the complete structure of the city. All students scoring high used DV1 to solve the problem, while only 20% of the students scoring low did it. Therefore, this problem allowed to discriminate very well the use of DV1 by the best visualizers, who are thus potential MG students.

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

<figure><img src="image_78.png" /><figcaption>Figure 2: Student E.E63031's solution to the problem of phase 3, grade 4</figcaption></figure>

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

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

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

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

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

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

<figure><img src="image_79.png" /><figcaption>Figure 3: Problem of phase 2, grade 5</figcaption></figure>
When solving the problem of Figure 3, students used *visual identification* (DV1) to extract information from the figure, identify the polygonal shapes that form the visible faces of each cube, and identify the cubes as simple objects and as parts of the columns. This descriptor was used by almost all students (94.4%), as it was not difficult for them to interpret the information. Once the elements of the figure were identified, the student used the *conservation of perception* to identify the cubes and count them. All the students scoring high used DV3.1 and DV3.2 to count the cubes and solve the problem. Also, 80% of the students scoring low evidenced DV3.1, so it was easy for them to identify the cubes that have a visible part or are deformed by the perspective. On the contrary, none of the students with low scores was able to identify the non-visible cubes (DV3.2). Figure 4 shows the solution by a student who drew separate representations of each element (columns) of the structure. Thus, the answer showed the three descriptors mentioned, because, after having identified the elements of the sculpture (DV1), the student used DV3.1 and DV3.2 to identify the partially and completely hidden cubes and count them.

<figure><img src="image_74.png" /><figcaption>Figure 4: Student E.E52002's solution to the problem of phase 2, grade 5</figcaption></figure>

### Type C problems: mentally disassemble or rearrange the elements that make up a structure

Martín's friends are playing with cubes that are held together with magnets. They have built the four colored shapes above and Martín has built the two larger shapes, A and B.

a) Is it possible, from Figure A, only by removing some cubes, to obtain each of the figures built by your friends? Justify your answer.

b) Is the above possible from Figure B? Justify your answer.

<figure><img src="image_75.png" /><figcaption>Figure 5: Problem of phase 3, grade 6</figcaption></figure>

When solving the problem in Figure 5, students found it easy to identify the cubes that form each structure in the statement, with their faces and edges. Therefore, almost all students (81.25%) used *visual identification* (DV1) and, therefore, this descriptor did not help to differentiate the best visualizers of grade 6, although, in other problems of lower grades, it did discriminate them very well. To determine the shapes of the colored structures and compare them with figures A and B, it is necessary to identify some cubes of each structure that are partially or completely hidden, making use of the *conservation of perception*. In this problem, almost all high-scoring students (85.7%) showed evidence of having used DV3.1 and DV3.2. Furthermore, 57.14% of the low-scoring students evidenced the use of DV3.1. Then, DV3.1 do not help to differentiate the best visualizers in this problem. On the other hand, DV3.2 helped more to differentiate students with higher visualization capacity, because it was used by fewer low-scoring students (42.8%). When asked to compare several
When solving the problem of Figure 3, students used *visual identification* (DV1) to extract information from the figure, identify the polygonal shapes that form the visible faces of each cube, and identify the cubes as simple objects and as parts of the columns. This descriptor was used by almost all students (94.4%), as it was not difficult for them to interpret the information. Once the elements of the figure were identified, the student used the *conservation of perception* to identify the cubes and count them. All the students scoring high used DV3.1 and DV3.2 to count the cubes and solve the problem. Also, 80% of the students scoring low evidenced DV3.1, so it was easy for them to identify the cubes that have a visible part or are deformed by the perspective. On the contrary, none of the students with low scores was able to identify the non-visible cubes (DV3.2). Figure 4 shows the solution by a student who drew separate representations of each element (columns) of the structure. Thus, the answer showed the three descriptors mentioned, because, after having identified the elements of the sculpture (DV1), the student used DV3.1 and DV3.2 to identify the partially and completely hidden cubes and count them.

<figure><img src="image_80.png" /><figcaption>Figure 4: Student E.E52002's solution to the problem of phase 2, grade 5</figcaption></figure>

### Type C problems: mentally disassemble or rearrange the elements that make up a structure

Martín's friends are playing with cubes that are held together with magnets. They have built the four colored shapes above and Martín has built the two larger shapes, A and B.

a) Is it possible, from Figure A, only by removing some cubes, to obtain each of the figures built by your friends? Justify your answer.

b) Is the above possible from Figure B? Justify your answer.

<figure><img src="image_81.png" /><figcaption>Figure 5: Problem of phase 3, grade 6</figcaption></figure>

When solving the problem in Figure 5, students found it easy to identify the cubes that form each structure in the statement, with their faces and edges. Therefore, almost all students (81.25%) used *visual identification* (DV1) and, therefore, this descriptor did not help to differentiate the best visualizers of grade 6, although, in other problems of lower grades, it did discriminate them very well. To determine the shapes of the colored structures and compare them with figures A and B, it is necessary to identify some cubes of each structure that are partially or completely hidden, making use of the *conservation of perception*. In this problem, almost all high-scoring students (85.7%) showed evidence of having used DV3.1 and DV3.2. Furthermore, 57.14% of the low-scoring students evidenced the use of DV3.1. Then, DV3.1 do not help to differentiate the best visualizers in this problem. On the other hand, DV3.2 helped more to differentiate students with higher visualization capacity, because it was used by fewer low-scoring students (42.8%). When asked to compare several
figures, students used *visual discrimination* (DV6) to compare some structures with others, by identifying visual similarities and differences between them. This descriptor was evidenced by almost all high-scoring students (85.7%) and only 14.28% of low-scoring students, so DV6 helped to differentiate students with higher visualization ability. Figure 6 presents the response of a student who showed these four descriptors. Consequently, among the descriptors necessary to solve this problem, DV6 is the one discriminating best the students with higher visualization capacity.

<figure><img src="image_76.png" /><figcaption>Figure 6: Student E.E63031's solution to the problem of phase 3, grade 6</figcaption></figure>

## CONCLUSIONS

Researchers in mathematics education have identified several capacities related to MG. The ability to visualize is necessary to achieve a complete mathematical education of students, although there is debate as to whether it is a characteristic ability of MG. In our study, we experimented with a set of 13 problems of three different types, administered in the OLCOMEP olympiad to participants in grades 2 and 4 to 6 of primary school. To correctly solve these problems, students had to use a variety of visualization skills. We have shown examples of the three types of problems posed, analyzed some solutions to those problems, and reported which visualization descriptors proved to be good identifiers of high visualization ability and potential MG students, as a large majority of the students who used them solved the problems requiring them well and obtained high scores in the olympiad.

Our study cannot provide generalizable answers, due to the small sample size, but it raises an important avenue for future research, consisting in analyzing: how the characteristics of different types of visualization problems influence the success of students with high and low visualization abilities; which visualization skills are more adequate to discriminate students who have high visualization abilities and, therefore, who are potential MG students.

## Acknowledgements

This research is part of the project PID2020-117395RB-I00, funded by MCIN/AEI/10.13039/501100011033.

## References

* Arcavi, A. (2003). The role of visual representations in the learning of mathematics. *Educational Studies in Mathematics*, 52(3), 215-241. https://doi.org/10.1023/A:1024312321077

* Bishop, A. J. (1983). Spatial abilities and mathematical thinking. In M. Zweng, T. Green, J. Kilpatrick, H. Pollak, & M. Suydam (Eds.), *Proceedings of the 4th I.C.M.E.* (pp. 176-178). Birkhauser.
figures, students used *visual discrimination* (DV6) to compare some structures with others, by identifying visual similarities and differences between them. This descriptor was evidenced by almost all high-scoring students (85.7%) and only 14.28% of low-scoring students, so DV6 helped to differentiate students with higher visualization ability. Figure 6 presents the response of a student who showed these four descriptors. Consequently, among the descriptors necessary to solve this problem, DV6 is the one discriminating best the students with higher visualization capacity.

<figure><img src="image_82.png" /><figcaption>Figure 6: Student E.E63031's solution to the problem of phase 3, grade 6</figcaption></figure>

## CONCLUSIONS

Researchers in mathematics education have identified several capacities related to MG. The ability to visualize is necessary to achieve a complete mathematical education of students, although there is debate as to whether it is a characteristic ability of MG. In our study, we experimented with a set of 13 problems of three different types, administered in the OLCOMEP olympiad to participants in grades 2 and 4 to 6 of primary school. To correctly solve these problems, students had to use a variety of visualization skills. We have shown examples of the three types of problems posed, analyzed some solutions to those problems, and reported which visualization descriptors proved to be good identifiers of high visualization ability and potential MG students, as a large majority of the students who used them solved the problems requiring them well and obtained high scores in the olympiad.

Our study cannot provide generalizable answers, due to the small sample size, but it raises an important avenue for future research, consisting in analyzing: how the characteristics of different types of visualization problems influence the success of students with high and low visualization abilities; which visualization skills are more adequate to discriminate students who have high visualization abilities and, therefore, who are potential MG students.

## Acknowledgements

This research is part of the project PID2020-117395RB-I00, funded by MCIN/AEI/10.13039/501100011033.

## References

* Arcavi, A. (2003). The role of visual representations in the learning of mathematics. *Educational Studies in Mathematics*, 52(3), 215-241. https://doi.org/10.1023/A:1024312321077

* Bishop, A. J. (1983). Spatial abilities and mathematical thinking. In M. Zweng, T. Green, J. Kilpatrick, H. Pollak, & M. Suydam (Eds.), *Proceedings of the 4th I.C.M.E.* (pp. 176-178). Birkhauser.
Cain, A. J. (2019). Visual thinking and simplicity of proof. *Philosophical Transactions A*, 377, 20180032.

Clements, D. H. (1999). Geometric and spatial thinking in young children. In J. V. Copley (Ed.), *Mathematics in the early years* (pp. 66-79). NCTM and National Association for the Education of Young Children.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), *Handbook of research on mathematics teaching and learning* (pp. 420-464). MacMillan.

Del Grande, J. (1990). Spatial sense. *The Arithmetic Teacher*, 37(6), 14-20.

Escrivá, M. T., Jaime, A., Gutiérrez, A., & Beltrán-Meneu, M. J. (2017). Geometría 3D y talento matemático. *UNO*, 77, 7-13.

Giaquinto, M. (2007). *Visual thinking in mathematics. An epistemological study*. Oxford University Press.

Gutiérrez, A. (1996). Visualization in 3-dimensional geometry: in search of a framework. In L. Puig, & A. Gutiérrez (Eds.), *Proceedings of the 20th International Conference of the P.M.E*. (Vol. 1, pp. 3-19). PME. https://www.uv.es/gutierre/archivos1/textospdf/Gut96c.pdf

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

Leikin, R. (2021). When practice needs more research: the nature and nurture of mathematical giftedness. *ZDM – Mathematics Education*, 53(7), 1579-1589. https://doi.org/10.1007/s11858-021-01276-9

Lowrie, T., Resnick, I., Harris, D., & Logan, T. (2020). In search of the mechanisms that enable transfer from spatial reasoning to mathematics understanding. *Mathematics Education Research Journal*, 32(2), 175-188. https://doi.org/10.1007/s13394-020-00336-9

Miller, R. (1990). Discovering mathematical talent. ERIC. https://www.eric digests.org/1994/talent.htm

Miragliotta, E., & Baccaglini-Frank, A. (2017). Visuo-spatial abilities and geometry: A first proposal of a theoretical framework for interpreting processes of visualization. In T. Dooley, & G. Gueudet (Eds.), *Proceedings of the 10th Congress of the European Society for Research in Mathematics Education* (pp. 3952-3959). ERME. http://www.mathematik.uni-dortmund.de/~erme/index.php?slab=proceedings

Mora, M., & Gutiérrez, A. (2021). Habilidades de visualización en niños de Primaria con alta capacidad matemática. In A. Gutiérrez et al. (Eds.), *Actas de las Jornadas Internacionales de Investigación y Práctica Docente en Alta Capacidad Matemática* (pp. 107-114). Universidad de La Rioja. https://www.uv.es/angel.gutierrez/archivos1/textospdf/MorGut21.pdf

Mora, M., Gutiérrez, A., & Jaime, A. (2023). Identification of pre-algebra problems of generalization that discriminate mathematical giftedness. Text of the presentation in the TWG3 of the *13th Congress of European Research in Mathematics Education* (CERME). To be published. CERME.

Presmeg, N. (1986). Visualisation and mathematical giftedness. *Educational Studies in Mathematics*, 17(3), 297-311. https://doi.org/10.1007/BF00305075

Ramírez, R. (2012). *Habilidades de visualización de los alumnos con talento matemático* (PhD dissertation). Universidad de Granada, Spain. Available in https://fqm193.ugr.es/produccion-cientifica/tesis/verowrddetalles/7461/descargar/

Ramírez, R., & Flores, P. (2017). Habilidades de visualización de estudiantes con talento matemático: Comparativa entre los test psicométricos y las habilidades de visualización manifestadas en tareas geométricas. *Enseñanza de las Ciencias*, 35(2), 179-196. http://dx.doi.org/10.5565/rev/ensciencias.2152

Van Garderen, D., Scheuermann, A., & Poch, A. (2014). Challenges students identified with a learning disability and as high-achieving experience when using diagrams as a visualization tool to solve mathematics word problems. *ZDM - Mathematics Education*, 46(1), 135-149. https://doi.org/10.1007/s11858-013-0519-1

Webb, R. M., Lubinski, D., & Benbow, C. P. (2007). Spatial ability: A neglected dimension in talent searches for intellectually precocious youth. *Journal of Educational Psychology*, 99(2), 397-420.
Cain, A. J. (2019). Visual thinking and simplicity of proof. *Philosophical Transactions A*, 377, 20180032.

Clements, D. H. (1999). Geometric and spatial thinking in young children. In J. V. Copley (Ed.), *Mathematics in the early years* (pp. 66-79). NCTM and National Association for the Education of Young Children.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), *Handbook of research on mathematics teaching and learning* (pp. 420-464). MacMillan.

Del Grande, J. (1990). Spatial sense. *The Arithmetic Teacher*, 37(6), 14-20.

Escrivá, M. T., Jaime, A., Gutiérrez, A., & Beltrán-Meneu, M. J. (2017). Geometría 3D y talento matemático. *UNO*, 77, 7-13.

Giaquinto, M. (2007). *Visual thinking in mathematics. An epistemological study*. Oxford University Press.

Gutiérrez, A. (1996). Visualization in 3-dimensional geometry: in search of a framework. In L. Puig, & A. Gutiérrez (Eds.), *Proceedings of the 20th International Conference of the P.M.E*. (Vol. 1, pp. 3-19). PME. https://www.uv.es/gutierre/archivos1/textospdf/Gut96c.pdf

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

Leikin, R. (2021). When practice needs more research: the nature and nurture of mathematical giftedness. *ZDM – Mathematics Education*, 53(7), 1579-1589. https://doi.org/10.1007/s11858-021-01276-9

Lowrie, T., Resnick, I., Harris, D., & Logan, T. (2020). In search of the mechanisms that enable transfer from spatial reasoning to mathematics understanding. *Mathematics Education Research Journal*, 32(2), 175-188. https://doi.org/10.1007/s13394-020-00336-9

Miller, R. (1990). Discovering mathematical talent. ERIC. https://www.eric digests.org/1994/talent.htm

Miragliotta, E., & Baccaglini-Frank, A. (2017). Visuo-spatial abilities and geometry: A first proposal of a theoretical framework for interpreting processes of visualization. In T. Dooley, & G. Gueudet (Eds.), *Proceedings of the 10th Congress of the European Society for Research in Mathematics Education* (pp. 3952-3959). ERME. http://www.mathematik.uni-dortmund.de/~erme/index.php?slab=proceedings

Mora, M., & Gutiérrez, A. (2021). Habilidades de visualización en niños de Primaria con alta capacidad matemática. In A. Gutiérrez et al. (Eds.), *Actas de las Jornadas Internacionales de Investigación y Práctica Docente en Alta Capacidad Matemática* (pp. 107-114). Universidad de La Rioja. https://www.uv.es/angel.gutierrez/archivos1/textospdf/MorGut21.pdf

Mora, M., Gutiérrez, A., & Jaime, A. (2023). Identification of pre-algebra problems of generalization that discriminate mathematical giftedness. Text of the presentation in the TWG3 of the *13th Congress of European Research in Mathematics Education* (CERME). To be published. CERME.

Presmeg, N. (1986). Visualisation and mathematical giftedness. *Educational Studies in Mathematics*, 17(3), 297-311. https://doi.org/10.1007/BF00305075

Ramírez, R. (2012). *Habilidades de visualización de los alumnos con talento matemático* (PhD dissertation). Universidad de Granada, Spain. Available in https://fqm193.ugr.es/produccion-cientifica/tesis/verowrddetalles/7461/descargar/

Ramírez, R., & Flores, P. (2017). Habilidades de visualización de estudiantes con talento matemático: Comparativa entre los test psicométricos y las habilidades de visualización manifestadas en tareas geométricas. *Enseñanza de las Ciencias*, 35(2), 179-196. http://dx.doi.org/10.5565/rev/ensciencias.2152

Van Garderen, D., Scheuermann, A., & Poch, A. (2014). Challenges students identified with a learning disability and as high-achieving experience when using diagrams as a visualization tool to solve mathematics word problems. *ZDM - Mathematics Education*, 46(1), 135-149. https://doi.org/10.1007/s11858-013-0519-1

Webb, R. M., Lubinski, D., & Benbow, C. P. (2007). Spatial ability: A neglected dimension in talent searches for intellectually precocious youth. *Journal of Educational Psychology*, 99(2), 397-420.
LESSON STUDY AS A CONTEXT FOR TEACHERS' LEARNING OF LANGUAGE
RESPONSIVE GEOMETRY TEACHING

Lisnet Mwadzaangati¹ and Jill Adler²

¹ University of Malawi, ² University of Witwatersrand

Our study focuses on language responsive geometry teaching (LRGT). We illuminate the dilemmas Malawian teachers confronted and the learning opportunities opened up during different stages of lesson study (LS) professional development that focused on similar triangles. Through a socio-cultural lens and related methodology, we used data from lesson planning, lessons and lesson reflection sessions to trace the teaching dilemmas that emerged and learning opportunities opened up. We show that in their planning and reflection discussions, the teachers deliberate when and how to name and represent the concept 'similar', thus questioning their routine practice of introducing a geometric concept by a definition. The discussions during the different LS stages provided teachers learning opportunities for improving LRGT practices of linking representations to the abstract mathematical concepts and how these are talked about.

INTRODUCTION

The teaching and learning of secondary school geometry is recognised as challenging in Malawi and there are calls for enhancing teaching quality through continuous professional development (PD) (Ministry of Education, Science and Technology [MoEST], 2020). We responded to this call by initiating a Lesson Study (LS) project focused on geometry with 10 Malawian secondary mathematics teachers from two schools. Originating in Japan over 100 years ago, LS has been widely taken up in many countries as effective PD for improving teacher knowledge and practice (Huang et al., 2019). Take up of LS is relatively new in Malawi and mainly focused at elementary mathematics (Fauskanger et al., 2019). According to research, when LS is introduced in a new context, structuring frameworks, also broadly known as theory-informed approaches, can scaffold teachers' entry in LS processes and enable systematic research of teacher learning through LS (Huang et al., 2019).

Malawian secondary mathematics teachers' challenges in teaching geometry are both in content and pedagogy (MoEST, 2020). We began the LS by conducting a two-day PD workshop with the secondary mathematics teachers. In the PD, we (1) discussed with the teachers the activities for promoting geometric reasoning using the mathematics teaching framework (MTF) a theoretically informed tool for working on teaching used in LS in a similar teaching and learning context to Malawi (Adler & Alshwaikh, 2019), and (2) introduced the teachers to LS mode of PD and its stages. Through studying teachers' participation in the first cycle, we confirmed that theory-informed LS supports adapting LS in a new context and reported on how LS can support teachers' learning of exemplification, a key teaching practice within the MTF (Mwadzaangati et al., 2022). We observed little attention to explanatory communication (a connected key teaching practice in the MTF) with several mathematical terms not elaborated fully, and learner talk was largely limited to reporting measurements but not expressing their thinking (Adler et al., 2022). In cycle 2, we aimed at working with the teachers to plan, enact and reflect on both how they used words to communicate geometric
LESSON STUDY AS A CONTEXT FOR TEACHERS’ LEARNING OF LANGUAGE
RESPONSIVE GEOMETRY TEACHING

Lisnet Mwadzaangati¹ and Jill Adler²

¹ University of Malawi, ² University of Witwatersrand

Our study focuses on language responsive geometry teaching (LRGT). We illuminate the dilemmas Malawian teachers confronted and the learning opportunities opened up during different stages of lesson study (LS) professional development that focused on similar triangles. Through a socio-cultural lens and related methodology, we used data from lesson planning, lessons and lesson reflection sessions to trace the teaching dilemmas that emerged and learning opportunities opened up. We show that in their planning and reflection discussions, the teachers deliberate when and how to name and represent the concept ‘similar’, thus questioning their routine practice of introducing a geometric concept by a definition. The discussions during the different LS stages provided teachers learning opportunities for improving LRGT practices of linking representations to the abstract mathematical concepts and how these are talked about.

INTRODUCTION

The teaching and learning of secondary school geometry is recognised as challenging in Malawi and there are calls for enhancing teaching quality through continuous professional development (PD) (Ministry of Education, Science and Technology [MoEST], 2020). We responded to this call by initiating a Lesson Study (LS) project focused on geometry with 10 Malawian secondary mathematics teachers from two schools. Originating in Japan over 100 years ago, LS has been widely taken up in many countries as effective PD for improving teacher knowledge and practice (Huang et al., 2019). Take up of LS is relatively new in Malawi and mainly focused at elementary mathematics (Fauskanger et al., 2019). According to research, when LS is introduced in a new context, structuring frameworks, also broadly known as theory-informed approaches, can scaffold teachers’ entry in LS processes and enable systematic research of teacher learning through LS (Huang et al., 2019).

Malawian secondary mathematics teachers' challenges in teaching geometry are both in content and pedagogy (MoEST, 2020). We began the LS by conducting a two-day PD workshop with the secondary mathematics teachers. In the PD, we (1) discussed with the teachers the activities for promoting geometric reasoning using the mathematics teaching framework (MTF) a theoretically informed tool for working on teaching used in LS in a similar teaching and learning context to Malawi (Adler & Alshwaikh, 2019), and (2) introduced the teachers to LS mode of PD and its stages. Through studying teachers' participation in the first cycle, we confirmed that theory-informed LS supports adapting LS in a new context and reported on how LS can support teachers' learning of exemplification, a key teaching practice within the MTF (Mwadzaangati et al., 2022). We observed little attention to explanatory communication (a connected key teaching practice in the MTF) with several mathematical terms not elaborated fully, and learner talk was largely limited to reporting measurements but not expressing their thinking (Adler et al., 2022). In cycle 2, we aimed at working with the teachers to plan, enact and reflect on both how they used words to communicate geometric
concepts, and how they engaged learners in expressing their geometric thinking, practices we associate with language responsive teaching (Prediger, 2022). Teachers from one school studied on the topic of similar triangles, a concept research has shown to be non-trivial (e.g. Seago et al., 2014).

The learners' challenges in similarity reported by the Malawian teachers are similar to those reported in literature including inability to: 1) understand and relate conditions of congruent and similar triangles; 2) identify corresponding pairs of equal angles and proportional sides in given congruent and similar figures; and 3) construct logical arguments for justifying congruency or similarity (Adler et al., 2022; Seago et al., 2014). Although use of diagrams that exemplify geometric objects (Mwadzaangati et al., 2022) and use of concrete representations is viewed as an effective approach to teaching of geometry, Seago et al. (2014) report that even with experiences of relevant diagrams and real objects, learners still struggle with meaningful geometric interpretation of observed figures. What is out of view in the studies on geometry and specifically on congruent and similar triangles are language responsive teaching (LRT) related issues and so mediation through talk that connects real or concrete objects with abstract ones. In this paper, we explore how LS became a context for the teachers' learning of LRT practices in geometry (hereafter called LRGT) through examining (1) what LRGT dilemmas did teachers confront when planning and teaching similarity? and (2) how did deliberating on these dilemmas through LS provide teachers learning opportunities in LRGT?

## THE MATHEMATICS TEACHING FRAMEWORK AND LANGUAGE RESPONSIVE GEOMETRY TEACHING

We take a social cultural theoretical orientation to mathematics teaching and learning rooted in Vygotsky (1978). The MTF is the ‘teaching’ version of the Mathematical Discourse in Instruction analytical framework developed for describing and evaluating the quality of mathematics made available in teaching (Adler & Alshwaikh, 2019; Adler & Ronda, 2015). It draws from key tenets of socio-cultural theory that view mathematics as a network of interrelated and well-organized scientific concepts, teaching and learning as goal directed with progression towards and in disciplinary ways of thinking teaching (Adler & Alshwaikh, 2019; Adler & Ronda, 2015; Vygotsky, 1978). Its starting point is that teaching is always about something, called the lesson goal, and the work of mathematics teaching is mediational. The MTF comprises three recognisable and inter-connected teaching practices, with key mediational means (bracketed) in mathematics classroom instruction: exemplification (examples, tasks, and their representational forms), explanatory communication (word use and justifying) and learner participation (what learners are invited to do, say and write). These mediational means work together to open (or close) opportunities for mathematical activity about the lesson goal. There is attention in the framework on the connections and coherence between these practices

Considering the focus of this paper, we zoom in here on explanatory communication (EC), and so the language practices emphasised in the MTGF and the representations in use. As indicated above, these practices are critical both for geometry teaching and learning and also in LRT in mathematics in general (Prediger, 2022). EC focuses on how we use words to (1) name and mediate mathematical objects, processes and procedures, while attending to movement between colloquial/everyday and technical/formal mathematical registers, and (2) how these are justified – with attention to whether these are statements and/or assertions or whether they are elaborated and then in what way. In Vygotskian terms, word use is important and linked with the relationship between the everyday and
concepts, and how they engaged learners in expressing their geometric thinking, practices we associate with language responsive teaching (Prediger, 2022). Teachers from one school studied on the topic of similar triangles, a concept research has shown to be non-trivial (e.g. Seago et al., 2014).

The learners' challenges in similarity reported by the Malawian teachers are similar to those reported in literature including inability to: 1) understand and relate conditions of congruent and similar triangles; 2) identify corresponding pairs of equal angles and proportional sides in given congruent and similar figures; and 3) construct logical arguments for justifying congruency or similarity (Adler et al., 2022; Seago et al., 2014). Although use of diagrams that exemplify geometric objects (Mwadzaangati et al., 2022) and use of concrete representations is viewed as an effective approach to teaching of geometry, Seago et al. (2014) report that even with experiences of relevant diagrams and real objects, learners still struggle with meaningful geometric interpretation of observed figures. What is out of view in the studies on geometry and specifically on congruent and similar triangles are language responsive teaching (LRT) related issues and so mediation through talk that connects real or concrete objects with abstract ones. In this paper, we explore how LS became a context for the teachers' learning of LRT practices in geometry (hereafter called LRGT) through examining (1) what LRGT dilemmas did teachers confront when planning and teaching similarity? and (2) how did deliberating on these dilemmas through LS provide teachers learning opportunities in LRGT?

## THE MATHEMATICS TEACHING FRAMEWORK AND LANGUAGE RESPONSIVE GEOMETRY TEACHING

We take a social cultural theoretical orientation to mathematics teaching and learning rooted in Vygotsky (1978). The MTF is the ‘teaching’ version of the Mathematical Discourse in Instruction analytical framework developed for describing and evaluating the quality of mathematics made available in teaching (Adler & Alshwaikh, 2019; Adler & Ronda, 2015). It draws from key tenets of socio-cultural theory that view mathematics as a network of interrelated and well-organized scientific concepts, teaching and learning as goal directed with progression towards and in disciplinary ways of thinking teaching (Adler & Alshwaikh, 2019; Adler & Ronda, 2015; Vygotsky, 1978). Its starting point is that teaching is always about something, called the lesson goal, and the work of mathematics teaching is mediational. The MTF comprises three recognisable and inter-connected teaching practices, with key mediational means (bracketed) in mathematics classroom instruction: exemplification (examples, tasks, and their representational forms), explanatory communication (word use and justifying) and learner participation (what learners are invited to do, say and write). These mediational means work together to open (or close) opportunities for mathematical activity about the lesson goal. There is attention in the framework on the connections and coherence between these practices

Considering the focus of this paper, we zoom in here on explanatory communication (EC), and so the language practices emphasised in the MTGF and the representations in use. As indicated above, these practices are critical both for geometry teaching and learning and also in LRT in mathematics in general (Prediger, 2022). EC focuses on how we use words to (1) name and mediate mathematical objects, processes and procedures, while attending to movement between colloquial/everyday and technical/formal mathematical registers, and (2) how these are justified – with attention to whether these are statements and/or assertions or whether they are elaborated and then in what way. In Vygotskian terms, word use is important and linked with the relationship between the everyday and
scientific concepts. Although the MTF clarifies what teachers need to consider as regarding word use and justifications (colloquial and formal word use and different kinds of justifications) it does not clarify how these link with representations used, hence the need for further study and probably the contribution of this paper on LRGT. We distinguished representations as either verbal or visual (diagrammatic or concrete or symbolic); and elaborated word use as movement between the everyday register, meaning-related school mathematics register and technical or formal register. MTF links with the work of other scholars working with content specific language practices (eg. Prediger, 2022). To answer our research questions, we were interested in the LRGT practices the teachers interacted with during their lesson planning and reflection sessions, the dilemmas that arose from the discussions and how these became learning opportunities for them.

## METHODOLOGY

Each LS cycle consisted of two initial planning sessions, followed by teaching lesson 1, reflection 1 and lesson planning 2, then lesson 2, followed by reflection 2 and lesson planning 3 (see Mwadzaangati et al., 2022 for detail). In this paper we report on the LRT issues from one school by conducting content analysis on transcripts of a) lesson planning sessions; b) lessons 1 and 2; and c) reflection 1 and 2 sessions. The lesson transcriptions capture line by line utterances together with illustrations of teacher activity in the classrooms. Our analysis involved reading the transcribed data several times and dividing it into lesson episodes related to different lesson foci. In this paper we focus on episode 2 where the notion of similarity and similar triangles was introduced, as this generated substantive dilemmas amongst the teachers related to word use and representational forms. In the lesson episode, we examined the learning opportunities opened up to the learners by analysing whether word use was in the everyday, or school mathematics, or formal register, and whether the related representations were verbal (words only) or visual (concrete manipulatives or diagrams or symbols), and how these were connected. In the lesson planning and reflection discussions, we examined what the teachers talked about in relation to word use and representations, the dilemmas implied in their talk and so opportunities opened for teachers learning.

## FINDINGS

The goal in both lessons was to establish the meaning of similar triangles, in relation to congruent triangles, and the proportionality of the corresponding sides. We begin by describing the LRGT practices of word use and representations deliberated by the teachers during their initial planning and the dilemmas confronted.

### LRGT practices and dilemmas discussed during the initial lesson planning sessions

The teachers began the planning session for the LS by analysing the content on 'similarity' from the textbooks they use in their schools. They noticed that the definition of similar triangles in some textbooks as 'figures with the same shape' is not complete and agreed to adapt the textbook definition as 'figures with the same shape but different size'. In addition, the teachers analysed whether the verbal and visual diagrammatic representations used in the textbooks support understanding of the definition of similar triangles. Both defining and representing are LRT practices (Prediger, 2022) and essential aspects in geometry (Seago et al. (2014), hence called LRGT practices in this paper. While these LRTG practices became the teachers' take-off points in both lessons, they also became a source of debate as they lead teachers to consider whether to ask learners to explore the concept of similarity
scientific concepts. Although the MTF clarifies what teachers need to consider as regarding word use and justifications (colloquial and formal word use and different kinds of justifications) it does not clarify how these link with representations used, hence the need for further study and probably the contribution of this paper on LRGT. We distinguished representations as either verbal or visual (diagrammatic or concrete or symbolic); and elaborated word use as movement between the everyday register, meaning-related school mathematics register and technical or formal register. MTF links with the work of other scholars working with content specific language practices (eg. Prediger, 2022). To answer our research questions, we were interested in the LRGT practices the teachers interacted with during their lesson planning and reflection sessions, the dilemmas that arose from the discussions and how these became learning opportunities for them.

## METHODOLOGY

Each LS cycle consisted of two initial planning sessions, followed by teaching lesson 1, reflection 1 and lesson planning 2, then lesson 2, followed by reflection 2 and lesson planning 3 (see Mwadzaangati et al., 2022 for detail). In this paper we report on the LRT issues from one school by conducting content analysis on transcripts of a) lesson planning sessions; b) lessons 1 and 2; and c) reflection 1 and 2 sessions. The lesson transcriptions capture line by line utterances together with illustrations of teacher activity in the classrooms. Our analysis involved reading the transcribed data several times and dividing it into lesson episodes related to different lesson foci. In this paper we focus on episode 2 where the notion of similarity and similar triangles was introduced, as this generated substantive dilemmas amongst the teachers related to word use and representational forms. In the lesson episode, we examined the learning opportunities opened up to the learners by analysing whether word use was in the everyday, or school mathematics, or formal register, and whether the related representations were verbal (words only) or visual (concrete manipulatives or diagrams or symbols), and how these were connected. In the lesson planning and reflection discussions, we examined what the teachers talked about in relation to word use and representations, the dilemmas implied in their talk and so opportunities opened for teachers learning.

## FINDINGS

The goal in both lessons was to establish the meaning of similar triangles, in relation to congruent triangles, and the proportionality of the corresponding sides. We begin by describing the LRGT practices of word use and representations deliberated by the teachers during their initial planning and the dilemmas confronted.

### LRGT practices and dilemmas discussed during the initial lesson planning sessions

The teachers began the planning session for the LS by analysing the content on 'similarity' from the textbooks they use in their schools. They noticed that the definition of similar triangles in some textbooks as 'figures with the same shape' is not complete and agreed to adapt the textbook definition as 'figures with the same shape but different size'. In addition, the teachers analysed whether the verbal and visual diagrammatic representations used in the textbooks support understanding of the definition of similar triangles. Both defining and representing are LRT practices (Prediger, 2022) and essential aspects in geometry (Seago et al. (2014), hence called LRGT practices in this paper. While these LRTG practices became the teachers' take-off points in both lessons, they also became a source of debate as they lead teachers to consider whether to ask learners to explore the concept of similarity
before they defined it, or whether it needed to be defined first as illustrated in extract 1. Note: that Tn stands for a particular teacher in all extracts.

223 T4: Okay, the first success criteria will be discussing examples of similar objects.

224 T5: Examples?

225 T4: Yes, examples of similar objects.

226 T3: How can we start discussing examples of similar triangles before defining similar objects?

227 T4: They should be defined from the examples of similar objects that they give. We can say, these are examples of similar objects, so what do you think is the definition of similar objects? The students will answer after giving examples of similar objects and seeing some of the characteristics of those similar objects.

228 T5: For me I also think after seeing and identifying examples of similar objects then we will go to defining similar objects after defining.

In extract 1, we notice that the teachers' discussions focus on language practices of representing similar objects (e.g. 223) and word use which include naming examples of similar objects and defining the concept 'similar' (e.g. 273, 275). The teachers had different views on whether to first ask students to define similar objects or give examples before showing them visual concrete representations. T4 and T5 suggested exploring and then naming/defining (223, 225, 227), while T3 questioned if students could give examples of similar objects before knowing its definition (226). From this discussion and what followed, we observed a LRGT dilemma between having learners experience the objects and then articulate the concept and vice versa. As we will see in teaching 1 below, the teachers finally agreed to first define the concept 'similar' before showing the examples (everyday representations).

## LRGT learning opportunities opened up for learners in classroom

Teaching 1, episode 1 (which we do not show here) began with teacher asking learners to say what they know about congruent triangles (how we define these), followed by naming conditions for congruency. Episode 2 proceeded as shown in column 1, Table 1. Note that T stands for teacher, S for student, MS for many students.

As evidenced in Table 1 during teaching 1, the teacher started from colloquial language register by asking learners what they know about the word 'similar'. In the learners responses, the language register continued to move between the colloquial (same appearance, name and colour e.g. 39, 41 and 42) and school mathematics register (same shape and size e.g. 40) even with the support of everyday objects (juice boxes and plates). The definition of similar triangles given by the teacher remained in the school mathematics register too (66). We notice that the idea of first defining the concept 'similar' then showing the everyday representations did not assist to move the discussion on the definition of the concept 'similar' to formal register.
before they defined it, or whether it needed to be defined first as illustrated in extract 1. Note: that Tn stands for a particular teacher in all extracts.

223 T4: Okay, the first success criteria will be discussing examples of similar objects.

224 T5: Examples?

225 T4: Yes, examples of similar objects.

226 T3: How can we start discussing examples of similar triangles before defining similar objects?

227 T4: They should be defined from the examples of similar objects that they give. We can say, these are examples of similar objects, so what do you think is the definition of similar objects? The students will answer after giving examples of similar objects and seeing some of the characteristics of those similar objects.

228 T5: For me I also think after seeing and identifying examples of similar objects then we will go to defining similar objects after defining.

In extract 1, we notice that the teachers' discussions focus on language practices of representing similar objects (e.g. 223) and word use which include naming examples of similar objects and defining the concept 'similar' (e.g. 273, 275). The teachers had different views on whether to first ask students to define similar objects or give examples before showing them visual concrete representations. T4 and T5 suggested exploring and then naming/defining (223, 225, 227), while T3 questioned if students could give examples of similar objects before knowing its definition (226). From this discussion and what followed, we observed a LRGT dilemma between having learners experience the objects and then articulate the concept and vice versa. As we will see in teaching 1 below, the teachers finally agreed to first define the concept 'similar' before showing the examples (everyday representations).

## LRGT learning opportunities opened up for learners in classroom

Teaching 1, episode 1 (which we do not show here) began with teacher asking learners to say what they know about congruent triangles (how we define these), followed by naming conditions for congruency. Episode 2 proceeded as shown in column 1, Table 1. Note that T stands for teacher, S for student, MS for many students.

As evidenced in Table 1 during teaching 1, the teacher started from colloquial language register by asking learners what they know about the word 'similar'. In the learners responses, the language register continued to move between the colloquial (same appearance, name and colour e.g. 39, 41 and 42) and school mathematics register (same shape and size e.g. 40) even with the support of everyday objects (juice boxes and plates). The definition of similar triangles given by the teacher remained in the school mathematics register too (66). We notice that the idea of first defining the concept 'similar' then showing the everyday representations did not assist to move the discussion on the definition of the concept 'similar' to formal register.
<table><caption>Table 1: Lesson episode 2, introducing similar triangles</caption><thead><tr><th>Teaching 1</th><th>Teaching 2</th></tr></thead><tbody><tr><td><p>T: Anyone who can define or explain</p><p>What you know about the word similar?</p><p>S1: Same appearance.</p><p>S2: Same shapes.</p><p>S3: Same size.</p><p>...</p><p>What do you notice about these shapes? (Two identical box juices).</p><p>S1: Same appearance.</p><p>S2: Same shapes and size.</p><p>S3: Same colour</p><p>S5: Same name</p><p>T: Okay, what about these shapes, what do you notice?</p><p>...</p><p>Are these plates similar?</p><p>MS: Yes/No</p><p>T: Can one stand and say why these plates are similar?</p><p>S1: They have same flowers.</p><p>T: Because they have same flowers.</p><p>S2: Because they have same shape and appearance.</p><p>T: Same shape and appearance.</p><p>S3: They are not similar</p><p>T: Why are you saying they are not similar?</p><p>S3: Because similar things have the same shape and size. So, if you can look at those plates, they do not have the same size.</p><p>T: Okay, similar objects are objects with the same shape...</p></td><td><p>T: Can we have two triangles where the corresponding angles are equal but the sizes</p><p>of the triangles are different?</p><p>MS: Yes/No!</p><p>T: What if we do something like this (Pastes diagram on board).</p><p>...</p><p>T: Let us compare angles in these two triangles ABC and ALM, do we have an angle in ALM which is equal to an angle in triangle ALM?</p><p>MS: yes!</p><p>T: Which angles?</p><p>...</p><p>139. T: Now two triangles which are the same in shape and size like these pairs of triangles we said they are congruent right?</p><p>MS: Yes</p><p>T: So how do we describe these two triangles, are there any words that you can use to describe two triangles with same shape but different size?</p><p>S4: Similar triangles</p><p>T: These are similar triangles ..., triangles which have correspondingly equal angles but the sizes are different.</p></td></tr></tbody></table>

After teaching 1, the teachers met to discuss the lesson. On LRGT practice related issues, they agreed that students were able to identify characteristics of similar objects in terms of shape and that the
<table><caption>Table 1: Lesson episode 2, introducing similar triangles</caption><thead><tr><th>Teaching 1</th><th>Teaching 2</th></tr></thead><tbody><tr><td><p>T: Anyone who can define or explain</p><p>What you know about the word similar?</p><p>S1: Same appearance.</p><p>S2: Same shapes.</p><p>S3: Same size.</p><p>...</p><p>What do you notice about these shapes? (Two identical box juices).</p><p>S1: Same appearance.</p><p>S2: Same shapes and size.</p><p>S3: Same colour</p><p>S5: Same name</p><p>T: Okay, what about these shapes, what do you notice?</p><p>...</p><p>Are these plates similar?</p><p>MS: Yes/No</p><p>T: Can one stand and say why these plates are similar?</p><p>S1: They have same flowers.</p><p>T: Because they have same flowers.</p><p>S2: Because they have same shape and appearance.</p><p>T: Same shape and appearance.</p><p>S3: They are not similar</p><p>T: Why are you saying they are not similar?</p><p>S3: Because similar things have the same shape and size. So, if you can look at those plates, they do not have the same size.</p><p>T: Okay, similar objects are objects with the same shape...</p></td><td><p>T: Can we have two triangles where the corresponding angles are equal but the sizes</p><p>of the triangles are different?</p><p>MS: Yes/No!</p><p>T: What if we do something like this (Pastes diagram on board).</p><p>...</p><p>T: Let us compare angles in these two triangles ABC and ALM, do we have an angle in ALM which is equal to an angle in triangle ALM?</p><p>MS: yes!</p><p>T: Which angles?</p><p>...</p><p>139. T: Now two triangles which are the same in shape and size like these pairs of triangles we said they are congruent right?</p><p>MS: Yes</p><p>T: So how do we describe these two triangles, are there any words that you can use to describe two triangles with same shape but different size?</p><p>S4: Similar triangles</p><p>T: These are similar triangles ..., triangles which have correspondingly equal angles but the sizes are different.</p></td></tr></tbody></table>

After teaching 1, the teachers met to discuss the lesson. On LRGT practice related issues, they agreed that students were able to identify characteristics of similar objects in terms of shape and that the
teaching materials were relevant for building the meaning of the word ‘similar’. However as also evidenced in the lesson episode, the teachers were concerned about lack of elaboration on the meaning of ‘similar triangles’, and lack of linking of the everyday objects and the formal concept of similar triangles. Therefore, during planning for teaching 2, the LRGT dilemma shifted to objects for illustrating congruency and similarity and how/whether to move from similar objects (the colloquial) to similar triangles (the formal) as shown in extract 2.

55 T3: So, you can now say when we started you were saying these objects are congruent, but now what do you say about these two objects the bigger and the smaller one so maybe they can mention that they have same shape and different size.

56 T5: Will learners be able to say that the two objects are similar?

57 T3: That's what I said because when we are talking about congruency we talk about triangles. But this time we are bringing the objects that are not triangles, that's why I was saying can we try improvising triangles? So that we talk about triangles, because you can go and talk about objects, ask what they notice, and it will come back to you like in first lesson to say.

58 T5: They have the same shapes, same size, same name, same colour.

59 T2: Same height.

60 T4: Time consuming also, it will be time consuming.

61 T3: But still if they say they have same shape, same size you can now say mathematically when objects have the same shape and size, what do you call them? May be because the most important thing is for them to mention and understand these properties.

62. T2: So, in answering that question, yes, any objects can be brought but it will be convenient if we show them triangular objects first.

In extract 2, the dilemma in focus is on representations to be used for introducing similar triangles.
The concern as learnt from teaching 1, we infer, is that if the teachers show everyday representations
and ask learners to name them, they will not give the required name (58) and that it will be time
wasting as they may end up only naming unnecessary everyday properties but not mentioning the
required word ‘similar’ triangles (60). Their dilemma is how to move to the formal register, so they
agreed to first use visual triangular representations as suggested by T2 (62) then show everyday
objects later after students understand the meaning of ‘similar triangles’. As teaching 2 shows in
Table 1, the language register started and moved between the formal (e.g. can we have two triangles
where the corresponding angles are equal) and school mathematics register (e.g. but the sizes of the
triangles are different) in 62. When the students disagreed (63), he used visual diagrammatic
representations of two triangles drawn on chart paper to support illustrating how it is possible to have
two equiangular triangles with different sizes (66) and concluded the episode by giving definition of
similar triangles which was in both formal and school mathematics register (143). In the lesson
reflection on teaching 2, the teachers were happy with the way similarity was introduced and agreed
that the move to concrete triangles, and engaging with whether two different triangles could be
equiangular was enabling for learners.
teaching materials were relevant for building the meaning of the word ‘similar’. However as also evidenced in the lesson episode, the teachers were concerned about lack of elaboration on the meaning of ‘similar triangles’, and lack of linking of the everyday objects and the formal concept of similar triangles. Therefore, during planning for teaching 2, the LRGT dilemma shifted to objects for illustrating congruency and similarity and how/whether to move from similar objects (the colloquial) to similar triangles (the formal) as shown in extract 2.

55 T3: So, you can now say when we started you were saying these objects are congruent, but now what do you say about these two objects the bigger and the smaller one so maybe they can mention that they have same shape and different size.

56 T5: Will learners be able to say that the two objects are similar?

57 T3: That's what I said because when we are talking about congruency we talk about triangles. But this time we are bringing the objects that are not triangles, that's why I was saying can we try improvising triangles? So that we talk about triangles, because you can go and talk about objects, ask what they notice, and it will come back to you like in first lesson to say.

58 T5: They have the same shapes, same size, same name, same colour.

59 T2: Same height.

60 T4: Time consuming also, it will be time consuming.

61 T3: But still if they say they have same shape, same size you can now say mathematically when objects have the same shape and size, what do you call them? May be because the most important thing is for them to mention and understand these properties.

62. T2: So, in answering that question, yes, any objects can be brought but it will be convenient if we show them triangular objects first.

In extract 2, the dilemma in focus is on representations to be used for introducing similar triangles.
The concern as learnt from teaching 1, we infer, is that if the teachers show everyday representations
and ask learners to name them, they will not give the required name (58) and that it will be time
wasting as they may end up only naming unnecessary everyday properties but not mentioning the
required word 'similar' triangles (60). Their dilemma is how to move to the formal register, so they
agreed to first use visual triangular representations as suggested by T2 (62) then show everyday
objects later after students understand the meaning of 'similar triangles'. As teaching 2 shows in
Table 1, the language register started and moved between the formal (e.g. can we have two triangles
where the corresponding angles are equal) and school mathematics register (e.g. but the sizes of the
triangles are different) in 62. When the students disagreed (63), he used visual diagrammatic
representations of two triangles drawn on chart paper to support illustrating how it is possible to have
two equiangular triangles with different sizes (66) and concluded the episode by giving definition of
similar triangles which was in both formal and school mathematics register (143). In the lesson
reflection on teaching 2, the teachers were happy with the way similarity was introduced and agreed
that the move to concrete triangles, and engaging with whether two different triangles could be
equiangular was enabling for learners.
## DISCUSSION AND CONCLUSION

We have analysed extracts from teachers’ lesson planning, lessons and their reflections on introducing and defining similar triangles. We notice that the teachers paid attention to LRGT practices of word use (naming and elaborating meanings) and linking these to representations, perhaps attempting to apply what they learnt in PD about LRGT. Of interest are two dilemmas for the teachers in regard to moving between naming concepts (and so the formal register) and representations to be used to bring out the mathematical talk. The first dilemma is whether to ask learners to explore a concept before they define it or define it first then ask learners to explore it. This dilemma relates to moving between the everyday and formal registers of LRT (Hardman, 2021) and confirms the Vygotskian view that language plays an important role on linking everyday concepts to scientific concepts (Vygotsky, 1978). When the teachers started from asking for the everyday meaning of the concept ‘similar’ then show everyday representations of similar objects, the definition of ‘similar objects’ remained in the everyday and school mathematics registers. This implies that the idea of starting with everyday language and everyday visual representations only opened learning opportunities for understanding the concept ‘similar’ in everyday and school mathematics language but did not support linking these to formal language.

The second dilemma which was observed during teaching 1 reflection was about how to link representations and formal language; how/whether to use everyday representations to link to formal language or how/whether to use geometric representations to link to formal language. The teachers had opportunity to realise that the representations used in teaching 1 did not support linking to formal language, and decided to change the representations and how to introduce the concept ‘similar’. Although the teachers only thought that the everyday objects were not suitable to start with because they lead to naming unnecessary aspects, we further add that the use of the three dimensional and circular objects complicated the teaching further because these did not help to solve the highlighted challenges of clarifying the critical aspects of proportionality of sides, equal corresponding angles and scale factor in similarity (Seago et al., 2014). This was evidenced in teaching 2, whereby when the teacher introduced ‘similarity’ with visual geometric representations, the aspect of equal angles emerged and the discussion moved between the formal and school mathematics with clear link between the visuals and the formal language. In later episodes, the geometric objects were also used to introduce the aspect of proportional sides and the teacher ended the lesson by bringing in the everyday representations and learners were able to talk about their similarity as same shape and different sizes. Agreeing with Hardman (2021) that for some concepts, linking the abstract and the everyday is effective when the teacher starts from the formal such as naming and defining similar triangles, later relating these to the everyday, for example showing or naming examples of similar objects. The findings also confirm with literature that the teaching and learning of mathematics involves mediating the complex relationships among linguistic, symbolic, visual forms of representation of mathematical knowledge and in geometry these are crucial for learners understanding (Mwadzaangati et al., 2022: Seago et al., 2014).

The opportunity for teachers’ learning that the interaction between the representations and the registers is important in geometry teaching was through LS processes of planning, teaching and reflecting. The dilemmas that the teachers experienced and discussing how to work around these during LS opened the teachers’ learning opportunities about LRGT. The teachers’ long debates and
DISCUSSION AND CONCLUSION

We have analysed extracts from teachers’ lesson planning, lessons and their reflections on introducing and defining similar triangles. We notice that the teachers paid attention to LRGT practices of word use (naming and elaborating meanings) and linking these to representations, perhaps attempting to apply what they learnt in PD about LRGT. Of interest are two dilemmas for the teachers in regard to moving between naming concepts (and so the formal register) and representations to be used to bring out the mathematical talk. The first dilemma is whether to ask learners to explore a concept before they define it or define it first then ask learners to explore it. This dilemma relates to moving between the everyday and formal registers of LRT (Hardman, 2021) and confirms the Vygotskian view that language plays an important role on linking everyday concepts to scientific concepts (Vygotsky, 1978). When the teachers started from asking for the everyday meaning of the concept ‘similar’ then show everyday representations of similar objects, the definition of ‘similar objects’ remained in the everyday and school mathematics registers. This implies that the idea of starting with everyday language and everyday visual representations only opened learning opportunities for understanding the concept ‘similar’ in everyday and school mathematics language but did not support linking these to formal language.

The second dilemma which was observed during teaching 1 reflection was about how to link representations and formal language; how/whether to use everyday representations to link to formal language or how/whether to use geometric representations to link to formal language. The teachers had opportunity to realise that the representations used in teaching 1 did not support linking to formal language, and decided to change the representations and how to introduce the concept ‘similar’. Although the teachers only thought that the everyday objects were not suitable to start with because they lead to naming unnecessary aspects, we further add that the use of the three dimensional and circular objects complicated the teaching further because these did not help to solve the highlighted challenges of clarifying the critical aspects of proportionality of sides, equal corresponding angles and scale factor in similarity (Seago et al., 2014). This was evidenced in teaching 2, whereby when the teacher introduced ‘similarity’ with visual geometric representations, the aspect of equal angles emerged and the discussion moved between the formal and school mathematics with clear link between the visuals and the formal language. In later episodes, the geometric objects were also used to introduce the aspect of proportional sides and the teacher ended the lesson by bringing in the everyday representations and learners were able to talk about their similarity as same shape and different sizes. Agreeing with Hardman (2021) that for some concepts, linking the abstract and the everyday is effective when the teacher starts from the formal such as naming and defining similar triangles, later relating these to the everyday, for example showing or naming examples of similar objects. The findings also confirm with literature that the teaching and learning of mathematics involves mediating the complex relationships among linguistic, symbolic, visual forms of representation of mathematical knowledge and in geometry these are crucial for learners understanding (Mwadzaangati et al., 2022: Seago et al., 2014).

The opportunity for teachers’ learning that the interaction between the representations and the registers is important in geometry teaching was through LS processes of planning, teaching and reflecting. The dilemmas that the teachers experienced and discussing how to work around these during LS opened the teachers’ learning opportunities about LRGT. The teachers’ long debates and
dilemmas in these LRGT practices of naming and representing imply that these practices are not easy
to deal with in a single LS cycle and confirms the need for continuous PDs in LRT.

Acknowledgement

This paper is based on postdoctoral fellowship work in the Wits Maths Connect Project at University of the Witwatersrand and in collaboration with the Faculty of Education at University of Malawi.
Any opinion, conclusion or recommendation expressed in this material is that of the authors.

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Hardman, J (2021). Vygotsky's decolonial pedagogical legacy in the 21st century: Back to the future. *Mind, Culture, and Activity, 28*(3), 219–233
https://doi.org/10.1080/10749039.2021.1941116

Huang, R., Gong, Z., & Han, X. (2019). Implementing mathematics teaching that promotes students’ understanding through theory-driven lesson study. In R. Huang, A. Takahashi, & J. P. da Ponte (Eds.), *Theory and practice of lesson study in mathematics* (pp. 605–631). Springer. https://doi.org/10.1007/978-3-030-04031-4_30

MoEST. (2020). *National Education Sector Investment Plan 2020-2030*. https://planipolis.iiep.unesco.org/en/2020/national-education-sector-investment-plan-2020-2030-nesip-7178

Mwadzaangati, L., Takker, S. & Adler, J. (2022). Teacher learning about exemplification in geometry through lesson study. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.). *Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 3, pp. 219-226).
PME

Prediger, S. (2022). Enhancing language for developing conceptual understanding: A research journey connecting different research approaches. J. Hodgen, E. Geraniou, G. Bolondi, & F. Ferretti (Eds.). *Proceedings of the Twelfth Congress of the European Society for Research in Mathematics Education* (pp. 8–33) CERME.

Seago, N. M., Jacobs, J. K., Heck, D. J., Nelson, C. L., & Malzahn, K. A. (2014). Impacting teachers’ understanding of geometric similarity: Results from field testing of the learning and teaching geometry professional development materials. *Professional Development in Education, 40*(4), 627–653.
https://doi.org/10.1080/19415257.2013.830144

Vygotsky, L. S. (1978). *Mind in society: Development of higher psychological processes*. Harvard University Press.
dilemmas in these LRGT practices of naming and representing imply that these practices are not easy
to deal with in a single LS cycle and confirms the need for continuous PDs in LRT.

Acknowledgement

This paper is based on postdoctoral fellowship work in the Wits Maths Connect Project at University of the Witwatersrand and in collaboration with the Faculty of Education at University of Malawi. Any opinion, conclusion or recommendation expressed in this material is that of the authors.

References

Adler, J., & Alshwaikh, J. (2019). A case of lesson study in South Africa. In R. Huang, A. Takahashi, & J. P. da Ponte (Eds.), *Theory and practice of lesson study in mathematics* (pp. 317–342). Springer International Publishing. https://doi.org/10.1007/978-3-030-04031-4_16

Adler, J., Mwadzaangati, L., & Takker, S. (2022). From defining as assertion to defining as explaining meaning: Teachers’ learning through theory-informed lesson study. *International Journal for Lesson & Learning Studies, 12* (1), 31–51. https://doi.org/10.1108/IJLLS-02-2022-0029

Adler, J., & Ronda, E. (2015). A framework for describing mathematics discourse in instruction and interpreting differences in teaching. *African Journal of Research in Mathematics, Science and Technology Education, 19*(3), 237–254. https://doi.org/10.1080/10288457.2015.1089677

Fauskanger, J., Jakobsen, A., & Kazima, M. (2019). Malawi mathematics teacher educators’ understanding of Lesson Study. *International Journal for Lesson and Learning Studies, 8*(1), 48–59. https://doi.org/10.1108/IJLLS-06-2018-0039

Hardman, J (2021). Vygotsky's decolonial pedagogical legacy in the 21st century: Back to the future. *Mind, Culture, and Activity, 28*(3), 219–233
https://doi.org/10.1080/10749039.2021.1941116

Huang, R., Gong, Z., & Han, X. (2019). Implementing mathematics teaching that promotes students’ understanding through theory-driven lesson study. In R. Huang, A. Takahashi, & J. P. da Ponte (Eds.), *Theory and practice of lesson study in mathematics* (pp. 605–631). Springer. https://doi.org/10.1007/978-3-030-04031-4_30

MoEST. (2020). *National Education Sector Investment Plan 2020-2030*. https://planipolis.iiep.unesco.org/en/2020/national-education-sector-investment-plan-2020-2030-nesip-7178

Mwadzaangati, L., Takker, S. & Adler, J. (2022). Teacher learning about exemplification in geometry through lesson study. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.). *Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 3, pp. 219-226). PME

Prediger, S. (2022). Enhancing language for developing conceptual understanding: A research journey connecting different research approaches. J. Hodgen, E. Geraniou, G. Bolondi, & F. Ferretti (Eds.). *Proceedings of the Twelfth Congress of the European Society for Research in Mathematics Education* (pp. 8–33) CERME.

Seago, N. M., Jacobs, J. K., Heck, D. J., Nelson, C. L., & Malzahn, K. A. (2014). Impacting teachers’ understanding of geometric similarity: Results from field testing of the learning and teaching geometry professional development materials. *Professional Development in Education, 40*(4), 627–653.
https://doi.org/10.1080/19415257.2013.830144

Vygotsky, L. S. (1978). *Mind in society: Development of higher psychological processes*. Harvard University Press.
# MULTI-PERSPECTIVITY: A ‘RED THREAD’ THROUGH DISCUSSIONS ON GEOMETRY FOR TEACHING AND LEARNING

Michael Neubrand

Carl von Ossietzky University, Oldenburg, Germany

*What makes geometry so special in school mathematics? — This paper discusses a comprehensive and multi-perspective structure within one can speak about geometry in the context of teaching and learning. It starts with some fundamental aspects providing a general frame. Then, three dimensions organize the field: General views of geometry; approaches to geometry; geometrical activities. Finally, a few remarks are given how to probe the idea of multi-perspectivity.*

The 9th ICMI Study of 1995 on “Perspectives on the Teaching of Geometry for the 21st Century” (Mammana & Villani, 1998) contains several chapters (esp. Chap. 6, Chap. 7) on evolutions, changes, trends of geometry such as changes in textbooks, curricula, technology, but also discusses deeper influences caused by the epistemology of mathematics, by the learning sciences, by social changes, etc. Constantly, the issue of the multi-perspectivity of geometry was highlighted, both, for geometry as a mathematical topic, and as a subject of school mathematics. Being so rich in perspectives seems to be characteristic for geometry (cf. Herbst et al., 2018; Graumann et al., 1996).

However, can we tap this prima facie vague idea of multi-perspectivity into a coherent, overarching, more or less systematic structure?

The “model” we propose sets a general frame for thinking about geometry in the context of teaching, learning, and education (first section), it states three dimensions to describe multi-perspectivity within a broad spectrum of geometry as a school topic (second section), and finally gives a few remarks to fields of probation of multi-perspectivity (third section).

## A GENERAL FRAME FOR EDUCATIONAL THINKING ABOUT GEOMETRY: GUIDELINES AND FUNDAMENTAL ASPECTS

A still remarkable contribution is the British 2001-Report on teaching and learning geometry (The Royal Society & Joint Mathematical Council, 2001). This report does not only proceed – as to expect for a policy document – to a list of comprehensive and action related recommendations, but considers, for that reason, fundamental issues of geometry in schools, esp. for the 11-19 years old. A preface by the chairman, Adrian Oldknow, sets the tone: Being aware of the long history of the topic and of the ubiquity of geometric images, forms, models in daily live, "... geometry should be one of the easiest branches of mathematics to teach. But this is not the case ..." (RS/JMC 2001, p VII). The reasons lie in some “pitfalls”, and these root in the concept highlighted in this paper, multi-perspectivity. Thus, we must be careful. Oldknow points first to the danger of “abundance”:

> [Geometry] suffers from an embarrassment of riches in terms of theories, results, techniques and applications. [...] We might refer to this, not unwelcome, problem as one of **abundance** [bold by author]. (RS/JMC 2001, p VIII).
# MULTI-PERSPECTIVITY: A ‘RED THREAD’ THROUGH DISCUSSIONS ON GEOMETRY FOR TEACHING AND LEARNING

Michael Neubrand

Carl von Ossietzky University, Oldenburg, Germany

*What makes geometry so special in school mathematics? — This paper discusses a comprehensive and multi-perspective structure within one can speak about geometry in the context of teaching and learning. It starts with some fundamental aspects providing a general frame. Then, three dimensions organize the field: General views of geometry; approaches to geometry; geometrical activities. Finally, a few remarks are given how to probe the idea of multi-perspectivity.*

The 9th ICMI Study of 1995 on “Perspectives on the Teaching of Geometry for the 21st Century” (Mammana & Villani, 1998) contains several chapters (esp. Chap. 6, Chap. 7) on evolutions, changes, trends of geometry such as changes in textbooks, curricula, technology, but also discusses deeper influences caused by the epistemology of mathematics, by the learning sciences, by social changes, etc. Constantly, the issue of the multi-perspectivity of geometry was highlighted, both, for geometry as a mathematical topic, and as a subject of school mathematics. Being so rich in perspectives seems to be characteristic for geometry (cf. Herbst et al., 2018; Graumann et al., 1996).

However, can we tap this prima facie vague idea of multi-perspectivity into a coherent, overarching, more or less systematic structure?

The “model” we propose sets a general frame for thinking about geometry in the context of teaching, learning, and education (first section), it states three dimensions to describe multi-perspectivity within a broad spectrum of geometry as a school topic (second section), and finally gives a few remarks to fields of probation of multi-perspectivity (third section).

## A GENERAL FRAME FOR EDUCATIONAL THINKING ABOUT GEOMETRY: GUIDELINES AND FUNDAMENTAL ASPECTS

A still remarkable contribution is the British 2001-Report on teaching and learning geometry (The Royal Society & Joint Mathematical Council, 2001). This report does not only proceed – as to expect for a policy document – to a list of comprehensive and action related recommendations, but considers, for that reason, fundamental issues of geometry in schools, esp. for the 11-19 years old. A preface by the chairman, Adrian Oldknow, sets the tone: Being aware of the long history of the topic and of the ubiquity of geometric images, forms, models in daily live, "... geometry should be one of the easiest branches of mathematics to teach. But this is not the case ..." (RS/JMC 2001, p VII). The reasons lie in some “pitfalls”, and these root in the concept highlighted in this paper, multi-perspectivity. Thus, we must be careful. Oldknow points first to the danger of “abundance”:

> [Geometry] suffers from an embarrassment of riches in terms of theories, results, techniques and applications. [...] We might refer to this, not unwelcome, problem as one of **abundance** [bold by author]. (RS/JMC 2001, p VIII).
Then, however, choices are necessary (in lessons, in the curriculum), and bring along new dangers:

> At one extreme there is a danger of choosing eclectically from this abundance in a way that leads to the teaching of a lot of apparently unconnected ‘bits’. At the other extreme there is a danger of developing a tightly organised body of knowledge which addresses only a very small part of geometry. Our challenge has been to combine breadth with both educational and mathematical coherence – a problem we refer to as **coherence**. (RS/JMC 2001, p VIII)

For curriculum construction, a further problem appears. In their school career students encounter a lot of material in geometry. Besides the personal overload one seeks to avoid by establishing “coherence”, there is also the mission of the educational administration to create for a certain topic a unified content with relations between the lower and the higher Grades. Oldknow says:

> We refer to this issue as one of **progression**. (RS/JMC 2001, p VIII).

The three terms abundance, coherence, and progression mark guidelines of educational thinking about geometry. They correspond to the three fundamental aspects described below: “Abundance” is an epistemological issue, since it states characteristics of geometry. “Coherence” appeals to pedagogy, since it is oriented to the personal learning, targets the construction of meaning, shows the longer chains of connections. “Progression” addresses the didactical intention of sensemaking curricula. These three aspects indicate the special focus when talking about a mathematical topic under educational viewpoints. It really makes a difference if we are speaking about geometry in the context of a sub-domain of mathematics, or about geometry as an essential part of mathematics in school, or even more general, about geometry for the purpose of building up a reflected view of “the world”. It therefore deserves attention to separate the aspects to speak about geometry.

## The epistemological aspect

Speaking about geometry as a mathematical topic should consider the multi-perspectivity of geometry itself. Thus, studies about historical developments, on the logical foundations, on mathematical standards in geometrical work, etc. are all welcome and fruitful, but in the educational context they always have to take into the consideration the origin of geometry as a human creation and the roots of geometry in human activity, be it cognitive, even mystic, aesthetical, technical, or in relation to the wider environment.

## The pedagogical aspect

Geometry as a school subject aims at education. In Germany we call that intention “Bildung” (Neubrand & Lengnink, 2023). “Bildung” does not address the content exhibited in the curriculum alone, but targets wider connections. For any content in school, one must admit the question in how far that content can contribute to human development. The German educator Heinz-Elmar Tenorth coined it as “Kultivierung der Lernfähigkeit” (Tenorth, 1994, S. 94 ff.), i.e. *cultivating the ability to learn*, and fostering the cognitive solution to any issue in our life. In that sense, “Bildung” is a target and a corrective for teaching and learning in school, even with respect to geometry.

## The didactical aspect

In the Anglo-Saxon tradition the term Curriculum means more than just the creation of syllabuses in school; in the continental tradition we prefer speaking of “Didaktik” (Westbury et al., 2000; Blum et al., 2019). In both traditions, school geometry is not only a set of contents (as important this is); rather, implementations must contain the use of that specific knowledge in contexts whatsoever
Then, however, choices are necessary (in lessons, in the curriculum), and bring along new dangers:

> At one extreme there is a danger of choosing eclectically from this abundance in a way that leads to the teaching of a lot of apparently unconnected ‘bits’. At the other extreme there is a danger of developing a tightly organised body of knowledge which addresses only a very small part of geometry. Our challenge has been to combine breadth with both educational and mathematical coherence – a problem we refer to as **coherence**. (RS/JMC 2001, p VIII)

For curriculum construction, a further problem appears. In their school career students encounter a lot of material in geometry. Besides the personal overload one seeks to avoid by establishing “coherence”, there is also the mission of the educational administration to create for a certain topic a unified content with relations between the lower and the higher Grades. Oldknow says:

> We refer to this issue as one of **progression**. (RS/JMC 2001, p VIII).

The three terms abundance, coherence, and progression mark guidelines of educational thinking about geometry. They correspond to the three fundamental aspects described below: “Abundance” is an epistemological issue, since it states characteristics of geometry. “Coherence” appeals to pedagogy, since it is oriented to the personal learning, targets the construction of meaning, shows the longer chains of connections. “Progression” addresses the didactical intention of sensemaking curricula. These three aspects indicate the special focus when talking about a mathematical topic under educational viewpoints. It really makes a difference if we are speaking about geometry in the context of a sub-domain of mathematics, or about geometry as an essential part of mathematics in school, or even more general, about geometry for the purpose of building up a reflected view of “the world”. It therefore deserves attention to separate the aspects to speak about geometry.

## The epistemological aspect

Speaking about geometry as a mathematical topic should consider the multi-perspectivity of geometry itself. Thus, studies about historical developments, on the logical foundations, on mathematical standards in geometrical work, etc. are all welcome and fruitful, but in the educational context they always have to take into the consideration the origin of geometry as a human creation and the roots of geometry in human activity, be it cognitive, even mystic, aesthetical, technical, or in relation to the wider environment.

## The pedagogical aspect

Geometry as a school subject aims at education. In Germany we call that intention “Bildung” (Neubrand & Lengnink, 2023). “Bildung” does not address the content exhibited in the curriculum alone, but targets wider connections. For any content in school, one must admit the question in how far that content can contribute to human development. The German educator Heinz-Elmar Tenorth coined it as “Kultivierung der Lernfähigkeit” (Tenorth, 1994, S. 94 ff.), i.e. *cultivating the ability to learn*, and fostering the cognitive solution to any issue in our life. In that sense, “Bildung” is a target and a corrective for teaching and learning in school, even with respect to geometry.

## The didactical aspect

In the Anglo-Saxon tradition the term Curriculum means more than just the creation of syllabuses in school; in the continental tradition we prefer speaking of “Didaktik” (Westbury et al., 2000; Blum et al., 2019). In both traditions, school geometry is not only a set of contents (as important this is); rather, implementations must contain the use of that specific knowledge in contexts whatsoever
(science, practice and customs in trade, craft, business, industry), a wide range of possible approaches,
and the acknowledgment of the experiences of the student.

THREE DIMENSIONS TO ORGANIZE DISCOURSES OF GEOMETRY FOR TEACHING AND LEARNING

To come from these fundamental, but rather abstract aspects (as an underlying frame) to the issues of
the daily agenda of geometry in school, and still not falling behind the multi-perspectivity strived for,
we differentiate three dimensions to organize discourses on geometry. Each dimension should follow
and explicate the three fundamental aspects, i.e. the epistemological, pedagogical, didactical foci.
They are to structure multi-perspectivity.

<figure><img src="image_77.png" /><figcaption>Figure 1: Three dimensions to organize discourses of geometry for teaching and learning</figcaption></figure>

General views of geometry

Under the sub-domains of school mathematics geometry is unique in showing so many views. This
opens a wide field of possibilities, still however, aware of the danger of abundance, and recalling that
pursuing coherence is always on the agenda. What one can learn is that none of the so may views
should be inferior or of less value, and equally, none of the views can stand alone. This stands behind
the statement from above that the problem of abundance is “not unwelcome” (RS/JMC 2001, p VIII).
The issue of multi-perspectivity of geometry itself can be traced back for long periods; an early source
is Artmann (1979), a recent witness of the idea, albeit with another focus, is Kusniak (2018); cf. also
the whole book of Herbst et al. (2018).

Here is a list, not claiming to be complete. Geometry can be viewed ...

* ... as a “ready to use” body of knowledge.

Of course, this view is, as a background, present whenever we discuss about geometry (as it would
be for all other parts of school mathematics).

* ... as field which gives a blueprint of “doing mathematics”.

In no other field of school mathematics, experiences of doing mathematics seem as accessible as in
geometry. There is the long history, but the central reason is that in geometry the way to the abstract
theory is not as technically demanding, as in some other fields (say, e.g., calculus). Geometry has a
wide range of theories; it could be formal, but there could be also, still strong and serious, theories
keeping open the appeal to the practices (Bender & Schreiber, 1980). One even can claim that
geometry is in itself a model for mathematics. Benno Artmann (1979) called it by the German word
“Vorbild” (literally “preset picture”, meaning something like a “template”); we list later some
(science, practice and customs in trade, craft, business, industry), a wide range of possible approaches,
and the acknowledgment of the experiences of the student.

THREE DIMENSIONS TO ORGANIZE DISCOURSES OF GEOMETRY FOR TEACHING AND LEARNING

To come from these fundamental, but rather abstract aspects (as an underlying frame) to the issues of
the daily agenda of geometry in school, and still not falling behind the multi-perspectivity strived for,
we differentiate three dimensions to organize discourses on geometry. Each dimension should follow
and explicate the three fundamental aspects, i.e. the epistemological, pedagogical, didactical foci.
They are to structure multi-perspectivity.

<figure><img src="image_83.png" /><figcaption>Figure 1: Three dimensions to organize discourses of geometry for teaching and learning</figcaption></figure>

General views of geometry

Under the sub-domains of school mathematics geometry is unique in showing so many views. This
opens a wide field of possibilities, still however, aware of the danger of abundance, and recalling that
pursuing coherence is always on the agenda. What one can learn is that none of the so may views
should be inferior or of less value, and equally, none of the views can stand alone. This stands behind
the statement from above that the problem of abundance is “not unwelcome” (RS/JMC 2001, p VIII).
The issue of multi-perspectivity of geometry itself can be traced back for long periods; an early source
is Artmann (1979), a recent witness of the idea, albeit with another focus, is Kusniak (2018); cf. also
the whole book of Herbst et al. (2018).

Here is a list, not claiming to be complete. Geometry can be viewed ...

* ... as a “ready to use” body of knowledge.

Of course, this view is, as a background, present whenever we discuss about geometry (as it would be for all other parts of school mathematics).

* ... as field which gives a blueprint of “doing mathematics”.

In no other field of school mathematics, experiences of doing mathematics seem as accessible as in
geometry. There is the long history, but the central reason is that in geometry the way to the abstract
theory is not as technically demanding, as in some other fields (say, e.g., calculus). Geometry has a
wide range of theories; it could be formal, but there could be also, still strong and serious, theories
keeping open the appeal to the practices (Bender & Schreiber, 1980). One even can claim that
geometry is in itself a model for mathematics. Benno Artmann (1979) called it by the German word
“Vorbild” (literally “preset picture”, meaning something like a “template”); we list later some
mathematical activities having authentical blueprints in geometry. Since one can hold the informal
level for long, geometry is a good area for “speaking about mathematics” (Neubrand, 2000).

* ... as a rich source of problems of a big variety.

Geometry opens a lot of possibilities of problems of different characters and wide-ranging difficulties. It goes from puzzles to severe problems. However, geometry problems often claim not only for a local solution but for embedding the problems into wider connections. This, among other reasons, is the potential of the classical (Euclidean) triangle geometry.

* ... as a basis to describe, plan, construct, realize technical equipment.

This is a very specific aspect of geometry. In the real world lots of questions with geometrical background must be tackled: Streets, tunnels, ramps, bridges; buildings; gears; etc. Geometry is not only in drawing and construction but in the deeper questions like stability or directions of forces. Geometry aims at understanding, not just description. Hahn (2012) gives many examples.

* ... as a basis to understand the space we live in.

Going beyond just constructing practical things, geometry enables us to conceptualize what we see
around us. In school we should use the full range of this view, from local orientation (maps, schematic
plans), to the geometry which guides us through the environment we live in (the neighborhood, the
earth, including weather, spreading of pollution), up to the space.

* ... as a cultural achievement, as a product of the development of mankind.

This is not meant as a source of anectodical stories alone. It strives what was said before under the pedagogical aspect (“Bildung”): Human development is in no field of school mathematics as clear as in geometry. Geometry is the origin of mathematics in the cultural history. The Latin term “more geometrico” is in the Western culture a metaphor for stringent thinking. But geometry is universal: I just point to Fukagawa & Rothman (2008; Japan) or to Gerdes (2010; Africa).

* ... as a rich supplier of forms for observation, interpretation, creation: visualization.

One can assume that this view of geometry is often unattended even neglected in schools, since there seem to be too less paths into the formal reign of mathematics. The opposite, however, is true. Geometry is a fundament, but also vice versa a product of visualization. Visualization as a concept strives aesthetics (Sinclair et al., 2007), it contains many far-reaching relations to arts (perspectivity, symmetries), it reaches out until the principles of the technical or even the evolutionary development (Hildebrandt & Tromba, 1996). Whatsoever, teaching and learning geometry should exploit that big stock of information, motivation, and launching platforms; see National Research Council (2006) for many suggestions.

**Approaches to geometry**

Multi-perspectivity in the general views opens multi-perspectivity in the approaches. No single approach fits it all or should be dominant; thus, teachers should be aware of the many possibilities. Each approach has its own dialectic. It can work for some students, or some topics; the didactical situation in the class calls for local decisions. And essentially, the term approach is too narrowly understood if we think about it only as a matter of motivation. The three dimensions influence each other. The approaches are so rich since geometry is so rich of views. The epistemological aspect triggers pedagogical orientations and didactical decisions.
mathematical activities having authentical blueprints in geometry. Since one can hold the informal
level for long, geometry is a good area for “speaking about mathematics” (Neubrand, 2000).

* ... as a rich source of problems of a big variety.

Geometry opens a lot of possibilities of problems of different characters and wide-ranging difficulties. It goes from puzzles to severe problems. However, geometry problems often claim not only for a local solution but for embedding the problems into wider connections. This, among other reasons, is the potential of the classical (Euclidean) triangle geometry.

* ... as a basis to describe, plan, construct, realize technical equipment.

This is a very specific aspect of geometry. In the real world lots of questions with geometrical
background must be tackled: Streets, tunnels, ramps, bridges; buildings; gears; etc. Geometry is not
only in drawing and construction but in the deeper questions like stability or directions of forces.
Geometry aims at understanding, not just description. Hahn (2012) gives many examples.

* ... as a basis to understand the space we live in.

Going beyond just constructing practical things, geometry enables us to conceptualize what we see
around us. In school we should use the full range of this view, from local orientation (maps, schematic
plans), to the geometry which guides us through the environment we live in (the neighborhood, the
earth, including weather, spreading of pollution), up to the space.

* ... as a cultural achievement, as a product of the development of mankind.

This is not meant as a source of anectodical stories alone. It strives what was said before under the
pedagogical aspect (“Bildung”): Human development is in no field of school mathematics as clear as
in geometry. Geometry is the origin of mathematics in the cultural history. The Latin term “more
geometrico” is in the Western culture a metaphor for stringent thinking. But geometry is universal: I
just point to Fukagawa & Rothman (2008; Japan) or to Gerdes (2010; Africa).

* ... as a rich supplier of forms for observation, interpretation, creation: visualization.

One can assume that this view of geometry is often unattended even neglected in schools, since there seem to be too less paths into the formal reign of mathematics. The opposite, however, is true. Geometry is a fundament, but also vice versa a product of visualization. Visualization as a concept strives aesthetics (Sinclair et al., 2007), it contains many far-reaching relations to arts (perspectivity, symmetries), it reaches out until the principles of the technical or even the evolutionary development (Hildebrandt & Tromba, 1996). Whatsoever, teaching and learning geometry should exploit that big stock of information, motivation, and launching platforms; see National Research Council (2006) for many suggestions.

**Approaches to geometry**

Multi-perspectivity in the general views opens multi-perspectivity in the approaches. No single
approach fits it all or should be dominant; thus, teachers should be aware of the many possibilities.
Each approach has its own dialectic. It can work for some students, or some topics; the didactical
situation in the class calls for local decisions. And essentially, the term approach is too narrowly
understood if we think about it only as a matter of motivation. The three dimensions influence each
other. The approaches are so rich since geometry is so rich of views. The epistemological aspect
triggers pedagogical orientations and didactical decisions.
Geometry can be approached (in the class, and equally by the individuals) ...

* ... by the relations to reality.

The condition, however, is to keep open that reality has many categories. It could start with the direct contact to things like the closer or wider environment, technical devices, facts from biology, architecture, etc. However, any approach needs goals going farther than motivation in the beginning. From the pedagogical aspect, an approach by reality should lead to deeper understanding. The German mathematics educator Heinrich Winter took the word “sublimation” to indicate that teaching geometry for “Bildung” should foster the ability to articulate the structural aspects of the world around us, it should make sensible for visual perception, transfer our observations conceptually, including a reflective attitude (Winter, 1997, p 29; my free translation).

* ... by the disclosure of the inner connections, by the wish to master a certain topic.

Too often approaches to geometry are thought of as to come from outside. But approaching can also come from inside. Wishing coherence is a universal human attitude. Thus, the drive to logical order, the wish to explain, in the sequel even to prove something, is not necessarily a sign for a “ready-to-use” geometry but can be guide and generate progress. It requires a metacognitive attitude in the class (Kaune, 2006), which can indeed be well realized by geometrical topics (e.g. Neubrand, 2000: the systematization of the set of quadrilaterals). Similarly, mastering something is human as well. But sheer repetition is not enough, practicing needs connection (see many papers of Erich Wittmann, 2021). It can be quite plain in geometry. Here is an “integrated exercise”: Look at all the various intersection this figure bears. What lengths of segments do appear?

<figure><img src="image_78.png" /><figcaption>Figure 2. A circle, an equilateral triangle, a square – and many intersections</figcaption></figure>

* ... by using materials, by handling geometrical devices, by measuring with instruments, by reflecting digital systems.

This approach is not a plea to return to the “old” ruler-and-compass times. The essence is that these approaches come via the material manifestation of geometrical concepts. A striking example is the phenomenon of “touching”, a concept reaching far into higher mathematics. Approaching geometry by devices and instruments is, furthermore, not meant as a subordination of digital approaches; rather, it claims for reflections about the differences, the advantages, and the pitfalls of each approach. The idea of “touching” is once more a good example: it calls for construction in the ruler-and-compass world, and often sticks to zoom-in / zoom-out strategies in the digital programs.

* ... through curiosity, exploration, investigation, by seeking for understanding.

Since geometry is as rich in views and contexts as described, individual approaches must be valued. Nothing is more personal than curiosity. Thus, geometry is greatly welcome as an extraordinary field
Geometry can be approached (in the class, and equally by the individuals) ...

* ... by the relations to reality.

The condition, however, is to keep open that reality has many categories. It could start with the direct contact to things like the closer or wider environment, technical devices, facts from biology, architecture, etc. However, any approach needs goals going farther than motivation in the beginning. From the pedagogical aspect, an approach by reality should lead to deeper understanding. The German mathematics educator Heinrich Winter took the word “sublimation” to indicate that teaching geometry for “Bildung” should foster the ability to articulate the structural aspects of the world around us, it should make sensible for visual perception, transfer our observations conceptually, including a reflective attitude (Winter, 1997, p 29; my free translation).

* ... by the disclosure of the inner connections, by the wish to master a certain topic.

Too often approaches to geometry are thought of as to come from outside. But approaching can also come from inside. Wishing coherence is a universal human attitude. Thus, the drive to logical order, the wish to explain, in the sequel even to prove something, is not necessarily a sign for a “ready-to-use” geometry but can be guide and generate progress. It requires a metacognitive attitude in the class (Kaune, 2006), which can indeed be well realized by geometrical topics (e.g. Neubrand, 2000: the systematization of the set of quadrilaterals). Similarly, mastering something is human as well. But sheer repetition is not enough, practicing needs connection (see many papers of Erich Wittmann, 2021). It can be quite plain in geometry. Here is an “integrated exercise”: Look at all the various intersection this figure bears. What lengths of segments do appear?

<figure><img src="image_84.png" /><figcaption>Figure 2. A circle, an equilateral triangle, a square – and many intersections</figcaption></figure>

* ... by using materials, by handling geometrical devices, by measuring with instruments, by reflecting digital systems.

This approach is not a plea to return to the “old” ruler-and-compass times. The essence is that these approaches come via the material manifestation of geometrical concepts. A striking example is the phenomenon of “touching”, a concept reaching far into higher mathematics. Approaching geometry by devices and instruments is, furthermore, not meant as a subordination of digital approaches; rather, it claims for reflections about the differences, the advantages, and the pitfalls of each approach. The idea of “touching” is once more a good example: it calls for construction in the ruler-and-compass world, and often sticks to zoom-in / zoom-out strategies in the digital programs.

* ... through curiosity, exploration, investigation, by seeking for understanding.

Since geometry is as rich in views and contexts as described, individual approaches must be valued. Nothing is more personal than curiosity. Thus, geometry is greatly welcome as an extraordinary field
in which exploration and investigation can be developed within a huge variety of difficulty levels, contexts, interests, etc. This makes geometry special, and in this way one can find the strongest signals that understanding is the final aim in school.

## Geometrical activities

The variety of approaches together with the pedagogical idea that learning is idiosyncratic in its nature produce the claim: Geometry becomes vivid if the students do it. There are many ways, but again no single activity marks the king's road. The more activities are done, the more options appear. One cannot do too much, rather it's dangerous to ignore or disregard possible activities.

Geometry allows activities like ...

* ... doing geometry by hand.

Sometimes one considers hands-on activities as the origin of thinking. At least in geometry we have a rich scale for that: folding, cutting, gluing, rolling, assembling, moving, etc. The impetus, then, to reflect these activities leads to understanding. It begins with noticing, it can continue with abbreviations, replacing an action symbolically, using a specific technique, etc., and on each stage with giving reasons for that what was done.

* ... drawing (with mechanical instruments, and within digital systems).

This is geometry's specialty. There should not be no verdict that the one is the more valuable than the other. Both represent geometry on the level of creating visible products and considering their manipulation.

* ... using numbers and calculation, realizing geometry with numbers.

Utilizing numbers, variables, formulas in the geometric context has many facets. It starts as early as with the geometric interpretation of numerical operations, patterns, and graphical manipulations; measurement is the next step. In this view, numbers and formulas express geometrical relations. However, there is also the other way round: Geometrical ideas contribute to the creation of analytic techniques: A “good model” of geometry requires realizations of concepts like location, distance, angles, volumes, and therefore one needs more than just the coordinates to build up Analytic Geometry (and, by that way, the door becomes open to generalize to higher dimensions). Anyway, geometry and numbers form a productive coexistence.

* ... visualization.

The central human activity of seeing, i.e. using the eyes, is specific for geometry. The activity of “look at and see” should be kept open in geometry lessons as long as possible. But then, two sides can be stated: Geometry calls for assuring oneself that the seen is what really happened; from there the road is open into argumentation at various levels. On the other side, geometry provides an arsenal for the active visualizing of facts, relations, operations. I called that double nature of visualization “contemplative” vs. “active” (Neubrand, 1987). In the era of digitalization, visualization becomes ubiquitous, and hence it increasingly plays its role in geometry education. Being aware of aesthetical categories forms a background for all that. (Sinclair et al., 2007).

* ... all the many typical mathematical working activities.

We already pointed to geometry as a “blueprint of mathematics”. Here is a list of what can happen authentically when teaching and learning geometry: clarifying of phenomena; ordering; establishing
in which exploration and investigation can be developed within a huge variety of difficulty levels, contexts, interests, etc. This makes geometry special, and in this way one can find the strongest signals that understanding is the final aim in school.

## Geometrical activities

The variety of approaches together with the pedagogical idea that learning is idiosyncratic in its nature produce the claim: Geometry becomes vivid if the students do it. There are many ways, but again no single activity marks the king's road. The more activities are done, the more options appear. One cannot do too much, rather it's dangerous to ignore or disregard possible activities.

Geometry allows activities like ...

* ... doing geometry by hand.

Sometimes one considers hands-on activities as the origin of thinking. At least in geometry we have a rich scale for that: folding, cutting, gluing, rolling, assembling, moving, etc. The impetus, then, to reflect these activities leads to understanding. It begins with noticing, it can continue with abbreviations, replacing an action symbolically, using a specific technique, etc., and on each stage with giving reasons for that what was done.

* ... drawing (with mechanical instruments, and within digital systems).

This is geometry's specialty. There should not be no verdict that the one is the more valuable than the other. Both represent geometry on the level of creating visible products and considering their manipulation.

* ... using numbers and calculation, realizing geometry with numbers.

Utilizing numbers, variables, formulas in the geometric context has many facets. It starts as early as with the geometric interpretation of numerical operations, patterns, and graphical manipulations; measurement is the next step. In this view, numbers and formulas express geometrical relations. However, there is also the other way round: Geometrical ideas contribute to the creation of analytic techniques: A “good model” of geometry requires realizations of concepts like location, distance, angles, volumes, and therefore one needs more than just the coordinates to build up Analytic Geometry (and, by that way, the door becomes open to generalize to higher dimensions). Anyway, geometry and numbers form a productive coexistence.

* ... visualization.

The central human activity of seeing, i.e. using the eyes, is specific for geometry. The activity of “look at and see” should be kept open in geometry lessons as long as possible. But then, two sides can be stated: Geometry calls for assuring oneself that the seen is what really happened; from there the road is open into argumentation at various levels. On the other side, geometry provides an arsenal for the active visualizing of facts, relations, operations. I called that double nature of visualization “contemplative” vs. “active” (Neubrand, 1987). In the era of digitalization, visualization becomes ubiquitous, and hence it increasingly plays its role in geometry education. Being aware of aesthetical categories forms a background for all that. (Sinclair et al., 2007).

* ... all the many typical mathematical working activities.

We already pointed to geometry as a “blueprint of mathematics”. Here is a list of what can happen authentically when teaching and learning geometry: clarifying of phenomena; ordering; establishing
relationships; uncovering hidden relations; going beyond what is known; broadening the horizon, generalizing, search for a thorough thought through an entire area, etc. These all are typical mathematical activities. It is up to the teacher to use that variety, and to reflect upon it (Neubrand, 2000), and geometry provides a rich substratum to let it grow.

## FIELDS OF PROBATION OF MULTI-PERSPECTIVITY

Multi-perspectivity must show its value as an overarching concept in several respects. Of course, curricula should reflect the many aspects a topic provides. Today, we often construct curricula along the competencies required for that field (see Hattermann et al., 2023, for the German way to do it in geometry). Then, multi-perspectivity calls for considerations beyond the common content-process dichotomy, since it respects the deeper aspects of the content itself. Not at least, multi-perspectivity is also an issue in setting up empirical studies for assessment, in the small (Neubrand, 2021) and the large scale (Neubrand, 2013). And above all, multi-perspectivity is a concept important enough to be an essential part of teacher education.

## References

* Artmann, B. (1979). *Elementargeometrie. Vorlesungsskript WS 1978/79*. Darmstadt: Technische Hochschule.

* Bender, P., & Schreiber, A. (1980). The principle of operative concept formation in geometry teaching. *Educational Studies in Mathematics* 11 (1), 59-90 (1980). https://doi.org/10.1007/BF00369160.

* Blum, W., Artigue, M., Mariotti, M., Sträßer, R., & Van den Heuvel-Panhuizen, M. (Eds) (2019). *European Traditions in Didactics of Mathematics*. Cham: Springer. https://doi.org/10.1007/978-3-030-05514-1.

* Fukagawa, H., & Rothman, T. (2008). *Sacred Mathematics: Japanese Temple Geometry*. Princeton: Princeton University Press. https://doi.org/10.2307/j.ctv1pzk68d.

* Gerdes, P. (2010). Exploration of technologies, emerging from African cultural practices, in mathematics (teacher) education. *ZDM Mathematics Education* 42 (1), 11-17. https://doi.org/10.1007/s11858-009-0208-2.

* Graumann, G., Hölzl, R., Krainer, K., Neubrand, M., & Struve, H. (1996). Tendenzen der Geometriedidaktik der letzten 20 Jahre. *Journal für Mathematik-Didaktik* 17 (3-4), 163-237. https://doi.org/10.1007/BF03338831.

* Hahn, A. (2012). *Mathematical Excursions to the World's Great Buildings*. Princeton University Press. https://doi.org/10.1515/9781400841998.

* Hattermann, M., Rezat, S., & Sträßer, R. (2023). Geometrie: Leitidee Raum und Form. In: R. Bruder et al. (Eds.), *Handbuch der Mathematikdidaktik, 2. Edition* (pp 201-242). Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-662-66604-3_7.

* Herbst, P., Cheah, U., Richard, P., & Jones, K. (Eds.) (2018). *International Perspectives on the Teaching and Learning of Geometry in Secondary Schools*. Cham: Springer. https://doi.org/10.1007/978-3-319-77476-3.

* Hildebrandt, S., & Tromba, A. (1996). *The Parsimonious Universe: Shape and Form in the Natural World*. Copernicus /Springer.

* Kaune, C. (2006). Reflection and metacognition in mathematics education: Tools for the improvement of teaching quality. *ZDM - Zentralblatt für Didaktik der Mathematik* 38(4), 350-360. https://doi.org/10.1007/BF02652795.

* Kuzniak, A. (2018). Thinking about the teaching of geometry through the lens of the theory of geometric working spaces. In: P. Herbst et al. (Eds.), *International Perspectives on the Teaching and Learning of Geometry in Secondary Schools* (pp 5-21). Cham: Springer. https://doi.org/10.1007/978-3-319-77476-3_2.
relationships; uncovering hidden relations; going beyond what is known; broadening the horizon, generalizing, search for a thorough thought through an entire area, etc. These all are typical mathematical activities. It is up to the teacher to use that variety, and to reflect upon it (Neubrand, 2000), and geometry provides a rich substratum to let it grow.

## FIELDS OF PROBATION OF MULTI-PERSPECTIVITY

Multi-perspectivity must show its value as an overarching concept in several respects. Of course, curricula should reflect the many aspects a topic provides. Today, we often construct curricula along the competencies required for that field (see Hattermann et al., 2023, for the German way to do it in geometry). Then, multi-perspectivity calls for considerations beyond the common content-process dichotomy, since it respects the deeper aspects of the content itself. Not at least, multi-perspectivity is also an issue in setting up empirical studies for assessment, in the small (Neubrand, 2021) and the large scale (Neubrand, 2013). And above all, multi-perspectivity is a concept important enough to be an essential part of teacher education.

## References

* Artmann, B. (1979). *Elementargeometrie. Vorlesungsskript WS 1978/79*. Darmstadt: Technische Hochschule.

* Bender, P., & Schreiber, A. (1980). The principle of operative concept formation in geometry teaching. *Educational Studies in Mathematics* 11 (1), 59-90 (1980). https://doi.org/10.1007/BF00369160.

* Blum, W., Artigue, M., Mariotti, M., Sträßer, R., & Van den Heuvel-Panhuizen, M. (Eds) (2019). *European Traditions in Didactics of Mathematics*. Cham: Springer. https://doi.org/10.1007/978-3-030-05514-1.

* Fukagawa, H., & Rothman, T. (2008). *Sacred Mathematics: Japanese Temple Geometry*. Princeton: Princeton University Press. https://doi.org/10.2307/j.ctv1pzk68d.

* Gerdes, P. (2010). Exploration of technologies, emerging from African cultural practices, in mathematics (teacher) education. *ZDM Mathematics Education* 42 (1), 11-17. https://doi.org/10.1007/s11858-009-0208-2.

* Graumann, G., Hölzl, R., Krainer, K., Neubrand, M., & Struve, H. (1996). Tendenzen der Geometriedidaktik der letzten 20 Jahre. *Journal für Mathematik-Didaktik* 17 (3-4), 163-237. https://doi.org/10.1007/BF03338831.

* Hahn, A. (2012). *Mathematical Excursions to the World's Great Buildings*. Princeton University Press. https://doi.org/10.1515/9781400841998.

* Hattermann, M., Rezat, S., & Sträßer, R. (2023). Geometrie: Leitidee Raum und Form. In: R. Bruder et al. (Eds.), *Handbuch der Mathematikdidaktik, 2. Edition* (pp 201-242). Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-662-66604-3_7.

* Herbst, P., Cheah, U., Richard, P., & Jones, K. (Eds.) (2018). *International Perspectives on the Teaching and Learning of Geometry in Secondary Schools*. Cham: Springer. https://doi.org/10.1007/978-3-319-77476-3.

* Hildebrandt, S., & Tromba, A. (1996). *The Parsimonious Universe: Shape and Form in the Natural World*. Copernicus /Springer.

* Kaune, C. (2006). Reflection and metacognition in mathematics education: Tools for the improvement of teaching quality. *ZDM - Zentralblatt für Didaktik der Mathematik* 38(4), 350-360. https://doi.org/10.1007/BF02652795.

* Kuzniak, A. (2018). Thinking about the teaching of geometry through the lens of the theory of geometric working spaces. In: P. Herbst et al. (Eds.), *International Perspectives on the Teaching and Learning of Geometry in Secondary Schools* (pp 5-21). Cham: Springer. https://doi.org/10.1007/978-3-319-77476-3_2.
Mammana, C., & Villani, V. (Eds.). (1998). *Perspectives on the teaching of Geometry for the 21st century. An ICMI Study*. Dordrecht: Kluwer. https://doi.org/10.1007/978-94-011-5226-6.

National Research Council (2006). *Learning to Think Spatially*. The National Academies Press. https://doi.org/10.17226/11019 / http://www.nap.edu.

Neubrand, M. (1987). Visualisieren: Beispiele zum darstellenden und operativen Charakter. *Der Mathematik-unterricht* 33 (4), 30-36.

Neubrand, M. (2000). Reflecting as a Didaktik construction: Speaking about mathematics in the mathematics classroom. In: I. Westbury, et al. (Eds.), *Teaching as a Reflective Practice: The German Didaktik Tradition* (pp 251-265). Lawrence Erlbaum Associates.

Neubrand, M. (2013). PISA mathematics in Germany: Extending the conceptual framework to enable a more differentiated assessment. In: M. Prenzel, M. Kobarg, K. Schöps & S. Rönnebeck (Eds.), *Research on PISA: Research outcomes of the PISA Research Conference 2009* (pp 39-49). Science+Business Media. https://doi.org/10.1007/978-94-007-4458-5_3.

Neubrand, M. (2021). Geometrie und geometrisches Denken: Orientierungen für den Inhalt und die didaktische Gestaltung des Geometrieunterrichts. In: É. Vásárhelyi & J. Sjuts (Hrsg.), *Theoretische und empirische Analysen zum geometrischen Denken* (pp 243-272). Münster: WTM-Verlag. https://doi.org/10.37626/GA9783959872003.0.14.

Neubrand, M., & Lengnink, K. (2023). Bildungstheoretische Grundlagen des Mathematikunterrichts. In: R. Bruder et al. (Eds.), *Handbuch der Mathematikdidaktik, 2. Edition* (pp 57-82). Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-662-66604-3_3.

Sinclair, N., Pimm, D., & Higginson, W. (Eds.) (2007). *Mathematics and the Aesthetic: New Approaches to an Ancient Affinity*. New York: Springer. https://doi.org/10.1007/978-0-387-38145-9.

Tenorth, H.-E. (1994). „Alle alles zu lehren“: Möglichkeiten und Perspektiven Allgemeiner Bildung. Darmstadt: Wissenschaftliche Buchgesellschaft.

The Royal Society & Joint Mathematical Council [RS/JMC] (2001). *Teaching and Learning geometry 11-19*. *Report of a Royal Society / Joint Mathematical Council working group*. The Royal Society. https://royalsociety.org/topics-policy/publications/2001/teaching-geometry/.

Westbury, I., Hopmann, St., & Riquarts, K. (Eds.) (2000). *Teaching as a Reflective Practice: The German Didaktik Tradition*. Lawrence Erlbaum Associates. https://doi.org/10.4324/9780203357781.

Winter, H. (1997). Mathematik als Schule der Anschauung oder: Allgemeinbildung im Mathematikunterricht des Gymnasiums. In: R. Biehler & H.-N. Jahnke (Eds.), *Mathematische Allgemeinbildung in der Kontroverse: Materialien eines Symposiums am 24. Juni 1996 im ZiF der Universität Bielefeld* (IDM Occasional Paper Nr. 163). Bielefeld: IDM - Institut für Didaktik der Mathematik der Universität.

Wittmann, E. (2021). *Connecting Mathematics and Mathematics Education. Collected Papers on Mathematics Education as a Design Science*. Springer. https://doi.org/10.1007/978-3-030-61570-3.
Mammana, C., & Villani, V. (Eds.). (1998). *Perspectives on the teaching of Geometry for the 21st century. An ICMI Study*. Dordrecht: Kluwer. https://doi.org/10.1007/978-94-011-5226-6.

National Research Council (2006). *Learning to Think Spatially*. The National Academies Press. https://doi.org/10.17226/11019 / http://www.nap.edu.

Neubrand, M. (1987). Visualisieren: Beispiele zum darstellenden und operativen Charakter. *Der Mathematik-unterricht* 33 (4), 30-36.

Neubrand, M. (2000). Reflecting as a Didaktik construction: Speaking about mathematics in the mathematics classroom. In: I. Westbury, et al. (Eds.), *Teaching as a Reflective Practice: The German Didaktik Tradition* (pp 251-265). Lawrence Erlbaum Associates.

Neubrand, M. (2013). PISA mathematics in Germany: Extending the conceptual framework to enable a more differentiated assessment. In: M. Prenzel, M. Kobarg, K. Schöps & S. Rönnebeck (Eds.), *Research on PISA: Research outcomes of the PISA Research Conference 2009* (pp 39-49). Science+Business Media. https://doi.org/10.1007/978-94-007-4458-5_3.

Neubrand, M. (2021). Geometrie und geometrisches Denken: Orientierungen für den Inhalt und die didaktische Gestaltung des Geometrieunterrichts. In: É. Vásárhelyi & J. Sjuts (Hrsg.), *Theoretische und empirische Analysen zum geometrischen Denken* (pp 243-272). Münster: WTM-Verlag. https://doi.org/10.37626/GA9783959872003.0.14.

Neubrand, M., & Lengnink, K. (2023). Bildungstheoretische Grundlagen des Mathematikunterrichts. In: R. Bruder et al. (Eds.), *Handbuch der Mathematikdidaktik, 2. Edition* (pp 57-82). Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-662-66604-3_3.

Sinclair, N., Pimm, D., & Higginson, W. (Eds.) (2007). *Mathematics and the Aesthetic: New Approaches to an Ancient Affinity*. New York: Springer. https://doi.org/10.1007/978-0-387-38145-9.

Tenorth, H.-E. (1994). „Alle alles zu lehren“: Möglichkeiten und Perspektiven Allgemeiner Bildung. Darmstadt: Wissenschaftliche Buchgesellschaft.

The Royal Society & Joint Mathematical Council [RS/JMC] (2001). *Teaching and Learning geometry 11-19*. *Report of a Royal Society / Joint Mathematical Council working group*. The Royal Society. https://royalsociety.org/topics-policy/publications/2001/teaching-geometry/.

Westbury, I., Hopmann, St., & Riquarts, K. (Eds.) (2000). *Teaching as a Reflective Practice: The German Didaktik Tradition*. Lawrence Erlbaum Associates. https://doi.org/10.4324/9780203357781.

Winter, H. (1997). Mathematik als Schule der Anschauung oder: Allgemeinbildung im Mathematikunterricht des Gymnasiums. In: R. Biehler & H.-N. Jahnke (Eds.), *Mathematische Allgemeinbildung in der Kontroverse: Materialien eines Symposiums am 24. Juni 1996 im ZiF der Universität Bielefeld* (IDM Occasional Paper Nr. 163). Bielefeld: IDM - Institut für Didaktik der Mathematik der Universität.

Wittmann, E. (2021). *Connecting Mathematics and Mathematics Education. Collected Papers on Mathematics Education as a Design Science*. Springer. https://doi.org/10.1007/978-3-030-61570-3.
# HOW TO TEACH GEOMETRY IN CONTINUITY ALONG SCHOOLING?

Marie-Jeanne Perrin-Glorian¹, Anne-Cécile Mathé², Valentina Celi³, Caroline Bulf ⁴

¹ LDAR, Université Paris-Cité, ² ACTé, Université Clermont Auvergne, ³ LaB-E3D (EA 7441), Université de Bordeaux, ⁴ LaB-E3D (EA 7441), Université de Bordeaux.

*This article reports on research, in the French context, into a possible continuum of teaching and learning plane geometry along compulsory schooling. It is based on an epistemological analysis of content, Duval's work on semiotic registers, and Brousseau's Theory of Didactic Situations, using a didactic engineering methodology.*

The ICME-13 survey (Sinclair et al., 2016) shows that there is currently little research that looks at both student learning in geometry and the teaching and training of teachers from the beginning of primary school through the middle of secondary school, when the use of mathematical evidence is expected. However, the plane geometry taught in compulsory education in France, as in many other countries, includes two main types of geometry, based on very different reference mathematical practices: a “material geometry” based on construction and validation with material tools and a “theoretical geometry” based on definitions and proof. Many studies have shown that supporting students in this evolution of practice is a real difficulty for teachers (Perrin-Glorian & Salin, 2010). Over the last twenty years, research has developed in France to reflect on the teaching of geometry in a continuity from grade 1 to grade 8 and to provide tools for teachers (Mathé et al., 2020; Guille-Biel Winder & Assude, 2023). Our aim is to question what these taught geometries cover, their epistemological foundations and their didactic stakes, and to search for a possible coherent progression of geometry throughout compulsory schooling. This research takes up some of the issues raised in Perrin-Glorian and Salin (2010) in a direction indicated by Sinclair et al. (2016). We restrict ourselves here to the field of plane geometry.

## THEORETICAL BACKGROUND

### Material geometry and theoretical geometry

Geometry encompasses several types of practice with different epistemological foundations. In the context of plane geometry, we call “material geometry” that geometry whose objective is practical in the graphical space of 2D drawings (freehand or constructed with geometric instruments): studying or tracing material geometric figures with material tools. The object of study is a drawing on a sheet of paper, or a computer screen and the mode of validation is material, such as tracing paper, geometric tools or by dragging on the computer screen. Nevertheless, geometric knowledge can be used to guide or justify the drawing: for example, to reproduce a parallelogram, it is not enough to know the dimensions of the sides, another information is needed, an angle or a diagonal. Knowing the lengths of the sides and a diagonal, spatial knowledge can help to place the elements correctly to get a parallelogram instead of a kite shaped quadrilateral. We call “theoretical geometry” a geometry whose objects are defined in a discursive form and whose validation is based on proof. Traditionally, in France as in many other countries, these two types of practices, are taught one after the other.
# HOW TO TEACH GEOMETRY IN CONTINUITY ALONG SCHOOLING?

Marie-Jeanne Perrin-Glorian¹, Anne-Cécile Mathé², Valentina Celi³, Caroline Bulf ⁴

¹ LDAR, Université Paris-Cité, ² ACTé, Université Clermont Auvergne, ³ LaB-E3D (EA 7441), Université de Bordeaux, ⁴ LaB-E3D (EA 7441), Université de Bordeaux.

*This article reports on research, in the French context, into a possible continuum of teaching and learning plane geometry along compulsory schooling. It is based on an epistemological analysis of content, Duval's work on semiotic registers, and Brousseau's Theory of Didactic Situations, using a didactic engineering methodology.*

The ICME-13 survey (Sinclair et al., 2016) shows that there is currently little research that looks at both student learning in geometry and the teaching and training of teachers from the beginning of primary school through the middle of secondary school, when the use of mathematical evidence is expected. However, the plane geometry taught in compulsory education in France, as in many other countries, includes two main types of geometry, based on very different reference mathematical practices: a “material geometry” based on construction and validation with material tools and a “theoretical geometry” based on definitions and proof. Many studies have shown that supporting students in this evolution of practice is a real difficulty for teachers (Perrin-Glorian & Salin, 2010). Over the last twenty years, research has developed in France to reflect on the teaching of geometry in a continuity from grade 1 to grade 8 and to provide tools for teachers (Mathé et al., 2020; Guille-Biel Winder & Assude, 2023). Our aim is to question what these taught geometries cover, their epistemological foundations and their didactic stakes, and to search for a possible coherent progression of geometry throughout compulsory schooling. This research takes up some of the issues raised in Perrin-Glorian and Salin (2010) in a direction indicated by Sinclair et al. (2016). We restrict ourselves here to the field of plane geometry.

## THEORETICAL BACKGROUND

### Material geometry and theoretical geometry

Geometry encompasses several types of practice with different epistemological foundations. In the context of plane geometry, we call “material geometry” that geometry whose objective is practical in the graphical space of 2D drawings (freehand or constructed with geometric instruments): studying or tracing material geometric figures with material tools. The object of study is a drawing on a sheet of paper, or a computer screen and the mode of validation is material, such as tracing paper, geometric tools or by dragging on the computer screen. Nevertheless, geometric knowledge can be used to guide or justify the drawing: for example, to reproduce a parallelogram, it is not enough to know the dimensions of the sides, another information is needed, an angle or a diagonal. Knowing the lengths of the sides and a diagonal, spatial knowledge can help to place the elements correctly to get a parallelogram instead of a kite shaped quadrilateral. We call “theoretical geometry” a geometry whose objects are defined in a discursive form and whose validation is based on proof. Traditionally, in France as in many other countries, these two types of practices, are taught one after the other.
However, although they use the same words, there is a rupture between these two ways of doing geometry in the nature of the objects of study, of the problems posed and in the methods of validation. Houdement and Kuzniak (2006) speak of a break between two paradigms: "Geometry I (natural)" and "Geometry II (Euclidean)". Our work is based on the idea that there are nevertheless continuity levers between these geometries.

**The role of drawing in geometry. Dimensional and instrumental deconstructions**

As Duval’s work (1998, 2014) shows, the theoretical geometry requires the articulation of several semiotic registers, mainly the verbal register and the graphic register. In a proof, the drawing can serve various functions, such as illustrating a statement, exploring multiple configurations for conjecture, adding plots to reveal relationships or isolate sub-figures, and so on. A major difficulty is that the visualization of figures in geometry is different from the usual way of looking at and talking about drawings. Duval (2014) shows how and why the relationship between visualization and discourse in a geometric activity is done at the level of figural units, which requires a dimensional deconstruction of shapes. The visualization of figures involves cognitive issues but is also closely related to geometric knowledge. Indeed, the objects of theoretical geometry are linked by propositions that translate into visual or instrumental characteristics of material figures. Thus, the material figures used to represent geometric objects defined in the verbal register may be the same as drawings studied for themselves or drawings representing objects in space. In fact, geometric statements refer to some relationships between objects that are often represented by figural units of dimension 1 or 0 while natural vision favors figural units of dimension 2. Therefore, the use of definitions and theorems requires a *dimensional deconstruction* of shapes, “the mathematical way of seeing” drawings composed of smaller figural units, related by geometric properties. Moreover, it is often necessary to use lines that are not drawn in the figure, which is a real difficulty (Mathé & Mithalal, 2019). Let's take an example.

<figure><img src="image_79.png" /><figcaption>Figure 1 - Example: reproduce the model figure from the trapezoid frame</figcaption></figure>

Instead of seeing a juxtaposition of three shapes, adopting a geometric vision of the figure on the left would mean deconstructing 2D shapes into sub-units and looking at the relationships between these sub-units. This requires to see networks of lines linked by intersections and angles, to extend a segment to reveal a square, to extend the arc of a circle to make a semicircle, to consider the perpendicularity of lines or the equality of lengths, to consider that the same point can be both the center of a circle and the vertex of a square, and so on. However, we hypothesize that an instrumented work in material geometry, under certain constraints that we will specify later in this text, can assist the students in constructing this mathematical relationship to drawings. Constructing or reproducing a figure with geometric tools can engage students in the instrumental deconstruction of shapes and the identification of a set of independent figural units, and of a sequence of actions performed through the use of instruments, making it possible to reconstruct either the object itself or a graphic representation of that object (Mithalal & Balacheff, 2019). Moreover, this work can allow the gradual emergence, within the graphic space, of a relationship to drawings and statements, that is close to that
However, although they use the same words, there is a rupture between these two ways of doing geometry in the nature of the objects of study, of the problems posed and in the methods of validation. Houdement and Kuzniak (2006) speak of a break between two paradigms: "Geometry I (natural)" and "Geometry II (Euclidean)". Our work is based on the idea that there are nevertheless continuity levers between these geometries.

**The role of drawing in geometry. Dimensional and instrumental deconstructions**

As Duval’s work (1998, 2014) shows, the theoretical geometry requires the articulation of several semiotic registers, mainly the verbal register and the graphic register. In a proof, the drawing can serve various functions, such as illustrating a statement, exploring multiple configurations for conjecture, adding plots to reveal relationships or isolate sub-figures, and so on. A major difficulty is that the visualization of figures in geometry is different from the usual way of looking at and talking about drawings. Duval (2014) shows how and why the relationship between visualization and discourse in a geometric activity is done at the level of figural units, which requires a dimensional deconstruction of shapes. The visualization of figures involves cognitive issues but is also closely related to geometric knowledge. Indeed, the objects of theoretical geometry are linked by propositions that translate into visual or instrumental characteristics of material figures. Thus, the material figures used to represent geometric objects defined in the verbal register may be the same as drawings studied for themselves or drawings representing objects in space. In fact, geometric statements refer to some relationships between objects that are often represented by figural units of dimension 1 or 0 while natural vision favors figural units of dimension 2. Therefore, the use of definitions and theorems requires a *dimensional deconstruction* of shapes, “the mathematical way of seeing” drawings composed of smaller figural units, related by geometric properties. Moreover, it is often necessary to use lines that are not drawn in the figure, which is a real difficulty (Mathé & Mithalal, 2019). Let’s take an example.

<figure><img src="image_85.png" /><figcaption>Figure 1 - Example: reproduce the model figure from the trapezoid frame</figcaption></figure>

Instead of seeing a juxtaposition of three shapes, adopting a geometric vision of the figure on the left would mean deconstructing 2D shapes into sub-units and looking at the relationships between these sub-units. This requires to see networks of lines linked by intersections and angles, to extend a segment to reveal a square, to extend the arc of a circle to make a semicircle, to consider the perpendicularity of lines or the equality of lengths, to consider that the same point can be both the center of a circle and the vertex of a square, and so on. However, we hypothesize that an instrumented work in material geometry, under certain constraints that we will specify later in this text, can assist the students in constructing this mathematical relationship to drawings. Constructing or reproducing a figure with geometric tools can engage students in the instrumental deconstruction of shapes and the identification of a set of independent figural units, and of a sequence of actions performed through the use of instruments, making it possible to reconstruct either the object itself or a graphic representation of that object (Mithalal & Balacheff, 2019). Moreover, this work can allow the gradual emergence, within the graphic space, of a relationship to drawings and statements, that is close to that
of theoretical geometry. For example, the reproduction of the figure, starting from the trapezoid frame (Figure 1, right) with a rectangular template and a compass makes it necessary to be aware of the geometric relationships between the trapezoid, and the square and the semicircle in order to make them appear. This fundamental learning process is not currently taught in regular classrooms.

## Material geometry and spatio-geometric knowledge

Berthelot and Salin (1998) show that geometric knowledge is not spontaneously transferred to solve problems in the 3D-space in which we move. They call spatial knowledge the knowledge used to solve these spatial problems and hypothesize that developing what they call “spatio-geometric knowledge”, derived from geometric knowledge in solving spatial problems in a modeling problematic, could help students link geometric knowledge and spatial knowledge. We consider these constructs in the 2D graphic space, and we hypothesize that some problems, which we try to characterize, can help students link and distinguish spatial knowledge and geometric knowledge.

By assembling shapes or reproducing figures, students may have the opportunity, if taught, to encounter the notions of alignment, intersection, angle, perpendicularity, or parallelism. Beyond the encounter with the cultural knowledge of geometric shapes, from the very beginning of school, through the manipulation of material objects, we can propose problems of placement of material objects or of tracing, which lead children to build spatio-geometric knowledge, on which geometric knowledge can then be based. For example, a square can be made by selecting from a set of pieces two isosceles rectangle triangles with hypotenuses of equal length; a square can be drawn by using a rectangular triangle template that is rotated. The equality of lengths can be checked by juxtaposition or superposition, that of angles by superposition (Celi, in Guille-Biel Winder & Assude, 2023, Ch.3, pp. 47-72). Later, spatio-geometric knowledge can be built by reproducing geometric drawings as the example in figure 1, provided that the conditions of this reproduction are properly chosen.

## Learning situations, situational and institutional knowledge

Our research is rooted in the TDS (Theory of Didactic Situations in Mathematics) (Brousseau, 1997). This theoretical framework is based on the idea that to know a mathematical concept is not only to be able to formulate a text of knowledge, but also to understand the reasons for this knowledge and at least some aspects of its usefulness and operationality in concrete situations. The aim of this theory is to identify favorable conditions for students to learn a given piece of mathematical knowledge. These conditions are seen as a *milieu* in which the subject pursues a goal. The student's interaction with this milieu is modeled by a *situation*.

Viewing learning as an adaptation to the constraints imposed on action in a situation leads one to consider a "double aspect of knowledge" (Margolinas & Bessot, 2021). The evolution of students' strategies in a situation reflects the evolution of their "situational knowledge". It is not always possible to express it. Learning is characterized by the stabilization of a certain amount of this knowledge as an adaptation to a particular situation. Some situational knowledge may be recognized as useful by an institution and then transformed into "institutional knowledge" in the process of institutionalization. Institutional knowledge is a text that results from the incorporation of situational knowledge into a shared institutional culture. In geometry, situational knowledge about the visualization of drawings is often not considered by school. The goal of developing this situational knowledge seems to us to be an important issue in material geometry, because we see it as a possible
of theoretical geometry. For example, the reproduction of the figure, starting from the trapezoid frame (Figure 1, right) with a rectangular template and a compass makes it necessary to be aware of the geometric relationships between the trapezoid, and the square and the semicircle in order to make them appear. This fundamental learning process is not currently taught in regular classrooms.

## Material geometry and spatio-geometric knowledge

Berthelot and Salin (1998) show that geometric knowledge is not spontaneously transferred to solve problems in the 3D-space in which we move. They call spatial knowledge the knowledge used to solve these spatial problems and hypothesize that developing what they call “spatio-geometric knowledge”, derived from geometric knowledge in solving spatial problems in a modeling problematic, could help students link geometric knowledge and spatial knowledge. We consider these constructs in the 2D graphic space, and we hypothesize that some problems, which we try to characterize, can help students link and distinguish spatial knowledge and geometric knowledge.

By assembling shapes or reproducing figures, students may have the opportunity, if taught, to encounter the notions of alignment, intersection, angle, perpendicularity, or parallelism. Beyond the encounter with the cultural knowledge of geometric shapes, from the very beginning of school, through the manipulation of material objects, we can propose problems of placement of material objects or of tracing, which lead children to build spatio-geometric knowledge, on which geometric knowledge can then be based. For example, a square can be made by selecting from a set of pieces two isosceles rectangle triangles with hypotenuses of equal length; a square can be drawn by using a rectangular triangle template that is rotated. The equality of lengths can be checked by juxtaposition or superposition, that of angles by superposition (Celi, in Guille-Biel Winder & Assude, 2023, Ch.3, pp. 47-72). Later, spatio-geometric knowledge can be built by reproducing geometric drawings as the example in figure 1, provided that the conditions of this reproduction are properly chosen.

## Learning situations, situational and institutional knowledge

Our research is rooted in the TDS (Theory of Didactic Situations in Mathematics) (Brousseau, 1997). This theoretical framework is based on the idea that to know a mathematical concept is not only to be able to formulate a text of knowledge, but also to understand the reasons for this knowledge and at least some aspects of its usefulness and operationality in concrete situations. The aim of this theory is to identify favorable conditions for students to learn a given piece of mathematical knowledge. These conditions are seen as a *milieu* in which the subject pursues a goal. The student's interaction with this milieu is modeled by a *situation*.

Viewing learning as an adaptation to the constraints imposed on action in a situation leads one to consider a "double aspect of knowledge" (Margolinas & Bessot, 2021). The evolution of students' strategies in a situation reflects the evolution of their "situational knowledge". It is not always possible to express it. Learning is characterized by the stabilization of a certain amount of this knowledge as an adaptation to a particular situation. Some situational knowledge may be recognized as useful by an institution and then transformed into "institutional knowledge" in the process of institutionalization. Institutional knowledge is a text that results from the incorporation of situational knowledge into a shared institutional culture. In geometry, situational knowledge about the visualization of drawings is often not considered by school. The goal of developing this situational knowledge seems to us to be an important issue in material geometry, because we see it as a possible
path from spatial knowledge to geometric knowledge. Helping teachers make connections between the strategies of constructing figures with instruments and the spatio-geometric knowledge involved and deriving the institutional knowledge of geometry from this situational knowledge also seems to us to be an important challenge.

## TO TEACH GEOMETRY IN CONTINUITY ALONG COMPULSORY SCHOOLING

We now come to our proposals for the management of a continuity from spatial to geometric knowledge throughout the school. They consist in identifying a field of practices and discourses that can be carried out on material figures in order to facilitate access to theoretical geometry.

### A fundamental situation in the geometric analysis of material figures: Figure reproduction

In TDS, a fundamental situation constitutes a generic situation that can be used to define several situations for didactic use (teaching situations) aimed at specific knowledge. Our work has led us to consider the reproduction of figures as a fundamental situation in the geometric analysis of figures. The generic problem consists in making a copy of a given material figure taken as a model, by means of instruments. The reproduction can be either identical, or of different size and/or orientation. Means of validation can evolve: first carried out by superimposing a solution on tracing paper on the produced figure, later by checking the properties that characterize the shape of the figure.

### Geometry of tracing

The geometry of tracing (GT) is the geometry generated by this fundamental situation. It is the geometry of making or reproducing figures with drawing instruments, without measuring instruments (there are no numbers in GT), but with constraints on the use of these instruments. We have called "figure restoration" the case where we ask to reproduce the model from a given starting figure as in Figure 1. An essential point in this geometry is to consider the knowledge required to reproduce the figure in relation to the instruments available. In fact, the instruments used condition the figural units and relationships on which the reproduction is based, and thus the way the figure must be seen. The model, the starting figure, the difference between them, and the available instruments are the main didactic variables of the situation.

There is a break between material geometry and theoretical geometry. However, the objects and relations of elementary geometry (straight line, angle, parallelism, etc.) reflect spatial properties that can be perceived by our senses. Drawing instruments produce visual characteristics that correspond to the representation of geometric properties. The word "tracing" evokes instruments and material figures, but GT as we understand it is different from material geometry. It corresponds to the search for a possible link in learning between drawing with instruments and the notions of abstract geometry. It translates the fact that the objects and axioms of Euclid's geometry can largely be seen as a theorization of the construction of figures, especially those figures that can be constructed with a ruler and a compass (by drawing straight lines and circles).

In order to establish the link between drawing with instruments and geometric properties, we consider "theoretical" drawing instruments insofar as they do not have the limitations of material instruments and have only one function: a *ruler* to draw straight lines, a *"length-transfer"* to carry a length on a line already drawn from a point on that line, a *"length-bisector"* to take half a length, a *"angle-*
path from spatial knowledge to geometric knowledge. Helping teachers make connections between the strategies of constructing figures with instruments and the spatio-geometric knowledge involved and deriving the institutional knowledge of geometry from this situational knowledge also seems to us to be an important challenge.

## TO TEACH GEOMETRY IN CONTINUITY ALONG COMPULSORY SCHOOLING

We now come to our proposals for the management of a continuity from spatial to geometric knowledge throughout the school. They consist in identifying a field of practices and discourses that can be carried out on material figures in order to facilitate access to theoretical geometry.

### A fundamental situation in the geometric analysis of material figures: Figure reproduction

In TDS, a fundamental situation constitutes a generic situation that can be used to define several situations for didactic use (teaching situations) aimed at specific knowledge. Our work has led us to consider the reproduction of figures as a fundamental situation in the geometric analysis of figures. The generic problem consists in making a copy of a given material figure taken as a model, by means of instruments. The reproduction can be either identical, or of different size and/or orientation. Means of validation can evolve: first carried out by superimposing a solution on tracing paper on the produced figure, later by checking the properties that characterize the shape of the figure.

### Geometry of tracing

The geometry of tracing (GT) is the geometry generated by this fundamental situation. It is the geometry of making or reproducing figures with drawing instruments, without measuring instruments (there are no numbers in GT), but with constraints on the use of these instruments. We have called "figure restoration" the case where we ask to reproduce the model from a given starting figure as in Figure 1. An essential point in this geometry is to consider the knowledge required to reproduce the figure in relation to the instruments available. In fact, the instruments used condition the figural units and relationships on which the reproduction is based, and thus the way the figure must be seen. The model, the starting figure, the difference between them, and the available instruments are the main didactic variables of the situation.

There is a break between material geometry and theoretical geometry. However, the objects and relations of elementary geometry (straight line, angle, parallelism, etc.) reflect spatial properties that can be perceived by our senses. Drawing instruments produce visual characteristics that correspond to the representation of geometric properties. The word "tracing" evokes instruments and material figures, but GT as we understand it is different from material geometry. It corresponds to the search for a possible link in learning between drawing with instruments and the notions of abstract geometry. It translates the fact that the objects and axioms of Euclid's geometry can largely be seen as a theorization of the construction of figures, especially those figures that can be constructed with a ruler and a compass (by drawing straight lines and circles).

In order to establish the link between drawing with instruments and geometric properties, we consider "theoretical" drawing instruments insofar as they do not have the limitations of material instruments and have only one function: a *ruler* to draw straight lines, a *"length-transfer"* to carry a length on a line already drawn from a point on that line, a *"length-bisector"* to take half a length, a *"angle-*
*transfer*” to reproduce an angle along a line with a point at which to place the vertex, a compass to draw circles.

In order to establish a geometric relationship with material figures, a precursor of that of theoretical geometry, it is necessary to respect certain rules of instrument use, which we have called the geometric use of instruments (Mathé et al., 2020; Mathé & Perrin-Glorian, in Guille-Biel Winder & Assude 2023, ch.1, pp. 3-34). To set up the ruler, you need two points, or an already drawn segment; to set up the length transfer, you need an already drawn line and a point on this line; to take the center of a segment, you transfer the length of this segment to the length bisector, then you transfer the half-length to the segment from one end; to set up the angle transfer (for example a template), you need a straight line and a point on this line: the top of the template is placed on the point and one side of the template is placed on the line, the angle is drawn on one side or the other; to draw parallel straight lines, drag one side of an angle template onto a straight line and draw on the other side. With these rules for the use of instruments, one can construct all the figures of plane geometry encountered in elementary and middle school. These functions can also be found in dynamic geometry. As the students’ knowledge progresses, the number of instruments can be reduced: the compass can soon replace the length-transfer, then the length-bisector, then the angle-transfer.

**From the geometry of tracing to the theoretical geometry**

To obtain proofs in theoretical geometry, we need definitions and theorems which, for the sake of overall coherence, must have an axiomatic basis. The question of the foundations of secondary school geometry was hotly debated from the 60s to the 80s. In France, the cases of congruence of triangles were abandoned and, in the 80s and 90s, geometric transformations were placed at the base. However, these tools are more accessible to students than transformations (Perrin & Perrin-Glorian, 2021). They require a less difficult dimensional deconstruction than transformations, which often require consideration of the points that define the lines (straight or circular) and the fact that a point is obtained by the intersection of two lines. Thus, the choice of an axiomatic foundation for secondary school geometry is quite important for students’ access to theoretical geometry.

**Dialectics of action, formulation, and validation around figure reproduction**

Of course, it would be reductive to approach the issues involved in teaching geometry solely through the instrumental dimension of geometric practice. This dimension is inseparable from other cognitive, semiotic and linguistic issues (Bulf et al., 2015). Moving from the geometric use of instruments to an initial conceptualization of spatio-geometric then geometric knowledge also requires linguistic work. Through multiple reformulations, networking, generalization (Sfard, 2001), it will also be a matter of supporting the students from a technical language (related to the use of instruments) to a geometric language (Petitfour, 2017). It is then a question of making the problems posed evolve towards the search for justifications and reasons for construction methods, and of moving the students from pragmatic proofs to intellectual proofs. Thus, we can see the emergence of different types of knowledge that can be linked to different types of situations for didactic use, derived from the fundamental situation of figure reproduction, based on the model of the dialectics of action, formulation and validation proposed by Brousseau (1997). In the next section, we briefly discuss how we have used our theoretical reflection to design learning situations that have been implemented, for more than twenty years now, in primary and early middle school classes.
*transfer*” to reproduce an angle along a line with a point at which to place the vertex, a compass to draw circles.

In order to establish a geometric relationship with material figures, a precursor of that of theoretical geometry, it is necessary to respect certain rules of instrument use, which we have called the geometric use of instruments (Mathé et al., 2020; Mathé & Perrin-Glorian, in Guille-Biel Winder & Assude 2023, ch.1, pp. 3-34). To set up the ruler, you need two points, or an already drawn segment; to set up the length transfer, you need an already drawn line and a point on this line; to take the center of a segment, you transfer the length of this segment to the length bisector, then you transfer the half-length to the segment from one end; to set up the angle transfer (for example a template), you need a straight line and a point on this line: the top of the template is placed on the point and one side of the template is placed on the line, the angle is drawn on one side or the other; to draw parallel straight lines, drag one side of an angle template onto a straight line and draw on the other side. With these rules for the use of instruments, one can construct all the figures of plane geometry encountered in elementary and middle school. These functions can also be found in dynamic geometry. As the students’ knowledge progresses, the number of instruments can be reduced: the compass can soon replace the length-transfer, then the length-bisector, then the angle-transfer.

**From the geometry of tracing to the theoretical geometry**

To obtain proofs in theoretical geometry, we need definitions and theorems which, for the sake of overall coherence, must have an axiomatic basis. The question of the foundations of secondary school geometry was hotly debated from the 60s to the 80s. In France, the cases of congruence of triangles were abandoned and, in the 80s and 90s, geometric transformations were placed at the base. However, these tools are more accessible to students than transformations (Perrin & Perrin-Glorian, 2021). They require a less difficult dimensional deconstruction than transformations, which often require consideration of the points that define the lines (straight or circular) and the fact that a point is obtained by the intersection of two lines. Thus, the choice of an axiomatic foundation for secondary school geometry is quite important for students’ access to theoretical geometry.

**Dialectics of action, formulation, and validation around figure reproduction**

Of course, it would be reductive to approach the issues involved in teaching geometry solely through the instrumental dimension of geometric practice. This dimension is inseparable from other cognitive, semiotic and linguistic issues (Bulf et al., 2015). Moving from the geometric use of instruments to an initial conceptualization of spatio-geometric then geometric knowledge also requires linguistic work. Through multiple reformulations, networking, generalization (Sfard, 2001), it will also be a matter of supporting the students from a technical language (related to the use of instruments) to a geometric language (Petitfour, 2017). It is then a question of making the problems posed evolve towards the search for justifications and reasons for construction methods, and of moving the students from pragmatic proofs to intellectual proofs. Thus, we can see the emergence of different types of knowledge that can be linked to different types of situations for didactic use, derived from the fundamental situation of figure reproduction, based on the model of the dialectics of action, formulation and validation proposed by Brousseau (1997). In the next section, we briefly discuss how we have used our theoretical reflection to design learning situations that have been implemented, for more than twenty years now, in primary and early middle school classes.
# TOWARDS LEARNING SITUATIONS

The progressions and situations that we are experimenting with in the classroom to organize the spatial to the geometric in school (Mathé et al., 2020; Mathé & Perrin-Glorian, in Guille-Biel Winder & Assude 2023, ch.1, pp. 3-34), articulate these different types of situations. We illustrate our argument with an *a priori* analysis<fnref n="14" /> of an example involving the square, to show in this very simple case the dialectic between situations of action, formulation, and validation on the reproduction of figures that we used in class on situations that we cannot describe here, for lack of space.

## Reproducing geometric figures as action situations

In the *action situations*, the students have a model figure, possibly a starting figure, and instruments. Problem solving is characterized by successful action. Students act directly on the *milieu*, which in turn provides them with *feedback*. They can test the validity of their reproduction strategies by comparing the material figure produced with the model to be reproduced (for example, by superimposing a solution provided on tracing paper). This feedback helps students to develop new, more operational strategies, for constructing new situational knowledge. For example, consider the problem of reproducing a square from two given consecutive sides (Figure 2). To do this, we need to draw the other two sides and the fourth vertex (point C) of the square. The knowledge required depends on the tools available. If you have a ruler and a right-angle transfer, without length transfer, you must think of the sides of the square as lines that can be extended and the vertex as the intersection of these lines. If you have a ruler and a compass, you must think of the equality of the sides and the missing point as the intersection of two circular arcs and think of the compass as a means of getting points at a fixed distance from a given point.

<figure><img src="image_80.png" /><figcaption>Figure 2 - Example: reproduce the square from two given consecutive sides</figcaption></figure>

Thus, the reproduction of the square leads to the emergence of properties of the square, depending on the instruments available. Nevertheless, the mere aim of materially reproducing a model figure does not make it necessary to formulate the geometric relations that could underpin action strategies.

## Formulation situations

*Formulation situations* are those that make the formulation of the implicit action model necessary and problematize it. Formulation is necessary when direct action on the *milieu* is prevented, for example because the model figure and the figure to be produced are distant in space or time. In the case of formulation situations for others, for example, a student has a model figure, and must produce a message that allows a classmate to reproduce the figure without having seen it. A key variable in these situations is the type of language used. It can be graphic (using diagrams) or verbal. If verbal, it can be oral or written. The verbal formulation itself covers different types of discourse. We can

<footnote n="14">¹⁴ In the TDS, an *a priori* analysis is an epistemological reflection that does not have a predictive but a causal meaning; it involves describing various possible (and therefore potentially reproductible) phenomena in the context of a classroom situation (Margolinas & Bessot, 2021).</footnote>
# TOWARDS LEARNING SITUATIONS

The progressions and situations that we are experimenting with in the classroom to organize the spatial to the geometric in school (Mathé et al., 2020; Mathé & Perrin-Glorian, in Guille-Biel Winder & Assude 2023, ch.1, pp. 3-34), articulate these different types of situations. We illustrate our argument with an *a priori* analysis<fnref n="14" /> of an example involving the square, to show in this very simple case the dialectic between situations of action, formulation, and validation on the reproduction of figures that we used in class on situations that we cannot describe here, for lack of space.

## Reproducing geometric figures as action situations

In the *action situations*, the students have a model figure, possibly a starting figure, and instruments. Problem solving is characterized by successful action. Students act directly on the *milieu*, which in turn provides them with *feedback*. They can test the validity of their reproduction strategies by comparing the material figure produced with the model to be reproduced (for example, by superimposing a solution provided on tracing paper). This feedback helps students to develop new, more operational strategies, for constructing new situational knowledge. For example, consider the problem of reproducing a square from two given consecutive sides (Figure 2). To do this, we need to draw the other two sides and the fourth vertex (point C) of the square. The knowledge required depends on the tools available. If you have a ruler and a right-angle transfer, without length transfer, you must think of the sides of the square as lines that can be extended and the vertex as the intersection of these lines. If you have a ruler and a compass, you must think of the equality of the sides and the missing point as the intersection of two circular arcs and think of the compass as a means of getting points at a fixed distance from a given point.

<figure><img src="image_86.png" /><figcaption>Figure 2 - Example: reproduce the square from two given consecutive sides</figcaption></figure>

Thus, the reproduction of the square leads to the emergence of properties of the square, depending on the instruments available. Nevertheless, the mere aim of materially reproducing a model figure does not make it necessary to formulate the geometric relations that could underpin action strategies.

## Formulation situations

*Formulation situations* are those that make the formulation of the implicit action model necessary and problematize it. Formulation is necessary when direct action on the *milieu* is prevented, for example because the model figure and the figure to be produced are distant in space or time. In the case of formulation situations for others, for example, a student has a model figure, and must produce a message that allows a classmate to reproduce the figure without having seen it. A key variable in these situations is the type of language used. It can be graphic (using diagrams) or verbal. If verbal, it can be oral or written. The verbal formulation itself covers different types of discourse. We can

<footnote n="14">¹⁴ In the TDS, an *a priori* analysis is an epistemological reflection that does not have a predictive but a causal meaning; it involves describing various possible (and therefore potentially reproductible) phenomena in the context of a classroom situation (Margolinas & Bessot, 2021).</footnote>
progressively impose constraints on the language to be mobilized, from a technical language that mentions the instruments to a geometric language that refers exclusively to the geometric objects and relations that are invoked.

In the example (Figure 2), in the case of verbal formulation situations for others, the first difficulty is to designate the elements of the figure: designation of points allows to designate segments, lines, etc. The second one is to designate the relationships between these elements. In the case students dispose of a ruler and a right-angle transfer and points are already designated, they must explain how to place their right-angle template and trace or to trace a line perpendicular to a side on a vertex. Technical language would be "place one side of your template along AD with the top on D and trace along the other side. Do the same along the side AB with the top on B. You get two lines that intersect in C". Geometric language would be "Trace the perpendicular to the AD side that passes through D and the perpendicular to the AB side through B. Their intersection is the point C." Using a ruler and a compass, the formulations could be: "Lay the compass on D and the mine on A and draw an arc. Do the same with the point on B. The arcs cut into C." or "Draw the circle with center D and radius AD, then the circle with center B and the same radius; these circles intersect at A and another point, that is C; trace the segments [CD] and [BC]. Thus, a formulation situation about this reproduction can confront students with the problem of naming and characterizing geometric objects and relationships.

The linguistic work involved in these situations, based on the reproduction, or instrumented construction of a figure, plays an essential role in the possible links between material geometry and theoretical geometry. On the one hand, there is a profound change in the object of the work, from the material figure to a text that allows the geometric definition of the represented object. In these situations, the need to construct a geometric language to designate the objects and relationships mobilized in the reproduction of a figure goes far beyond questions of lexicon. These situations confront students with the question of how to characterize geometric objects, that is, what information is necessary and sufficient to define a figure.

## Validation situations and start proving

In action and formulation situations related to the reproduction of figures, the validation of drawing strategies or formulations intended to communicate these strategies is based on pragmatic evidence. The *validation situations* must make the students' geometric activity evolve towards the search for justifications and reasons for constructions and involve them in intellectual proofs.

In the example (Figure 2), questions can be asked like: "Does this message allow, for sure, to get a square and why?" This can be an opportunity to generate counterexamples to invalidate a message or to produce arguments to prove its validity, for instance: "it's a square because we have constructed a quadrilateral with three right angles and it has two consecutive sides of the same length"; "we have constructed a quadrilateral with four sides of the same length and one right angle". It can lead students towards a proof approach, based on the construction of the square using geometric instruments.

## DISCUSSION: FROM LEARNING SITUATIONS TO TEACHING SITUATIONS

Our observations in class for many years are encouraging. However, it is difficult for primary school teachers to manage the complex set of didactic variables mentioned or to perceive the potential stakes of the Geometry of Tracing. The diffusion of this way of teaching geometry in regular classes requires new research with collaboration between teachers and researchers. In France, such research has now
progressively impose constraints on the language to be mobilized, from a technical language that mentions the instruments to a geometric language that refers exclusively to the geometric objects and relations that are invoked.

In the example (Figure 2), in the case of verbal formulation situations for others, the first difficulty is to designate the elements of the figure: designation of points allows to designate segments, lines, etc. The second one is to designate the relationships between these elements. In the case students dispose of a ruler and a right-angle transfer and points are already designated, they must explain how to place their right-angle template and trace or to trace a line perpendicular to a side on a vertex. Technical language would be "place one side of your template along AD with the top on D and trace along the other side. Do the same along the side AB with the top on B. You get two lines that intersect in C". Geometric language would be "Trace the perpendicular to the AD side that passes through D and the perpendicular to the AB side through B. Their intersection is the point C." Using a ruler and a compass, the formulations could be: "Lay the compass on D and the mine on A and draw an arc. Do the same with the point on B. The arcs cut into C." or "Draw the circle with center D and radius AD, then the circle with center B and the same radius; these circles intersect at A and another point, that is C; trace the segments [CD] and [BC]. Thus, a formulation situation about this reproduction can confront students with the problem of naming and characterizing geometric objects and relationships.

The linguistic work involved in these situations, based on the reproduction, or instrumented construction of a figure, plays an essential role in the possible links between material geometry and theoretical geometry. On the one hand, there is a profound change in the object of the work, from the material figure to a text that allows the geometric definition of the represented object. In these situations, the need to construct a geometric language to designate the objects and relationships mobilized in the reproduction of a figure goes far beyond questions of lexicon. These situations confront students with the question of how to characterize geometric objects, that is, what information is necessary and sufficient to define a figure.

## Validation situations and start proving

In action and formulation situations related to the reproduction of figures, the validation of drawing strategies or formulations intended to communicate these strategies is based on pragmatic evidence. The *validation situations* must make the students' geometric activity evolve towards the search for justifications and reasons for constructions and involve them in intellectual proofs.

In the example (Figure 2), questions can be asked like: "Does this message allow, for sure, to get a square and why?" This can be an opportunity to generate counterexamples to invalidate a message or to produce arguments to prove its validity, for instance: "it's a square because we have constructed a quadrilateral with three right angles and it has two consecutive sides of the same length"; "we have constructed a quadrilateral with four sides of the same length and one right angle". It can lead students towards a proof approach, based on the construction of the square using geometric instruments.

## DISCUSSION: FROM LEARNING SITUATIONS TO TEACHING SITUATIONS

Our observations in class for many years are encouraging. However, it is difficult for primary school teachers to manage the complex set of didactic variables mentioned or to perceive the potential stakes of the Geometry of Tracing. The diffusion of this way of teaching geometry in regular classes requires new research with collaboration between teachers and researchers. In France, such research has now
begun in some institutional places that make this work possible, bringing together primary or
secondary school teachers, and researchers to collaborate and experiment in the classroom, even to
produce resources (Guille-Biel Winder & Assude, 2023). These collaborative spaces seem to be
particularly conducive to changing practices, as evidenced by a number of works that go in this
direction (cf. ICMI Study 25 "Teachers of Mathematics Working and Learning in Collaborative
Groups"). We emphasize the importance of developing now the research that allows to describe the
professional teaching actions that can help teachers to build this new contract.

References

Berthelot, R., & Salin, M. H. (1998). The role of pupil's spatial knowledge in the elementary teaching of geometry. In Mammama, C. and Villani, V. *Perspectives on the teaching of geometry in the 21st century*. *An ICMI study* (pp.71-78). Kluwer Academic Publishers.

Bulf, C., Mathé, A. C., & Mithalal, J. (2015). Langage et construction de connaissances dans une situation de résolution de problèmes en géométrie. *Recherches en Didactique des Mathématiques*, 35(1), 7-36.

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

Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammama & V. Villani (Eds.), *Perspectives on the teaching of geometry in the 21st century. An ICMI study* (pp. 37-52). Kluwer Academic Publishers.

Duval, R. (2014). The first crucial point in geometry learning: Visualization. *Mediterranean Journal for Research in Mathematics Education, 13*(1-2), 1-28.

Guille-Biel Winder, C., & Assude, T. (Eds.) (2023). *Articulations between sensitive space, graphic space, geometric space. Resources, practices and training*. ISTE Science Publishing.

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

Margolinas, C., & Bessot, A. (2021). *Guy Brousseau Unit, ICMI AMOR*, https://www.mathunion.org/icmi/awards/amor/guy-brousseau-unit

Mathé, A. C., Barrier, T., & Perrin-Glorian M. J. (2020). *Enseigner la géométrie élémentaire, enjeux, ruptures et continuités*. Les Sciences de l'éducation aujourd'hui, éditions Academia – L'Harmattan.

Mathé, A. C., & Mithalal, J. (2019). L'usage des dessins et le rôle du langage en géométrie: quelques enjeux pour l'enseignement. In S. Coppé, E. Roditi, et al. (Eds.), *Nouvelles perspectives en didactique: géométrie, évaluation des apprentissages mathématiques* (pp.47-86). La Pensée sauvage.

Mithalal, J., & Balacheff, N. (2019). The instrumental deconstruction as a link between drawing and geometrical figure. *Educational Studies in Mathematics, 100*(2), 161–176.

Perrin, D., & Perrin-Glorian, M. J. (Eds) (2021). *Enseigner la géométrie au cycle 4. Comparer des triangles pour démontrer*. IREM, Université Paris-Cité.

Perrin-Glorian, M. J., & Salin, M. H. (2010). Didactique de la géométrie. Peut-on commencer à faire le point ? In Coulange L. & Hache C. (Eds) *Actes du séminaire national de didactique des mathématiques. Année 2009* (pp.47-82). IREM, Université Paris-Diderot.

Petitfour, E. (2017). Outils théoriques d'analyse de l'action instrumentée, au service de l'étude de difficultés d'élèves dyspraxiques en géométrie. *Recherches en Didactique des Mathématiques*, 37(3-2), 247-288.

Sinclair, N., Bartolini Bussi, M. G., de Villiers, M., Jones, K., Kortenkamp, U., Leung., A., & Owens, K. (2016). Recent research on geometry education: an ICME-13 survey team report. *ZDM Mathematics Education, 48*, 691–719

Sfard, A. (2001) There is more to discourse than meets the ears: Learning from mathematical communication things that we have not known before. *Educational Studies in Mathematics, 46*(1/3), 13-57.
begun in some institutional places that make this work possible, bringing together primary or
secondary school teachers, and researchers to collaborate and experiment in the classroom, even to
produce resources (Guille-Biel Winder & Assude, 2023). These collaborative spaces seem to be
particularly conducive to changing practices, as evidenced by a number of works that go in this
direction (cf. ICMI Study 25 “Teachers of Mathematics Working and Learning in Collaborative
Groups”). We emphasize the importance of developing now the research that allows to describe the
professional teaching actions that can help teachers to build this new contract.

References

Berthelot, R., & Salin, M. H. (1998). The role of pupil's spatial knowledge in the elementary teaching of geometry. In Mammama, C. and Villani, V. *Perspectives on the teaching of geometry in the 21st century*. *An ICMI study* (pp.71-78). Kluwer Academic Publishers.

Bulf, C., Mathé, A. C., & Mithalal, J. (2015). Langage et construction de connaissances dans une situation de résolution de problèmes en géométrie. *Recherches en Didactique des Mathématiques*, 35(1), 7-36.

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

Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammama & V. Villani (Eds.), *Perspectives on the teaching of geometry in the 21st century. An ICMI study* (pp. 37-52). Kluwer Academic Publishers.

Duval, R. (2014). The first crucial point in geometry learning: Visualization. *Mediterranean Journal for Research in Mathematics Education, 13*(1-2), 1-28.

Guille-Biel Winder, C., & Assude, T. (Eds.) (2023). *Articulations between sensitive space, graphic space, geometric space. Resources, practices and training*. ISTE Science Publishing.

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

Margolinas, C., & Bessot, A. (2021). *Guy Brousseau Unit, ICMI AMOR*, https://www.mathunion.org/icmi/awards/amor/guy-brousseau-unit

Mathé, A. C., Barrier, T., & Perrin-Glorian M. J. (2020). *Enseigner la géométrie élémentaire, enjeux, ruptures et continuités*. Les Sciences de l'éducation aujourd'hui, éditions Academia – L'Harmattan.

Mathé, A. C., & Mithalal, J. (2019). L'usage des dessins et le rôle du langage en géométrie: quelques enjeux pour l'enseignement. In S. Coppé, E. Roditi, et al. (Eds.), *Nouvelles perspectives en didactique: géométrie, évaluation des apprentissages mathématiques* (pp.47-86). La Pensée sauvage.

Mithalal, J., & Balacheff, N. (2019). The instrumental deconstruction as a link between drawing and geometrical figure. *Educational Studies in Mathematics, 100*(2), 161–176.

Perrin, D., & Perrin-Glorian, M. J. (Eds) (2021). *Enseigner la géométrie au cycle 4. Comparer des triangles pour démontrer*. IREM, Université Paris-Cité.

Perrin-Glorian, M. J., & Salin, M. H. (2010). Didactique de la géométrie. Peut-on commencer à faire le point ? In Coulange L. & Hache C. (Eds) *Actes du séminaire national de didactique des mathématiques. Année 2009* (pp.47-82). IREM, Université Paris-Diderot.

Petitfour, E. (2017). Outils théoriques d'analyse de l'action instrumentée, au service de l'étude de difficultés d'élèves dyspraxiques en géométrie. *Recherches en Didactique des Mathématiques*, 37(3-2), 247-288.

Sinclair, N., Bartolini Bussi, M. G., de Villiers, M., Jones, K., Kortenkamp, U., Leung., A, & Owens, K. (2016). Recent research on geometry education: an ICME-13 survey team report. *ZDM Mathematics Education, 48*, 691–719

Sfard, A. (2001) There is more to discourse than meets the ears: Learning from mathematical communication things that we have not known before. *Educational Studies in Mathematics*, 46(1/3), 13-57.
# APPROACH TO TEACHING GEOMETRY TO DYSPRAXIC STUDENTS

Edith Petitfour

LDAR, Univ Rouen Normandie

The aim of this contribution is to present an overview of an approach to teaching geometry to dyspraxic fifth and sixth-grade students. These students face some important difficulties in using geometrical instruments (ruler, compass, set square) with required precision. We've developed a dyadic work system, based on the preserved skills of dyspraxic students, to allow them to study geometry without drawing geometric figures being an obstacle. We present this work system, its theoretical background, an implementation in a pencil-and-paper environment and the possibilities offered by a human interaction simulator to contribute to teaching geometry to dyspraxic students.

## INTRODUCTION

In French primary school and at the beginning of secondary school (students aged 9-12 years), the teaching and learning of geometry is based primarily on situations requiring the use of instruments (ruler, compass and set square) in a pencil-and-paper environment. Tracing with instruments to reproduce, to represent or to construct geometric figures is an injunction of the school curriculum, and is supposed to contribute to the conceptualization of geometric concepts. However, this teaching method, which uses manipulatives, is not suitable for all students, and in particular for students with dyspraxia, as they cannot perform the skilled movements necessary for such academic tasks: their motor performances are slower, less accurate and more variable (Elbasan & Kayihan, 2012). These students invariably fail to complete tasks in geometry, whereas they do have the core abilities – reasoning, language and memory – that would otherwise enable them to conceptualize geometric notions.

Our research aims at proposing alternative ways of teaching geometry, accessible to all students, and in particular for those who have difficulties in manipulation in order to meet the objectives of the geometry curriculum. In the first part of this paper, we present a dyadic system for teaching geometry considering the handicap of dyspraxic students and their preserved capabilities. We next present an example of an implementation of this work system in a pencil-and-paper environment with two students (dyspraxic student and standard student). Finally, we present the possibilities offered by a human interaction simulator to contribute to teaching geometry to dyspraxic students.

## A WORK SYSTEM FOR TEACHING GEOMETRY

The dyadic system (Petitfour, 2016) is based on the joint activity of two students, an *instructor* and a *constructor*, who interact to solve a geometric construction problem. This system is anchored in a conception of learning as a social phenomenon (Vygotsky, 1978): the formation of mathematical concepts is carried out through social interactions, in joint work around the resolution of a problem. The work takes place through free exchanges on possible construction techniques, the implementation of a chosen technique in the “instructor-constructor dyad” and discussions on the validity of what is produced. This dyadic work system is based on the language dimension of geometric activity, in order
# APPROACH TO TEACHING GEOMETRY TO DYSPRAXIC STUDENTS

Edith Petitfour

LDAR, Univ Rouen Normandie

The aim of this contribution is to present an overview of an approach to teaching geometry to dyspraxic fifth and sixth-grade students. These students face some important difficulties in using geometrical instruments (ruler, compass, set square) with required precision. We've developed a dyadic work system, based on the preserved skills of dyspraxic students, to allow them to study geometry without drawing geometric figures being an obstacle. We present this work system, its theoretical background, an implementation in a pencil-and-paper environment and the possibilities offered by a human interaction simulator to contribute to teaching geometry to dyspraxic students.

## INTRODUCTION

In French primary school and at the beginning of secondary school (students aged 9-12 years), the teaching and learning of geometry is based primarily on situations requiring the use of instruments (ruler, compass and set square) in a pencil-and-paper environment. Tracing with instruments to reproduce, to represent or to construct geometric figures is an injunction of the school curriculum, and is supposed to contribute to the conceptualization of geometric concepts. However, this teaching method, which uses manipulatives, is not suitable for all students, and in particular for students with dyspraxia, as they cannot perform the skilled movements necessary for such academic tasks: their motor performances are slower, less accurate and more variable (Elbasan & Kayihan, 2012). These students invariably fail to complete tasks in geometry, whereas they do have the core abilities – reasoning, language and memory – that would otherwise enable them to conceptualize geometric notions.

Our research aims at proposing alternative ways of teaching geometry, accessible to all students, and in particular for those who have difficulties in manipulation in order to meet the objectives of the geometry curriculum. In the first part of this paper, we present a dyadic system for teaching geometry considering the handicap of dyspraxic students and their preserved capabilities. We next present an example of an implementation of this work system in a pencil-and-paper environment with two students (dyspraxic student and standard student). Finally, we present the possibilities offered by a human interaction simulator to contribute to teaching geometry to dyspraxic students.

## A WORK SYSTEM FOR TEACHING GEOMETRY

The dyadic system (Petitfour, 2016) is based on the joint activity of two students, an *instructor* and a *constructor*, who interact to solve a geometric construction problem. This system is anchored in a conception of learning as a social phenomenon (Vygotsky, 1978): the formation of mathematical concepts is carried out through social interactions, in joint work around the resolution of a problem. The work takes place through free exchanges on possible construction techniques, the implementation of a chosen technique in the “instructor-constructor dyad” and discussions on the validity of what is produced. This dyadic work system is based on the language dimension of geometric activity, in order
to capitalize the preserved capabilities of dyspraxic students. It aims to replace instrumented action situations for dyspraxic student with situations that are as close to action as possible, but without any actual action, while producing similar effects in terms of geometric learning.

The construction proceeds as follows. The instructor expresses his/her *prior intention* (Searle, 1983) through language instructions to the constructor, accompanied by gestures. The constructor carries out the instrumented actions requested while the instructor observes the concrete implementation and obtains precise feedback. Let's specify the language, gestures and feedback specific to this work.

## Geometric technical language

The instructor uses a technical language (Petitfour, 2016) to formulate his/her plan for the instrumented actions. In the field of geometry, the technical language relates to the use of instruments in connection with the geometric properties they convey. It is used to express the relationship between geometric lines and a given instrument in order to produce a new line. For a given instrumented action, the instructor first mentions the instrument to be used, then gives its position by describing the relations between the instrument and graphic objects already present and finally indicates the location of the line. For example, drawing a circle of center A and radius AB (geometric language) corresponds to the following instructions in technical language: take the compass, put the point of the compass on point A and the pencil lead on point B, draw the circle. This language must not be confused with “manipulative language” that expresses corporeal action (e.g., “Hold the compass by the top, not by the legs”, “Press more on the point of the compass than on the pencil lead”).

## Mathematical gestures

Following McNeill (1992) in that gestures, together with language, help constitute thought, we propose to associate with oral language the use of mathematical gestures, i.e. any body movement, spontaneous or deliberate, carrying a meaning related to mathematics. In geometry, deictic gestures (Figure 1a) can, for example, link elements of the drawing (straight line, point) with parts of the instrument (side or vertex of the right angle of the set square); mimetic gestures (Figure 1b) can evoke the handling of the instrument (turning the compass); iconic gestures (Figure 1c) can represent a geometric relationship or a geometric object (triangle).

<figure><img src="image_81.png" /><figcaption>Figure 1: Example of mathematical gestures</figcaption></figure>

These mathematical gestures can be produced by dyspraxic students: they require less precision than actions and they offer the advantage of expressing the planned drawing without leaving a graphic
to capitalize the preserved capabilities of dyspraxic students. It aims to replace instrumented action situations for dyspraxic student with situations that are as close to action as possible, but without any actual action, while producing similar effects in terms of geometric learning.

The construction proceeds as follows. The instructor expresses his/her *prior intention* (Searle, 1983) through language instructions to the constructor, accompanied by gestures. The constructor carries out the instrumented actions requested while the instructor observes the concrete implementation and obtains precise feedback. Let's specify the language, gestures and feedback specific to this work.

## Geometric technical language

The instructor uses a technical language (Petitfour, 2016) to formulate his/her plan for the instrumented actions. In the field of geometry, the technical language relates to the use of instruments in connection with the geometric properties they convey. It is used to express the relationship between geometric lines and a given instrument in order to produce a new line. For a given instrumented action, the instructor first mentions the instrument to be used, then gives its position by describing the relations between the instrument and graphic objects already present and finally indicates the location of the line. For example, drawing a circle of center A and radius AB (geometric language) corresponds to the following instructions in technical language: take the compass, put the point of the compass on point A and the pencil lead on point B, draw the circle. This language must not be confused with “manipulative language” that expresses corporeal action (e.g., “Hold the compass by the top, not by the legs”, “Press more on the point of the compass than on the pencil lead”).

## Mathematical gestures

Following McNeill (1992) in that gestures, together with language, help constitute thought, we propose to associate with oral language the use of mathematical gestures, i.e. any body movement, spontaneous or deliberate, carrying a meaning related to mathematics. In geometry, deictic gestures (Figure 1a) can, for example, link elements of the drawing (straight line, point) with parts of the instrument (side or vertex of the right angle of the set square); mimetic gestures (Figure 1b) can evoke the handling of the instrument (turning the compass); iconic gestures (Figure 1c) can represent a geometric relationship or a geometric object (triangle).

<figure><img src="image_87.png" /><figcaption>Figure 1: Example of mathematical gestures</figcaption></figure>

These mathematical gestures can be produced by dyspraxic students: they require less precision than actions and they offer the advantage of expressing the planned drawing without leaving a graphic