G., Hidayanti, M., Lita, Dede, S. N., Cahyaningsih, U., & Suciawati, V. (2021). Mathematical knowledge for early childhood teaching: A deep insight on how pre-service teachers prepare mathematical activities. *Journal of Physics: Conference Series*, **1778**(1). https://doi.org/10.1088/1742-6596/1778/1/012017

Kinzer, C., Gerhardt, K., & Coca, N. (2016). Building a case for blocks as kindergarten mathematics learning tools. *Early Childhood Education Journal*, **44**, 389-402.

Maier, A., & Benz, C. (2020). “A triangle is like a tent”: children’s conception of geometric shapes. In *Mathematics education in the early years: Results from the POEM4 Conference, 2018* (pp. 229-248). Springer International Publishing.

NCTM. (2000). *Principles and standards for school mathematics*. National Council of Teachers of Mathematics.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational Studies in Mathematics*, **12**(2), 151-169. https://doi.org/10.1007/BF00305619

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

Bernabeu, M. (2022). “Tips” para la enseñanza-aprendizaje de las figuras geométricas. *Números. Revista de Didáctica de las Matemáticas, 110*, 113-128.

Bernabeu, M., & Llinares, S. (2017). Comprensión de las figuras geométricas en niños de 6-9 años. *Educación Matemática, 29*(2), 9-35. https://doi.org/10.24844/EM2902.01

Bernabeu, M., Moreno, M., & Llinares, S. (2019). El conocimiento geométrico de los/las estudiantes para maestro/a de educación infantil a través de una tarea de clasificación de cuadriláteros. In R. Roig-Vila, (Ed.), *Investigación e innovación en la enseñanza superior: Nuevos contextos, nuevas ideas* (pp. 34-44). Octaedro.

Casey, B. M., Andrews, N., Schindler, H., Kersh, J. E., Samper, A., & Copley, J. (2008). The development of spatial skills through interventions involving block building activities. *Cognition and Instruction, 26*(3), 269-309.

Clements, D., Swaminathan, S., Hannibal, M., & Sarama, J. (1999). Young children's concepts of shape. *Journal for research in Mathematics Education, 19*(2), 212-219. https://doi.org/10.2307/749610

Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.), *Exploiting mental imagery with computers in mathematics education* (pp. 142-157). Springer. https://doi.org/10.1007/978-3-642-57771-0_10

Duval, R. (2017). *Understanding the mathematical way of thinking - The registers of semiotic representations*. Springer International Publishing.

Fischbein, E. (1993). The theory of figural concepts. *Educational Studies in Mathematics, 24*(2), 139-162. https://doi.org/10.1007/BF01273689

Herbst, P., Fujita, T., Halverscheid, S., & Weiss, M. (2017). *The learning and teaching of geometry in secondary schools: A modeling perspective*. Routledge.

Jatisunda, M. G., Hidayanti, M., Lita, Dede, S. N., Cahyaningsih, U., & Suciawati, V. (2021). Mathematical knowledge for early childhood teaching: A deep insight on how pre-service teachers prepare mathematical activities. *Journal of Physics: Conference Series, 1778*(1). https://doi.org/10.1088/1742-6596/1778/1/012017

Kinzer, C., Gerhardt, K., & Coca, N. (2016). Building a case for blocks as kindergarten mathematics learning tools. *Early Childhood Education Journal, 44*, 389-402.

Maier, A., & Benz, C. (2020). “A triangle is like a tent”: children’s conception of geometric shapes. In *Mathematics education in the early years: Results from the POEM4 Conference, 2018* (pp. 229-248). Springer International Publishing.

NCTM. (2000). *Principles and standards for school mathematics*. National Council of Teachers of Mathematics.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational Studies in Mathematics, 12*(2), 151-169. https://doi.org/10.1007/BF00305619

Verdine, B. N., Lucca, K. R., Golinkoff, R. M., Hirsh-Pasek, K., & Newcombe, N. S. (2016). The shape of things: The origin of young children’s knowledge of the names and properties of geometric forms. *Journal of Cognition and Development, 17*(1), 142-161. https://doi.org/10.1080/15248372.2015.1016610
# KNOWLEDGE AND REASONING IN CIRCULATION DURING A SITUATION OF FIGURES REPRODUCTION BY FOLDING

Sylvie Blanquart¹, Claire Guille-Biel Winder², Édith Petitfour³

¹ Lab-E3D, University of Bordeaux, ² ADEF, Aix-Marseille University, ³ LDAR, Univ Rouen Normandie

This work is part of a research project aimed at identifying the evolution of knowledge and reasoning of French elementary school students (aged 7-8) in the implementation of a situation involving the reproduction of figures by folding. The analysis of a verbalization phase is operationalized using a semiotic approach. It reveals reasoning based on spatial knowledge, as well as the circulation of geometric knowledge, but also highlights numerous semiotic dissonances.

This work is part of a research project that aims to identify the evolution of knowledge and reasoning of French elementary school students (aged 7-8) during the implementation of a figure reproduction situation by folding, the ‘Pliox situation’ (Guille-Biel Winder, 2014). Based on several experiments, Guille-Biel Winder (2021) highlighted the importance of the teacher's language during collective phases, in relations between students and the milieu (Brousseau, 1997). We take a semiotic approach to this study, focusing here on a verbalization phase. After outlining our theoretical and methodological bases, we present the ‘Pliox situation’. We then provide elements for a priori and a posteriori analyses of the selected phase.

## THEORETICAL AND METHODOLOGICAL BACKGROUND

In the context of geometry, we begin by drawing on the Duval’s work, in particular the notion of figural unit (0D, 1D, 2D, 3D), and the apprehension levels of geometric figures - perceptive, operative (mereologic, optic, place) and discursive apprehensions (Duval, 1999). This work also falls within the general framework of the theory of didactic situations (TDS) that takes into account both knowledge to be taught and learned, and conditions of its appearance (Brousseau, 1997). In what follows, we return more specifically to the notion of verbalization.

### Notion of verbalization

Within the TDS framework, the knowledge modeling is based on situations of action, formulation and validation (Brousseau, 1997). A process, organized around different dialectics (and not a fixed succession of situations), enables the construction of knowledge. This process is not necessarily linear and, in its contingency, depends on the management of the didactic sequence planned by the teacher. For this reason, we use the term ‘phase’ to refer to a stage clearly identified or organized by the teacher in the implementation of a situation. Based on the work of Blanquart (2020), we conceive of verbalization phase as an intermediary between action phase and formulation phase. When knowledge is verbalized, its function is to designate precisely the (instrumented) actions performed (or to be performed), or to describe their purpose. This action is contextualized, situated in time and space, and marked by the one acting. However, we differentiate between verbalization and effective action, because, from the point of view of knowledge, verbalization already enables a first reflexive
# KNOWLEDGE AND REASONING IN CIRCULATION DURING A SITUATION OF FIGURES REPRODUCTION BY FOLDING

Sylvie Blanquart¹, Claire Guille-Biel Winder², Édith Petitfour³

¹ Lab-E3D, University of Bordeaux, ² ADEF, Aix-Marseille University, ³ LDAR, Univ Rouen Normandie

This work is part of a research project aimed at identifying the evolution of knowledge and reasoning of French elementary school students (aged 7-8) in the implementation of a situation involving the reproduction of figures by folding. The analysis of a verbalization phase is operationalized using a semiotic approach. It reveals reasoning based on spatial knowledge, as well as the circulation of geometric knowledge, but also highlights numerous semiotic dissonances.

This work is part of a research project that aims to identify the evolution of knowledge and reasoning of French elementary school students (aged 7-8) during the implementation of a figure reproduction situation by folding, the ‘Pliox situation’ (Guille-Biel Winder, 2014). Based on several experiments, Guille-Biel Winder (2021) highlighted the importance of the teacher’s language during collective phases, in relations between students and the milieu (Brousseau, 1997). We take a semiotic approach to this study, focusing here on a verbalization phase. After outlining our theoretical and methodological bases, we present the ‘Pliox situation’. We then provide elements for a priori and a posteriori analyses of the selected phase.

## THEORETICAL AND METHODOLOGICAL BACKGROUND

In the context of geometry, we begin by drawing on the Duval’s work, in particular the notion of figural unit (0D, 1D, 2D, 3D), and the apprehension levels of geometric figures - perceptive, operative (mereologic, optic, place) and discursive apprehensions (Duval, 1999). This work also falls within the general framework of the theory of didactic situations (TDS) that takes into account both knowledge to be taught and learned, and conditions of its appearance (Brousseau, 1997). In what follows, we return more specifically to the notion of verbalization.

### Notion of verbalization

Within the TDS framework, the knowledge modeling is based on situations of action, formulation and validation (Brousseau, 1997). A process, organized around different dialectics (and not a fixed succession of situations), enables the construction of knowledge. This process is not necessarily linear and, in its contingency, depends on the management of the didactic sequence planned by the teacher. For this reason, we use the term ‘phase’ to refer to a stage clearly identified or organized by the teacher in the implementation of a situation. Based on the work of Blanquart (2020), we conceive of verbalization phase as an intermediary between action phase and formulation phase. When knowledge is verbalized, its function is to designate precisely the (instrumented) actions performed (or to be performed), or to describe their purpose. This action is contextualized, situated in time and space, and marked by the one acting. However, we differentiate between verbalization and effective action, because, from the point of view of knowledge, verbalization already enables a first reflexive
feedback on action: the subject verbalizes for others (or for himself). To that end, he/she selects the information to be communicated, the order in which he/she states it, and the signs useful for this communication (oral language, gestures, written symbols). The subject must take into account an interlocutor who can provide him/her feedback on the form of the message, and who is therefore part of his/her environment. He/she must be able to point to the objects he/she is talking about and their relationships (e.g. in geometry, through a geometric technical language (Petitfour, 2016)). This can encourage the acquisition of linguistic knowledge and the production of reasoning, and thus help to link visualization and discursive activity, which is essential in the field of geometry. We define reasoning as any voluntary action or formulation by a subject that enables progress towards a defined goal, constructed with his/her knowledge, and which he/she can make explicit, even summarily (Blanquart, 2020). The analysis of reasoning is, in particular, based on semiotic analysis.

## Semiotic approach

Drawing on Vygotskian sociocultural theory of learning (Vygotsky, 1978) and following Radford (2009), we consider that in mathematics, ‘thinking does not occur solely in the head but also in and through a sophisticated semiotic coordination of speech, body, gestures, symbols and tools’ (ibid., p. 113). We then rely on the concept of semiotic bundle (Arzarello, 2006) to grasp the multimodality of the learning process and to consider the dynamic development of interactions between the different systems of signs, such as language, coordinated with embodied aspects (gestures, body postures, glances, actions with tools) and written (words, drawings, mathematical symbols). We use a semiotic table (Petitfour & Houdement, 2022) to reveal the multimodality of teaching and learning phenomena and to interpret the different signs activated during interactions between the actors in the classroom (e.g. Figure 6). Finally, we use the term *semiotic dissonance* (ibid.) to name a discrepancy between two interpretations of the same signs, or a quirky personal interpretation of a mathematical sign.

## THE ‘PLIOX SITUATION’

The study takes place in the context of the development of the ‘Pliox situation’, a situation involving reproduction of figures by folding. In order to dissociate the geometrical shape from the artifact (Rabardel, 1995), we design by ‘Pliox’ a square sheet of paper, shared by four colored square areas (yellow, red, green and blue) (Figure 1). The generic ‘Pliox situation’ consists in a didactic situation (according to Brousseau sense): it is the reproduction of a model figure by folding a Pliox. The possible folds correspond to the symmetric axes of square (diagonals and medians) of the so-called ‘secondary squares’ (i.e. the four colored squares) (Figure 1(c)).

<figure><img src="image_131.png" /><figcaption>Figure 1: *Recto* (a) and *verso* (b) of the Pliox; possible folds (c)</figcaption></figure>

The ‘Pliox situation’ can be seen as an origami in the sensitive space, that is to say it corresponds to a spatial problem. The Pliox is accessible to vision; turning, returning, folding and unfolding the Pliox are the possible manipulations (3D gestures with tactile and visual coordination); all movements of the Pliox in regard to the subject are possible, as well as those of the subject; the subject is outside
feedback on action: the subject verbalizes for others (or for himself). To that end, he/she selects the information to be communicated, the order in which he/she states it, and the signs useful for this communication (oral language, gestures, written symbols). The subject must take into account an interlocutor who can provide him/her feedback on the form of the message, and who is therefore part of his/her environment. He/she must be able to point to the objects he/she is talking about and their relationships (e.g. in geometry, through a geometric technical language (Petitfour, 2016)). This can encourage the acquisition of linguistic knowledge and the production of reasoning, and thus help to link visualization and discursive activity, which is essential in the field of geometry. We define reasoning as any voluntary action or formulation by a subject that enables progress towards a defined goal, constructed with his/her knowledge, and which he/she can make explicit, even summarily (Blanquart, 2020). The analysis of reasoning is, in particular, based on semiotic analysis.

## Semiotic approach

Drawing on Vygotskian sociocultural theory of learning (Vygotsky, 1978) and following Radford (2009), we consider that in mathematics, ‘thinking does not occur solely in the head but also in and through a sophisticated semiotic coordination of speech, body, gestures, symbols and tools’ (ibid., p. 113). We then rely on the concept of semiotic bundle (Arzarello, 2006) to grasp the multimodality of the learning process and to consider the dynamic development of interactions between the different systems of signs, such as language, coordinated with embodied aspects (gestures, body postures, glances, actions with tools) and written (words, drawings, mathematical symbols). We use a semiotic table (Petitfour & Houdement, 2022) to reveal the multimodality of teaching and learning phenomena and to interpret the different signs activated during interactions between the actors in the classroom (e.g. Figure 6). Finally, we use the term *semiotic dissonance* (ibid.) to name a discrepancy between two interpretations of the same signs, or a quirky personal interpretation of a mathematical sign.

## THE ‘PLIOX SITUATION’

The study takes place in the context of the development of the ‘Pliox situation’, a situation involving reproduction of figures by folding. In order to dissociate the geometrical shape from the artifact (Rabardel, 1995), we design by ‘Pliox’ a square sheet of paper, shared by four colored square areas (yellow, red, green and blue) (Figure 1). The generic ‘Pliox situation’ consists in a didactic situation (according to Brousseau sense): it is the reproduction of a model figure by folding a Pliox. The possible folds correspond to the symmetric axes of square (diagonals and medians) of the so-called ‘secondary squares’ (i.e. the four colored squares) (Figure 1(c)).

<figure><img src="image_138.png" /><figcaption>Figure 1: *Recto* (a) and *verso* (b) of the Pliox; possible folds (c)</figcaption></figure>

The ‘Pliox situation’ can be seen as an origami in the sensitive space, that is to say it corresponds to a spatial problem. The Pliox is accessible to vision; turning, returning, folding and unfolding the Pliox are the possible manipulations (3D gestures with tactile and visual coordination); all movements of the Pliox in regard to the subject are possible, as well as those of the subject; the subject is outside
the considered space, which contains the object. Therefore the corresponding space is the 3D micro-
space. Moreover, the ‘Pliox situation’ can be considered according to a geometrical point of view.
The Pliox is indeed very flat, so that it can be related to a representation of a two-dimensional
geometrical figure (possibly shared into other sub-figures). In primary school, we believe that a figure
is a drawing whose properties can be specified within the theoretical geometrical framework by
considering it as a representation of a geometrical figure (Duval et al., 2005). In this case, a figure
can be obtained by juxtaposition or superposition of shapes. Consequently, we consider reproduction
of figures by folding a Pliox as geometrical problems. The progress of the situation consists of several
sessions. The first one is devoted to the manufacturing of the artifact by students, the others to
reproduction of several figures by folding. We focus here on the reproduction of model figure (a) (see
Figure 2(a)).

A PRIORI ANALYSIS

According to Berthelot & Salin (1995), ‘the usual teaching geometry tasks call up skills from usual microspace relations’ (p. 23). The ‘Pliox situation’ involves indeed spatial knowledge (because of the orientation and the positions of figures that appear into model figures), but also geometrical one (mainly about the square, the rectangle, the right-angled triangle and their relationships, and specific straight lines (diagonals, medians)). Moreover, two points of view on a same figure can be involved in this situation: what is immediately recognized (the colored zones and the global form according to Gestalt Theory) and the figures who can be identified after taking account of some folds. For this case, knowledge and recognition of elementary two-dimensional figural units require perceptual apprehension and operative (mereologic, optic and place) apprehension (Duval, 1999)<fnref n="16" />. Reproducing figure (a) requires identification of the external figure (isosceles right-angled triangle) and to take care of the three colored zones (yellow, red and green) and their relative positions (but not necessarily to recognize them as triangles or square). This model highlights a relationship between square and isosceles right-angled triangle: dividing a square into two parts can lead to two isosceles right-angled triangles. The reproduction also requires folding according to a diagonal of the square. Hence the folds allow recognizing two red triangles, a yellow and red triangle, but also a red and green square or a yellow, red and green rectangle, ... (Figure 2(b)).

<figure><img src="image_132.png" /><figcaption>Figure 2: A model figure (a) and various decompositions (b)</figcaption></figure>

The ‘Pliox situation’ is an action situation (Brousseau, 1997) within which different phases are set
up: (1) presentation of the model figure; (2) short collective analysis of the model figure; (3)
individual reproduction of the model figure by folding; (4) collective highlighting of procedures and
validation; (5) mathematical synthesis. We focus here on phase (2) that corresponds, for our sense, to
a verbalization phase. During this phase, the mereologic decomposition induced by color as well as
the spatial organization of the figural units can be highlighted. The request to justify the identification
of a figure leads gradually the students to become aware of some properties (the number of sides and
vertices of the square, of the rectangle...); some relationships between some of the shapes can also

<footnote n="16"><sup>16</sup> For more details, see (Guille-Biel Winder, 2014).</footnote>
the considered space, which contains the object. Therefore the corresponding space is the 3D micro-
space. Moreover, the ‘Pliox situation’ can be considered according to a geometrical point of view.
The Pliox is indeed very flat, so that it can be related to a representation of a two-dimensional
geometrical figure (possibly shared into other sub-figures). In primary school, we believe that a figure
is a drawing whose properties can be specified within the theoretical geometrical framework by
considering it as a representation of a geometrical figure (Duval et al., 2005). In this case, a figure
can be obtained by juxtaposition or superposition of shapes. Consequently, we consider reproduction
of figures by folding a Pliox as geometrical problems. The progress of the situation consists of several
sessions. The first one is devoted to the manufacturing of the artifact by students, the others to
reproduction of several figures by folding. We focus here on the reproduction of model figure (a) (see
Figure 2(a)).

A PRIORI ANALYSIS

According to Berthelot & Salin (1995), ‘the usual teaching geometry tasks call up skills from usual microspace relations’ (p. 23). The ‘Pliox situation’ involves indeed spatial knowledge (because of the orientation and the positions of figures that appear into model figures), but also geometrical one (mainly about the square, the rectangle, the right-angled triangle and their relationships, and specific straight lines (diagonals, medians)). Moreover, two points of view on a same figure can be involved in this situation: what is immediately recognized (the colored zones and the global form according to Gestalt Theory) and the figures who can be identified after taking account of some folds. For this case, knowledge and recognition of elementary two-dimensional figural units require perceptual apprehension and operative (mereologic, optic and place) apprehension (Duval, 1999)<fnref n="16" />. Reproducing figure (a) requires identification of the external figure (isosceles right-angled triangle) and to take care of the three colored zones (yellow, red and green) and their relative positions (but not necessarily to recognize them as triangles or square). This model highlights a relationship between square and isosceles right-angled triangle: dividing a square into two parts can lead to two isosceles right-angled triangles. The reproduction also requires folding according to a diagonal of the square. Hence the folds allow recognizing two red triangles, a yellow and red triangle, but also a red and green square or a yellow, red and green rectangle, ... (Figure 2(b)).

<figure><img src="image_139.png" /><figcaption>Figure 2: A model figure (a) and various decompositions (b)</figcaption></figure>

The ‘Pliox situation’ is an action situation (Brousseau, 1997) within which different phases are set
up: (1) presentation of the model figure; (2) short collective analysis of the model figure; (3)
individual reproduction of the model figure by folding; (4) collective highlighting of procedures and
validation; (5) mathematical synthesis. We focus here on phase (2) that corresponds, for our sense, to
a verbalization phase. During this phase, the mereologic decomposition induced by color as well as
the spatial organization of the figural units can be highlighted. The request to justify the identification
of a figure leads gradually the students to become aware of some properties (the number of sides and
vertices of the square, of the rectangle...); some relationships between some of the shapes can also

<footnote n="16">¹⁶ For more details, see (Guille-Biel Winder, 2014).</footnote>
been highlighted; the spatial vocabulary (over, down, behind...) and the geometric vocabulary (square, triangle, diagonal, point...) can appear as necessary to the communication. Let's see what knowledge and reasoning can be identified when implementing this phase in a classroom.

## A POSTERIOI ANALYSIS

The sequence takes place in a class of 23 students (14 boys, 9 girls) aged 8-9. A beginner teacher with a scientific background, who had done university work on the ‘Pliox situation’, giving us access to her preparations and stated intentions, led it. The sequence (6 sessions) was filmed by the teacher, then transcribed by us. Each session was split into episodes (unit on the task, on the nature of the activity). Session 1 was devoted to diagnostic assessment. Session 2 aimed to build the Pliox; it led to instrumental recognition of the square and rectangle (checking right angles with a square and equal lengths with a graduated ruler); students reproduced various model figures in later sessions. The analyzed verbalization phase (session 3, episode 2) corresponds to the presentation and analysis of the model figure (a) that is the students' first reproduction work. During this phase, the model figure is displayed on the blackboard in this orientation. The teacher asks the students to say what they see, and progressively invites them to show on the Pliox the figurative elements they are talking about. Based on the synopsis, we identified within three extracts of episode 2 ‘remarkable events’ (Leutenegger, 2009) i.e. significant in terms of students' knowledge and/or reasoning. We present and analyze them in what follows, supplementing them when necessary with elements of analysis from other sessions of the sequence.

Figures apprehension

In the first extract (1'02" to 2'43"), a student locates and names triangles, differentiating them by their relative size ('one large', 'two small triangles') while outlining them with her finger. Another pupil, Clara, then announces 'a square on top'. She outlines it, while other students dispute: one says it's not a square, another that 'you can see it's a rhombus'. Clara explains how she recognized a square by making a rotating gesture with her hand, while saying 'Well, you put it a little upside down, like this'. We can see here that the students all rely on their perceptive understanding of figures to recognize and name the three colored juxtaposed sub-figures as well as the overall figure they compose. Clara's intervention reveals a line of reasoning that demonstrates mastery of spatial knowledge: a geometric figure does not change its name (and therefore its nature) according to its orientation on the board; we can choose the orientation of a figure (by mentally rotating it) to better recognize it in comparison with the 'prototypical figure' stored in memory. As shown by the students' reactions to Clara's proposal ('it's not a square', 'it's a rhombus'), and as confirmed by the teacher in her professional writing, many of the students in the class do not possess this knowledge at this point of the session, since they consider the orientation of the red figure to be decisive in naming it square or rhombus. This is how Mat explained it during the diagnostic evaluation (session 1), based on an actual rotation of the PLIOX (Figure 3) (on this figure, T indicates the teacher). Furthermore, to recognize the square, his (erroneous) reasoning was based on the horizontality and verticality of the figure's sides, showing these directions and expressing it by 'seeing straight' in a spatial sense (Figure 4).
been highlighted; the spatial vocabulary (over, down, behind...) and the geometric vocabulary (square, triangle, diagonal, point...) can appear as necessary to the communication. Let's see what knowledge and reasoning can be identified when implementing this phase in a classroom.

## A POSTERIOI ANALYSIS

The sequence takes place in a class of 23 students (14 boys, 9 girls) aged 8-9. A beginner teacher with a scientific background, who had done university work on the ‘Pliox situation’, giving us access to her preparations and stated intentions, led it. The sequence (6 sessions) was filmed by the teacher, then transcribed by us. Each session was split into episodes (unit on the task, on the nature of the activity). Session 1 was devoted to diagnostic assessment. Session 2 aimed to build the Pliox; it led to instrumental recognition of the square and rectangle (checking right angles with a square and equal lengths with a graduated ruler); students reproduced various model figures in later sessions. The analyzed verbalization phase (session 3, episode 2) corresponds to the presentation and analysis of the model figure (a) that is the students' first reproduction work. During this phase, the model figure is displayed on the blackboard in this orientation. The teacher asks the students to say what they see, and progressively invites them to show on the Pliox the figurative elements they are talking about. Based on the synopsis, we identified within three extracts of episode 2 ‘remarkable events’ (Leutenegger, 2009) i.e. significant in terms of students' knowledge and/or reasoning. We present and analyze them in what follows, supplementing them when necessary with elements of analysis from other sessions of the sequence.

Figures apprehension

In the first extract (1'02" to 2'43"), a student locates and names triangles, differentiating them by their relative size ('one large', 'two small triangles') while outlining them with her finger. Another pupil, Clara, then announces 'a square on top'. She outlines it, while other students dispute: one says it's not a square, another that 'you can see it's a rhombus'. Clara explains how she recognized a square by making a rotating gesture with her hand, while saying 'Well, you put it a little upside down, like this'. We can see here that the students all rely on their perceptive understanding of figures to recognize and name the three colored juxtaposed sub-figures as well as the overall figure they compose. Clara's intervention reveals a line of reasoning that demonstrates mastery of spatial knowledge: a geometric figure does not change its name (and therefore its nature) according to its orientation on the board; we can choose the orientation of a figure (by mentally rotating it) to better recognize it in comparison with the 'prototypical figure' stored in memory. As shown by the students' reactions to Clara's proposal ('it's not a square', 'it's a rhombus'), and as confirmed by the teacher in her professional writing, many of the students in the class do not possess this knowledge at this point of the session, since they consider the orientation of the red figure to be decisive in naming it square or rhombus. This is how Mat explained it during the diagnostic evaluation (session 1), based on an actual rotation of the PLIOX (Figure 3) (on this figure, T indicates the teacher). Furthermore, to recognize the square, his (erroneous) reasoning was based on the horizontality and verticality of the figure's sides, showing these directions and expressing it by 'seeing straight' in a spatial sense (Figure 4).
<figure><img src="image_133.png" /><figcaption>Figure 3: Square and rhombus recognition (Mat, session 1)</figcaption></figure>

<figure><img src="image_134.png" /><figcaption>Figure 4: Spatial characterization of the square (Mat, session 1)</figcaption></figure>

Thus, the students' reasoning (right or wrong) to recognize this figure involves spatial knowledge without explicit reference to geometric properties of the square.

## Vocabulary and geometric knowledge

In the second extract (2'43" to 4'26"), which follows a disagreement between students about the name of the red figure, the teacher asks them to recall what they saw in session 2 about the square. One student talks about 'faces and sides', another about 'faces and edges'. The teacher discards these terms associated with solids. Luis says there are three sides in the square, but points to three of the square vertices on the Pliox. The teacher invalidates this proposal by asking one pupil to show the sides of this figure (the pupil first points to them, and, at the teacher's request, follows them with her finger), then another to show the vertices (which are pointed to). The teacher then summarizes the square characteristics: four sides and four vertices. In this extract, we identify the students' use of geometric terms specific to the characterization of a solid ('faces', 'edges') and therefore not suitable for the description of a plane figure. Maybe this misuse of terms is linked to specificity of the Pliox as a two-sided material object. Moreover, Luis' intervention highlights another lack of geometric knowledge. On the one hand, he announces an incorrect number of sides for the square (three instead of four), and on the other, he uses the term 'side' that is at odds with the designated graphic object (a vertex). This same semiotic dissonance will be repeated later (session 4) for another student who asserts that the square has four sides (a correct assertion), but points to the vertices to show which sides he's talking about. We can therefore conclude that, even if students do settle into a geometric language register to express properties of the square, the terms chosen are not always appropriate and the meaning given to the terms is not always adequate.

## Difficulties with right angles

In the next extract (4'26" to 7'03"), Katia points out that the square also has four right angles. She points to a side to indicate a right angle (figure 5, 4'52"), and answers the teacher that this can be checked with the ruler. Following the teacher's negative response, she immediately suggests the set square. She then positions the set square on the Pliox (figure 5, 5'28") and points to the vertex of the
<figure><img src="image_140.png" /><figcaption>Figure 3: Square and rhombus recognition (Mat, session 1)</figcaption></figure>

<figure><img src="image_141.png" /><figcaption>Figure 4: Spatial characterization of the square (Mat, session 1)</figcaption></figure>

Thus, the students' reasoning (right or wrong) to recognize this figure involves spatial knowledge without explicit reference to geometric properties of the square.

## Vocabulary and geometric knowledge

In the second extract (2'43" to 4'26"), which follows a disagreement between students about the name of the red figure, the teacher asks them to recall what they saw in session 2 about the square. One student talks about 'faces and sides', another about 'faces and edges'. The teacher discards these terms associated with solids. Luis says there are three sides in the square, but points to three of the square vertices on the Pliox. The teacher invalidates this proposal by asking one pupil to show the sides of this figure (the pupil first points to them, and, at the teacher's request, follows them with her finger), then another to show the vertices (which are pointed to). The teacher then summarizes the square characteristics: four sides and four vertices. In this extract, we identify the students' use of geometric terms specific to the characterization of a solid ('faces', 'edges') and therefore not suitable for the description of a plane figure. Maybe this misuse of terms is linked to specificity of the Pliox as a two-sided material object. Moreover, Luis' intervention highlights another lack of geometric knowledge. On the one hand, he announces an incorrect number of sides for the square (three instead of four), and on the other, he uses the term 'side' that is at odds with the designated graphic object (a vertex). This same semiotic dissonance will be repeated later (session 4) for another student who asserts that the square has four sides (a correct assertion), but points to the vertices to show which sides he's talking about. We can therefore conclude that, even if students do settle into a geometric language register to express properties of the square, the terms chosen are not always appropriate and the meaning given to the terms is not always adequate.

## Difficulties with right angles

In the next extract (4'26" to 7'03"), Katia points out that the square also has four right angles. She points to a side to indicate a right angle (figure 5, 4'52"), and answers the teacher that this can be checked with the ruler. Following the teacher's negative response, she immediately suggests the set square. She then positions the set square on the Pliox (figure 5, 5'28") and points to the vertex of the
right angle of the set square (figure 5, 5'29"), saying 'There's one there'. The teacher asks her to be more precise ('Position it well, look, you have to put it on the edge like we did yesterday'). Katia readjusts the set square and specifies that the angle is right. She then changes the orientation of the Pliox and quickly checks a second angle with the right positioning of the square (figure 5, 5'45").

<figure><img src="image_135.png" /><figcaption>Figure 5: Instrumented verification (Katia, session 3)</figcaption></figure>

Katia then struggles to position the set square to check a third angle. The teacher (T) comes to her help: she positions the tool herself and leaves it to Katia to say whether the angle is right or not (figure 6).

<figure><img src="image_136.png" /><figcaption>Figure 6: Positioning the set square (Katia and Teacher, session 3)</figcaption></figure>

Katia doesn't recognize that the angle is right-angled, so the teacher (T) helps her to spot it (figure 7). The teacher then asks Katia for the number of right angles. When she replies 'three', the teacher sets up the set square to check the fourth angle of the figure. Katia doesn't recognize it as right, so the teacher asserts that it is, relying on the verification work done the day before.

<figure><img src="image_137.png" /><figcaption>Figure 7: Checking the third right angle (Katia and Teacher, session 3)</figcaption></figure>

In this episode, Katia states a correct geometric property of the square – it has four right angles – but as in episode 2, we observe a semiotic dissonance between the terms used ('right angle') and the graphic object pointed to (side) (Figure 5, 4'52"). It's possible that Katia associates the qualifier 'right' with the straightness of the square's sides<fnref n="17" />, since she initially proposes the ruler as an instrument for verifying the property. Her remark 'And here too, it's straight', made as she follows with the finger

<footnote n="17">¹⁷ In French, the same word ('droit') means both 'straight' and 'right'.</footnote>
right angle of the set square (figure 5, 5'29"), saying 'There's one there'. The teacher asks her to be more precise ('Position it well, look, you have to put it on the edge like we did yesterday'). Katia readjusts the set square and specifies that the angle is right. She then changes the orientation of the Pliox and quickly checks a second angle with the right positioning of the square (figure 5, 5'45").

<figure><img src="image_142.png" /><figcaption>Figure 5: Instrumented verification (Katia, session 3)</figcaption></figure>

Katia then struggles to position the set square to check a third angle. The teacher (T) comes to her help: she positions the tool herself and leaves it to Katia to say whether the angle is right or not (figure 6).

<figure><img src="image_143.png" /><figcaption>Figure 6: Positioning the set square (Katia and Teacher, session 3)</figcaption></figure>

Katia doesn't recognize that the angle is right-angled, so the teacher (T) helps her to spot it (figure 7). The teacher then asks Katia for the number of right angles. When she replies 'three', the teacher sets up the set square to check the fourth angle of the figure. Katia doesn't recognize it as right, so the teacher asserts that it is, relying on the verification work done the day before.

<figure><img src="image_144.png" /><figcaption>Figure 7: Checking the third right angle (Katia and Teacher, session 3)</figcaption></figure>

In this episode, Katia states a correct geometric property of the square – it has four right angles – but as in episode 2, we observe a semiotic dissonance between the terms used ('right angle') and the graphic object pointed to (side) (Figure 5, 4'52"). It's possible that Katia associates the qualifier 'right' with the straightness of the square's sides<fnref n="17" />, since she initially proposes the ruler as an instrument for verifying the property. Her remark 'And here too, it's straight', made as she follows with the finger

<footnote n="17">¹⁷ In French, the same word ('droit') means both 'straight' and 'right'.</footnote>
along one side of the square (Figure 7, 6'18"), corroborates this hypothesis. It's also possible that 'straight' is used in the spatial sense of verticality, as Mat did in session 1 (Figure 4, 3'39"). Since the ruler as an instrument for verifying a right angle doesn't suit the teacher, Katia suggests the square. The use of this instrument leads her to look for right angles. Let's note that Katia points to the vertex of the angle (object 0D) to designate the right angle (object 2D) (Figure 5, 5'29"), as does the teacher (Figure 6, 6'01"). This assimilation of an angle to its vertex perhaps contributes to Katia's difficulty in positioning the set square correctly on the angle to be checked. For the first positioning, the teacher notes a lack of precision: the edges of the right angle of the set square are not adjusted on the sides of the angle to be checked, only the vertex of the right angle of the square is positioned on the vertex of the angle to be checked. We later note a lack of technical knowledge (Petitfour, 2016) for the third positioning: Katia places just one side of the right angle of the square on one side of the square (Figure 6, 5'51" and 5'56"). Furthermore, she does not correctly interpret the nature of an angle (right or not) when the square is correctly positioned, as we can see when checking the last two angles of the figure (Figure 7, 6'13").

## SYNTHESIS AND CONCLUSION

In this verbalization phase, we noted difficulties in recognizing a square in a non-prototypical situation. This finding resonates with the works of Battista (2007) and Marchand (2020). We observe that the reasoning (right or wrong) produced by the students is spontaneously based on spatial knowledge, perhaps encouraged by the artifact's manipulative possibilities. In our view, these spatial reasoning deserve to be discussed in class, or even institutionalized, as they contribute to the construction of geometric knowledge (Blanquart, 2023). Following on from the work of Berthelot and Salin (1995), this study highlights the importance of taking into account the interweaving of the spatial and the geometric knowledge. It also raises the question of the need to formulate spatial knowledge, while remaining vigilant to the polysemy of certain terms (such as the word 'droit' in French in the last episode, that means 'right' or 'straight'). In addition, the teacher's request to associate, through gestures, the oral and material or graphic designations of geometric objects enabled us to identify numerous semiotic dissonances. These show that the use of a geometric lexicon by students is no guarantee of the acquisition of geometric knowledge. This highlights the value of semiotic analysis for the researcher, and suggests the importance for the teacher of not focusing exclusively on oral language or vocabulary.

## Acknowledgements

The research reported here has grown in the research context of AMPIRIC (Innovation, Research and Teaching Cluster for Education, Aix-Marseille University), and, in particular, the J2MA project, financially supported by the *Caisse des Dépôts et des Consignations*.

## References

Arzarello, F. (2006). Semiosis as multimodal process. In L. Radford, & B. D'Amore (Eds.), *Sémiotique, culture et pensée mathématique, Número especial, Revista Latinoamerica de Investigación en Matemática Educativa* (pp. 267–299).

Battista, M. T. (2007). The development of geometric and spatial thinking. In F.K. Lester, Jr. (Ed.), *Second handbook of research on mathematics teaching and learning* (pp. 843-908). NCTM.
along one side of the square (Figure 7, 6'18"), corroborates this hypothesis. It's also possible that 'straight' is used in the spatial sense of verticality, as Mat did in session 1 (Figure 4, 3'39"). Since the ruler as an instrument for verifying a right angle doesn't suit the teacher, Katia suggests the square. The use of this instrument leads her to look for right angles. Let's note that Katia points to the vertex of the angle (object 0D) to designate the right angle (object 2D) (Figure 5, 5'29"), as does the teacher (Figure 6, 6'01"). This assimilation of an angle to its vertex perhaps contributes to Katia's difficulty in positioning the set square correctly on the angle to be checked. For the first positioning, the teacher notes a lack of precision: the edges of the right angle of the set square are not adjusted on the sides of the angle to be checked, only the vertex of the right angle of the square is positioned on the vertex of the angle to be checked. We later note a lack of technical knowledge (Petitfour, 2016) for the third positioning: Katia places just one side of the right angle of the square on one side of the square (Figure 6, 5'51" and 5'56"). Furthermore, she does not correctly interpret the nature of an angle (right or not) when the square is correctly positioned, as we can see when checking the last two angles of the figure (Figure 7, 6'13").

## SYNTHESIS AND CONCLUSION

In this verbalization phase, we noted difficulties in recognizing a square in a non-prototypical situation. This finding resonates with the works of Battista (2007) and Marchand (2020). We observe that the reasoning (right or wrong) produced by the students is spontaneously based on spatial knowledge, perhaps encouraged by the artifact's manipulative possibilities. In our view, these spatial reasoning deserve to be discussed in class, or even institutionalized, as they contribute to the construction of geometric knowledge (Blanquart, 2023). Following on from the work of Berthelot and Salin (1995), this study highlights the importance of taking into account the interweaving of the spatial and the geometric knowledge. It also raises the question of the need to formulate spatial knowledge, while remaining vigilant to the polysemy of certain terms (such as the word 'droit' in French in the last episode, that means 'right' or 'straight'). In addition, the teacher's request to associate, through gestures, the oral and material or graphic designations of geometric objects enabled us to identify numerous semiotic dissonances. These show that the use of a geometric lexicon by students is no guarantee of the acquisition of geometric knowledge. This highlights the value of semiotic analysis for the researcher, and suggests the importance for the teacher of not focusing exclusively on oral language or vocabulary.

## Acknowledgements

The research reported here has grown in the research context of AMPIRIC (Innovation, Research and Teaching Cluster for Education, Aix-Marseille University), and, in particular, the J2MA project, financially supported by the *Caisse des Dépôts et des Consignations*.

## References

Arzarello, F. (2006). Semiosis as multimodal process. In L. Radford, & B. D'Amore (Eds.), *Sémiotique, culture et pensée mathématique, Número especial, Revista Latinoamerica de Investigación en Matemática Educativa* (pp. 267–299).

Battista, M. T. (2007). The development of geometric and spatial thinking. In F.K. Lester, Jr. (Ed.), *Second handbook of research on mathematics teaching and learning* (pp. 843-908). NCTM.
Berthelot, R., & Salin, M.H. (1995). The role of pupils'space experience in the elementary teaching of geometry. In C. Mammana (Ed.). *Pre-proceedings of the ICMI Study Perspectives on the teaching of geometry for the 21st century* (pp. 23-27). University of Catania.

Blanquart, S. (2020). *Raisonnements géométriques d'élèves de cycle 3, duos de situations, rôle de l'enseignant*. (Thèse de doctorat). Université de Paris.

Blanquart, S. (2023). Geometric reasoning in grades 4 to 6, the teacher's role: methodological overview and results. In C. Guille-Biel Winder, & T. Assude (Eds.), *Articulations between tangible space, graphical space and geometrical space, ressources practices and training* (pp. 221-230). Iste Wiley.

Brousseau, G. (1997). *Theory of didactical situations in mathematics. Didactique des mathématiques 1970 - 1990*. Kluwer Academic Publishers.

Duval, R. (1999). Representation, Vision and Vizualisation: cognitive functions in mathematical thinking. Basic issues fore learning. In *Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education* (pp. 3-26).

Duval, R., Godin, M., & Perrin-Glorian, M. J. (2005). Reproduction de figures à l'école élémentaire. In C. Castela, & C. Houdement (Eds.), *Actes du séminaire national de didactique des mathématiques 2004* (pp. 5-89). Irem de Paris 7.

Guille-Biel Winder C. (2014). Étude d'usne situation de reproduction de figures par pliage en cycle 2: le PLIOX. *Annales de Didactique et de Sciences Cognitives, 19*, 103-128.

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. *Recherches en Didactique des Mathématiques, 41*(1), 55-96.

Houdement, C., & Kuzniak, A. (2006). Paradigmes géométriques et enseignement de la géométrie. *Annales de Didactique et Sciences Cognitives, 11*, 175-193.

Leutenegger, F. (2009). *Le temps d'instruire. Approche clinique et expérimentale du didactique ordinaire en mathématiques*. Peter Lang.

Marchand, P. (2020). Quelques assises pour valoriser le développement des connaissances spatiales à l'école primaire. *Recherches en Didactique des Mathématiques, 40*(2), 135-178.

Petitfour, E. (2016). Teaching geometry to visual-spatial dyspraxic pupils. In Paper presented at the 13th international congress on mathematical education. https://hal.archives-ouvertes.fr/hal-02119848/document.

Petitfour, E., & Houdement, C. (2022). The didactic effects of semiotic microphenomena in mathematics. In C. Houdement, C. de Hosson, & C. Hache (Eds.), *Semiotic approaches in science didactics* (pp. 209-246). ISTE Wiley.

Rabardel, P. (1995). *Les hommes et les technologies. Approche cognitive des instruments contemporains*. Armand Colin.

Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. *Educational Studies in Mathematics, 70*(2), 111-126.

Vygotsky, L. (1978). *Mind in society: the development of higher psychological processes*. Harvard University Press.
Berthelot, R., & Salin, M.H. (1995). The role of pupils'space experience in the elementary teaching of geometry. In C. Mammana (Ed.). *Pre-proceedings of the ICMI Study Perspectives on the teaching of geometry for the 21st century* (pp. 23-27). University of Catania.

Blanquart, S. (2020). *Raisonnements géométriques d'élèves de cycle 3, duos de situations, rôle de l'enseignant.* (Thèse de doctorat). Université de Paris.

Blanquart, S. (2023). Geometric reasoning in grades 4 to 6, the teacher's role: methodological overview and results. In C. Guille-Biel Winder, & T. Assude (Eds.), *Articulations between tangible space, graphical space and geometrical space, ressources practices and training* (pp. 221-230). Iste Wiley.

Brousseau, G. (1997). *Theory of didactical situations in mathematics. Didactique des mathématiques 1970 - 1990*. Kluwer Academic Publishers.

Duval, R. (1999). Representation, Vision and Vizualisation: cognitive functions in mathematical thinking. Basic issues fore learning. In *Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education* (pp. 3-26).

Duval, R., Godin, M., & Perrin-Glorian, M. J. (2005). Reproduction de figures à l'école élémentaire. In C. Castela, & C. Houdement (Eds.), *Actes du séminaire national de didactique des mathématiques 2004* (pp. 5-89). Irem de Paris 7.

Guille-Biel Winder C. (2014). Étude d'usne situation de reproduction de figures par pliage en cycle 2: le PLIOX. *Annales de Didactique et de Sciences Cognitives, 19*, 103–128.

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. *Recherches en Didactique des Mathématiques, 41*(1), 55–96.

Houdement, C., & Kuzniak, A. (2006). Paradigmes géométriques et enseignement de la géométrie. *Annales de Didactique et Sciences Cognitives, 11*, 175–193.

Leutenegger, F. (2009). *Le temps d'instruire. Approche clinique et expérimentale du didactique ordinaire en mathématiques*. Peter Lang.

Marchand, P. (2020). Quelques assises pour valoriser le développement des connaissances spatiales à l'école primaire. *Recherches en Didactique des Mathématiques, 40*(2), 135–178.

Petitfour, E. (2016). Teaching geometry to visual-spatial dyspraxic pupils. In Paper presented at the 13th international congress on mathematical education. https://hal.archives-ouvertes.fr/hal-02119848/document.

Petitfour, E., & Houdement, C. (2022). The didactic effects of semiotic microphenomena in mathematics. In C. Houdement, C. de Hosson, & C. Hache (Eds.), *Semiotic approaches in science didactics* (pp. 209–246). ISTE Wiley.

Rabardel, P. (1995). *Les hommes et les technologies. Approche cognitive des instruments contemporains*. Armand Colin.

Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. *Educational Studies in Mathematics, 70*(2), 111–126.

Vygotsky, L. (1978). *Mind in society: the development of higher psychological processes*. Harvard University Press.
# THE DIGITAL INTERACTIVE MATHEMATICAL MAP FOR GEOMETRY

Matthias Brandl¹, Urs Hackstein¹, Mirela Vinerean², Yvonne Liljekvist²

¹ University of Passau, Faculty of Computer Science and Mathematics, Didactics of Mathematics, Passau, Germany, ² Karlstad University, Faculty of Health, Science and Technology, Department of Mathematics and Computer Science, Karlstad, Sweden

We present results from the design-based research and development process of a digital didactical tool connecting secondary and tertiary mathematics education among others by historical aspects and elements from Mathematics, Didactics, History, Literature and Technology especially for prospective mathematics teachers: Digital Interactive Mathematical Maps (DIMM), developed at the University of Passau, Germany. The DIMM for Geometry was integrated into a second semester course on Euclidean and non-Euclidean geometry in mathematics education at Karlstad University, Sweden, designed collaboratively by university mathematics lecturers and researchers in mathematics education from the cooperating universities. While, according to the results of evaluation, the (perceived) usefulness of individual functionalities of the digital tool was confirmed, the use of the DIMM in the courses seems to promote favorable beliefs about mathematics, too.

## DIGITAL INTERACTIVE MATHEMATICAL MAPS (DIMM)

In this article, we present a new digital tool for teaching geometry and other areas of mathematics. Within this tool we combine a historical-genetic didactical approach with a technological tool for the visualization of historical and biographical aspects of mathematics and mathematicians, respectively (Brandl, 2009; Brandl & Vinerean, 2023; Datzmann et al., 2020; Przybilla et al., 2022; Vinerean et al., in press), to support students with a meaningful connection regarding the development of mathematical knowledge by “the trials, the triumphs, the tribulations” (I.I. Rabi in Holton et al., 1970, as cited in Kubli, 2002, p. 57) of the people behind this science. One of the most important things is here the humanistic element in the learning process: “Portraying scientists as human beings and giving students the opportunity to become effectively involved in the story of science are worthy goals in themselves” (Klassen, 2006, p. 51), whereas in many common school books “only the bare decontextualized scientific facts, theories, and laws along with the exemplar problems that demonstrate them” (ibid., p. 33) are presented. The motivation is therefore to overcome historically developed fragmentation of mathematical contents in curriculum at higher secondary schools and to offer an integration of the world of higher mathematics in the sense of Felix Klein (Klein, 2016/1924: “double discontinuity”), which is still a problem today (Hefendehl-Hebeker, 2013; Pinto & Cooper, 2022; Winsløw & Grønbæk, 2014). The double discontinuity implies that teachers often base their school teaching on their own pre-university experiences and do not rely on their academic mathematics knowledge gained at university (Bauer & Partheil, 2009; Hefendehl-Hebeker, 2013). Mathematics teaching becomes therefore fragmented, and students have fewer opportunities to notice connections and develop conceptual understanding (Winsløw & Grønbæk, 2014). To overcome Klein's double discontinuity, students are required to notice connections between the ‘axiomatic-
# THE DIGITAL INTERACTIVE MATHEMATICAL MAP FOR GEOMETRY

Matthias Brandl¹, Urs Hackstein¹, Mirela Vinerean², Yvonne Liljekvist²

¹ University of Passau, Faculty of Computer Science and Mathematics, Didactics of Mathematics, Passau, Germany, ² Karlstad University, Faculty of Health, Science and Technology, Department of Mathematics and Computer Science, Karlstad, Sweden

We present results from the design-based research and development process of a digital didactical tool connecting secondary and tertiary mathematics education among others by historical aspects and elements from Mathematics, Didactics, History, Literature and Technology especially for prospective mathematics teachers: Digital Interactive Mathematical Maps (DIMM), developed at the University of Passau, Germany. The DIMM for Geometry was integrated into a second semester course on Euclidean and non-Euclidean geometry in mathematics education at Karlstad University, Sweden, designed collaboratively by university mathematics lecturers and researchers in mathematics education from the cooperating universities. While, according to the results of evaluation, the (perceived) usefulness of individual functionalities of the digital tool was confirmed, the use of the DIMM in the courses seems to promote favorable beliefs about mathematics, too.

## DIGITAL INTERACTIVE MATHEMATICAL MAPS (DIMM)

In this article, we present a new digital tool for teaching geometry and other areas of mathematics. Within this tool we combine a historical-genetic didactical approach with a technological tool for the visualization of historical and biographical aspects of mathematics and mathematicians, respectively (Brandl, 2009; Brandl & Vinerean, 2023; Datzmann et al., 2020; Przybilla et al., 2022; Vinerean et al., in press), to support students with a meaningful connection regarding the development of mathematical knowledge by “the trials, the triumphs, the tribulations” (I.I. Rabi in Holton et al., 1970, as cited in Kubli, 2002, p. 57) of the people behind this science. One of the most important things is here the humanistic element in the learning process: “Portraying scientists as human beings and giving students the opportunity to become effectively involved in the story of science are worthy goals in themselves” (Klassen, 2006, p. 51), whereas in many common school books “only the bare decontextualized scientific facts, theories, and laws along with the exemplar problems that demonstrate them” (ibid., p. 33) are presented. The motivation is therefore to overcome historically developed fragmentation of mathematical contents in curriculum at higher secondary schools and to offer an integration of the world of higher mathematics in the sense of Felix Klein (Klein, 2016/1924: “double discontinuity”), which is still a problem today (Hefendehl-Hebeker, 2013; Pinto & Cooper, 2022; Winsløw & Grønbæk, 2014). The double discontinuity implies that teachers often base their school teaching on their own pre-university experiences and do not rely on their academic mathematics knowledge gained at university (Bauer & Partheil, 2009; Hefendehl-Hebeker, 2013). Mathematics teaching becomes therefore fragmented, and students have fewer opportunities to notice connections and develop conceptual understanding (Winsløw & Grønbæk, 2014). To overcome Klein's double discontinuity, students are required to notice connections between the ‘axiomatic-
formal' world of university mathematics and the 'perceptual-symbolic' or 'conceptual-embodied'
world of school mathematics (cf. Tall, 2008).

The digital tool “Digital Interactive Mathematical Maps” (DIMM) that we present here is based on first ideas in Brandl (2009) and has been developed at the professorship for Didactics of Mathematics at the University of Passau (Germany). It was tested in different teaching and learning contexts at Karlstad University (Sweden) and University of Passau. The DIMM are freely accessible<fnref n="18" /> and offer a visual representation of the historical development of mathematical knowledge in time as well as the interrelation of different knowledge items according to each other. The core of the maps are three-dimensional nets, starting from one initial problem and expanding like a tree from node to node. The nodes represent pieces of mathematical knowledge, often a famous discovery resulting in a central theorem (see Figure 1). One dimension of the maps represents time, while the other two dimensions represent inner-mathematical dependency and relationship. The relationship between the nodes is computed by an algorithm from graph theory using certain weights given when putting a new content into the database (for details e.g. see Przybilla et al., 2021, 2022). Clicking on a node opens the corresponding content on a linked timeline, where milestones of the considered mathematical area are clearly and consecutively displayed (see e.g. Figure 10 in Brandl & Vinerean, 2023).

<figure><img src="image_138.png" /><figcaption>Figure 6: Screenshot of the Geometry Map (14 September 2023) and preview of a node</figcaption></figure>

Functionalities of the DIMM

Each node of the map or element of the timeline, respectively, comprises information about the
mathematical discovery and the person responsible for that, flanked by providing biographical and
factual information via texts, links to further publicly available sites and free videos. Where feasible
and suitable, these links to webpages contain besides instructive texts to go deeper in the material
also instructive tasks or problems from well-known mathematics competitions such as
Bundeswettbewerb Mathematik and Landeswettbewerb Mathematik Bayern from Germany or the
European Girls' Mathematical Olympiad. These contents open up the possibility to use the DIMM
also in courses for gifted students and for the preparation for mathematical competitions.

<footnote n="18">¹⁸ The DIMM are freely accessible under the address https://math-map.fim.uni-passau.de.</footnote>
formal' world of university mathematics and the 'perceptual-symbolic' or 'conceptual-embodied'
world of school mathematics (cf. Tall, 2008).

The digital tool “Digital Interactive Mathematical Maps” (DIMM) that we present here is based on first ideas in Brandl (2009) and has been developed at the professorship for Didactics of Mathematics at the University of Passau (Germany). It was tested in different teaching and learning contexts at Karlstad University (Sweden) and University of Passau. The DIMM are freely accessible<fnref n="18" /> and offer a visual representation of the historical development of mathematical knowledge in time as well as the interrelation of different knowledge items according to each other. The core of the maps are three-dimensional nets, starting from one initial problem and expanding like a tree from node to node. The nodes represent pieces of mathematical knowledge, often a famous discovery resulting in a central theorem (see Figure 1). One dimension of the maps represents time, while the other two dimensions represent inner-mathematical dependency and relationship. The relationship between the nodes is computed by an algorithm from graph theory using certain weights given when putting a new content into the database (for details e.g. see Przybilla et al., 2021, 2022). Clicking on a node opens the corresponding content on a linked timeline, where milestones of the considered mathematical area are clearly and consecutively displayed (see e.g. Figure 10 in Brandl & Vinerean, 2023).

<figure><img src="image_145.png" /><figcaption>Figure 6: Screenshot of the Geometry Map (14 September 2023) and preview of a node</figcaption></figure>

Functionalities of the DIMM

Each node of the map or element of the timeline, respectively, comprises information about the
mathematical discovery and the person responsible for that, flanked by providing biographical and
factual information via texts, links to further publicly available sites and free videos. Where feasible
and suitable, these links to webpages contain besides instructive texts to go deeper in the material
also instructive tasks or problems from well-known mathematics competitions such as
Bundeswettbewerb Mathematik and Landeswettbewerb Mathematik Bayern from Germany or the
European Girls' Mathematical Olympiad. These contents open up the possibility to use the DIMM
also in courses for gifted students and for the preparation for mathematical competitions.

<footnote n="18"><sup>18</sup> The DIMM are freely accessible under the address https://math-map.fim.uni-passau.de.</footnote>
The content elements of the map or the timeline, respectively, can be collected by a shopping cart system and exported for further use in lessons or tasks. Other functionalities are so-called “horizontal cuts” and “vertical cuts”. A “horizontal cut” is a projection into the plane of all nodes selected in a certain period of time and possibly filtered by a specific topic. It allows for an optimal visualization of inner-mathematical relationships between the different content-nodes (Figure 2 right). Contrarily, flanking this functionality is the “vertical cut” mechanism resulting in a reduction of all nodes to those representing the historical development of the mathematical idea or result in form of a two-dimensional directed graph (Figure 2 left). Clicking on the nodes then allows for an efficient and meaningful connected way through the genesis of the mathematical aspect and its explorers. Furthermore, a summary of all nodes being part of the evolutionary line can be created and downloaded, in order to be distributed to other learners and worked with in offline mode.

<figure><img src="image_139.png" /><figcaption>Figure 7: Exemplary vertical (left) and horizontal (right) cut (14 September 2023)</figcaption></figure>

## THE DIMM IN GEOMETRY EDUCATION

There are several possibilities to use the DIMM in teaching. One possibility is their use in combination with elements from narrative didactics as the offered content elements of the map nodes serve as providers of basis bricks for the construction of motivational narratives for school lessons (see Brandl & Vinerean, 2023). Especially, the vertical cut function of the map can help to create narratives covering several points in time. The “main goal of the vertical cut is to promote an accurate picture of mathematics as an *emerging science*. It is a consequence of its very nature that mistakes and misunderstandings can and probably will occur when *doing* or *creating* mathematics.” (Przybilla et al., 2022, p. 4796). A further possibility offers the integration of competition tasks for a certain amount of nodes. By a way through the map as described in Table 1, mathematically gifted students can work on the tasks in a manner that on the one hand fits their special needs and on the
The content elements of the map or the timeline, respectively, can be collected by a shopping cart system and exported for further use in lessons or tasks. Other functionalities are so-called “horizontal cuts” and “vertical cuts”. A “horizontal cut” is a projection into the plane of all nodes selected in a certain period of time and possibly filtered by a specific topic. It allows for an optimal visualization of inner-mathematical relationships between the different content-nodes (Figure 2 right). Contrarily, flanking this functionality is the “vertical cut” mechanism resulting in a reduction of all nodes to those representing the historical development of the mathematical idea or result in form of a two-dimensional directed graph (Figure 2 left). Clicking on the nodes then allows for an efficient and meaningful connected way through the genesis of the mathematical aspect and its explorers. Furthermore, a summary of all nodes being part of the evolutionary line can be created and downloaded, in order to be distributed to other learners and worked with in offline mode.

<figure><img src="image_146.png" /><figcaption>Figure 7: Exemplary vertical (left) and horizontal (right) cut (14 September 2023)</figcaption></figure>

## THE DIMM IN GEOMETRY EDUCATION

There are several possibilities to use the DIMM in teaching. One possibility is their use in combination with elements from narrative didactics as the offered content elements of the map nodes serve as providers of basis bricks for the construction of motivational narratives for school lessons (see Brandl & Vinerean, 2023). Especially, the vertical cut function of the map can help to create narratives covering several points in time. The “main goal of the vertical cut is to promote an accurate picture of mathematics as an *emerging science*. It is a consequence of its very nature that mistakes and misunderstandings can and probably will occur when *doing* or *creating* mathematics.” (Przybilla et al., 2022, p. 4796). A further possibility offers the integration of competition tasks for a certain amount of nodes. By a way through the map as described in Table 1, mathematically gifted students can work on the tasks in a manner that on the one hand fits their special needs and on the
other hand connects different contents with each other via historical development (Brandl & Kaiser, 2022; Brandl & Szabo, 2019, 2023):

<table><caption>Table 1: Way through the Geometry DIMM with tasks from competitions (as presented in a similar version in Brandl & Szabo, 2023, as Table 1)</caption><thead><tr><td>Year</td><td>Mathematician</td><td>Mathematical Content</td></tr></thead><tbody><tr><td>90 AD</td><td>Menelaus</td><td>Theorem of Menelaus</td></tr><tr><td>320 BC</td><td>Euclid</td><td>Elements of Euclid</td></tr><tr><td>520 BC</td><td>Pythagoras</td><td>Congruence</td></tr><tr><td>580 BC</td><td>Thales</td><td>Intercept Theorem</td></tr><tr><td>1850 BC</td><td>Moscow Papyrus</td><td>Area of a triangle</td></tr></tbody></table>

## METHODOLOGICAL APPROACH: QUALITATIVE EVALUATION OF A GEOMETRY COURSE

In a second semester course on Euclidean and non-Euclidean geometry in mathematics education at Karlstad University (Sweden), the centuries-long discussion about Euclid's parallel postulate was picked as a suitable example, which also connects school mathematics (Euclidean geometry) with university mathematics (non-Euclidean geometry). The course was part of the second semester of the teacher-training program (300 ECTS credits) and could also be attended by teachers who wished to receive further mathematical training. Mathematics teachers and researchers of mathematics education from Karlstad University and the University of Passau had designed the components of the course cooperatively prior to the study (Brandl & Vinerean, 2023; Przybilla et al., 2022; Vinerean et al., in press).

The mathematical content was flanked by the necessity for fostering competencies with regard to the evaluation of digital learning tools. In sum, 44 students (27 teachers & 17 prospective teachers) actively took part in the course. Weekly lectures were followed up by weekly work assignments<fnref n="19" />. They concerned the timeline (TL), narrative didactics as an area of application of the contents in the map, the three-dimensional mathematical map (3D Map), the vertical cut (VC), the horizontal cut (HC), the discussion around the parallel postulates, and a general evaluation of the didactical tool (cf. ibid.).

For the technical evaluation of the tool we used the technology acceptance model (TAM) of Davis (1985), which is widely used in the community and well supported both theoretically and empirically (cf. Scherer et al., 2019). The goal was to analyze students' answers to questions on the perceived usefulness and ease of use of the technology (cf. Przybilla et al., 2022; Vinerean et al., in press). Therefore we added quizzes in connection with the weekly assignments concerning the functionalities of the map. Using the TAM as described in Przybilla et al. (2022), we addressed two major beliefs - *perceived usefulness* and *perceived ease of use* - and the fact that the design features in the tool (in

<footnote n="19">¹⁹ The weekly assignments are freely available at https://docs.google.com/document/d/e/2PACX-1vTq_w6A2MqresPr5PQLGujDt3NIS6i7aiL7ShsmaR841TtI-hrsyTM2zLY6UEhbULBumIx0zEtTOz6V/pub.</footnote>
other hand connects different contents with each other via historical development (Brandl & Kaiser, 2022; Brandl & Szabo, 2019, 2023):

<table><caption>Table 1: Way through the Geometry DIMM with tasks from competitions (as presented in a similar version in Brandl & Szabo, 2023, as Table 1)</caption><thead><tr><td>Year</td><td>Mathematician</td><td>Mathematical Content</td></tr></thead><tbody><tr><td>90 AD</td><td>Menelaus</td><td>Theorem of Menelaus</td></tr><tr><td>320 BC</td><td>Euclid</td><td>Elements of Euclid</td></tr><tr><td>520 BC</td><td>Pythagoras</td><td>Congruence</td></tr><tr><td>580 BC</td><td>Thales</td><td>Intercept Theorem</td></tr><tr><td>1850 BC</td><td>Moscow Papyrus</td><td>Area of a triangle</td></tr></tbody></table>

## METHODOLOGICAL APPROACH: QUALITATIVE EVALUATION OF A GEOMETRY COURSE

In a second semester course on Euclidean and non-Euclidean geometry in mathematics education at Karlstad University (Sweden), the centuries-long discussion about Euclid's parallel postulate was picked as a suitable example, which also connects school mathematics (Euclidean geometry) with university mathematics (non-Euclidean geometry). The course was part of the second semester of the teacher-training program (300 ECTS credits) and could also be attended by teachers who wished to receive further mathematical training. Mathematics teachers and researchers of mathematics education from Karlstad University and the University of Passau had designed the components of the course cooperatively prior to the study (Brandl & Vinerean, 2023; Przybilla et al., 2022; Vinerean et al., in press).

The mathematical content was flanked by the necessity for fostering competencies with regard to the evaluation of digital learning tools. In sum, 44 students (27 teachers & 17 prospective teachers) actively took part in the course. Weekly lectures were followed up by weekly work assignments<fnref n="19" />. They concerned the timeline (TL), narrative didactics as an area of application of the contents in the map, the three-dimensional mathematical map (3D Map), the vertical cut (VC), the horizontal cut (HC), the discussion around the parallel postulates, and a general evaluation of the didactical tool (cf. ibid.).

For the technical evaluation of the tool we used the technology acceptance model (TAM) of Davis (1985), which is widely used in the community and well supported both theoretically and empirically (cf. Scherer et al., 2019). The goal was to analyze students' answers to questions on the perceived usefulness and ease of use of the technology (cf. Przybilla et al., 2022; Vinerean et al., in press). Therefore we added quizzes in connection with the weekly assignments concerning the functionalities of the map. Using the TAM as described in Przybilla et al. (2022), we addressed two major beliefs - *perceived usefulness* and *perceived ease of use* - and the fact that the design features in the tool (in

<footnote n="19">¹⁹ The weekly assignments are freely available at https://docs.google.com/document/d/e/2PACX-1vTq_w6A2MqresPr5PQLGujDt3NIS6i7aiL7ShsmaR841TtI-hrsyTM2zLY6UEhbULBumIx0zEtTOz6V/pub.</footnote>
our case, the individual components and functionalities of the map: 3D Map, TL, VC and HC) directly
influence the two beliefs (Figure 3).

Regarding the evaluation of elements from narrative didactics, the methodological procedures are
explained in detail in Brandl & Vinerean (2023, pp. 9 – 10) being executed in two ways:

A) Formulation of a short historical-orientated narrative motivation for a free-of-choice topic from the school curriculum by using information provided in the nodes or the timeline of the Digital Interactive Mathematical Map

B) Description of the historical evolvement of non-Euclidean geometry from Euclidean geometry by using information provided in the nodes or the timeline of the Digital Interactive Mathematical Map

<figure><img src="image_140.png" /><figcaption>Figure 3: Adaption of the TAM (Davis, 1985) for the technical evaluation of the DIMM</figcaption></figure>

RESULTS

The students largely agreed or strongly agreed with the *perceived ease of use* (cf. Table 2 in Vinerean et al., in press): 85% considered that the timeline is simple to use, 77% that the selection function is useful, 74% that the preview function helps to navigate through the map, and 84% that bookmarking videos is useful.

Similarly, the students largely agreed or strongly agreed with the *perceived usefulness of the tool* both as students and as future teachers: 85% confirmed that they use the search function and the filter function to find a node in the map that interest them. In addition, 93% of the students agreed or strongly agreed that the ‘filter date’ functionality helps them to get deeper insights about the development of mathematics and the circumstances of this development, and 75% that they can imagine themselves in the future, when learning new content at university, informing themselves about its origins and development. However, 22% could not decide on the last statement, suggesting that one intervention might not be enough to make a real change in the attitude of the students.
our case, the individual components and functionalities of the map: 3D Map, TL, VC and HC) directly
influence the two beliefs (Figure 3).

Regarding the evaluation of elements from narrative didactics, the methodological procedures are
explained in detail in Brandl & Vinerean (2023, pp. 9 – 10) being executed in two ways:

A) Formulation of a short historical-orientated narrative motivation for a free-of-choice topic from the school curriculum by using information provided in the nodes or the timeline of the Digital Interactive Mathematical Map

B) Description of the historical evolvement of non-Euclidean geometry from Euclidean geometry by using information provided in the nodes or the timeline of the Digital Interactive Mathematical Map

<figure><img src="image_147.png" /><figcaption>Figure 3: Adaption of the TAM (Davis, 1985) for the technical evaluation of the DIMM</figcaption></figure>

RESULTS

The students largely agreed or strongly agreed with the *perceived ease of use* (cf. Table 2 in Vinerean et al., in press): 85% considered that the timeline is simple to use, 77% that the selection function is useful, 74% that the preview function helps to navigate through the map, and 84% that bookmarking videos is useful.

Similarly, the students largely agreed or strongly agreed with the *perceived usefulness of the tool* both as students and as future teachers: 85% confirmed that they use the search function and the filter function to find a node in the map that interest them. In addition, 93% of the students agreed or strongly agreed that the ‘filter date’ functionality helps them to get deeper insights about the development of mathematics and the circumstances of this development, and 75% that they can imagine themselves in the future, when learning new content at university, informing themselves about its origins and development. However, 22% could not decide on the last statement, suggesting that one intervention might not be enough to make a real change in the attitude of the students.
When asked in their role as future teachers, 91% of the students agreed or strongly agreed that they see the vertical cut functionality as a useful and clear way to show historical developments, and 93% that they see the horizontal cut functionality as a useful and clear way to show thematical connections. In addition, 82% confirmed that they think that knowledge of the historical lines of development of mathematical concepts will help them in their future profession as a teacher. However, 18% could not decide on the last statement, leaving room for improvement in the way the map influence them in this direction. Additionally, we cite an excerpt of the conclusion from Vinerean et al. (in press):

> Overall, few of the students (6%) agreed or strongly agreed that they had previously experienced the historical-genetic principle (i.e., before the course). Of the students, 22% agreed or strongly agreed that they had experienced the teaching of mathematics at the university as oriented on the historical-genetic principle. [...] Overall, the students largely agreed or strongly agreed with the perceived usefulness of the tool both as students and as (becoming) teachers.

Regarding the analysis of the evaluation part concerning narrative didactics we decided on an approach via a single-case study and a qualitative analysis of the text. Detailed results can be found in Brandl & Vinerean (2023), where also an exemplary excerpt from a student's assignment and answer to the feedback possibilities on the narrative didactics aspect is presented in order to illustrate successful and not successful use of the desired elements. "Student 5" gives a short introduction concerning the motivation and intention to use narrative elements and pieces of biographies collected with the help of the map in an imaginary mathematics lesson (ibid., p. 10, translated from Swedish to English by Brandl and Vinerean):

> To introduce 4-5 graders to coordinates and the number plane I think that the story about Descartes seeing the flies on the ceiling can really help to visualize and understand it. By now, the students should already be familiar with the number line. To expand on that knowledge I will tell a very abridged version of the story of Descartes and tie it to previous discoveries by other people.

A further result of a qualitative content analysis of the students' answers indicated the following with respect to influence on beliefs related to the nature of mathematics (Vinerean et al., 2023, p. 575):

> [T]he map seems to open up for viewing mathematics as an emerging science. [...] Dynamic understanding of mathematics can be seen in, for instance, students' reasoning about the struggle when mathematicians develop mathematical concepts. [...] Most students show a process-related and application-related orientation in relation to their future role as teachers when using the map. [...] Overall, students' reflections suggest that the use of the digital tool Interactive Mathematical Map(s) promotes favorable beliefs related to the nature of mathematics.

Finally, we present some results concerning students' experience with the combination of the two scaffolding tools narrative didactics and Digital Interactive Mathematical Map (with emphases on narrative didactics) from the evaluating quizzes in Przybilla et al. (2022), Brandl & Vinerean (2023) and Vinerean et al. (in press): Almost 80% of the students found the contents in the map useful to write a narrative motivation that could be used in their teaching; only half of the students had experienced teaching with narrative elements before. Moreover, 95% of the students would like to incorporate narrative elements in their future teaching, and almost 70% of the students would appreciate narrative approaches in university courses. In addition, students find it "rewarding [...] to be able to produce a text oneself!", and most of the students found the information in the map relevant and easy to use for the assignment purpose and for their own understanding.
When asked in their role as future teachers, 91% of the students agreed or strongly agreed that they see the vertical cut functionality as a useful and clear way to show historical developments, and 93% that they see the horizontal cut functionality as a useful and clear way to show thematical connections. In addition, 82% confirmed that they think that knowledge of the historical lines of development of mathematical concepts will help them in their future profession as a teacher. However, 18% could not decide on the last statement, leaving room for improvement in the way the map influence them in this direction. Additionally, we cite an excerpt of the conclusion from Vinerean et al. (in press):

> Overall, few of the students (6%) agreed or strongly agreed that they had previously experienced the historical-genetic principle (i.e., before the course). Of the students, 22% agreed or strongly agreed that they had experienced the teaching of mathematics at the university as oriented on the historical-genetic principle. [...] Overall, the students largely agreed or strongly agreed with the perceived usefulness of the tool both as students and as (becoming) teachers.

Regarding the analysis of the evaluation part concerning narrative didactics we decided on an approach via a single-case study and a qualitative analysis of the text. Detailed results can be found in Brandl & Vinerean (2023), where also an exemplary excerpt from a student's assignment and answer to the feedback possibilities on the narrative didactics aspect is presented in order to illustrate successful and not successful use of the desired elements. "Student 5" gives a short introduction concerning the motivation and intention to use narrative elements and pieces of biographies collected with the help of the map in an imaginary mathematics lesson (ibid., p. 10, translated from Swedish to English by Brandl and Vinerean):

> To introduce 4-5 graders to coordinates and the number plane I think that the story about Descartes seeing the flies on the ceiling can really help to visualize and understand it. By now, the students should already be familiar with the number line. To expand on that knowledge I will tell a very abridged version of the story of Descartes and tie it to previous discoveries by other people.

A further result of a qualitative content analysis of the students' answers indicated the following with respect to influence on beliefs related to the nature of mathematics (Vinerean et al., 2023, p. 575):

> [T]he map seems to open up for viewing mathematics as an emerging science. [...] Dynamic understanding of mathematics can be seen in, for instance, students' reasoning about the struggle when mathematicians develop mathematical concepts. [...] Most students show a process-related and application-related orientation in relation to their future role as teachers when using the map. [...] Overall, students' reflections suggest that the use of the digital tool Interactive Mathematical Map(s) promotes favorable beliefs related to the nature of mathematics.

Finally, we present some results concerning students' experience with the combination of the two scaffolding tools narrative didactics and Digital Interactive Mathematical Map (with emphases on narrative didactics) from the evaluating quizzes in Przybilla et al. (2022), Brandl & Vinerean (2023) and Vinerean et al. (in press): Almost 80% of the students found the contents in the map useful to write a narrative motivation that could be used in their teaching; only half of the students had experienced teaching with narrative elements before. Moreover, 95% of the students would like to incorporate narrative elements in their future teaching, and almost 70% of the students would appreciate narrative approaches in university courses. In addition, students find it "rewarding [...] to be able to produce a text oneself!", and most of the students found the information in the map relevant and easy to use for the assignment purpose and for their own understanding.
## SUMMARY

To summarize, in this article, we present a digital tool, Digital Interactive Mathematical Maps (DIMM), for teaching geometry and other areas of mathematics. It brings together mathematics itself with didactical methods, historical events, narrative elements, and technological aspects. Both the logico-scientific and the narrative mode of teaching are addressed (Brandl & Vinerean, 2023; Bruner, 1986). The tool allows for more successful learning processes as it visualize connections between various contexts. The connections serve as anchor points in a scaffolding structure (Brandl & Vinerean, 2023). Evaluations of usefulness of the map and its bridging teaching concept showed to be promising when striving for a holistic design of mathematics lessons, that is, bringing together areas of science and arts, as well as dimensions of rational thought and emotional feelings in a learning process. Further, the map seemed to promote both a process-oriented and an application-oriented approach as well as favorable beliefs, such as mathematics being an emerging science promoting a view of *doing* mathematics, in which an open error culture can be established (Przybilla et al., 2022).

## Acknowledgement

This project is part of the “Qualitätsoffensive Lehrerbildung”, a joint initiative of the Federal Government and the Länder of Germany which aims to improve the quality of teacher training. The program is funded by the Federal Ministry of Education and Research (BMBF, funding reference 01JA1924). This project was also supported by the research center Science, Mathematics and Engineering Educational Research (SMEER) at Karlstad University.

## References

- Bauer, T., & Partheil, U. (2009). Schnittstellenmodule in der Lehramtsausbildung im Fach Mathematik. *Math. Semesterber.*, 56, 85–103.

- Brandl, M. (2009). The vibrating string – an initial problem for modern mathematics; historical and didactical aspects. In I. Witzke (Ed.), *Mathematical Practice and Development throughout History: Proceedings of the 18th Novembertagung on the History, Philosophy and Didactics of Mathematics* (pp. 95–114). Logos.

- Brandl, M., & Kaiser, T. (2022). Digital mathematical maps. In S. A. Chamberlin (Ed.), On the Road to Mathematical Expertise and Innovation, Proceedings of the 12th International Conference on Mathematical Creativity and Giftedness (MCG 12, September 25 – 28, 2022) (pp. 297–299), Conference Proceedings in Mathematics Education, 8. WTM. https://doi.org/10.37626/GA9783959872263.0

- Brandl, M., & Szabo, A. (2023, September). *Digital Learning Environments to Support Autonomous Learning Processes of Mathematically Creative and Gifted Students* [Paper presentation]. 26th International Conference on Interactive Collaborative Learning (ICL2023), Madrid, Spain.

- Brandl, M., & Vinerean, M. (2023). Narrative Didactics in Mathematics Education: Results from a University Geometry Course. *Open Education Studies*, 5(1), 20220186. https://doi.org/10.1515/edu-2022-0186

- Bruner, J. (1986). *Actual Minds, Possible Worlds*. Harvard University Press.

- Datzmann, A., Przybilla, J., Brandl, M., & Kaiser, T. (2020). New teaching techniques aiming to connect school and university mathematics in geometry. In A. Donevska-Todorova, E. Faggiano, J. Trgalova, Z. Lavicza, R. Weinhandl, A. Clark-Wilson & H.-G. Weigand (Eds.), *Proceedings of the Tenth ERME Topic Conference (ETC10) on Mathematics Education in the Digital Age (MEDA), 16-18 September 2020 in Linz, Austria* (pp. 37–44). Johannes Kepler University. https://hal.archives-ouvertes.fr/hal-02932218/document
## SUMMARY

To summarize, in this article, we present a digital tool, Digital Interactive Mathematical Maps (DIMM), for teaching geometry and other areas of mathematics. It brings together mathematics itself with didactical methods, historical events, narrative elements, and technological aspects. Both the logico-scientific and the narrative mode of teaching are addressed (Brandl & Vinerean, 2023; Bruner, 1986). The tool allows for more successful learning processes as it visualize connections between various contexts. The connections serve as anchor points in a scaffolding structure (Brandl & Vinerean, 2023). Evaluations of usefulness of the map and its bridging teaching concept showed to be promising when striving for a holistic design of mathematics lessons, that is, bringing together areas of science and arts, as well as dimensions of rational thought and emotional feelings in a learning process. Further, the map seemed to promote both a process-oriented and an application-oriented approach as well as favorable beliefs, such as mathematics being an emerging science promoting a view of *doing* mathematics, in which an open error culture can be established (Przybilla et al., 2022).

## Acknowledgement

This project is part of the “Qualitätsoffensive Lehrerbildung”, a joint initiative of the Federal Government and the Länder of Germany which aims to improve the quality of teacher training. The program is funded by the Federal Ministry of Education and Research (BMBF, funding reference 01JA1924). This project was also supported by the research center Science, Mathematics and Engineering Educational Research (SMEER) at Karlstad University.

## References

- Bauer, T., & Partheil, U. (2009). Schnittstellenmodule in der Lehramtsausbildung im Fach Mathematik. *Math. Semesterber.*, 56, 85–103.

- Brandl, M. (2009). The vibrating string – an initial problem for modern mathematics; historical and didactical aspects. In I. Witzke (Ed.), *Mathematical Practice and Development throughout History: Proceedings of the 18th Novembertagung on the History, Philosophy and Didactics of Mathematics* (pp. 95–114). Logos.

- Brandl, M., & Kaiser, T. (2022). Digital mathematical maps. In S. A. Chamberlin (Ed.), On the Road to Mathematical Expertise and Innovation, Proceedings of the 12th International Conference on Mathematical Creativity and Giftedness (MCG 12, September 25 – 28, 2022) (pp. 297–299), Conference Proceedings in Mathematics Education, 8. WTM. https://doi.org/10.37626/GA9783959872263.0

- Brandl, M., & Szabo, A. (2023, September). *Digital Learning Environments to Support Autonomous Learning Processes of Mathematically Creative and Gifted Students* [Paper presentation]. 26th International Conference on Interactive Collaborative Learning (ICL2023), Madrid, Spain.

- Brandl, M., & Vinerean, M. (2023). Narrative Didactics in Mathematics Education: Results from a University Geometry Course. *Open Education Studies*, 5(1), 20220186. https://doi.org/10.1515/edu-2022-0186

- Bruner, J. (1986). *Actual Minds, Possible Worlds*. Harvard University Press.

- Datzmann, A., Przybilla, J., Brandl, M., & Kaiser, T. (2020). New teaching techniques aiming to connect school and university mathematics in geometry. In A. Donevska-Todorova, E. Faggiano, J. Trgalova, Z. Lavicza, R. Weinhandl, A. Clark-Wilson & H.-G. Weigand (Eds.), *Proceedings of the Tenth ERME Topic Conference (ETC10) on Mathematics Education in the Digital Age (MEDA), 16-18 September 2020 in Linz, Austria* (pp. 37–44). Johannes Kepler University. https://hal.archives-ouvertes.fr/hal-02932218/document
Davis, F. D. (1985). *A technology acceptance model for empirically testing new end-user information systems: Theory and results* [Doctoral dissertation, Massachusetts Institute of Technology].
http://hdl.handle.net/1721.1/15192

Hefendehl-Hebeker, L. (2013). Doppelte Diskontinuität oder die Chance der Brückenschläge. In C. Ableitinger, J. Kramer, & S. Prediger (Eds.), *Zur doppelten Diskontinuität in der Gymnasiallehrerbildung. Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik* (pp. 1–15). Springer Spektrum. https://doi.org/10.1007/978-3-658-01360-8_1

Holton, G., Rutherford, J. & Watson, F. (1970). *The Project Physics Course*. Holt, Rinehart & Winston.

Klassen, S. (2006). A theoretical framework for contextual science teaching. *Interchange*, 37, 1–2, 31–61.

Klein, F. (2016). *Elementary Mathematics from higher standpoint. Volume I: Arithmetic Algebra Analysis*. (G. Schubring, Trans.) Springer. (Original work published 1924). https://doi.org/10.1007/978-3-662-49442-4

Kubli, F. (2002). *Plädoyer für Erzählungen im Physikunterricht: Geschichte und Geschichten als Verstehenshilfen – Ergebnisse einer Untersuchung* (2nd ed). Aulis Deubner.

Pinto, A., & Cooper, J. (2022). The road not taken – Investigating affordances of infinitesimal calculus for enriching the repertoire of secondary mathematics teachers. *International Journal of Research in Undergraduate Mathematics Education, 8*(2), 318–338. https://doi.org/10.1007/s40753-021-00161-w

Przybilla, J., Brandl, M., Vinerean, M., & Liljekvist, Y. (2021). Interactive Mathematical Maps – A contextualized way of meaningful Learning. In G. A. Nortvedt, N. F. Buchholtz, J. Fauskanger, F. Hreinsdóttir, M. Hähkiöniemi, B. E. Jessen, J. Kurvits, Y. Liljekvist, M. Misfeldt, M. Naalsund, H. K. Nilsen, G. Pálsdóttir, P. Portaankorva-Koivisto, J. Radišić, & A. Wernberg (Eds.), *Bringing Nordic mathematics education into the future. Preceedings of NORMA 20. The ninth Nordic Conference on Mathematics Education*. Oslo, 2021 (pp. 209–216). (Skrifter från Svensk Förening för MatematikDidaktisk Forskning; No. 14). Svensk förening för matematikdidaktisk forskning (SMDF).
https://www.uv.uio.no/ils/english/about/events/2021/norma/proceedings/

Przybilla, J., Brandl, M., Vinerean, M., & Liljekvist, Y. (2022). Digital mathematical maps – results from iterative research cycles. In J. Hodgen, E. Geraniou, G. Bolondi & F. Ferretti. (Eds.), *Proceedings of the Twelfth Congress of the European Society for Research in Mathematics Education (CERME12)* (pp. 4793–4800). ERME. https://hal.archives-ouvertes.fr/hal-03754749/

Scherer, R., Siddiq, F., & Tondeur, J. (2019). The technology acceptance model (TAM): A meta-analytic structural equation modeling approach to explaining teachers’ adoption of digital technology in education. *Computers & Education, 128*, 13–35. https://doi.org/10.1016/j.compedu.2018.09.009

Tall, D. (2008). The transition to formal thinking in mathematics. *Mathematics Education Research Journal*, 20(2), 5–24. https://doi.org/10.1007/BF03217474

Vinerean, M., Brandl, M., & Liljekvist, Y. (2023). Promoting favourable beliefs of prospective math teachers concerning the nature of mathematics by using Interactive Mathematical Maps. In M. Trigueros, B. Barquero, R. Hochmuth, & J. Peters (Eds.), *Proceedings of the Fourth conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022)* (pp. 574–575). Hannover: University of Hannover and INDRUM.
https://hal.science/INDRUM2022/public/INDRUM2022_Proceedings.pdf.

Vinerean, M., Liljekvist, Y., Brandl, M., & Przybilla, J. (in press). Didactical usefulness of Interactive Mathematical Maps - Designing activities supporting student teachers' learning. In M. Misfeldt, U. T. Jankvist, E. Geraniou, I. H. Hojsted, A. L. Tamborg, R. J. E. Coopman, & B. E. Jessen (Eds.), *Nordic Studies in Mathematics Education – Nordisk Matematikkdidaktikk, NOMAD – Thematic Issue Digital resources in mathematics education*. NCM.

Winsløw, C., & Grønbæk, N. (2014). Klein's double discontinuity revisited: contemporary challenges for universities preparing teachers to teach calculus. *Recherches en Didactique des Mathématiques*, 34(1), 59–86. https://www.researchgate.net/publication/243963587
Davis, F. D. (1985). *A technology acceptance model for empirically testing new end-user information systems: Theory and results* [Doctoral dissertation, Massachusetts Institute of Technology].
http://hdl.handle.net/1721.1/15192

Hefendehl-Hebeker, L. (2013). Doppelte Diskontinuität oder die Chance der Brückenschläge. In C. Ableitinger, J. Kramer, & S. Prediger (Eds.), *Zur doppelten Diskontinuität in der Gymnasiallehrerbildung. Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik* (pp. 1–15). Springer Spektrum. https://doi.org/10.1007/978-3-658-01360-8_1

Holton, G., Rutherford, J. & Watson, F. (1970). *The Project Physics Course*. Holt, Rinehart & Winston.

Klassen, S. (2006). A theoretical framework for contextual science teaching. *Interchange*, 37, 1–2, 31–61.

Klein, F. (2016). *Elementary Mathematics from higher standpoint. Volume I: Arithmetic Algebra Analysis*. (G. Schubring, Trans.) Springer. (Original work published 1924). https://doi.org/10.1007/978-3-662-49442-4

Kubli, F. (2002). *Plädoyer für Erzählungen im Physikunterricht: Geschichte und Geschichten als Verstehenshilfen – Ergebnisse einer Untersuchung* (2nd ed). Aulis Deubner.

Pinto, A., & Cooper, J. (2022). The road not taken – Investigating affordances of infinitesimal calculus for enriching the repertoire of secondary mathematics teachers. *International Journal of Research in Undergraduate Mathematics Education, 8*(2), 318–338. https://doi.org/10.1007/s40753-021-00161-w

Przybilla, J., Brandl, M., Vinerean, M., & Liljekvist, Y. (2021). Interactive Mathematical Maps – A contextualized way of meaningful Learning. In G. A. Nortvedt, N. F. Buchholtz, J. Fauskanger, F. Hreinsdóttir, M. Hähkiöniemi, B. E. Jessen, J. Kurvits, Y. Liljekvist, M. Misfeldt, M. Naalsund, H. K. Nilsen, G. Pálsdóttir, P. Portaankorva-Koivisto, J. Radišić, & A. Wernberg (Eds.), *Bringing Nordic mathematics education into the future. Preceedings of NORMA 20. The ninth Nordic Conference on Mathematics Education*. Oslo, 2021 (pp. 209–216). (Skrifter från Svensk Förening för MatematikDidaktisk Forskning; No. 14). Svensk förening för matematikdidaktisk forskning (SMDF). https://www.uv.uio.no/ils/english/about/events/2021/norma/proceedings/

Przybilla, J., Brandl, M., Vinerean, M., & Liljekvist, Y. (2022). Digital mathematical maps – results from iterative research cycles. In J. Hodgen, E. Geraniou, G. Bolondi & F. Ferretti. (Eds.), *Proceedings of the Twelfth Congress of the European Society for Research in Mathematics Education (CERME12)* (pp. 4793–4800). ERME. https://hal.archives-ouvertes.fr/hal-03754749/

Scherer, R., Siddiq, F., & Tondeur, J. (2019). The technology acceptance model (TAM): A meta-analytic structural equation modeling approach to explaining teachers’ adoption of digital technology in education. *Computers & Education*, 128, 13–35. https://doi.org/10.1016/j.compedu.2018.09.009

Tall, D. (2008). The transition to formal thinking in mathematics. *Mathematics Education Research Journal*, 20(2), 5–24. https://doi.org/10.1007/BF03217474

Vinerean, M., Brandl, M., & Liljekvist, Y. (2023). Promoting favourable beliefs of prospective math teachers concerning the nature of mathematics by using Interactive Mathematical Maps. In M. Trigueros, B. Barquero, R. Hochmuth, & J. Peters (Eds.), *Proceedings of the Fourth conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022)* (pp. 574–575). Hannover: University of Hannover and INDRUM. https://hal.science/INDRUM2022/public/INDRUM2022_Proceedings.pdf.

Vinerean, M., Liljekvist, Y., Brandl, M., & Przybilla, J. (in press). Didactical usefulness of Interactive Mathematical Maps - Designing activities supporting student teachers' learning. In M. Misfeldt, U. T. Jankvist, E. Geraniou, I. H. Hojsted, A. L. Tamborg, R. J. E. Coopman, & B. E. Jessen (Eds.), *Nordic Studies in Mathematics Education – Nordisk Matematikkdidaktikk, NOMAD – Thematic Issue Digital resources in mathematics education*. NCM.

Winsløw, C., & Grønbæk, N. (2014). Klein's double discontinuity revisited: contemporary challenges for universities preparing teachers to teach calculus. *Recherches en Didactique des Mathématiques*, 34(1), 59–86. https://www.researchgate.net/publication/243963587
# GRIDS AS OBJECTS AND TOOLS FOR THE GEOMETRY CURRICULUM

Cathy Bruce¹, Nathalie Sinclair², Jessica Bodnar¹, Nicholas Jackiw²

¹ Trent University, Canada, ² Simon Fraser University, Canada

> *Although not typically considered as basic objects of the geometry curriculum, grids are significant objects-to-think-with that, we argue, simultaneously require and enable geometric seeing, acting and reasoning. Grids function both as intrinsically rich geometric structures and as geometric ways of structuring the world. We situate our work within the research on spatial reasoning. We also draw on theories of embodied cognition that emphasize the importance of bodily actions, and gestures in particular, for exploring and communicating geometric ideas. Through two examples, we show the complexity and conceptual richness of grids. One example involves young learners engaging with paper-and-pencil grid tasks. The other describes a multi-touch app that enables users to create and manipulate grids for ordering or analyzing their surroundings. Although the examples are quite different in nature, together they suggest how working with grids could constitute a significant component of the geometry curriculum that would not only support students' spatial reasoning, but also equip them with a powerful tool for seeing, acting and thinking.*

## LEVERAGING SPATIAL REASONING TO THINK-WITH AND ACT-ON GRIDS AS DYNAMIC SPATIAL STRUCTURES

In many countries including Canada and the United States, geometry is increasingly overshadowed as a curriculum topic by arithmetic and early algebra. Even topics that are geometric in nature are typically treated numerically or quantitatively. Much of the focus on geometry is devoted to learning the names of shapes—what Duval (2006) calls geometry as botany. Despite the preponderance of *numerical* approaches in the curriculum—and their perceived significance to real world situations—research over the past decade has shown a stunning correlation between *spatial* reasoning and mathematical achievement. Further, research has also shown that spatial reasoning is malleable: it can be improved with practice. Several Canadian jurisdictions (Ontario and British Columbia) have thus begun emphasizing spatial reasoning in the curriculum. It is in this spirit, of “spatializing” the curriculum, that Francis et al. (2023) propose number lines and mutable grids as tools that are spatial in nature and relevant across the curriculum (including in geometry, number, and measurement strands). In this paper, we focus on the grid, which we put forth as an object-to-think-with (to use Papert’s (1980) term), that functions both as a geometrically rich structure in its own right and as a geometric tool that can be used to structure, and mathematise, the world.

Grids are used in coordinate systems, but also in 2D and 3D measurement, in geometric transformations, and in multiplication. In these cases, grids often appear incidentally, as background fields or frames of reference rather than as objects defined, emphasized, or analyzed in their own right. Grids, their attributes, and applications can all be introduced more explicitly, as tools, or as ways of producing structures. Placing a grid on a 2D shape, for example, can decompose it into equally sized units that enable measurement. Gridding a drawing can make it easier to copy and/or enlarge. Imposing a grid provides a structure that invites qualitative, proportional focused thinking
# GRIDS AS OBJECTS AND TOOLS FOR THE GEOMETRY CURRICULUM

Cathy Bruce¹, Nathalie Sinclair², Jessica Bodnar¹, Nicholas Jackiw²

¹ Trent University, Canada, ² Simon Fraser University, Canada

> *Although not typically considered as basic objects of the geometry curriculum, grids are significant objects-to-think-with that, we argue, simultaneously require and enable geometric seeing, acting and reasoning. Grids function both as intrinsically rich geometric structures and as geometric ways of structuring the world. We situate our work within the research on spatial reasoning. We also draw on theories of embodied cognition that emphasize the importance of bodily actions, and gestures in particular, for exploring and communicating geometric ideas. Through two examples, we show the complexity and conceptual richness of grids. One example involves young learners engaging with paper-and-pencil grid tasks. The other describes a multi-touch app that enables users to create and manipulate grids for ordering or analyzing their surroundings. Although the examples are quite different in nature, together they suggest how working with grids could constitute a significant component of the geometry curriculum that would not only support students' spatial reasoning, but also equip them with a powerful tool for seeing, acting and thinking.*

# LEVERAGING SPATIAL REASONING TO THINK-WITH AND ACT-ON GRIDS AS DYNAMIC SPATIAL STRUCTURES

In many countries including Canada and the United States, geometry is increasingly overshadowed as a curriculum topic by arithmetic and early algebra. Even topics that are geometric in nature are typically treated numerically or quantitatively. Much of the focus on geometry is devoted to learning the names of shapes—what Duval (2006) calls geometry as botany. Despite the preponderance of *numerical* approaches in the curriculum—and their perceived significance to real world situations—research over the past decade has shown a stunning correlation between *spatial* reasoning and mathematical achievement. Further, research has also shown that spatial reasoning is malleable: it can be improved with practice. Several Canadian jurisdictions (Ontario and British Columbia) have thus begun emphasizing spatial reasoning in the curriculum. It is in this spirit, of “spatializing” the curriculum, that Francis et al. (2023) propose number lines and mutable grids as tools that are spatial in nature and relevant across the curriculum (including in geometry, number, and measurement strands). In this paper, we focus on the grid, which we put forth as an object-to-think-with (to use Papert’s (1980) term), that functions both as a geometrically rich structure in its own right and as a geometric tool that can be used to structure, and mathematise, the world.

Grids are used in coordinate systems, but also in 2D and 3D measurement, in geometric transformations, and in multiplication. In these cases, grids often appear incidentally, as background fields or frames of reference rather than as objects defined, emphasized, or analyzed in their own right. Grids, their attributes, and applications can all be introduced more explicitly, as tools, or as ways of producing structures. Placing a grid on a 2D shape, for example, can decompose it into equally sized units that enable measurement. Gridding a drawing can make it easier to copy and/or enlarge. Imposing a grid provides a structure that invites qualitative, proportional focused thinking
rather than (necessarily) numerical thinking. For example, placing a grid on a human body makes it easy to see how large the head is compared to the whole body.

The work we present in this paper builds on prior studies that Bruce, Sinclair and the Spatial Reasoning Study Group (comprised of researchers from Australia, Canada and the United States) have carried out on the development of young children’s spatial reasoning and its role in mathematics learning (Sinclair & Bruce, 2015). It also extends the work of Jackiw and Sinclair (2014) who have designed gesture-driven multi-touch apps for young learners to create and manipulate numbers.

## GRIDS AS SPATIAL OBJECTS

We see grids as spatial tools. They can be tangible physical objects such as graph paper or even lines drawn through the sand, or they can be imagined objects that serve as mental constructions allowing us to conceptualize mathematical metaphors such as continuous and infinite space, parallel lines and infinite points along those lines (Lakoff & Nuñez, 2000). They can be laid beneath or on top of other objects; they can be stretched, skewed, rotated and even wrapped around other surfaces. They can be scaled up and down, zoomed in and out, stretched and twisted, parallel or convergent. Structured grids typically feature cells of identical shapes, but we also find unstructured non-uniform grids, polar grids, perspective grids of broad variety, spherical and space-packing and other 3D grids, and many more uncommon members of the grid world.

Despite the possibilities that grids enable, the traditional square grid that students encounter throughout their geometry learning is presented as static in nature, with the perspective and structure being fixed. Interestingly, what is left unexplored in most school geometry is the conceptual underpinning that grid lines represent divisions in continuous space (Lakoff & Núñez, 2000), which is of concern given that the fundamental properties of grid structures have been reported as being difficult for students to generalize (Battista et al., 1998, Outhred & Mitchelmore, 2000). Grids can help us organize and partition space, track and describe movements and locations in coding and mapping, as well as compare objects and distances visually. The particular way in which grids structure space provides a system for thinking about many fundamental ideas in geometry, algebra and in mathematics more broadly. It follows that students would benefit from having explicit opportunities to develop a robust relationship with grids as spatial objects to think-with and act-on.

While textbooks and worksheets usually present square, static grids that have already been made, digital technologies can make it easier for students to create their own grids and to explore grids in terms of their anatomy (geometric properties) and their flexibility (how they can be applied and manipulated in various contexts) in making sense of space. In our explorations with children, we have used both static canonical grids with low-tech materials such as black grid lines on clear plastic sheets as well as dynamic grids that can be rotated, shifted, stretched and compressed using iPads and software developed by Jackiw (in press). In both cases, we are interested in how learners act-on and think-with grids, paying particular attention to the way they see grids and the gestures they make to describe, create and manipulate grids.

## INSTANTIATIONS OF GRIDS AS OBJECTS-TO-THINK-WITH AND ACT-ON

We now turn to two cases from our research that point to the potential conceptual richness and power of grids. We focus not only on grids as visual objects, but also on the gestures involved in acting on them. We posit that new ways of moving and acting (of moving one's fingers, for example) produce
rather than (necessarily) numerical thinking. For example, placing a grid on a human body makes it easy to see how large the head is compared to the whole body.

The work we present in this paper builds on prior studies that Bruce, Sinclair and the Spatial Reasoning Study Group (comprised of researchers from Australia, Canada and the United States) have carried out on the development of young children’s spatial reasoning and its role in mathematics learning (Sinclair & Bruce, 2015). It also extends the work of Jackiw and Sinclair (2014) who have designed gesture-driven multi-touch apps for young learners to create and manipulate numbers.

## GRIDS AS SPATIAL OBJECTS

We see grids as spatial tools. They can be tangible physical objects such as graph paper or even lines drawn through the sand, or they can be imagined objects that serve as mental constructions allowing us to conceptualize mathematical metaphors such as continuous and infinite space, parallel lines and infinite points along those lines (Lakoff & Nuñez, 2000). They can be laid beneath or on top of other objects; they can be stretched, skewed, rotated and even wrapped around other surfaces. They can be scaled up and down, zoomed in and out, stretched and twisted, parallel or convergent. Structured grids typically feature cells of identical shapes, but we also find unstructured non-uniform grids, polar grids, perspective grids of broad variety, spherical and space-packing and other 3D grids, and many more uncommon members of the grid world.

Despite the possibilities that grids enable, the traditional square grid that students encounter throughout their geometry learning is presented as static in nature, with the perspective and structure being fixed. Interestingly, what is left unexplored in most school geometry is the conceptual underpinning that grid lines represent divisions in continuous space (Lakoff & Núñez, 2000), which is of concern given that the fundamental properties of grid structures have been reported as being difficult for students to generalize (Battista et al., 1998, Outhred & Mitchelmore, 2000). Grids can help us organize and partition space, track and describe movements and locations in coding and mapping, as well as compare objects and distances visually. The particular way in which grids structure space provides a system for thinking about many fundamental ideas in geometry, algebra and in mathematics more broadly. It follows that students would benefit from having explicit opportunities to develop a robust relationship with grids as spatial objects to think-with and act-on.

While textbooks and worksheets usually present square, static grids that have already been made, digital technologies can make it easier for students to create their own grids and to explore grids in terms of their anatomy (geometric properties) and their flexibility (how they can be applied and manipulated in various contexts) in making sense of space. In our explorations with children, we have used both static canonical grids with low-tech materials such as black grid lines on clear plastic sheets as well as dynamic grids that can be rotated, shifted, stretched and compressed using iPads and software developed by Jackiw (in press). In both cases, we are interested in how learners act-on and think-with grids, paying particular attention to the way they see grids and the gestures they make to describe, create and manipulate grids.

## INSTANTIATIONS OF GRIDS AS OBJECTS-TO-THINK-WITH AND ACT-ON

We now turn to two cases from our research that point to the potential conceptual richness and power of grids. We focus not only on grids as visual objects, but also on the gestures involved in acting on them. We posit that new ways of moving and acting (of moving one's fingers, for example) produce
new ways of thinking, which is in line with theories of embodied cognition that refute the Cartesian mind-body binary (e.g., Nemirovsky et al., 2013; de Freitas & Sinclair, 2014). Given the importance that gestures play both in mathematics learning and communicating (Edwards et al., 2017), we are particularly interested in both spontaneous and intentionally designed gestural experiences.

## Case one: Early encounters with a static, square grid

In an effort to capture some of the earliest conceptions of rectilinear grids, Bodnar (2023) conducted a series of exploratory one-to-one interviews with children ages 3 to 7. Emir, age 5, was shown a 5x5 square grid that had some of the cells covered with coloured paper. He was asked to think of the coloured area as a blanket that was covering some of the squares on the 5x5 grid and that his job was to determine how many squares were “hiding under the blanket” (see Fig. 1a). When shown the 5x5 grid with a 3x3 coloured ‘blanket’, Emir explained how he was thinking about the ‘covered area’:

Interviewer: How many squares are hiding under the blanket?

Emir: Nine

Interviewer: How did you know it was nine?

Emir: I was thinking one here, two here, three here, (points to where each cell might be in the top row that is covered) four here, five here, six here (points to where the second row of cells would be), but then there was another row I didn't really notice... Nine! Cause six plus three is nine...is this whole square.

Emir tapped each cell with his pointer finger as he counted squares and said, “one here, two here” and so on across the first two rows in sequence, which may suggest that he sees the relationship between cells as non-continuous. That is, each cell was independent and was not bound to a higher structural system, such as a row. Interestingly, Emir paused his count after the first two rows, then used his left hand to cover the area of six squares that he had just counted, and then smoothly ran his right pointer finger back and forth horizontally along the bottom row. He did not count each cell in this bottom row but rather named and gestured to the whole row as a unit of three, related cells. After having some more time to consider this grid, Emir said: “this whole square” while he used both hands to cover the 3x3 area. The square of nine came together as one whole unit, one whole shape that Emir dis-embedded from the surrounding 5x5 grid.

<figure><img src="image_141.png" /><figcaption>Figure 1: (a) Emir breaks apart a 3x3 covered grid into an area of six with the addition of a row of three to reach 9; (b) Emir's attempt to complete a 3x3 grid structure</figcaption></figure>

In another task, the interviewer asked Emir to complete a square 3x3 grid that was partially generated (adapted from Battista et al., 1998), as in Fig 1b. Emir was asked first to give an estimate of how many square tiles he thought it would take to cover the entire grid (showing one plastic tile equivalent to one cell as a visual point of reference). He estimated it would require 10 tiles. After giving his
new ways of thinking, which is in line with theories of embodied cognition that refute the Cartesian mind-body binary (e.g., Nemirovsky et al., 2013; de Freitas & Sinclair, 2014). Given the importance that gestures play both in mathematics learning and communicating (Edwards et al., 2017), we are particularly interested in both spontaneous and intentionally designed gestural experiences.

## Case one: Early encounters with a static, square grid

In an effort to capture some of the earliest conceptions of rectilinear grids, Bodnar (2023) conducted a series of exploratory one-to-one interviews with children ages 3 to 7. Emir, age 5, was shown a 5x5 square grid that had some of the cells covered with coloured paper. He was asked to think of the coloured area as a blanket that was covering some of the squares on the 5x5 grid and that his job was to determine how many squares were “hiding under the blanket” (see Fig. 1a). When shown the 5x5 grid with a 3x3 coloured ‘blanket’, Emir explained how he was thinking about the ‘covered area’:

Interviewer: How many squares are hiding under the blanket?

Emir: Nine

Interviewer: How did you know it was nine?

Emir: I was thinking one here, two here, three here, (points to where each cell might be in the top row that is covered) four here, five here, six here (points to where the second row of cells would be), but then there was another row I didn't really notice... Nine! Cause six plus three is nine...is this whole square.

Emir tapped each cell with his pointer finger as he counted squares and said, “one here, two here” and so on across the first two rows in sequence, which may suggest that he sees the relationship between cells as non-continuous. That is, each cell was independent and was not bound to a higher structural system, such as a row. Interestingly, Emir paused his count after the first two rows, then used his left hand to cover the area of six squares that he had just counted, and then smoothly ran his right pointer finger back and forth horizontally along the bottom row. He did not count each cell in this bottom row but rather named and gestured to the whole row as a unit of three, related cells. After having some more time to consider this grid, Emir said: “this whole square” while he used both hands to cover the 3x3 area. The square of nine came together as one whole unit, one whole shape that Emir dis-embedded from the surrounding 5x5 grid.

<figure><img src="image_148.png" /><figcaption>Figure 1: (a) Emir breaks apart a 3x3 covered grid into an area of six with the addition of a row of three to reach 9; (b) Emir's attempt to complete a 3x3 grid structure</figcaption></figure>

In another task, the interviewer asked Emir to complete a square 3x3 grid that was partially generated (adapted from Battista et al., 1998), as in Fig 1b. Emir was asked first to give an estimate of how many square tiles he thought it would take to cover the entire grid (showing one plastic tile equivalent to one cell as a visual point of reference). He estimated it would require 10 tiles. After giving his
estimate he was then asked to “complete the grid” with a pencil. Emir began to draw individual cells within the grid. He started in the top left corner by connecting the partial grid line indicators provided to make one ‘square’ cell. He then continued in a one-by-one approach, generating the cells as a series of discrete squares with no evidence of continuity or relationship.

In comparing Emir’s responses to the two tasks, we can see that in the first, Emir had some reasonable strategies for thinking about a series of 9 cells in the formation of 3 rows of 3 cells each, as evidenced in his gestures and language. However, when it was Emir’s job to construct a grid with limited information (a perimeter and some partial partition lines), Emir opted for a single cell approach, with no particular attention to rows or columns. We observed this same phenomenon consistently: young children found the blanket task far more accessible compared to the grid construction task. Emir does not appear to see the grid as a structure in which cells are components of both rows and columns that are produced by vertical and horizontal lines that intersect. This suggests that the grid is a complex visual and haptic object for young learners that may require intentional pedagogical mediation, such as shifting learners’ attention from the 2D spaces produced by gridlines to the 1D lines themselves, which Duval (2006) calls dimensional decomposition.

## Case two: Griddler application

The Griddler (Jackiw, in press) is an app we are developing to explore ideas about grids outside of plotting and early-number contexts. The broad purpose of the app is to let users superimpose dynamic grids on their worlds, and to use these grids to assess geometric and spatial patterns and relationships present in those worlds. The specific design has evolved through cycles of user evaluation. Users, in this case, are considered without particular prerequisite age or educational background; and while both the app’s emphasis on “gridding one’s world” and the specific gestures it encourages users to apply are potentially provocative pedagogically, the app features no explicit pedagogic questions or formal mathematical language in its (minimal) interface. Rather, the intended broader learning design of the software in any pedagogic context puts such potential questions and language in conversations outside the tool itself, where they can follow a trajectory meaningful to a specific learner rather than to some entire class of generic “app users.”

In choosing a view of “their world” to ‘begrid’, users may opt alternately to manipulate *what is directly in front of them* (represented in the app by the live video feed of their device’s camera); or *what is important, precious or interesting* to them (represented by the collection of images they’ve stored on their camera’s photo roll), or finally *where they are*, represented by a re-scalable map showing their position in their neighbourhood, city, or continent. Finally, a last view lets a teacher suggest a canonical image for exploration (based on a small set of images provided for this purpose), so that multiple users, with multiple devices, can explore a common image. This fourth view breaks the premise of strict user-centricity in favour of a group-centricity that supports whole-class exploration or precise instructional moments.

Imposing a dynamic grid on any of these world views is accomplished through a set of touch-screen gestures. A first finger touch on any world view causes a square grid to sprout and expand from the touched location, rapidly covering the entire view (Fig. 2a). Subsequent dragging with one finger translates this grid so that the cells line up as the user wishes over the image (2b). Two fingers simultaneously dragged twist and shrink or enlarge the grid, allowing the user to quickly resize the
estimate he was then asked to “complete the grid” with a pencil. Emir began to draw individual cells within the grid. He started in the top left corner by connecting the partial grid line indicators provided to make one ‘square’ cell. He then continued in a one-by-one approach, generating the cells as a series of discrete squares with no evidence of continuity or relationship.

In comparing Emir’s responses to the two tasks, we can see that in the first, Emir had some reasonable strategies for thinking about a series of 9 cells in the formation of 3 rows of 3 cells each, as evidenced in his gestures and language. However, when it was Emir’s job to construct a grid with limited information (a perimeter and some partial partition lines), Emir opted for a single cell approach, with no particular attention to rows or columns. We observed this same phenomenon consistently: young children found the blanket task far more accessible compared to the grid construction task. Emir does not appear to see the grid as a structure in which cells are components of both rows and columns that are produced by vertical and horizontal lines that intersect. This suggests that the grid is a complex visual and haptic object for young learners that may require intentional pedagogical mediation, such as shifting learners’ attention from the 2D spaces produced by gridlines to the 1D lines themselves, which Duval (2006) calls dimensional decomposition.

## Case two: Griddler application

The Griddler (Jackiw, in press) is an app we are developing to explore ideas about grids outside of plotting and early-number contexts. The broad purpose of the app is to let users superimpose dynamic grids on their worlds, and to use these grids to assess geometric and spatial patterns and relationships present in those worlds. The specific design has evolved through cycles of user evaluation. Users, in this case, are considered without particular prerequisite age or educational background; and while both the app’s emphasis on “gridding one’s world” and the specific gestures it encourages users to apply are potentially provocative pedagogically, the app features no explicit pedagogic questions or formal mathematical language in its (minimal) interface. Rather, the intended broader learning design of the software in any pedagogic context puts such potential questions and language in conversations outside the tool itself, where they can follow a trajectory meaningful to a specific learner rather than to some entire class of generic “app users.”

In choosing a view of “their world” to ‘begrid’, users may opt alternately to manipulate *what is directly in front of them* (represented in the app by the live video feed of their device’s camera); or *what is important, precious or interesting* to them (represented by the collection of images they’ve stored on their camera’s photo roll), or finally *where they are*, represented by a re-scalable map showing their position in their neighbourhood, city, or continent. Finally, a last view lets a teacher suggest a canonical image for exploration (based on a small set of images provided for this purpose), so that multiple users, with multiple devices, can explore a common image. This fourth view breaks the premise of strict user-centricity in favour of a group-centricity that supports whole-class exploration or precise instructional moments.

Imposing a dynamic grid on any of these world views is accomplished through a set of touch-screen gestures. A first finger touch on any world view causes a square grid to sprout and expand from the touched location, rapidly covering the entire view (Fig. 2a). Subsequent dragging with one finger translates this grid so that the cells line up as the user wishes over the image (2b). Two fingers simultaneously dragged twist and shrink or enlarge the grid, allowing the user to quickly resize the
square grid to exactly match any visible feature of the view (2c). Introducing a third finger (perhaps on a spare hand) stretches the square grid—pegged down in two places by the prior fingers—to expand into a rectangular one, or shears it into a grid of oblique parallelograms. A fourth finger further deforms the original grid into a receding perspective grid. In Figure 2, a user rapidly begrids the child’s head, and observes 1:5 ratio of head:body height.

<figure><img src="image_142.png" /><figcaption>Figure 2: (a) Touching the screen to create a grid; (b) One-finger dragging to translate; (c) Two-finger dragging to rotate and dilate</figcaption></figure>

This gestural language for grid manipulation draws on our previous work in pedagogically provocative gesture design (e.g. Jackiw & Sinclair, 2014) and reflects several design goals for users' interactions with, and early conceptualizations of, “grid experience”. First, it positions all possible grids in a dynamic, continuous evolution of an initial square grid, rather than in a typology of fundamentally separate grids. Learners begin with a conceptually simple, generic structure and gradually modify it, through direct manipulation, into more specific forms appropriate to gridding specific aspects of their world. Second, it associates unique physical gestures with different types of manipulations, offering embodied and haptic instantiations of operations that are also experienced visually (in their effect on the grid) and propositionally (in any conversation one has around the grid, such as “move it over here”). Third, it presents a pedagogically laddered trajectory of gestures which begin with the obvious, accessible act of dragging, for basic operations, and develops in a structured fashion (adding “one more finger” at a time) into less obvious or less familiar gestures for more advanced operations, revealing new capabilities as users gain proficiencies with more elementary operations. Fourth, as an overall grammar of grid transformation, it features strong underlying mathematical coherence: if a finger position on the screen represents an idealized geometric point on the plane, then a finger dragging from one location to another represents a point-to-point mapping. Transformationally, such a single point-to-point mapping defines a translation of the plane: the grid *moves*. Two point-to-point mappings define a similarity: with two fingers, the grid also *dilates* and *rotates*. Three point-to-point mappings (equivalently, two triangles) define an affinity (the grid now also *shears*) and the mapping of two convex quadrilaterals (four points, four fingers) defines a perspectivity (gridlines now *converge*).

The impact of these four design goals is, of course, not easily isolatable into four distinct dimensions of user experience. Instead, they operate together, and interact with users’ prior touch-screen experience as well. For example, most newcomers interpret *The Griddler’s* one-finger and two finger transformations as already-familiar gestures with expected behaviours: “dragging” and “pinching” are common manipulation skills in diverse mobile apps. Applying them to fit a grid cell exactly around some feature of an image may require their use at a more precise or thoughtful level than
square grid to exactly match any visible feature of the view (2c). Introducing a third finger (perhaps on a spare hand) stretches the square grid—pegged down in two places by the prior fingers—to expand into a rectangular one, or shears it into a grid of oblique parallelograms. A fourth finger further deforms the original grid into a receding perspective grid. In Figure 2, a user rapidly begrids the child’s head, and observes 1:5 ratio of head:body height.

<figure><img src="image_149.png" /><figcaption>Figure 2: (a) Touching the screen to create a grid; (b) One-finger dragging to translate; (c) Two-finger dragging to rotate and dilate</figcaption></figure>

This gestural language for grid manipulation draws on our previous work in pedagogically provocative gesture design (e.g. Jackiw & Sinclair, 2014) and reflects several design goals for users' interactions with, and early conceptualizations of, “grid experience”. First, it positions all possible grids in a dynamic, continuous evolution of an initial square grid, rather than in a typology of fundamentally separate grids. Learners begin with a conceptually simple, generic structure and gradually modify it, through direct manipulation, into more specific forms appropriate to gridding specific aspects of their world. Second, it associates unique physical gestures with different types of manipulations, offering embodied and haptic instantiations of operations that are also experienced visually (in their effect on the grid) and propositionally (in any conversation one has around the grid, such as “move it over here”). Third, it presents a pedagogically laddered trajectory of gestures which begin with the obvious, accessible act of dragging, for basic operations, and develops in a structured fashion (adding “one more finger” at a time) into less obvious or less familiar gestures for more advanced operations, revealing new capabilities as users gain proficiencies with more elementary operations. Fourth, as an overall grammar of grid transformation, it features strong underlying mathematical coherence: if a finger position on the screen represents an idealized geometric point on the plane, then a finger dragging from one location to another represents a point-to-point mapping. Transformationally, such a single point-to-point mapping defines a translation of the plane: the grid *moves*. Two point-to-point mappings define a similarity: with two fingers, the grid also *dilates* and *rotates*. Three point-to-point mappings (equivalently, two triangles) define an affinity (the grid now also *shears*) and the mapping of two convex quadrilaterals (four points, four fingers) defines a perspectivity (gridlines now *converge*).

The impact of these four design goals is, of course, not easily isolatable into four distinct dimensions of user experience. Instead, they operate together, and interact with users’ prior touch-screen experience as well. For example, most newcomers interpret *The Griddler’s* one-finger and two finger transformations as already-familiar gestures with expected behaviours: “dragging” and “pinching” are common manipulation skills in diverse mobile apps. Applying them to fit a grid cell exactly around some feature of an image may require their use at a more precise or thoughtful level than
applying them simply to “zoom in” on an image, but they remain accessible starting rungs of the
laddered trajectory of gesture. By contrast, the next ladder rung—the three-finger drag, which is
easiest with two hands—falls outside most users’ gestural vernacular. This physical unfamiliarity
creates a mild “access gate” keeping the conceptually unfamiliar idea of the parallelogram grid
somewhat beyond users’ naïve grid manipulations. It also perhaps promotes a more intentional,
reflective consideration of the gesture’s effect once it is encountered or discovered. In our piloting of
*The Griddler*, where most new users already know the impact of one- and two-fingered grid
transformations, we see them having to actively develop proficiency in controlling or predicting the
effect of three-finger transformations. In *The Griddler*, this intentional design mapping of gestural
affordance to mathematical definition makes gestures not only communicative but also functional
epistemically (Jackiw & Sinclair, 2017).

DISCUSSION

Recently there are some strong examples of the incorporation of spatial reasoning within and beyond geometry-focused learning experiences (see Lowrie & Logan, 2023; Mulligan et al., 2020; Moss et al., 2015; Ng & Sinclair, 2015) as a dynamic visual-spatial approach. These studies recognise both the fundamental role that imagery plays in mathematical thinking, even outside of geometry, and the many models, representations and diagrams that can be used to mediate visual-spatial conceptualisations of mathematical concepts. Our focus on the grid contributes to these efforts, but extends prior work on spatializing concepts to also consider spatializing tools. We argue that grids function both epistemologically (as objects to be learned through acting-on them) and ontologically (as tools to use—and hence objects-to-think-with) in ways that cut across much of the curriculum.

In the two cases described above, we highlighted the potential of explicitly leveraging grids for
mathematics thinking as an underpinning spatial structure that supports children in their spatial sense
– their making sense of space and their geometry thinking. In this discussion, we focus on two
implications that challenge typical conceptions of how geometry is treated in school mathematics.
The first is that the application of objects-to-think-with and act-on have the effect of folding together
geometric thinking and action instead of artificially splitting them apart as is typical classroom
practice in North America and other parts of the globe. The second implication is that we can
reconceive of a geometry curriculum in a world where objects become the basis for exploration of
mathematics principles – that grids, for example, as dynamic tools and spatial structures, can be the
curriculum starting point and the site of investigation, with the mathematics tumbling strategically
from these investigations. With Hansen (2009), “…we take the position that *geometry is about
grasping space*, not only with the eyes and hands, but also with the mind” (p. 240).

Spatial objects as an opportunity to fold thinking and action together

In a push against the artificial disconnect between conceptual thinking and procedural actions of school mathematics, researchers have become interested in pedagogical artefacts that can fold together thinking and doing (Bruce et al., 2023; Niss & Højgaard, 2019). In the previous example of Emir, the very act of partitioning a figure to make a square grid appears to involve reasoning spatially about concepts such as equal spacing, continuousness, intersecting space and the spatial integration of cells, as well as density. In the example of *The Griddler*, we see that placing and moving two fingers on a touch surface can facilitate thinking about and acting on scaling and rotation of structured
applying them simply to “zoom in” on an image, but they remain accessible starting rungs of the
laddered trajectory of gesture. By contrast, the next ladder rung—the three-finger drag, which is
easiest with two hands—falls outside most users’ gestural vernacular. This physical unfamiliarity
creates a mild “access gate” keeping the conceptually unfamiliar idea of the parallelogram grid
somewhat beyond users’ naïve grid manipulations. It also perhaps promotes a more intentional,
reflective consideration of the gesture’s effect once it is encountered or discovered. In our piloting of
*The Griddler*, where most new users already know the impact of one- and two-fingered grid
transformations, we see them having to actively develop proficiency in controlling or predicting the
effect of three-finger transformations. In *The Griddler*, this intentional design mapping of gestural
affordance to mathematical definition makes gestures not only communicative but also functional
epistemically (Jackiw & Sinclair, 2017).

DISCUSSION

Recently there are some strong examples of the incorporation of spatial reasoning within and beyond geometry-focused learning experiences (see Lowrie & Logan, 2023; Mulligan et al., 2020; Moss et al., 2015; Ng & Sinclair, 2015) as a dynamic visual-spatial approach. These studies recognise both the fundamental role that imagery plays in mathematical thinking, even outside of geometry, and the many models, representations and diagrams that can be used to mediate visual-spatial conceptualisations of mathematical concepts. Our focus on the grid contributes to these efforts, but extends prior work on spatializing concepts to also consider spatializing tools. We argue that grids function both epistemologically (as objects to be learned through acting-on them) and ontologically (as tools to use—and hence objects-to-think-with) in ways that cut across much of the curriculum.

In the two cases described above, we highlighted the potential of explicitly leveraging grids for
mathematics thinking as an underpinning spatial structure that supports children in their spatial sense
– their making sense of space and their geometry thinking. In this discussion, we focus on two
implications that challenge typical conceptions of how geometry is treated in school mathematics.
The first is that the application of objects-to-think-with and act-on have the effect of folding together
geometric thinking and action instead of artificially splitting them apart as is typical classroom
practice in North America and other parts of the globe. The second implication is that we can
reconceive of a geometry curriculum in a world where objects become the basis for exploration of
mathematics principles – that grids, for example, as dynamic tools and spatial structures, can be the
curriculum starting point and the site of investigation, with the mathematics tumbling strategically
from these investigations. With Hansen (2009), “…we take the position that *geometry is about
grasping space*, not only with the eyes and hands, but also with the mind” (p. 240).

Spatial objects as an opportunity to fold thinking and action together

In a push against the artificial disconnect between conceptual thinking and procedural actions of school mathematics, researchers have become interested in pedagogical artefacts that can fold together thinking and doing (Bruce et al., 2023; Niss & Højgaard, 2019). In the previous example of Emir, the very act of partitioning a figure to make a square grid appears to involve reasoning spatially about concepts such as equal spacing, continuousness, intersecting space and the spatial integration of cells, as well as density. In the example of *The Griddler*, we see that placing and moving two fingers on a touch surface can facilitate thinking about and acting on scaling and rotation of structured
grids to fit various contexts while maintaining the integrity of the shape of the cells. Acting on the grid by placing more fingers on the grid enables the eyes, hands and mind to explore affine and perspective transformations.

## Reconceiving geometry curriculum

As a thought experiment, we consider how this focus on objects-to-think-with and act-on might reposition geometry curriculum for young students as an active dynamic learning space, unbounded by typical textbook-driven approaches of naming and sorting canonical shapes. We conceive of a geometry curriculum where the objects explicitly reveal and support student conceptions of geometric principles and ideas as a primary goal rather than simply as the means to understanding a rule. In other words, exploration with grids *is* the mathematics from which we can draw key ideas about geometric principles, configurations, manipulations and uses of grids. In such a curriculum the goal can be flipped: rather than having the curriculum goal of teaching children that rectangles have four sides and four right angles, the creation and manipulation of dynamic rectangular figures made with a tool such as *The Geometer's Sketchpad* (Jackiw, 1991), and grids applied and manipulated using a tool such as *The Griddler*, become a highly focused and exploratory method for revealing the ‘rules’ they follow in space and how they can help us think mathematically. Such a repositioning of dynamic play with grids as a curriculum objective to draw out geometric principles and spatial reasoning, notably requires a high level of teacher confidence and understanding.

## Classroom implications

In building teaching facility for a geometry curriculum that is truly exploratory and dynamic in nature, researchers, geometers and teachers will need to work together. As researchers, our efforts are squarely focused in this arena. The research described (in case 1) will enable us to design lessons that support students’ structural awareness. Similarly, the tool described (in case 2) can provide a basis for classroom activities involve gridding (some of which we have already developed and will share at the conference). In both cases however, efforts will be needed to prevent use of these resources from relapsing into procedural approaches giving rise to the botanical metaphor used by Duval—which in our case would involve simply naming and classifying static grids and their parts. We think such efforts might shift teachers’ conceptions of geometry away from a rigid set of given concepts and theorems towards what Herbst et al. (2017) describe as more flexible ways of modelling and understanding shape and space.

## Acknowledgements

Collaborative and independent funding from the Social Sciences and Humanities Research Council of Canada have enabled this work and supported the design and development of The Griddler.

## References

* Battista, M., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. V. A. (1998). Students’ spatial structuring of 2D arrays of squares. *Journal for Research in Mathematics Education, 29*(5), 503–532.

* Bodnar, J. (2023). *Young children's spatial conceptions of two-dimensional grid structures*. [Unpublished master's thesis]. Trent University.

* Bruce, C. D., Flynn, T., Yearly, S., & Hawes, Z. (2023). Leveraging number lines and unit fractions to build student understanding: Insights from a mixed-methods study. *Canadian Journal of Science, Mathematics and Technology Education, 23*(2), 322–339.
grids to fit various contexts while maintaining the integrity of the shape of the cells. Acting on the grid by placing more fingers on the grid enables the eyes, hands and mind to explore affine and perspective transformations.

## Reconceiving geometry curriculum

As a thought experiment, we consider how this focus on objects-to-think-with and act-on might reposition geometry curriculum for young students as an active dynamic learning space, unbounded by typical textbook-driven approaches of naming and sorting canonical shapes. We conceive of a geometry curriculum where the objects explicitly reveal and support student conceptions of geometric principles and ideas as a primary goal rather than simply as the means to understanding a rule. In other words, exploration with grids *is* the mathematics from which we can draw key ideas about geometric principles, configurations, manipulations and uses of grids. In such a curriculum the goal can be flipped: rather than having the curriculum goal of teaching children that rectangles have four sides and four right angles, the creation and manipulation of dynamic rectangular figures made with a tool such as *The Geometer's Sketchpad* (Jackiw, 1991), and grids applied and manipulated using a tool such as *The Griddler*, become a highly focused and exploratory method for revealing the 'rules' they follow in space and how they can help us think mathematically. Such a repositioning of dynamic play with grids as a curriculum objective to draw out geometric principles and spatial reasoning, notably requires a high level of teacher confidence and understanding.

## Classroom implications

In building teaching facility for a geometry curriculum that is truly exploratory and dynamic in nature, researchers, geometers and teachers will need to work together. As researchers, our efforts are squarely focused in this arena. The research described (in case 1) will enable us to design lessons that support students' structural awareness. Similarly, the tool described (in case 2) can provide a basis for classroom activities involve gridding (some of which we have already developed and will share at the conference). In both cases however, efforts will be needed to prevent use of these resources from relapsing into procedural approaches giving rise to the botanical metaphor used by Duval—which in our case would involve simply naming and classifying static grids and their parts. We think such efforts might shift teachers' conceptions of geometry away from a rigid set of given concepts and theorems towards what Herbst et al. (2017) describe as more flexible ways of modelling and understanding shape and space.

## Acknowledgements

Collaborative and independent funding from the Social Sciences and Humanities Research Council of Canada have enabled this work and supported the design and development of The Griddler.

## References

* Battista, M., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. V. A. (1998). Students' spatial structuring of 2D arrays of squares. *Journal for Research in Mathematics Education, 29*(5), 503–532.

* Bodnar, J. (2023). *Young children's spatial conceptions of two-dimensional grid structures*. [Unpublished master's thesis]. Trent University.

* Bruce, C. D., Flynn, T., Yearly, S., & Hawes, Z. (2023). Leveraging number lines and unit fractions to build student understanding: Insights from a mixed-methods study. *Canadian Journal of Science, Mathematics and Technology Education, 23*(2), 322–339.
de Freitas, E., & Sinclair, N. (2014). *Mathematics and the body: Material entanglements in the classroom*. Cambridge University Press.

Duval, R. (2006). A cognitive analysis of problems of comprehension in the learning of mathematics. *Educational Studies in Mathematics, 61*, 103–131.

Edwards, L., Ferrara, F. & Moore-Russo, D. (2017). *Emerging perspectives on gesture and embodiment in mathematics*. Information Age Publishing.

Francis, K., Davis, B., Bruce, C., & Sinclair, N. (2023). Emergent technologies for developing mathematical objects-to-think-with. In B. Pepin, G. Gueudet, & J. Choppin (Eds.), *Handbook of Digital Resources in Mathematics Education*. Springer.

Hansen, V.L. (2009). Changes and trends in geometry curricula; Section 1. In C. Mammana, & V. Villanie (Eds.), *Perspectives on Teaching Geometry for the 21st Century*. Kluwer Academic Publishers.

Herbst, P., Fujita, T., Halverscheid, S., & Weiss, M. (2017). *The teaching and learning of geometry in secondary schools: A modeling perspective*. Impact series. Routledge.

Jackiw, N. (1991). *The Geometer's Sketchpad*. Software application. Key Curriculum Press.

Jackiw, N., & Sinclair, N. (2014). *TouchCounts*. iPad application. The Tangible Mathematics Project. Burnaby, BC.

Jackiw, N., & Sinclair, N. (2017). TouchCounts and gesture design. In T. Hammond, A. Adler & M. Prasad (Eds), *Frontiers in Pen and Touch* (pp. 51-64). Springer.

Jackiw, N. (in press) *The Griddler*, iPhone/iPad application. The Tangible Mathematics Project. Burnaby, BC.

Lakoff, G., & Núñez, R. (2000). *Where mathematics comes from: How the embodied mind brings mathematics into being*. Basic Books.

Lowrie, T., & Logan, T. (2023). Spatial visualization supports students’ math: mechanisms for spatial transfer. *Journal of Intelligence, 11*(6), 127.

Moss, J., Hawes, Z., Naqvi, S., & Caswell, B. (2015). Adapting Japanese lesson study to enhance the teaching and learning of geometry and spatial reasoning in early years classrooms: a case study. *ZDM Mathematics Education, 15*(3).

Mulligan, J., Oslington, G., & English, L. (2020). Supporting early mathematical development through a “pattern and structure” intervention program. *ZDM, 52*(4), 663-676.

Nemirovsky, R., Kelton, M. L., & Rhodehamel, B. (2013). Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit. *Journal for Research in Mathematics Education, 44*(2), 372–415.

Ng, O., & Sinclair, N. (2015). “Area without numbers”: Using touchscreen dynamic geometry to reason about shape. *Canadian Journal of Science, Mathematics and Technology Education, 15*(1), 84-101.

Niss, M., & Højgaard, T. (2019). Mathematical competencies revisited. *Educational Studies in Mathematics, 102*(1), 9-28.

Outhred, L. N., & Mitchelmore, M. C. (2000). Young children’s intuitive understanding of rectangular area measurement. *Journal for Research in Mathematics Education, 31*(2), 144–167.

Papert, S. (1980). *Mindstorms: children, computers, and powerful ideas*. Basic Books.

Sinclair, N., & Bruce, C. D. (2015). New opportunities in geometry education at the primary school. *ZDM, 47*, 319-329.
de Freitas, E., & Sinclair, N. (2014). *Mathematics and the body: Material entanglements in the classroom*. Cambridge University Press.

Duval, R. (2006). A cognitive analysis of problems of comprehension in the learning of mathematics. *Educational Studies in Mathematics, 61*, 103–131.

Edwards, L., Ferrara, F. & Moore-Russo, D. (2017). *Emerging perspectives on gesture and embodiment in mathematics*. Information Age Publishing.

Francis, K., Davis, B., Bruce, C., & Sinclair, N. (2023). Emergent technologies for developing mathematical objects-to-think-with. In B. Pepin, G. Gueudet, & J. Choppin (Eds.), *Handbook of Digital Resources in Mathematics Education*. Springer.

Hansen, V.L. (2009). Changes and trends in geometry curricula; Section 1. In C. Mammana, & V. Villanie (Eds.), *Perspectives on Teaching Geometry for the 21st Century*. Kluwer Academic Publishers.

Herbst, P., Fujita, T., Halverscheid, S., & Weiss, M. (2017). *The teaching and learning of geometry in secondary schools: A modeling perspective*. Impact series. Routledge.

Jackiw, N. (1991). *The Geometer's Sketchpad*. Software application. Key Curriculum Press.

Jackiw, N., & Sinclair, N. (2014). *TouchCounts*. iPad application. The Tangible Mathematics Project. Burnaby, BC.

Jackiw, N., & Sinclair, N. (2017). TouchCounts and gesture design. In T. Hammond, A. Adler & M. Prasad (Eds), *Frontiers in Pen and Touch* (pp. 51-64). Springer.

Jackiw, N. (in press) *The Griddler*, iPhone/iPad application. The Tangible Mathematics Project. Burnaby, BC.

Lakoff, G., & Núñez, R. (2000). *Where mathematics comes from: How the embodied mind brings mathematics into being*. Basic Books.

Lowrie, T., & Logan, T. (2023). Spatial visualization supports students’ math: mechanisms for spatial transfer. *Journal of Intelligence, 11*(6), 127.

Moss, J., Hawes, Z., Naqvi, S., & Caswell, B. (2015). Adapting Japanese lesson study to enhance the teaching and learning of geometry and spatial reasoning in early years classrooms: a case study. *ZDM Mathematics Education, 15*(3).

Mulligan, J., Oslington, G., & English, L. (2020). Supporting early mathematical development through a “pattern and structure” intervention program. *ZDM, 52*(4), 663-676.

Nemirovsky, R., Kelton, M. L., & Rhodehamel, B. (2013). Playing mathematical instruments: Emerging perceptuomotor integration with an interactive mathematics exhibit. *Journal for Research in Mathematics Education, 44*(2), 372–415.

Ng, O., & Sinclair, N. (2015). “Area without numbers”: Using touchscreen dynamic geometry to reason about shape. *Canadian Journal of Science, Mathematics and Technology Education, 15*(1), 84-101.

Niss, M., & Højgaard, T. (2019). Mathematical competencies revisited. *Educational Studies in Mathematics, 102*(1), 9-28.

Outhred, L. N., & Mitchelmore, M. C. (2000). Young children’s intuitive understanding of rectangular area measurement. *Journal for Research in Mathematics Education, 31*(2), 144–167.

Papert, S. (1980). *Mindstorms: children, computers, and powerful ideas*. Basic Books.

Sinclair, N., & Bruce, C. D. (2015). New opportunities in geometry education at the primary school. *ZDM, 47*, 319-329.
# VIRTUAL ENVIRONMENT FOR SPATIAL KNOWLEDGE: OPPORTUNITIES AND LEARNING OBJECTIVES

Sylvia Coutat, Jean-Luc Dorier

Geneva University

This text presents a research project in which a didactical engineering on spatial orientation using a virtual environment has been conceived and experimented in five classes with the same students over 3 years in grades 2 to 4. After a general presentation of the project and some theoretical elements, an example of a task on path planning is presented and analyzed in order to illustrate some possibilities of the technological environment.

## RESEARCH CONTEXT

Nowadays it is practically impossible to organize our daily lives without technology. Indeed, technologies are taking up more and more space in several of our actions, including spatial orientation when traveling. Clearly less and less people still uses a map to move around a city, on foot or by car, since Global Positioning Systems (GPS) are now easily accessible and have totally changed our orientation skills (Ruginski & al., 2019). Although children rarely use GPS when traveling, three-dimensional technologies are nevertheless present in many videogames and have inevitably an influence on their construction of space orientation skills (Spence & Feng, 2010). These are important issues regarding the evolution of spatial orientation and more generally our representation of space. As educators, and researchers in education, we would like to better estimate their effect on the learning of spatial knowledge and see how we could use virtual environments and possibly virtual reality to design school activities more meaningful for students at the beginning of primary school regarding spatial orientation within the mathematics curriculum.

These reflections motivated a research project called SPAGEO<fnref n="20" /> which connects three research fields: cognitive psychology, educational technologies, and mathematics didactics. Each of these fields contributed to the whole project which combines three facets each being led by one the three components. The first facet, led by the cognitive psychologists aims to document the spatial abilities of a large population of children aged 7 to 10. The second facet, led by the mathematics didacticians aims to design a teaching sequence over several levels of primary school on spatial orientation using a virtual environment. The third, led by the educational technologists aims to conceive and develop of a virtual environment, in the shape of a city on a computer, that we called SPAGEO City. A demo of the immersive view is available on the website<fnref n="21" />. It provides access to the city's various neighborhoods, including the city center and the residential area (Coutat & al., to be published).

In this paper, we will only refer to some aspects of the second facet, on which we were the leading team. Our focus was first to conceive and experiment some spatial orientation activities and show

<footnote n="20">²⁰ « Rethinking the links between spatial knowledge and geometry in primary education through virtual environments » funded by the Swiss National Science Foundation – 2019-2023 (Subside n°100019_188947/1).</footnote>

<footnote n="21">²¹ https://tecfa.unige.ch/tecfa/research/spageo/</footnote>
# VIRTUAL ENVIRONMENT FOR SPATIAL KNOWLEDGE: OPPORTUNITIES AND LEARNING OBJECTIVES

Sylvia Coutat, Jean-Luc Dorier

Geneva University

This text presents a research project in which a didactical engineering on spatial orientation using a virtual environment has been conceived and experimented in five classes with the same students over 3 years in grades 2 to 4. After a general presentation of the project and some theoretical elements, an example of a task on path planning is presented and analyzed in order to illustrate some possibilities of the technological environment.

## RESEARCH CONTEXT

Nowadays it is practically impossible to organize our daily lives without technology. Indeed, technologies are taking up more and more space in several of our actions, including spatial orientation when traveling. Clearly less and less people still uses a map to move around a city, on foot or by car, since Global Positioning Systems (GPS) are now easily accessible and have totally changed our orientation skills (Ruginski & al., 2019). Although children rarely use GPS when traveling, three-dimensional technologies are nevertheless present in many videogames and have inevitably an influence on their construction of space orientation skills (Spence & Feng, 2010). These are important issues regarding the evolution of spatial orientation and more generally our representation of space. As educators, and researchers in education, we would like to better estimate their effect on the learning of spatial knowledge and see how we could use virtual environments and possibly virtual reality to design school activities more meaningful for students at the beginning of primary school regarding spatial orientation within the mathematics curriculum.

These reflections motivated a research project called SPAGEO<fnref n="20" /> which connects three research fields: cognitive psychology, educational technologies, and mathematics didactics. Each of these fields contributed to the whole project which combines three facets each being led by one the three components. The first facet, led by the cognitive psychologists aims to document the spatial abilities of a large population of children aged 7 to 10. The second facet, led by the mathematics didacticians aims to design a teaching sequence over several levels of primary school on spatial orientation using a virtual environment. The third, led by the educational technologists aims to conceive and develop of a virtual environment, in the shape of a city on a computer, that we called SPAGEO City. A demo of the immersive view is available on the website<fnref n="21" />. It provides access to the city's various neighborhoods, including the city center and the residential area (Coutat & al., to be published).

In this paper, we will only refer to some aspects of the second facet, on which we were the leading team. Our focus was first to conceive and experiment some spatial orientation activities and show

<footnote n="20">²⁰ « Rethinking the links between spatial knowledge and geometry in primary education through virtual environments » funded by the Swiss National Science Foundation – 2019-2023 (Subside n°100019_188947/1).</footnote>

<footnote n="21">²¹ https://tecfa.unige.ch/tecfa/research/spageo/</footnote>
what can be done with the help of a virtual city. We led our experiment in all the classes of two schools in Geneva, around 100 students, that we followed over three years, while in Grades 2, 3 and 4 (7 to 10 years old)<fnref n="22" />. For each year, a didactic engineering of about six to eight sessions of 1 to 1.5 hours each was designed and tested. Of course, we worked in close collaboration with the technologists in order to implement in SPAGEO city, the different aspects and tools that our activities required. Moreover, the collaboration with the psychologists was important in order to have a more precise idea of our students' spatial abilities. In this sense, the development of the virtual environment has been thought out using the competences from the three components. The discussions and interactions were focused on the opportunities and challenges that could be used by the technology to achieve our research and teaching objectives taking into account the various works on spatial abilities by psychologists.

This text presents a digital technology for teaching and learning spatial knowledge with a specific focus the opportunities and challenges that are presented by the use of technology in geometry education.

## SPATIAL KNOWLEDGE IN GEOMETRY

In this study, the mathematics at stake is related to spatial knowledge. Berthelot and Salin (1999) in their study on the teaching of space and geometry defined spatial knowledge as “the knowledge that allows a subject to control his relations to sensitive space” (Our translation from the original). Marchand (2020) refined this definition by referring to mental transformations or manipulations that a subject uses to anticipate or control objects from the sensible space (Our translation from the original):

> We refer to the knowledge which, physically (concrete action) or mentally (MI [Mental Imageries]), leads the learner to control, anticipate and communicate the states, transformations or deformations of objects in the sensitive space, the graphic space and, hypothetically, geometric space (Marchand, 2020, p. 142).

This mental consideration appears also in cognitive psychology, with spatial abilities or skills which are constructed using different factors like spatial visualization, spatial orientation, mental rotation, and spatial navigation (Carroll, 1993). Using spatial knowledge and spatial abilities in the analyses can contribute to a better understanding of tasks related to spatial knowledge or geometry (Coutat & al., 2022; Coutat & Berney, 2023). In our research we are particularly interested in spatial navigation, as the ability to locate and move in real or virtual space by using perception and comprehension of the organization of this space (Pick & al., 1999). According to Seigel and White (1975) three stages of developmental spatial knowledge are identified: landmark knowledge, route knowledge and survey knowledge. Landmark knowledge comprises environmental objects perceived and recognized, they can be points of reference, global (seen from far) or local (for more precise movement). The use and recognition of landmarks are essential features of spatial representations. They can become strategical elements of a route, to or from which a person moves (Golledge, 1999). Indeed, landmarks can be used to identify a strategic position (start or arrival) where specific actions may be needed. In this sense, route knowledge is made of the relations a subject is able to construct between landmarks. Duroisin (2015) considers itineraries like an ordered succession of decision points linked together by

<footnote n="22">²² In Grades 3 and 4 there were 5 classes which reduced to 4 in Grade 5.</footnote>
what can be done with the help of a virtual city. We led our experiment in all the classes of two
schools in Geneva, around 100 students, that we followed over three years, while in Grades 2, 3 and
4 (7 to 10 years old)<fnref n="22" />. For each year, a didactic engineering of about six to eight sessions of 1 to 1.5
hours each was designed and tested. Of course, we worked in close collaboration with the
technologists in order to implement in SPAGEO city, the different aspects and tools that our activities
required. Moreover, the collaboration with the psychologists was important in order to have a more
precise idea of our students' spatial abilities. In this sense, the development of the virtual environment
has been thought out using the competences from the three components. The discussions and
interactions were focused on the opportunities and challenges that could be used by the technology
to achieve our research and teaching objectives taking into account the various works on spatial
abilities by psychologists.

This text presents a digital technology for teaching and learning spatial knowledge with a specific focus the opportunities and challenges that are presented by the use of technology in geometry education.

SPATIAL KNOWLEDGE IN GEOMETRY

In this study, the mathematics at stake is related to spatial knowledge. Berthelot and Salin (1999) in their study on the teaching of space and geometry defined spatial knowledge as “the knowledge that allows a subject to control his relations to sensitive space” (Our translation from the original). Marchand (2020) refined this definition by referring to mental transformations or manipulations that a subject uses to anticipate or control objects from the sensible space (Our translation from the original):

We refer to the knowledge which, physically (concrete action) or mentally (MI [Mental Imageries]), leads the learner to control, anticipate and communicate the states, transformations or deformations of objects in the sensitive space, the graphic space and, hypothetically, geometric space (Marchand, 2020, p. 142).

This mental consideration appears also in cognitive psychology, with spatial abilities or skills which are constructed using different factors like spatial visualization, spatial orientation, mental rotation, and spatial navigation (Carroll, 1993). Using spatial knowledge and spatial abilities in the analyses can contribute to a better understanding of tasks related to spatial knowledge or geometry (Coutat & al., 2022; Coutat & Berney, 2023). In our research we are particularly interested in spatial navigation, as the ability to locate and move in real or virtual space by using perception and comprehension of the organization of this space (Pick & al., 1999). According to Seigel and White (1975) three stages of developmental spatial knowledge are identified: landmark knowledge, route knowledge and survey knowledge. Landmark knowledge comprises environmental objects perceived and recognized, they can be points of reference, global (seen from far) or local (for more precise movement). The use and recognition of landmarks are essential features of spatial representations. They can become strategical elements of a route, to or from which a person moves (Golledge, 1999). Indeed, landmarks can be used to identify a strategic position (start or arrival) where specific actions may be needed. In this sense, route knowledge is made of the relations a subject is able to construct between landmarks. Duroisin (2015) considers itineraries like an ordered succession of decision points linked together by

<footnote n="22">²² In Grades 3 and 4 there were 5 classes which reduced to 4 in Grade 5.</footnote>