## CONCLUSIONS

Experience has shown that the teaching and learning of geometry in classrooms with didactic resources can be limited by different aspects. On the one hand, the resource itself may constitute an obstacle by presenting mathematical or didactic restrictions. An example, from our experience, is the case of the introduction of trigonometric ratios through the use of fixed geoplanes. To overcome this physical limitation, we have combined it with another of a technological nature. Likewise, it merits accompanying the material with an adequate didactic analysis, planning and reflection by the teacher.

It is essential to remember the exploratory need for certain resources, such as geoplanes, polyforms, circular sectors and tangrams. This exploratory power lies in “letting touch” and “letting do”. Although, as has been indicated, it is convenient to implement oriented instructions, it is important that the teacher offers each student a previous time of personal manipulation of the resource. On the other hand, an aspect that also highlights the work with manipulatives in the teaching and learning of geometry is the potential to put geometric concepts into practice not only during the implementation of a didactic sequence or during the game but also during the previous process of elaboration. This shows the usefulness of giving students the opportunity to develop their own learning materials. During elaboration, students make guesses (correct and incorrect) through trial and error, interrelate concepts, take measurements, and pose new questions.

In addition, users may constitute another limitation. On the one hand, students may have difficulties for manipulation, either due to motor issues or prior knowledge. But it is also important to note that access to resources starts with teachers. Knowing them, manipulating them, analyzing them... These are aspects to be taken into account also in the initial and continuous training of teachers. The mere fact of presenting a manipulative resource to the class does not disoblige the teacher to be present with all his senses and to have an objective or intentionality with the activity. Precisely, as the experience of working in various concrete schools has marked us, with the mere fact of manipulating a resource you do not learn. The exchange and sharing of productions between students collaborates a lot. It is not only about using the resource, but requires a specialized look from whoever is in charge, with awareness of how and for what. Indeed, we have been noticing in the territory different levels of specificity of use of the resource: absent, dependent, intermittent and critical. Hence the need to consider the resource as a scaffold that does not generate dependence every time one have to work with the concept, becoming an obstacle. In critical use, to which we aspire as extension teachers and researchers, it also comes into play to know when to dispense with the resource.

Specifically, in the PM, the content “didactic resources” is present in the four years of the program, in each of the annual curricular spaces of PPD (I to IV). They are integrated into didactic sequences that the students themselves develop, as well as in microclasses in which future teachers play the roles of teachers and students. The referenced ReMatEd projects are integrated, year after year, by a minimum of 10 future teachers in Mathematics (a requirement of the University, according to the relevance of living this type of experiences) and the majority of the teachers of PPD also integrates ReMatEd. There is, in this sense, a community established in this regard with an approach aimed at documentary genesis processes (Gueudet & Trouche, 2009).

From ReMatEd we have gone through different contexts: from closets full of abandoned resources and without knowing their existence by the members of the institution, to schools with the intention
of using them but with ignorance of how or the economic inability to have copies. The proposal of
workshops located with teachers, either for the construction or for the incorporation of the resources
to their classes, the implementations with our accompaniment, and the resource kits, as well as
teaching how to assemble them, were some of the strategies carried out to overcome these difficulties.

Thus, in the present work we seek to document some of these attempts to address possible ways to
achieve the objectives in terms of learning geometry in classrooms and contexts with limitations, both
socioeconomic and training. Finally, we highlight as a key the amalgam between the extension and
research work located for the formation of the team. The extension has allowed a close link with the
environment and the investigative gaze has empowered to take these experiences as an input for
reflections, for their subsequent critical analysis in order to make contributions to the community.

References

Alegre, H., Domínguez, E., Landaluce, N. & Pípolo, S. (2018). Materiales didácticos en la enseñanza de la Matemática. In N. Sgreccia (Ed.). *Procesos de aAcompañamiento en la formación inicial y continua de profesores en matemática* (pp.153-174). Fahren House.

Alsina, C., Burgués, C. & Fortuny, J. (1988). *Materiales para construir la geometría*. Síntesis.

Atiyah, M. (2001). Mathematics in the 20th Century: Geometry versus Algebra. *Mathematics Today*, **37**(2), 47-49.

Corica, A.R. & Marín, E. (2014). Actividad de estudio e investigación para la enseñanza de nociones de geometría. *Números*, **85**, 91-114.

Corbetta, S. (2022). *Desarrollo sostenible. Capacitación en Ambiente. Ley Yolanda 27592*. Ministerio de Ambiente y Desarrollo Sostenible.

Dominguez, E. (2021). *Uso del Tangram como recurso didáctico en la formación inicial de Profesores en Matemática. El caso de la UNR* [Master’s Thesis]. UNR.

Gueudet, G. & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? *Educational Studies in Mathematics*, **71**(3), 199-218.

Gutiérrez, A. (2010). Introducción al Seminario I sobre Enseñanza y Aprendizaje de la Geometría. In M. Moreno, A. Estrada, J. Carrillo y T. Sierra (Eds.). *Investigación en Educación Matemática XIV* (pp.17-19). SEIEM.

Hierro, L.M. & Seller, E.P. (2020). El aprendizaje basado en el juego como herramienta socioeducativa en contextos comunitarios vulnerables. *Prisma Social*, (30), 88-114.

Jones, K., Fujita, T. & Ding, L. (2006). Informing the pedagogy for geometry: learning from teaching approaches in China and Japan. *Proceedings of the British Society for Research into Learning Mathematics*, **26**(2), 110-114.

Sgreccia, N., Mengarelli, M.S., Schaefer, L. & Menna, C. (2019). Hacer Matemática con Recursos Didácticos y Juegos Educativos. *Novedades Educativas*, **31**(339), 25-33.

Villarroel, S. & Sgreccia, N. (2012). Enseñanza de la Geometría en Secundaria. Caracterización de materiales didácticos concretos y habilidades. *Unión*, **8**(29), 59-84.
of using them but with ignorance of how or the economic inability to have copies. The proposal of workshops located with teachers, either for the construction or for the incorporation of the resources to their classes, the implementations with our accompaniment, and the resource kits, as well as teaching how to assemble them, were some of the strategies carried out to overcome these difficulties.

Thus, in the present work we seek to document some of these attempts to address possible ways to achieve the objectives in terms of learning geometry in classrooms and contexts with limitations, both socioeconomic and training. Finally, we highlight as a key the amalgam between the extension and research work located for the formation of the team. The extension has allowed a close link with the environment and the investigative gaze has empowered to take these experiences as an input for reflections, for their subsequent critical analysis in order to make contributions to the community.

## References

Alegre, H., Domínguez, E., Landaluce, N. & Pípolo, S. (2018). Materiales didácticos en la enseñanza de la Matemática. In N. Sgreccia (Ed.). *Procesos de aAcompañamiento en la formación inicial y continua de profesores en matemática* (pp.153-174). Fahren House.

Alsina, C., Burgués, C. & Fortuny, J. (1988). *Materiales para construir la geometría*. Síntesis.

Atiyah, M. (2001). Mathematics in the 20th Century: Geometry versus Algebra. *Mathematics Today*, 37(2), 47-49.

Corica, A.R. & Marín, E. (2014). Actividad de estudio e investigación para la enseñanza de nociones de geometría. *Números*, 85, 91-114.

Corbetta, S. (2022). *Desarrollo sostenible. Capacitación en Ambiente. Ley Yolanda 27592*. Ministerio de Ambiente y Desarrollo Sostenible.

Dominguez, E. (2021). *Uso del Tangram como recurso didáctico en la formación inicial de Profesores en Matemática. El caso de la UNR* [Master’s Thesis]. UNR.

Gueudet, G. & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? *Educational Studies in Mathematics*, 71(3), 199-218.

Gutiérrez, A. (2010). Introducción al Seminario I sobre Enseñanza y Aprendizaje de la Geometría. In M. Moreno, A. Estrada, J. Carrillo y T. Sierra (Eds.). *Investigación en Educación Matemática XIV* (pp.17-19). SEIEM.

Hierro, L.M. & Seller, E.P. (2020). El aprendizaje basado en el juego como herramienta socioeducativa en contextos comunitarios vulnerables. *Prisma Social*, (30), 88-114.

Jones, K., Fujita, T. & Ding, L. (2006). Informing the pedagogy for geometry: learning from teaching approaches in China and Japan. *Proceedings of the British Society for Research into Learning Mathematics*, 26(2), 110-114.

Sgreccia, N., Mengarelli, M.S., Schaefer, L. & Menna, C. (2019). Hacer Matemática con Recursos Didácticos y Juegos Educativos. *Novedades Educativas*, 31(339), 25-33.

Villarroel, S. & Sgreccia, N. (2012). Enseñanza de la Geometría en Secundaria. Caracterización de materiales didácticos concretos y habilidades. *Unión*, 8(29), 59-84.
# HOW DO TEACHERS COORDINATE STUDENTS' EMPIRICAL PERCEPTION AND LOGICAL REASONING IN 3D DYNAMIC GEOMETRY ENVIRONMENTS: CHINESE AND FRENCH CASES

Mingyu Shao¹, Jana Trgalova², Luc Trouche³

¹ South China Normal University, Guangzhou, China; ² Haute Ecole Pédagogique du Canton de Vaud, Lausanne, Switzerland; ³ Ecole Normale Supérieure de Lyon, Lyon, France

This paper presents a part of a French-Sino PhD project. It investigates mathematics teachers' practices, while using 3D dynamic geometry environments (DGE), for coordinating students' empirical perception and logical reasoning, and the teachers' justification for their coordination behavior. Adopting a qualitative approach in a multiple-case study, we select two teachers both in France and in China and develop a framework to analyze teachers' coordination behavior in lessons with 3D DGEs. The analysis reveals several patterns of coordination behavior, which are unevenly distributed among the four teachers' cases. Subsequently, the teachers' justifications are identified and contrasted. The findings illustrate how teachers' coordination patterns could be influenced by institutional, pedagogical, material and temporal factors. Finally, some implications for teacher education are drawn, along with a cross-cultural perspective for future research on 3D geometry teaching with technology.

## INTRODUCTION

This paper addresses teachers' didactical challenge when they use 3D DGEs in upper secondary school. 3D DGEs refer to digital files or websites constructed with a dynamic geometry software like Cabri 3D or GeoGebra. They enable users to create and manipulate 3D dynamic models, giving access to different viewpoints and to feedback (Hohenwarter & Jones, 2007). While 3D DGE bring new opportunities for teaching 3D geometry, they create a challenge for teachers for balancing students' empirical perception (e.g., visual and sensory motor) and logical reasoning (e.g., inductive and theoretical deductive reasoning). At the beginning, researchers always highlighted a progression from empirical perception and inductive reasoning to theoretical deductive reasoning in 3D DGEs (Accascina & Rogora, 2006). But later researchers argued the rationality of perceptual and inductive approaches to geometric knowledge, considering cultural, institutional, temporal and material contexts of teaching (Nardi et al., 2012). We call *coordination behavior* teachers' instructional behavior considering both aspects in students' activity – empirical perception and logical reasoning – in mathematics lessons with 3D DGE, and *coordination patterns* the regularities in their coordination behavior. This paper reports representative *coordination patterns* of French and Chinese teachers unveiled in Shao's PhD study (2022), situated in a Sino-French Cooperation Project which offers two advantages: a defamiliarization mechanism helping researchers re-see teaching practices in their original cultural context; an analytic process evidencing variations in factors like didactical traditions, institutional contexts and available materials, likely to impact teachers' practice.

In this paper, we do not preset criteria for what constitutes an appropriate coordination pattern, but rather try to understand teachers’ context and difficulties leading them to that pattern. After unveiling
# HOW DO TEACHERS COORDINATE STUDENTS' EMPIRICAL PERCEPTION AND LOGICAL REASONING IN 3D DYNAMIC GEOMETRY ENVIRONMENTS: CHINESE AND FRENCH CASES

Mingyu Shao¹, Jana Trgalova², Luc Trouche³

¹ South China Normal University, Guangzhou, China; ² Haute Ecole Pédagogique du Canton de Vaud, Lausanne, Switzerland; ³ Ecole Normale Supérieure de Lyon, Lyon, France

This paper presents a part of a French-Sino PhD project. It investigates mathematics teachers' practices, while using 3D dynamic geometry environments (DGE), for coordinating students' empirical perception and logical reasoning, and the teachers' justification for their coordination behavior. Adopting a qualitative approach in a multiple-case study, we select two teachers both in France and in China and develop a framework to analyze teachers' coordination behavior in lessons with 3D DGEs. The analysis reveals several patterns of coordination behavior, which are unevenly distributed among the four teachers' cases. Subsequently, the teachers' justifications are identified and contrasted. The findings illustrate how teachers' coordination patterns could be influenced by institutional, pedagogical, material and temporal factors. Finally, some implications for teacher education are drawn, along with a cross-cultural perspective for future research on 3D geometry teaching with technology.

## INTRODUCTION

This paper addresses teachers' didactical challenge when they use 3D DGEs in upper secondary school. 3D DGEs refer to digital files or websites constructed with a dynamic geometry software like Cabri 3D or GeoGebra. They enable users to create and manipulate 3D dynamic models, giving access to different viewpoints and to feedback (Hohenwarter & Jones, 2007). While 3D DGE bring new opportunities for teaching 3D geometry, they create a challenge for teachers for balancing students' empirical perception (e.g., visual and sensory motor) and logical reasoning (e.g., inductive and theoretical deductive reasoning). At the beginning, researchers always highlighted a progression from empirical perception and inductive reasoning to theoretical deductive reasoning in 3D DGEs (Accascina & Rogora, 2006). But later researchers argued the rationality of perceptual and inductive approaches to geometric knowledge, considering cultural, institutional, temporal and material contexts of teaching (Nardi et al., 2012). We call *coordination behavior* teachers' instructional behavior considering both aspects in students' activity – empirical perception and logical reasoning – in mathematics lessons with 3D DGE, and *coordination patterns* the regularities in their coordination behavior. This paper reports representative *coordination patterns* of French and Chinese teachers unveiled in Shao's PhD study (2022), situated in a Sino-French Cooperation Project which offers two advantages: a defamiliarization mechanism helping researchers re-see teaching practices in their original cultural context; an analytic process evidencing variations in factors like didactical traditions, institutional contexts and available materials, likely to impact teachers' practice.

In this paper, we do not preset criteria for what constitutes an appropriate coordination pattern, but rather try to understand teachers’ context and difficulties leading them to that pattern. After unveiling
several coordination patterns, we identify influencing factors from the justifications that teachers proposed for their behavior or decision-making, to answer the research question: “how do factors influencing teachers’ coordination patterns vary from one country to another?”

## LITERATURE REVIEW AND THEORETICAL FRAMEWORK

Coordination behavior, a notion specifically proposed in our study, is described by Shao (2022) as by adapting Toulmin (2003)'s diagram of argumentation (Fig. 1). This diagram is now widely used to model informal argumentation processes in mathematics instructional settings. Our adapted framework of coordination behavior encompasses several specifications to some of its components, aiming to reflect a mode of interaction between students' perception and logical reasoning. We also integrate, from research literature (Conner et al., 2014; Accascina & Rogora, 2006), types of teacher support, including questions (e.g., *requesting an idea, requesting elaboration*), supportive actions (e.g., *directing, promoting, evaluating*), and typical usages of 3D DGEs (e.g., *dragging for a better view, dragging for contrast*). These reflect teachers' strategies to foster students' empirical perception and logical reasoning. To save space, we do not elaborate on them, but directly present, in the Results section, the coordination patterns obtained by applying this diagram to data analysis.

<figure><img src="image_169.png" /><figcaption>Figure 1: Components in Toulmin's diagram (normal) and our specifications (italics)</figcaption></figure>

Besides unveiling teachers’ coordination patterns, this study attempts to identify influences on the patterns. To do this, we identify a *priori* a range of such influencing factors: the epistemological factors (a mathematical theorem or definition), pedagogical ones (a pedagogical principle), institutional ones (recommendation in a curriculum or textbook), empirical ones (teacher’s personal teaching experience), or evaluative ones (personally held view or belief) from Nardi et al. (2012); the temporal, material, and cultural contexts from Herbst & Chazan (2011). In our analysis, we refer to these factors but remain open to other influencing factors emerging from data.

## METHODS

We adopt a qualitative approach to a multiple-case study, involving two French and two Chinese teachers who use 3D DGEs in upper secondary 3D geometry lessons (3 lessons of French teachers and 6 lessons of Chinese teachers). The four teachers are experienced in, or at least keen on, using 3D DGEs; thus their cases are supposed to evidence fruitful coordination patterns and allow for a maximum exploration of the focal question within the time limit. For each teacher, the data collected include lesson videos – to faithfully record teachers’ coordination behavior – and a follow-up interview – to elicit teacher’s justification and influencing factors. The interview is conducted after
several coordination patterns, we identify influencing factors from the justifications that teachers proposed for their behavior or decision-making, to answer the research question: “how do factors influencing teachers’ coordination patterns vary from one country to another?”

## LITERATURE REVIEW AND THEORETICAL FRAMEWORK

Coordination behavior, a notion specifically proposed in our study, is described by Shao (2022) as by adapting Toulmin (2003)'s diagram of argumentation (Fig. 1). This diagram is now widely used to model informal argumentation processes in mathematics instructional settings. Our adapted framework of coordination behavior encompasses several specifications to some of its components, aiming to reflect a mode of interaction between students' perception and logical reasoning. We also integrate, from research literature (Conner et al., 2014; Accascina & Rogora, 2006), types of teacher support, including questions (e.g., *requesting an idea, requesting elaboration*), supportive actions (e.g., *directing, promoting, evaluating*), and typical usages of 3D DGEs (e.g., *dragging for a better view, dragging for contrast*). These reflect teachers' strategies to foster students' empirical perception and logical reasoning. To save space, we do not elaborate on them, but directly present, in the Results section, the coordination patterns obtained by applying this diagram to data analysis.

<figure><img src="image_176.png" /><figcaption>Figure 1: Components in Toulmin's diagram (normal) and our specifications (italics)</figcaption></figure>

Besides unveiling teachers’ coordination patterns, this study attempts to identify influences on the patterns. To do this, we identify a *priori* a range of such influencing factors: the epistemological factors (a mathematical theorem or definition), pedagogical ones (a pedagogical principle), institutional ones (recommendation in a curriculum or textbook), empirical ones (teacher’s personal teaching experience), or evaluative ones (personally held view or belief) from Nardi et al. (2012); the temporal, material, and cultural contexts from Herbst & Chazan (2011). In our analysis, we refer to these factors but remain open to other influencing factors emerging from data.

## METHODS

We adopt a qualitative approach to a multiple-case study, involving two French and two Chinese teachers who use 3D DGEs in upper secondary 3D geometry lessons (3 lessons of French teachers and 6 lessons of Chinese teachers). The four teachers are experienced in, or at least keen on, using 3D DGEs; thus their cases are supposed to evidence fruitful coordination patterns and allow for a maximum exploration of the focal question within the time limit. For each teacher, the data collected include lesson videos – to faithfully record teachers’ coordination behavior – and a follow-up interview – to elicit teacher’s justification and influencing factors. The interview is conducted after
an initial analysis of the empirical and logical reasoning activities in the lesson videos, and we question teachers' decision-making related to the activities and their justifications. A qualitative content analysis is performed on the lesson videos with the adapted diagram of argumentation. The videos are divided into episodes by the tasks or subtasks in the lesson. Each episode is modelled by a diagram of argumentation. Then the episodes are grouped by the basic structure of their diagrams – combination of argumentation components. As a result, each group of episodes shares the same basic structure in the diagrams and corresponds to one coordination pattern. Then, the interview scripts related to each group are gathered, from which teachers' justifications and influencing factors on their coordination patterns are identified.

Since a range of influencing factors have been identified in literature, we focus on those relevant to teachers' coordination and briefly describe their states in France and China (Table 1).

<table><caption>Table 1: Factors likely to influence teachers' actions in coordination</caption><thead><tr><th>Factors</th><th>France</th><th>China</th></tr></thead><tbody><tr><td>Epistemological</td><td>Modern mathematics system, including axiomatic Euclidean geometry, established early in France but late in China</td><td></td></tr><tr><td>Institutional</td><td>Both curricula take mathematical proof at the heart, and highlight development of spatial visualization, deductive reasoning and task-solving abilities; Exam requirements vary from one task to another</td><td></td></tr><tr><td>Pedagogical</td><td>Didactical tradition of highlighting rigorous proof and pure mathematics system (Gueudet et al., 2017); Theory of didactical situations (Brousseau, 2002)</td><td>Didactical tradition of highlighting classroom efficiency and “two basis” (Xu et al., 2013); Theory of teaching with variations (Gu et al., 2004)</td></tr><tr><td>Empirical and evaluative</td><td>Teacher's personal teaching and learning experience, views, values or beliefs, different among individuals</td><td></td></tr><tr><td>Temporal</td><td>Tight teaching schedule; Flexible way of managing time for one lesson</td><td>Tight teaching schedule; Well-structured time for one lesson</td></tr><tr><td>Material</td><td>A set of well-structured 3D DGEs offering a progression of 3D geometry tasks, GeoGebra tools, and feedback</td><td>3D DGEs mostly need to be designed by teachers themselves</td></tr></tbody></table>

## DATA ANALYSIS AND FINDINGS

This section presents representative coordination patterns emerging from data, and the corresponding teachers' justifications. The italics in the following text are elements in the new behavior framework.

### Coordination pattern 1: From empirical perception to theoretical deductive reasoning

This pattern is characterized by a chain of *perceptual data – non-absolute qualifier – claim* and a chain of *factual data – deductive warrant – absolute qualifier – claim*. That means teachers move students from a perceptual to a theoretical approach to confirm a conclusion. When the first claim is incorrect, teachers propose a *rebuttal* (i.e., rebut students' biased perception) by *dragging for a better view* or *evoking feedback* in 3D DGE. This pattern is identified when Dora helps students construct
an initial analysis of the empirical and logical reasoning activities in the lesson videos, and we question teachers' decision-making related to the activities and their justifications. A qualitative content analysis is performed on the lesson videos with the adapted diagram of argumentation. The videos are divided into episodes by the tasks or subtasks in the lesson. Each episode is modelled by a diagram of argumentation. Then the episodes are grouped by the basic structure of their diagrams – combination of argumentation components. As a result, each group of episodes shares the same basic structure in the diagrams and corresponds to one coordination pattern. Then, the interview scripts related to each group are gathered, from which teachers' justifications and influencing factors on their coordination patterns are identified.

Since a range of influencing factors have been identified in literature, we focus on those relevant to teachers' coordination and briefly describe their states in France and China (Table 1).

<table><caption>Table 1: Factors likely to influence teachers' actions in coordination</caption><thead><tr><th>Factors</th><th>France</th><th>China</th></tr></thead><tbody><tr><td>Epistemological</td><td>Modern mathematics system, including axiomatic Euclidean geometry, established early in France but late in China</td><td></td></tr><tr><td>Institutional</td><td>Both curricula take mathematical proof at the heart, and highlight development of spatial visualization, deductive reasoning and task-solving abilities; Exam requirements vary from one task to another</td><td></td></tr><tr><td>Pedagogical</td><td>Didactical tradition of highlighting rigorous proof and pure mathematics system (Gueudet et al., 2017); Theory of didactical situations (Brousseau, 2002)</td><td>Didactical tradition of highlighting classroom efficiency and “two basis” (Xu et al., 2013); Theory of teaching with variations (Gu et al., 2004)</td></tr><tr><td>Empirical and evaluative</td><td>Teacher's personal teaching and learning experience, views, values or beliefs, different among individuals</td><td></td></tr><tr><td>Temporal</td><td>Tight teaching schedule; Flexible way of managing time for one lesson</td><td>Tight teaching schedule; Well-structured time for one lesson</td></tr><tr><td>Material</td><td>A set of well-structured 3D DGEs offering a progression of 3D geometry tasks, GeoGebra tools, and feedback</td><td>3D DGEs mostly need to be designed by teachers themselves</td></tr></tbody></table>

## DATA ANALYSIS AND FINDINGS

This section presents representative coordination patterns emerging from data, and the corresponding teachers' justifications. The italics in the following text are elements in the new behavior framework.

### Coordination pattern 1: From empirical perception to theoretical deductive reasoning

This pattern is characterized by a chain of *perceptual data – non-absolute qualifier – claim* and a chain of *factual data – deductive warrant – absolute qualifier – claim*. That means teachers move students from a perceptual to a theoretical approach to confirm a conclusion. When the first claim is incorrect, teachers propose a *rebuttal* (i.e., rebut students' biased perception) by *dragging for a better view* or *evoking feedback* in 3D DGE. This pattern is identified when Dora helps students construct
intersection of planes (EID) and (ABC) in a 3D DGE (Fig. 2). At first, student S1 claims that lines (EI) and (AD) intersect to produce one point in the target intersection, drawing on perceptual effects on the screen. Dora lets S1 construct the intersection, evoking the 3D DGE internal feedback of “nothing happens”. Then Dora drags for a better view in the 3D DGE (Fig. 3), which makes S1 uncertain about her claim and propose a new one: (EI) and (AB) intersect. Dora constantly asks for S1’s explanation (*request elaboration*) and *evaluates* her response only when she says the two lines are “both in plane (EAB)”. Here, S1 probably refers to the geometric theorem she learned earlier – two lines in space are either secant or parallel when they are coplanar, and it is clear that Dora prefers students to utilize theorems rather than empirical perception.

<figure><img src="image_170.png" /><figcaption>Figure 2: S2’s screen at the beginning</figcaption></figure>

<figure><img src="image_171.png" /><figcaption>Figure 3: Dora drags for a better view in 3D DGE</figcaption></figure>

## Coordination pattern 2: Combine directed empirical perception with geometric theorem

This pattern is characterized by a chain of *perceptual data – deductive warrant – absolute qualifier – claim*. That means teachers lead students to combine empirical evidence in 3D DGE with theorems and make them certain of the conclusion derived. They always *direct* students to explore perceptual data that can satisfy the precondition of a theorem. Then they *promote* students to get a claim by combining the perceptual data with the theorem (deductive warrant).

<figure><img src="image_172.png" /><figcaption>Figure 4: The cube overlaps with its initial position</figcaption></figure>

<figure><img src="image_173.png" /><figcaption>Figure 5: Huang simulates the rotation of the cube</figcaption></figure>

This pattern is identified when Huang helps students determine a section plane that forms equal angles with all edges of a cube (part of a multiple-choice task in his lessons, Fig. 4). Huang shows the configuration in Fig.4 and reminds students of a theorem: “a figure always keeps its shape and size while being rotated”. Then he rotates the cube around the diagonal (AG) with a slider, showing that [AE], [AB] and [AD] would respectively coincide with [AB], [AD] and [AE] (Fig. 5). Combining the “coinciding effect” and the theorem, Huang leads students to, and makes them certain of, a conclusion that the three triangles ABO, ADO and AEO are congruent, and thus the angles between plane (EDB) and [AE], [AB], [AD] are all equal.
intersection of planes (EID) and (ABC) in a 3D DGE (Fig. 2). At first, student S1 claims that lines (EI) and (AD) intersect to produce one point in the target intersection, drawing on perceptual effects on the screen. Dora lets S1 construct the intersection, evoking the 3D DGE internal feedback of “nothing happens”. Then Dora drags for a better view in the 3D DGE (Fig. 3), which makes S1 uncertain about her claim and propose a new one: (EI) and (AB) intersect. Dora constantly asks for S1’s explanation (*request elaboration*) and *evaluates* her response only when she says the two lines are “both in plane (EAB)”. Here, S1 probably refers to the geometric theorem she learned earlier – two lines in space are either secant or parallel when they are coplanar, and it is clear that Dora prefers students to utilize theorems rather than empirical perception.

<figure><img src="image_177.png" /><figcaption>Figure 2: S2’s screen at the beginning</figcaption></figure>

<figure><img src="image_178.png" /><figcaption>Figure 3: Dora drags for a better view in 3D DGE</figcaption></figure>

## Coordination pattern 2: Combine directed empirical perception with geometric theorem

This pattern is characterized by a chain of *perceptual data – deductive warrant – absolute qualifier – claim*. That means teachers lead students to combine empirical evidence in 3D DGE with theorems and make them certain of the conclusion derived. They always *direct* students to explore perceptual data that can satisfy the precondition of a theorem. Then they *promote* students to get a claim by combining the perceptual data with the theorem (deductive warrant).

<figure><img src="image_179.png" /><figcaption>Figure 4: The cube overlaps with its initial position</figcaption></figure>

<figure><img src="image_180.png" /><figcaption>Figure 5: Huang simulates the rotation of the cube</figcaption></figure>

This pattern is identified when Huang helps students determine a section plane that forms equal angles with all edges of a cube (part of a multiple-choice task in his lessons, Fig. 4). Huang shows the configuration in Fig.4 and reminds students of a theorem: “a figure always keeps its shape and size while being rotated”. Then he rotates the cube around the diagonal (AG) with a slider, showing that [AE], [AB] and [AD] would respectively coincide with [AB], [AD] and [AE] (Fig. 5). Combining the “coinciding effect” and the theorem, Huang leads students to, and makes them certain of, a conclusion that the three triangles ABO, ADO and AEO are congruent, and thus the angles between plane (EDB) and [AE], [AB], [AD] are all equal.
Coordination pattern 3: Focus on empirical perception

This pattern is characterized by a chain of *perceptual data* – *absolute qualifier* – *claim*. That means teachers make students certain of a conclusion only by fostering their empirical perception. Perceptual data concern multiple observable examples, which are sometimes evoked by teachers with the modality of *dragging for contrast*, sometimes provided by students under teachers’ questions of *requesting elaboration*. With the perceptual data, teachers directly propose a conclusion about which students generally hold no doubt. This pattern is identified when teachers help students understand in which condition the 2D representation can faithfully represent properties in space (Fig. 6), or when they teach axioms about relative positions in space.

<figure><img src="image_174.png" /><figcaption>Figure 6: Dora drags to contrast a bottom angle (∠JFA) (a) and a front angle (∠BGF) in a 2D representation (b)</figcaption></figure>

Coordination pattern 4: Combine empirical perception with inductive reasoning

This pattern is characterized by a chain of *perceptual data* – *inductive warrant* – *absolute qualifier* – *claim*. That means teachers lead students to combine inductive reasoning with perceptual evidence and make them certain of the generalized conclusion. Perceptual data mainly include evidence that teachers evoked *dragging for separation* and *dragging for a better view* in 3D DGE.

<figure><img src="image_175.png" /><figcaption>Figure 7: Huang drags to help students to evidence the invariant aspect – P always projected onto the bisector of ∠ADC</figcaption></figure>

<caption>Figure 8: Huang drags for a “better view” to observe the position of projection of P</caption>

The pattern is identified when Huang teaches students a conclusion about a figural pattern: for any pyramid P-ADC situated in a cube, with P being in the diagonal face of the cube, the orthogonal projection of P to the base plane always falls on the bisector of ∠ADC (Fig. 8). Huang uses the two dragging modalities (Fig. 7-8) and guides students to generalize the conclusion by inductive reasoning.
Coordination pattern 3: Focus on empirical perception

This pattern is characterized by a chain of *perceptual data* – *absolute qualifier* – *claim*. That means teachers make students certain of a conclusion only by fostering their empirical perception. Perceptual data concern multiple observable examples, which are sometimes evoked by teachers with the modality of *dragging for contrast*, sometimes provided by students under teachers’ questions of *requesting elaboration*. With the perceptual data, teachers directly propose a conclusion about which students generally hold no doubt. This pattern is identified when teachers help students understand in which condition the 2D representation can faithfully represent properties in space (Fig. 6), or when they teach axioms about relative positions in space.

<figure><img src="image_181.png" /><figcaption>Figure 6: Dora drags to contrast a bottom angle (∠JFA) (a) and a front angle (∠BGF) in a 2D representation (b)</figcaption></figure>

Coordination pattern 4: Combine empirical perception with inductive reasoning

This pattern is characterized by a chain of *perceptual data* – *inductive warrant* – *absolute qualifier* – *claim*. That means teachers lead students to combine inductive reasoning with perceptual evidence and make them certain of the generalized conclusion. Perceptual data mainly include evidence that teachers evoked *dragging for separation* and *dragging for a better view* in 3D DGE.

<figure><img src="image_182.png" /><figcaption>Figure 7: Huang drags to help students to evidence the invariant aspect – P always projected onto the bisector of ∠ADC</figcaption></figure>

<caption>Figure 8: Huang drags for a “better view” to observe the position of projection of P</caption>

The pattern is identified when Huang teaches students a conclusion about a figural pattern: for any pyramid P-ADC situated in a cube, with P being in the diagonal face of the cube, the orthogonal projection of P to the base plane always falls on the bisector of ∠ADC (Fig. 8). Huang uses the two dragging modalities (Fig. 7-8) and guides students to generalize the conclusion by inductive reasoning.
Coordination pattern 5: Empirical perception as auxiliary support for deductive reasoning

In this pattern, characterized by a chain of factual data – deductive warrant – absolute qualifier– claim, the teacher support aiming to elicit chain components involves perceptual effects serving as auxiliary support for theoretical deductive reasoning. This pattern is observed when teachers use 3D DGEs, gestures or real objects to illustrate the structure of a 3D task figure or a geometric theorem.

In summary, five coordination patterns are identified, which differ in the interaction mode between theoretical and perceptual activities, and in the role played by 3D DGE. Table 2 shows frequencies of the coordination patterns in each teacher’s case. Next we contrast patterns and related justifications of the French and Chinese teachers.

<table><caption>Table 2: Coordination pattern frequencies by teachers</caption><thead><tr><th>Episodes involving coordination pattern</th><th>Dora</th><th>Sonia</th><th>Huang</th><th>Xia</th></tr></thead><tbody><tr><td>1. From perception to theoretical deductive reasoning</td><td>32</td><td>16</td><td>0</td><td>2</td></tr><tr><td>2. Combine directed perception with geometric theorem</td><td>12</td><td>2</td><td>42</td><td>0</td></tr><tr><td>3. Focus on perception</td><td>4</td><td>10</td><td>0</td><td>18</td></tr><tr><td>4. Combine perception with inductive reasoning</td><td>0</td><td>0</td><td>4</td><td>0</td></tr><tr><td>5. Perception as auxiliary support for theoretical deductive reasoning</td><td>5</td><td>0</td><td>28</td><td>14</td></tr><tr><td>Total Episodes</td><td>53</td><td>28</td><td>74</td><td>34</td></tr></tbody></table>

Patterns 1 and 5, fostering students’ theoretical deductive reasoning, occur frequently in all teachers. French teachers most frequently move students from empirical perception to theoretical deductive reasoning (pattern 1). This corresponds to the fact that in two of their three lessons, every student has access to a computer. In the interviews, both Dora and Sonia express their wish to let students learn by autonomy in the 3D DGEs, designed by a researcher-teacher team in France and aiming to create a “milieu” for students to interact with and learn from feedback. Here we identify pedagogical influence on Dora and Sonia’s coordination patterns (Theory of didactical situations). Dora and Sonia also express the pedagogical influence of the French didactic tradition of highlighting rigorous proof:

Dora: I want them to reason all the time, or more precisely, to appropriate the theorem.

Sonia: There are two things to make an intersection, 'construct' and 'justify'... they should know how to do, and meanwhile know why to do this (Sonia).

For the two Chinese teachers, pattern 5 is more frequently observed than pattern 1. The 3D DGEs they use are all designed by the teachers themselves, and in pattern 5, they choose to control them throughout their lessons, providing perceptual support only when students struggle with theoretical deductive reasoning. Both Huang and Xia think that in this way, the lessons can go “more efficiently”. Furthermore, the corresponding tasks are all short-essay tasks in exams requiring students to rigorously demonstrate their reasoning. Huang and Xia express concern for the exam requirements:

Huang: They won’t have any aid except for the paper, pens and drawings in the exam. If I present them the dynamic models at very first, they still cannot manage it when back to paper.

Therefore, we identify here institutional influence (exam requirements for particular 3D geometry tasks) and pedagogical influence (didactical tradition of highlighting efficiency in the classroom).

Notably, Dora and Sonia mention that the availability of well-prepared 3D DGEs leads them to allow students exploring tasks in autonomy in a computer room. In Dora’s lesson in an ordinary classroom,
Coordination pattern 5: Empirical perception as auxiliary support for deductive reasoning

In this pattern, characterized by a chain of factual data – deductive warrant – absolute qualifier–claim, the teacher support aiming to elicit chain components involves perceptual effects serving as auxiliary support for theoretical deductive reasoning. This pattern is observed when teachers use 3D DGEs, gestures or real objects to illustrate the structure of a 3D task figure or a geometric theorem.

In summary, five coordination patterns are identified, which differ in the interaction mode between theoretical and perceptual activities, and in the role played by 3D DGE. Table 2 shows frequencies of the coordination patterns in each teacher’s case. Next we contrast patterns and related justifications of the French and Chinese teachers.

<table><caption>Table 2: Coordination pattern frequencies by teachers</caption><thead><tr><th>Episodes involving coordination pattern</th><th>Dora</th><th>Sonia</th><th>Huang</th><th>Xia</th></tr></thead><tbody><tr><td>1. From perception to theoretical deductive reasoning</td><td>32</td><td>16</td><td>0</td><td>2</td></tr><tr><td>2. Combine directed perception with geometric theorem</td><td>12</td><td>2</td><td>42</td><td>0</td></tr><tr><td>3. Focus on perception</td><td>4</td><td>10</td><td>0</td><td>18</td></tr><tr><td>4. Combine perception with inductive reasoning</td><td>0</td><td>0</td><td>4</td><td>0</td></tr><tr><td>5. Perception as auxiliary support for theoretical deductive reasoning</td><td>5</td><td>0</td><td>28</td><td>14</td></tr><tr><td>Total Episodes</td><td>53</td><td>28</td><td>74</td><td>34</td></tr></tbody></table>

Patterns 1 and 5, fostering students’ theoretical deductive reasoning, occur frequently in all teachers. French teachers most frequently move students from empirical perception to theoretical deductive reasoning (pattern 1). This corresponds to the fact that in two of their three lessons, every student has access to a computer. In the interviews, both Dora and Sonia express their wish to let students learn by autonomy in the 3D DGEs, designed by a researcher-teacher team in France and aiming to create a “milieu” for students to interact with and learn from feedback. Here we identify pedagogical influence on Dora and Sonia’s coordination patterns (Theory of didactical situations). Dora and Sonia also express the pedagogical influence of the French didactic tradition of highlighting rigorous proof:

Dora: I want them to reason all the time, or more precisely, to appropriate the theorem.

Sonia: There are two things to make an intersection, 'construct' and 'justify'... they should know how to do, and meanwhile know why to do this (Sonia).

For the two Chinese teachers, pattern 5 is more frequently observed than pattern 1. The 3D DGEs they use are all designed by the teachers themselves, and in pattern 5, they choose to control them throughout their lessons, providing perceptual support only when students struggle with theoretical deductive reasoning. Both Huang and Xia think that in this way, the lessons can go “more efficiently”. Furthermore, the corresponding tasks are all short-essay tasks in exams requiring students to rigorously demonstrate their reasoning. Huang and Xia express concern for the exam requirements:

Huang: They won’t have any aid except for the paper, pens and drawings in the exam. If I present them the dynamic models at very first, they still cannot manage it when back to paper.

Therefore, we identify here institutional influence (exam requirements for particular 3D geometry tasks) and pedagogical influence (didactical tradition of highlighting efficiency in the classroom).

Notably, Dora and Sonia mention that the availability of well-prepared 3D DGEs leads them to allow students exploring tasks in autonomy in a computer room. In Dora’s lesson in an ordinary classroom,
only pattern 5 is observed: Dora controls 3D DGEs and guides students' work on paper. Here the material factor (3D DGE availability) emerges as another influence on teachers' coordination patterns.

The other three patterns either focus on fostering students' empirical perception (patterns 3 and 4) or combine students' perceptual activities with incomplete theoretical deductive reasoning (pattern 2). These patterns are unevenly spread over the four teachers, among which Huang's case is very specific: although pattern 3 is not observed, he is the only teacher mobilizing pattern 4 and he shows preference for pattern 2 much more than the other three teachers. The geometry tasks corresponding to the two patterns are small exam tasks, i.e., multiple-choice or fill-in-blank tasks that only require a final answer. In pattern 4, Huang keeps students at the level of rational conjecture derived from empirical perception and inductive reasoning, which is enough to correctly answer the corresponding task. When evoking this pattern, Huang attaches importance to developing students' visualization in space and the ability of formulating rational conjectures, and expresses his concern for exam requirements:

**Huang:** Students should not spend too much time on such a task, it takes up a small portion of the total marks of the exam. If they have a good vision in space and can “guess” the correct answer on that basis, I will say OK, it’s enough.

Similarly, pattern 2 is identified in Huang's case, when he explains small exam tasks that involve complex 3D figural structures. Huang encourages students to take perceptual evidence in 3D DGEs as a fact to replace a part of theoretical deductive reasoning, because he believes that the empirical perceptual support in 3D DGEs could allow them to “accumulate images that can be remobilized back to the paper” and to “improve their task-solving efficiency.” Here, we identify institutional (exam requirements for particular 3D geometry tasks) and evaluative (Huang’s value in 3D geometry teaching and views on advantages of 3D DGEs) influences on coordination patterns.

Pattern 2 is also identified in Dora and Sonia, but their justification is more influenced by temporal factors and the content of the tasks at stake. They always value students' theoretical activities but compromise with perceptual support from 3D DGEs due to complex task figures and time constraints:

**Dora:** The students have a good vision, but they did not dare to go. It's my role to reassure them that it's the right direction... if I don't help them out, they would be stuck and not learn much from this lesson... if we had three hours, I would leave them to try by themselves.

Pattern 3 is identified in both French teachers and Xia. For these teachers, students only need to “understand” but not “prove” a geometric axiom or rule, that is why they focus on fostering students’ empirical perception when helping students to understand a rule. As can be seen, the three teachers have similar personal views on advantages of 3D DGEs in illustrating geometric axioms or rules. From that we identify evaluative influences on their coordination patterns.

## DISCUSSION AND CONCLUSION

To conclude, the teachers, while using 3D DGEs, show different coordination patterns and they refer to different factors to justify them. Firstly, pedagogical (e.g., didactical tradition in a country) and evaluative factors (e.g., teachers’ views on the educational value of 3D DGEs) have significant influence on the four teachers’ behavior. The evaluative factors differ among individual teachers, and we observe more differences within groups than between groups of teachers. Secondly, the two
only pattern 5 is observed: Dora controls 3D DGEs and guides students' work on paper. Here the material factor (3D DGE availability) emerges as another influence on teachers' coordination patterns.

The other three patterns either focus on fostering students' empirical perception (patterns 3 and 4) or combine students' perceptual activities with incomplete theoretical deductive reasoning (pattern 2). These patterns are unevenly spread over the four teachers, among which Huang's case is very specific: although pattern 3 is not observed, he is the only teacher mobilizing pattern 4 and he shows preference for pattern 2 much more than the other three teachers. The geometry tasks corresponding to the two patterns are small exam tasks, i.e., multiple-choice or fill-in-blank tasks that only require a final answer. In pattern 4, Huang keeps students at the level of rational conjecture derived from empirical perception and inductive reasoning, which is enough to correctly answer the corresponding task. When evoking this pattern, Huang attaches importance to developing students' visualization in space and the ability of formulating rational conjectures, and expresses his concern for exam requirements:

**Huang:** Students should not spend too much time on such a task, it takes up a small portion of the total marks of the exam. If they have a good vision in space and can “guess” the correct answer on that basis, I will say OK, it’s enough.

Similarly, pattern 2 is identified in Huang's case, when he explains small exam tasks that involve complex 3D figural structures. Huang encourages students to take perceptual evidence in 3D DGEs as a fact to replace a part of theoretical deductive reasoning, because he believes that the empirical perceptual support in 3D DGEs could allow them to “accumulate images that can be remobilized back to the paper” and to “improve their task-solving efficiency.” Here, we identify institutional (exam requirements for particular 3D geometry tasks) and evaluative (Huang’s value in 3D geometry teaching and views on advantages of 3D DGEs) influences on coordination patterns.

Pattern 2 is also identified in Dora and Sonia, but their justification is more influenced by temporal factors and the content of the tasks at stake. They always value students' theoretical activities but compromise with perceptual support from 3D DGEs due to complex task figures and time constraints:

**Dora:** The students have a good vision, but they did not dare to go. It's my role to reassure them that it's the right direction... if I don't help them out, they would be stuck and not learn much from this lesson... if we had three hours, I would leave them to try by themselves.

Pattern 3 is identified in both French teachers and Xia. For these teachers, students only need to “understand” but not “prove” a geometric axiom or rule, that is why they focus on fostering students’ empirical perception when helping students to understand a rule. As can be seen, the three teachers have similar personal views on advantages of 3D DGEs in illustrating geometric axioms or rules. From that we identify evaluative influences on their coordination patterns.

## DISCUSSION AND CONCLUSION

To conclude, the teachers, while using 3D DGEs, show different coordination patterns and they refer to different factors to justify them. Firstly, pedagogical (e.g., didactical tradition in a country) and evaluative factors (e.g., teachers’ views on the educational value of 3D DGEs) have significant influence on the four teachers’ behavior. The evaluative factors differ among individual teachers, and we observe more differences within groups than between groups of teachers. Secondly, the two
French teachers refer more to material and temporal factors in their justifications, whereas the two Chinese teachers refer more to institutional ones, especially exam requirements. Finally, the content of 3D geometry tasks is a new factor emerging from data. This study can be related to Herbst & Chazan's (2011) study that confronts teachers with others' teaching episodes in which some norms are breached to elicit teachers' practical rationality, i.e., their dispositions towards norms of customary teaching. Here, we adopt an alternative technique – confronting teachers with their own teaching episodes – to elicit their practical rationality. Furthermore, this study attests the interest of a cross-cultural approach for deepening our understanding of what constitute a “rational” teaching practice (coordination pattern) and what factors lead teachers to that pattern. The diverse rational coordination patterns form a repertoire of practice resources that teachers can mobilize according to their didactic objectives. The intricate web of factors that have different influences on different group of teachers are supposed to help teacher educators to design more targeted training programs.

## References

Accascina, G., & Rogora, E. (2006). Using Cabri3D diagrams for teaching geometry. *International Journal for Technology in Mathematics Education, 13*(1), 11–22.

Brousseau, G. (2002). *Theory of didactical situations in mathematics: didactique des mathématiques, 1970–1990*. Springer Netherlands.

Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Teacher support for collective argumentation: A framework for examining how teachers support students’ engagement in mathematical activities. *Educational Studies in Mathematics, 86*(3), 401–429.

Gu, L., Hunag, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In *How Chinese Learn Mathematics: Vol. Volume 1* (pp. 309–347). World Scientific.

Gueudet, G., Bueno-Ravel, L., Modeste, S., & Trouche, L. (2017). Curriculum in France: A National Frame in Transition. In D. Thompson et al. (Eds.), *International Perspectives on Mathematics Curriculum* (pp. 41–70). International Age Publishing.

Hohenwarter, M., & Jones, K. (2007). Ways of linking geometry and algebra, the case of Geogebra. *Proceedings of the British Society for Research into Learning Mathematics, 27*(3), 126–131.

Herbst, P., & Chazan, D. (2011). Research on Practical Rationality: Studying the justification of actions in mathematics teaching. *The Mathematics Enthusiast, 8*(3), 405–462

Nardi, E., Biza, I., & Zachariades, T. (2012). ‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation. *Educational Studies in Mathematics, 79*(2), 157–173.

Shao, M. (2022). How do teachers support students’ coordination of perception and logical reasoning in 3D dynamic geometry environments: French and Chinese cases [Phd thesis, Ecole normale supérieure deyon; East China normal university].

Toulmin, S. E. (2003). *The Uses of Argument* (2nd ed.). Cambridge: Cambridge University Press; Cambridge Core.

Xu, B., Kong, Q., Yu, P., & Su, H. (2013). Chinese mathematics classroom teaching. In *Mathematics education in china: Tradition and reality* (pp. 73–120). Jiangsu Education Publishing House.
French teachers refer more to material and temporal factors in their justifications, whereas the two Chinese teachers refer more to institutional ones, especially exam requirements. Finally, the content of 3D geometry tasks is a new factor emerging from data. This study can be related to Herbst & Chazan's (2011) study that confronts teachers with others' teaching episodes in which some norms are breached to elicit teachers' practical rationality, i.e., their dispositions towards norms of customary teaching. Here, we adopt an alternative technique – confronting teachers with their own teaching episodes – to elicit their practical rationality. Furthermore, this study attests the interest of a cross-cultural approach for deepening our understanding of what constitute a “rational” teaching practice (coordination pattern) and what factors lead teachers to that pattern. The diverse rational coordination patterns form a repertoire of practice resources that teachers can mobilize according to their didactic objectives. The intricate web of factors that have different influences on different group of teachers are supposed to help teacher educators to design more targeted training programs.

## References

Accascina, G., & Rogora, E. (2006). Using Cabri3D diagrams for teaching geometry. *International Journal for Technology in Mathematics Education, 13*(1), 11–22.

Brousseau, G. (2002). *Theory of didactical situations in mathematics: didactique des mathématiques, 1970–1990*. Springer Netherlands.

Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Teacher support for collective argumentation: A framework for examining how teachers support students’ engagement in mathematical activities. *Educational Studies in Mathematics, 86*(3), 401–429.

Gu, L., Hunag, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In *How Chinese Learn Mathematics: Vol. Volume 1* (pp. 309–347). World Scientific.

Gueudet, G., Bueno-Ravel, L., Modeste, S., & Trouche, L. (2017). Curriculum in France: A National Frame in Transition. In D. Thompson et al. (Eds.), *International Perspectives on Mathematics Curriculum* (pp. 41–70). International Age Publishing.

Hohenwarter, M., & Jones, K. (2007). Ways of linking geometry and algebra, the case of Geogebra. *Proceedings of the British Society for Research into Learning Mathematics, 27*(3), 126–131.

Herbst, P., & Chazan, D. (2011). Research on Practical Rationality: Studying the justification of actions in mathematics teaching. *The Mathematics Enthusiast, 8*(3), 405–462

Nardi, E., Biza, I., & Zachariades, T. (2012). ‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation. *Educational Studies in Mathematics, 79*(2), 157–173.

Shao, M. (2022). How do teachers support students’ coordination of perception and logical reasoning in 3D dynamic geometry environments: French and Chinese cases [Phd thesis, Ecole normale supérieure deyon; East China normal university].

Toulmin, S. E. (2003). *The Uses of Argument* (2nd ed.). Cambridge: Cambridge University Press; Cambridge Core.

Xu, B., Kong, Q., Yu, P., & Su, H. (2013). Chinese mathematics classroom teaching. In *Mathematics education in china: Tradition and reality* (pp. 73–120). Jiangsu Education Publishing House.
# ANALOGIES: A WAY TO PROMOTE THE LEARNING OF PROOF IN 3D GEOMETRY USING DYNAMIC GEOMETRY ENVIRONMENTS

Camilo Sua, Angel Gutiérrez, Adela Jaime

Universidad de Valencia, Departamento de Didáctica de la Matemática

Research on the influence of dynamic geometry environments on the teaching and learning of 3D geometry is embryonic. Some proposals that use these dynamic environments have involved analogy as a methodological approach to studying geometric objects in space through comparison with objects on the plane so that some properties of plane geometry are taken as the basis of the exploration and formulation of properties in space. Adhering to this line of work and as part of ongoing doctoral research, we present in this document some ideas about the role that analogy plays in students' mathematical activity when they solve construction-and-proof problems with the help of a 2D and 3D dynamic geometry environment. We emphasize the relationships between analogy, elaboration of proofs, and the use of a digital environment when students' productions are analyzed.

## INTRODUCTION

The impact and benefits of dynamic geometry environments (hereafter, DGE) in the development of different processes of mathematical activity, including learning to prove, are significant (Sinclair & Robutti, 2013). However, the focus and development of the research has been mainly on 2D configurations, with the attention paid to 3D geometry or the use of 3D-DGE not being similar (Gutiérrez & Jaime, 2015). And, in the schools, 3D geometry is scarcely studied in depth. Some arguments that support this decision allude to the complexity of the study of 3D objects when plane representations of them are involved, since it is difficult to correctly visualize or represent the properties of these objects in these representations (Parzysz, 1988). However, mathematically gifted students can benefit of the challenges provided by 3D geometry problems and, in particular, proof problems. Then, in our research, we analyze the learning by mathematically gifted students because it is an aspect on which there is scarce literature (Jaime & Gutiérrez, 2017).

Some proposals for the study of 3D geometry have involved 3D-DGE and analogies between 2D and 3D objects as a method to recognize similarities and differences between analogous objects in both domains (Echeverry et al., 2018; Ferrarello et al., 2020). We are determined to provide results along this same path. We are conducting a doctoral research to analyze the learning of proof by mathematically gifted students in 3D geometry with the support of 2D and 3D GeoGebra. This research focuses on the establishment and use of analogies between 2D and 3D objects when students solve construction-and-proof problems.

This document aims to show how the use of analogy, as a driver of students’ mathematical activity when entering the world of 3D geometry, with the support of a 3D-DGE, promotes the learning of mathematical proof. We analyze the sequence of problems we posed to a sample of mathematically gifted students and the results of its implementation. We focus on the relationships built between analogy, the use of 3D-DGE, and the elaboration of proofs.
# ANALOGIES: A WAY TO PROMOTE THE LEARNING OF PROOF IN 3D GEOMETRY USING DYNAMIC GEOMETRY ENVIRONMENTS

Camilo Sua, Angel Gutiérrez, Adela Jaime

Universidad de Valencia, Departamento de Didáctica de la Matemática

Research on the influence of dynamic geometry environments on the teaching and learning of 3D geometry is embryonic. Some proposals that use these dynamic environments have involved analogy as a methodological approach to studying geometric objects in space through comparison with objects on the plane so that some properties of plane geometry are taken as the basis of the exploration and formulation of properties in space. Adhering to this line of work and as part of ongoing doctoral research, we present in this document some ideas about the role that analogy plays in students' mathematical activity when they solve construction-and-proof problems with the help of a 2D and 3D dynamic geometry environment. We emphasize the relationships between analogy, elaboration of proofs, and the use of a digital environment when students' productions are analyzed.

## INTRODUCTION

The impact and benefits of dynamic geometry environments (hereafter, DGE) in the development of different processes of mathematical activity, including learning to prove, are significant (Sinclair & Robutti, 2013). However, the focus and development of the research has been mainly on 2D configurations, with the attention paid to 3D geometry or the use of 3D-DGE not being similar (Gutiérrez & Jaime, 2015). And, in the schools, 3D geometry is scarcely studied in depth. Some arguments that support this decision allude to the complexity of the study of 3D objects when plane representations of them are involved, since it is difficult to correctly visualize or represent the properties of these objects in these representations (Parzysz, 1988). However, mathematically gifted students can benefit of the challenges provided by 3D geometry problems and, in particular, proof problems. Then, in our research, we analyze the learning by mathematically gifted students because it is an aspect on which there is scarce literature (Jaime & Gutiérrez, 2017).

Some proposals for the study of 3D geometry have involved 3D-DGE and analogies between 2D and 3D objects as a method to recognize similarities and differences between analogous objects in both domains (Echeverry et al., 2018; Ferrarello et al., 2020). We are determined to provide results along this same path. We are conducting a doctoral research to analyze the learning of proof by mathematically gifted students in 3D geometry with the support of 2D and 3D GeoGebra. This research focuses on the establishment and use of analogies between 2D and 3D objects when students solve construction-and-proof problems.

This document aims to show how the use of analogy, as a driver of students’ mathematical activity when entering the world of 3D geometry, with the support of a 3D-DGE, promotes the learning of mathematical proof. We analyze the sequence of problems we posed to a sample of mathematically gifted students and the results of its implementation. We focus on the relationships built between analogy, the use of 3D-DGE, and the elaboration of proofs.
# THEORETICAL BACKGROUND

## Analogy: A way to extend ideas

An *analogy* is a relationship of similarity between two domains through specific objects and relationships. A domain is a representation of certain aspects of a situation, model, problem, conceptual structure, etc. (Schlimm, 2008). By establishing a set of relationships in one domain and taking them to another domain, in which these relationships are valid, analogy allows the formulation of hypotheses and the simplification of complex mental operations in the second domain, when these are performed in the first domain that is already known (Fishbein, 2002).

Making analogies is also a product of human activity, with several benefits (Richland & Simms, 2015). This process demands recognizing corresponding conceptual structures between different domains and their similarities or differences, thus advancing from a comparison based on superficial aspects or characteristics to one based on relationships between those domains. It means that this process has implications for the learning of mathematics (English, 1997) since new objects and relationships in an unknown domain can be discovered by extending ideas from a known domain.

## Learning of proof and support of dynamic geometry environments

To analyze the learning of proof, we consider different strategies of the students when they validate mathematical statements. We consider that students, as learners of this practice, use with understanding theoretical elements, modes of reasoning, and forms of communication that are valid within the frame of this activity. We find support for this need in the approaches of Stylianides et al. (2016), making explicit the different nature of the ways of validating. In this sense, we understand the proof as an empirical or deductive mathematical argument, a sequence of connected assertions for or against a mathematical statement.

In our study, we emphasize the learning of proof through construction-and-proof problems. These problems ask i) to create on the DGE a geometric figure having some properties required by the problem that must be preserved under dragging, and ii) to prove that the procedure used to create the figure is correct by explaining and validating the way of construction (Mariotti, 2019). The statement to be provided is that the sequence of actions of the construction produced a figure that fits the conditions of the problem.

The tools provided by a DGE are related to theoretical elements of Euclidean geometry. When students use these tools to construct geometric objects, personal meanings are produced, thanks to the dependency relationships they discover and verify through dragging. Using DGE tools to solve different problems, personal meanings can progressively become mathematical meanings incorporating theoretical elements. Bartolini-Bussi and Mariotti (2008) call this relationship the semiotic potential of an artifact. Therefore, solving construction-and-proof problems allows students to take advantage of the possibilities of DGE, the logical system that underlies it, and evoke theoretical meanings of the tools they have used in the solutions (Mariotti, 2019).

# METHODOLOGICAL CONSIDERATIONS

The content of this paper is part of a doctoral research adopting case study methodology. The research design considered a hypothetical learning trajectory (Simon & Tzur, 2004), whose objective was the learning of proof by mathematically gifted students. To achieve this objective, we developed a
# THEORETICAL BACKGROUND

## Analogy: A way to extend ideas

An *analogy* is a relationship of similarity between two domains through specific objects and relationships. A domain is a representation of certain aspects of a situation, model, problem, conceptual structure, etc. (Schlimm, 2008). By establishing a set of relationships in one domain and taking them to another domain, in which these relationships are valid, analogy allows the formulation of hypotheses and the simplification of complex mental operations in the second domain, when these are performed in the first domain that is already known (Fishbein, 2002).

Making analogies is also a product of human activity, with several benefits (Richland & Simms, 2015). This process demands recognizing corresponding conceptual structures between different domains and their similarities or differences, thus advancing from a comparison based on superficial aspects or characteristics to one based on relationships between those domains. It means that this process has implications for the learning of mathematics (English, 1997) since new objects and relationships in an unknown domain can be discovered by extending ideas from a known domain.

## Learning of proof and support of dynamic geometry environments

To analyze the learning of proof, we consider different strategies of the students when they validate mathematical statements. We consider that students, as learners of this practice, use with understanding theoretical elements, modes of reasoning, and forms of communication that are valid within the frame of this activity. We find support for this need in the approaches of Stylianides et al. (2016), making explicit the different nature of the ways of validating. In this sense, we understand the proof as an empirical or deductive mathematical argument, a sequence of connected assertions for or against a mathematical statement.

In our study, we emphasize the learning of proof through construction-and-proof problems. These problems ask i) to create on the DGE a geometric figure having some properties required by the problem that must be preserved under dragging, and ii) to prove that the procedure used to create the figure is correct by explaining and validating the way of construction (Mariotti, 2019). The statement to be provided is that the sequence of actions of the construction produced a figure that fits the conditions of the problem.

The tools provided by a DGE are related to theoretical elements of Euclidean geometry. When students use these tools to construct geometric objects, personal meanings are produced, thanks to the dependency relationships they discover and verify through dragging. Using DGE tools to solve different problems, personal meanings can progressively become mathematical meanings incorporating theoretical elements. Bartolini-Bussi and Mariotti (2008) call this relationship the semiotic potential of an artifact. Therefore, solving construction-and-proof problems allows students to take advantage of the possibilities of DGE, the logical system that underlies it, and evoke theoretical meanings of the tools they have used in the solutions (Mariotti, 2019).

# METHODOLOGICAL CONSIDERATIONS

The content of this paper is part of a doctoral research adopting case study methodology. The research design considered a hypothetical learning trajectory (Simon & Tzur, 2004), whose objective was the learning of proof by mathematically gifted students. To achieve this objective, we developed a
sequence of 18 construction-and-proof problems that involved the equidistance relationship and the analogy between 2D and 3D domains. Equidistance offers an opportunity to observe solution strategies guided by theoretical and perceptual aspects, as well as similarities and differences between the geometric objects of the compared domains. To solve the problems, students only used GeoGebra. This software was used because it provided simultaneity between the 2D and 3D configurations, an aspect that we consider useful to support the establishment and use of analogies.

The sequence brought students closer to the perpendicular bisector and circle and their 3D analogs (sphere and bisector plane) to construct these objects, define each object as a locus, and provide valuable properties to solve subsequent problems. Some problems required the construction, first in 2D and then in 3D, of geometric objects that satisfied equidistance properties (e.g., construct an equilateral triangle given its side). Other problems required constructing a 2D object and an analogous 3D object (e.g., finding the center of a circle in 2D and then the center of a sphere in 3D).

Four Spanish mathematically gifted students participated, aged between 11 and 14 years, belonging to different school grades and with different academic experiences. We recognize the students as mathematically gifted because they had participated in workshops on general and mathematical giftedness. The students had used GeoGebra 2D previously but not to carry out robust constructions in a context like the ones described. His knowledge of GeoGebra 3D and spatial geometry was scarce. The characteristics of the students were suitable to the process of solution of each problem to provide them with useful conceptual and instrumental elements to solve subsequent problems and for them to mobilize their knowledge of 2D geometry to the 3D domain, where it would be amplified and deepened by converting it into support for the mathematical activity deployed there.

For each problem, the students had to solve it and then discuss the solution and justify its correctness to the teacher (the first author of this paper), who led the conversation. The teacher's participation was relevant; his questioning of the students' productions promoted the ideas and arguments to advance from empirical and perceptual approaches to deductive and theoretical ones. The sessions were held by video conference and recorded in audio and video after informed approval by the students and their parents. As students were in different school grades and had different previous knowledge, the teaching sessions were organized as individual clinical interviews.

We present three moments in the sequence and what we expected the students to do in each one regarding the analogies developed, the use of GeoGebra tools, and the elaboration of proofs. Student names have been changed by pseudonyms. We contrast these considerations with the students' productions when implementing the sequence. Supported by this information, we discussed the articulation between the three elements that intervened in the sequence design.

# THREE MOMENTS OF THE SEQUENCE

## First moment: discovery of new geometric objects

The first problems were solved in 2D and 3D GeoGebra. These problems aimed to introduce the circle, perpendicular bisector, and their 3D analogs (sphere and bisector plane) as loci. Drag and trace functions were relevant. This characterization would be the basis of the analogies established between the circle, sphere, perpendicular bisector, and bisector plane. In instrumental terms, the recognition of tools and schemes that would allow each object to be constructed robustly was anticipated. As in this phase, geometric objects and properties were discovered, deductive proofs were not expected.
sequence of 18 construction-and-proof problems that involved the equidistance relationship and the analogy between 2D and 3D domains. Equidistance offers an opportunity to observe solution strategies guided by theoretical and perceptual aspects, as well as similarities and differences between the geometric objects of the compared domains. To solve the problems, students only used GeoGebra. This software was used because it provided simultaneity between the 2D and 3D configurations, an aspect that we consider useful to support the establishment and use of analogies.

The sequence brought students closer to the perpendicular bisector and circle and their 3D analogs (sphere and bisector plane) to construct these objects, define each object as a locus, and provide valuable properties to solve subsequent problems. Some problems required the construction, first in 2D and then in 3D, of geometric objects that satisfied equidistance properties (e.g., construct an equilateral triangle given its side). Other problems required constructing a 2D object and an analogous 3D object (e.g., finding the center of a circle in 2D and then the center of a sphere in 3D).

Four Spanish mathematically gifted students participated, aged between 11 and 14 years, belonging to different school grades and with different academic experiences. We recognize the students as mathematically gifted because they had participated in workshops on general and mathematical giftedness. The students had used GeoGebra 2D previously but not to carry out robust constructions in a context like the ones described. His knowledge of GeoGebra 3D and spatial geometry was scarce. The characteristics of the students were suitable to the process of solution of each problem to provide them with useful conceptual and instrumental elements to solve subsequent problems and for them to mobilize their knowledge of 2D geometry to the 3D domain, where it would be amplified and deepened by converting it into support for the mathematical activity deployed there.

For each problem, the students had to solve it and then discuss the solution and justify its correctness to the teacher (the first author of this paper), who led the conversation. The teacher's participation was relevant; his questioning of the students' productions promoted the ideas and arguments to advance from empirical and perceptual approaches to deductive and theoretical ones. The sessions were held by video conference and recorded in audio and video after informed approval by the students and their parents. As students were in different school grades and had different previous knowledge, the teaching sessions were organized as individual clinical interviews.

We present three moments in the sequence and what we expected the students to do in each one regarding the analogies developed, the use of GeoGebra tools, and the elaboration of proofs. Student names have been changed by pseudonyms. We contrast these considerations with the students' productions when implementing the sequence. Supported by this information, we discussed the articulation between the three elements that intervened in the sequence design.

# THREE MOMENTS OF THE SEQUENCE

## First moment: discovery of new geometric objects

The first problems were solved in 2D and 3D GeoGebra. These problems aimed to introduce the circle, perpendicular bisector, and their 3D analogs (sphere and bisector plane) as loci. Drag and trace functions were relevant. This characterization would be the basis of the analogies established between the circle, sphere, perpendicular bisector, and bisector plane. In instrumental terms, the recognition of tools and schemes that would allow each object to be constructed robustly was anticipated. As in this phase, geometric objects and properties were discovered, deductive proofs were not expected.
Instead, the use of empirical and perceptual evidence was expected, as well as the description of each object or set of points in terms of its appearance.

The students quickly incorporated the circle and perpendicular bisector into their activity since they knew them in advance. His school experiences also allowed the sphere to be introduced in 3D as a substitute for the circle and support equidistance concerning a point. This did not happen with the bisector plane. The students were unaware of this object, which provoked different exploration strategies when solving the problem that involved it.

Mario relied on the trace and drag function to move points of the perpendicular bisector out of the XY plane and drag them on it, controlling this action with the dotted line that drag function provides in GeoGebra 3D (Figure 1a). He used the labels of the distances of the displaced point concerning the two fixed points to validate his strategy inductively and said: *Point C continues going through the perpendicular bisector... by a perpendicular, a vertical line, then it will always be at the same distance.* Hector realized the sufficiency of changing the height of the points of the perpendicular bisector (z component of its coordinates) to preserve equidistance in 3D, so he saw in vertical dragging a way to obtain points that solved the problem, as he explained: *As high as point D is, up or down the perpendicular bisector, there will always be the same distance.* To prove equidistance, he referred to the line determined by the displaced point and the midpoint of fixed points as a perpendicular bisector, without being aware that this lacked sufficient theoretical support (Figure 1b). Both students characterized the set of points obtained as a perpendicular plane to the XY plane.

<figure><img src="image_176.png" /><figcaption>Figure 1. Discovering the bisector plane</figcaption></figure>

Rafa anticipated a plane as a locus and the solution to the problem (Figure 1c). Like Hector, Rafa obtained this plane by dragging the points of the perpendicular bisector vertically, and he justified his actions: *We have moved point C on the Z axis, A and B are at the same height, so it does not vary (the distances between A, C and B, C remain the same).* However, Rafa also explained that it was possible if the configuration taken as a base (perpendicular bisector and points that determine it) were contained in a horizontal plane. Like Hector, he proved the validity of his construction by referring to the line determined by each dragged point and the midpoint of the fixed points as a perpendicular bisector. Juan developed a different strategy. From the beginning, he recognized the existence of infinite perpendicular bisectors of a segment in 3D, so he constructed one of those lines and rotated it to determine all the solutions to the problem: *There are infinitely many perpendiculars to segment AB. I could draw any one and move point C on it.* The proof of his construction took advantage of the fact that the line constructed was the perpendicular bisector of the segment. The set of bisectors was characterized as a plane.
Instead, the use of empirical and perceptual evidence was expected, as well as the description of each object or set of points in terms of its appearance.

The students quickly incorporated the circle and perpendicular bisector into their activity since they knew them in advance. His school experiences also allowed the sphere to be introduced in 3D as a substitute for the circle and support equidistance concerning a point. This did not happen with the bisector plane. The students were unaware of this object, which provoked different exploration strategies when solving the problem that involved it.

Mario relied on the trace and drag function to move points of the perpendicular bisector out of the XY plane and drag them on it, controlling this action with the dotted line that drag function provides in GeoGebra 3D (Figure 1a). He used the labels of the distances of the displaced point concerning the two fixed points to validate his strategy inductively and said: *Point C continues going through the perpendicular bisector... by a perpendicular, a vertical line, then it will always be at the same distance.* Hector realized the sufficiency of changing the height of the points of the perpendicular bisector (z component of its coordinates) to preserve equidistance in 3D, so he saw in vertical dragging a way to obtain points that solved the problem, as he explained: *As high as point D is, up or down the perpendicular bisector, there will always be the same distance.* To prove equidistance, he referred to the line determined by the displaced point and the midpoint of fixed points as a perpendicular bisector, without being aware that this lacked sufficient theoretical support (Figure 1b). Both students characterized the set of points obtained as a perpendicular plane to the XY plane.

<figure><img src="image_183.png" /><figcaption>Figure 1. Discovering the bisector plane</figcaption></figure>

Rafa anticipated a plane as a locus and the solution to the problem (Figure 1c). Like Hector, Rafa obtained this plane by dragging the points of the perpendicular bisector vertically, and he justified his actions: *We have moved point C on the Z axis, A and B are at the same height, so it does not vary (the distances between A, C and B, C remain the same).* However, Rafa also explained that it was possible if the configuration taken as a base (perpendicular bisector and points that determine it) were contained in a horizontal plane. Like Hector, he proved the validity of his construction by referring to the line determined by each dragged point and the midpoint of the fixed points as a perpendicular bisector. Juan developed a different strategy. From the beginning, he recognized the existence of infinite perpendicular bisectors of a segment in 3D, so he constructed one of those lines and rotated it to determine all the solutions to the problem: *There are infinitely many perpendiculars to segment AB. I could draw any one and move point C on it.* The proof of his construction took advantage of the fact that the line constructed was the perpendicular bisector of the segment. The set of bisectors was characterized as a plane.
## Second moment: substitution of 2D objects with 3D objects

The second block of problems was intended to use the geometric objects introduced. Students had to build objects with specific characteristics, first on the plane and then in space. Replacing the plane with space was intended to exchange the circle and perpendicular bisector for the sphere and bisector plane when necessary. Working in 3D would take advantage of strategies for constructing objects and proving properties worked out on the plane.

The substitution of 2D objects for 3D objects was seen, for example, in the strategies of some students to construct points in space that are equidistant from two fixed points. Hector and Rafa, aware of the lack of a tool that would allow constructing the perpendicular bisector or the bisector plane, adapted the procedure to construct the perpendicular bisector on paper with a ruler and compass, replacing the circles with spheres and using one of the points of their intersection and the midpoint of the given points to determine a perpendicular bisector (Figure 2a), or three points of the intersection to determine the bisector plane (Figure 2b). For them, replacing circles with spheres, as long as they preserve the equidistance to a fixed point, supports the validity of their construction. Hector argued: *I have done the same as in 2D, instead of circles I used spheres... I have created the perpendicular bisector.*

<figure><img src="image_177.png" /><figcaption>Figure 2. Construction of perpendicular bisector and bisector plane in 3D</figcaption></figure>

When the problem conditions did not suggest the replacement of 2D objects with 3D objects, the students used the configuration represented in 2D as the basis of their exploration in 3D. An example of this was obtaining the set of points in 3D that are equidistant from the points of a circle. Previously, the center of the circle had been constructed in 2D. Mario combined vertical dragging and the trace of the center of the circle to determine a perpendicular line to the XY plane as a solution to the problem (Figure 2c): *It is a circle where the center is at a different height.* The other students anticipated this locus, ignoring the vertical dragging and constructing the perpendicular line. Mario proved his construction based on the conservation of equidistance under vertical dragging. Hector combined empirical evidence and theoretical elements to prove it. Juan and Rafa defined the perpendicularity relationship between a line and a plane, which allowed them to prove the equidistance.

## Third moment: sophisticate use of objects and properties

The final problems requested constructions only in 3D, leaving behind the simultaneous work with the plane and the use of analogies. At this sequence stage, robust constructions of geometric objects and fluidity in using their properties to solve problems were expected. The proofs developed should be deductive or have features very close to these.
## Second moment: substitution of 2D objects with 3D objects

The second block of problems was intended to use the geometric objects introduced. Students had to build objects with specific characteristics, first on the plane and then in space. Replacing the plane with space was intended to exchange the circle and perpendicular bisector for the sphere and bisector plane when necessary. Working in 3D would take advantage of strategies for constructing objects and proving properties worked out on the plane.

The substitution of 2D objects for 3D objects was seen, for example, in the strategies of some students to construct points in space that are equidistant from two fixed points. Hector and Rafa, aware of the lack of a tool that would allow constructing the perpendicular bisector or the bisector plane, adapted the procedure to construct the perpendicular bisector on paper with a ruler and compass, replacing the circles with spheres and using one of the points of their intersection and the midpoint of the given points to determine a perpendicular bisector (Figure 2a), or three points of the intersection to determine the bisector plane (Figure 2b). For them, replacing circles with spheres, as long as they preserve the equidistance to a fixed point, supports the validity of their construction. Hector argued: *I have done the same as in 2D, instead of circles I used spheres... I have created the perpendicular bisector.*

<figure><img src="image_184.png" /><figcaption>Figure 2. Construction of perpendicular bisector and bisector plane in 3D</figcaption></figure>

When the problem conditions did not suggest the replacement of 2D objects with 3D objects, the students used the configuration represented in 2D as the basis of their exploration in 3D. An example of this was obtaining the set of points in 3D that are equidistant from the points of a circle. Previously, the center of the circle had been constructed in 2D. Mario combined vertical dragging and the trace of the center of the circle to determine a perpendicular line to the XY plane as a solution to the problem (Figure 2c): *It is a circle where the center is at a different height.* The other students anticipated this locus, ignoring the vertical dragging and constructing the perpendicular line. Mario proved his construction based on the conservation of equidistance under vertical dragging. Hector combined empirical evidence and theoretical elements to prove it. Juan and Rafa defined the perpendicularity relationship between a line and a plane, which allowed them to prove the equidistance.

## Third moment: sophisticate use of objects and properties

The final problems requested constructions only in 3D, leaving behind the simultaneous work with the plane and the use of analogies. At this sequence stage, robust constructions of geometric objects and fluidity in using their properties to solve problems were expected. The proofs developed should be deductive or have features very close to these.
The students' actions when solving the problems reflected a purposeful use of 3D objects and robust constructions of them. This was the case of Rafa, who used 3D objects in the construction process and proved it based on their properties, leaving behind the reference to vertical drag or the analogy between these objects and their corresponding ones on the plane. Juan and Hector, in addition to Rafa's action, explained some constructions in similar ways, both based on an analogy between the plane and space. An example of this is what was made when they constructed the center of a given sphere. They constructed chords of the sphere and their bisector planes (Figure 3a), stating that the point of intersection of the planes was the center of the sphere by simultaneously equidistant from the ends of the chords. Juan explained that the choice of the bisector planes was based on an experience in the plane, in which they used a perpendicular bisector of chords in the circle to construct its center (Figure 3b): *In 2D, we used the intersection of the perpendicular bisectors of points in the circle. In this case (3D), instead of perpendicular bisectors, we used the bisector plane.*

<figure><img src="image_178.png" /><figcaption>Figure 3. Construction of perpendicular bisector and bisector plane in 3D</figcaption></figure>

Mario's actions do not allow us to generalize the ideas presented above. Although Mario knew procedures for constructing geometric objects that solved problems and he was sure about the result obtained, he could not develop a deductive proof of its construction. As the vertical dragging of points marked his experience moving objects from plane to space, he inductively established these results as properties that, although true, did not have theoretical support. He determined the center of the sphere with the help of bisector planes, as did Juan and Hector. When he solved another problem that asked to construct an equidistant point from four non-coplanar points, he used lines containing points that were equidistant from the points of the circles determined by three of the given points and their intersection (Figure 3c). In both cases, the robustness of his constructions was accompanied by gaps in the elaborate proofs.

## DISCUSSION AND CONCLUSIONS

<figure><img src="image_179.png" /><figcaption>Figure 4. Moments of construction of perpendicular bisector and bisector plane in 3D</figcaption></figure>
The students' actions when solving the problems reflected a purposeful use of 3D objects and robust constructions of them. This was the case of Rafa, who used 3D objects in the construction process and proved it based on their properties, leaving behind the reference to vertical drag or the analogy between these objects and their corresponding ones on the plane. Juan and Hector, in addition to Rafa's action, explained some constructions in similar ways, both based on an analogy between the plane and space. An example of this is what was made when they constructed the center of a given sphere. They constructed chords of the sphere and their bisector planes (Figure 3a), stating that the point of intersection of the planes was the center of the sphere by simultaneously equidistant from the ends of the chords. Juan explained that the choice of the bisector planes was based on an experience in the plane, in which they used a perpendicular bisector of chords in the circle to construct its center (Figure 3b): *In 2D, we used the intersection of the perpendicular bisectors of points in the circle. In this case (3D), instead of perpendicular bisectors, we used the bisector plane.*

<figure><img src="image_185.png" /><figcaption>Figure 3. Construction of perpendicular bisector and bisector plane in 3D</figcaption></figure>

Mario's actions do not allow us to generalize the ideas presented above. Although Mario knew procedures for constructing geometric objects that solved problems and he was sure about the result obtained, he could not develop a deductive proof of its construction. As the vertical dragging of points marked his experience moving objects from plane to space, he inductively established these results as properties that, although true, did not have theoretical support. He determined the center of the sphere with the help of bisector planes, as did Juan and Hector. When he solved another problem that asked to construct an equidistant point from four non-coplanar points, he used lines containing points that were equidistant from the points of the circles determined by three of the given points and their intersection (Figure 3c). In both cases, the robustness of his constructions was accompanied by gaps in the elaborate proofs.

## DISCUSSION AND CONCLUSIONS

<figure><img src="image_186.png" /><figcaption>Figure 4. Moments of construction of perpendicular bisector and bisector plane in 3D</figcaption></figure>
Figure 4 shows a diagram with the trajectory described previously. Each moment is represented through a triad in which the analogy, the use of the DGE, and the elaboration of proofs are articulated. The articulation derived from the hypothetical learning trajectory is presented in blue. In red is the articulation of the elements at each moment, according to the results obtained.

In the first moment, the combined use of vertical dragging and a locus in the plane became the basis for exploring and discovering analogous objects in space. Although we expected that the dragging of points would be an expected action, we did not expect the ways some students used it to guarantee the conservation of equidistance. This relationship led them to formulate proofs for equidistance in space, characterizing the discovered object in 3D through the 2D object taken as a base.

The relationship between vertical dragging and equidistance is a personal meaning promoted by using DGE functions. This meaning evolved in the second moment of the sequence when the characteristics of one of the problems led the students to replace dragging points with the construction of the perpendicular line to the plane. In this way, vertical dragging became a vehicle for establishing the relationship between perpendicularity and equidistance in 3D (mathematical meaning). Determining the meaning of the perpendicularity between a line and a plane allowed some students to deductively prove compliance of equidistance and not offer empirical proof as an explanation.

The absence of some construction tools favored the analogy between the circle and the sphere, an aspect considered when the sequence was designed. However, the procedures of some students to construct the perpendicular bisector and the bisector plane in 3D were not part of our assumption. These students simplified 3D mental operations to build these objects by recovering and adapting to the 3D-DGE construction protocols with a ruler and compass. Proving these constructions led to evoking the corresponding proofs in 2D, a valid strategy for them since the circle and sphere shared the property of equidistance.

The third moment shows a more mature state of transition in which the analogy begins to disappear. The use of geometric objects is observed, endowed with useful properties for constructing and elaborating deductive proofs. The analogy, if used, allows ideas to be extended from one domain to another by recovering construction strategies in the plane that are then adapted to the characteristics of the 3D configuration, thereby advancing the resolution of the problem in the latter domain.

Mario's actions offer a different picture. The use of vertical dragging in the exploration and characterization of sets of points had a greater presence when compared with the actions of his colleagues. Mario transformed vertical dragging into a tool for construction and proof when he worked in 3D, but the inductive nature with which this tool was developed did not allow the level of his proofs to be similar to that of his colleagues. Although the analogy allowed him to solve problems in 3D, his journey to get to this point was slower. This was evident in the episodes in which other students anticipated locus while he resorted to vertical dragging in search of solutions or used the discovered geometric objects without being aware of why they provided a solution to the problem.

Construction-and-proof problems provide a scenario in which the need to construct objects with specific properties leads to the need to articulate the available tools and knowledge. In the context of 3D geometry, there is also a search and connection with what is known about 2D geometry when the tools provided by 3D-DGE do not allow replicating the actions executed in the plane. Analogy can support these processes but requires the design of appropriate problems that promote it. As we have
Figure 4 shows a diagram with the trajectory described previously. Each moment is represented through a triad in which the analogy, the use of the DGE, and the elaboration of proofs are articulated. The articulation derived from the hypothetical learning trajectory is presented in blue. In red is the articulation of the elements at each moment, according to the results obtained.

In the first moment, the combined use of vertical dragging and a locus in the plane became the basis for exploring and discovering analogous objects in space. Although we expected that the dragging of points would be an expected action, we did not expect the ways some students used it to guarantee the conservation of equidistance. This relationship led them to formulate proofs for equidistance in space, characterizing the discovered object in 3D through the 2D object taken as a base.

The relationship between vertical dragging and equidistance is a personal meaning promoted by using DGE functions. This meaning evolved in the second moment of the sequence when the characteristics of one of the problems led the students to replace dragging points with the construction of the perpendicular line to the plane. In this way, vertical dragging became a vehicle for establishing the relationship between perpendicularity and equidistance in 3D (mathematical meaning). Determining the meaning of the perpendicularity between a line and a plane allowed some students to deductively prove compliance of equidistance and not offer empirical proof as an explanation.

The absence of some construction tools favored the analogy between the circle and the sphere, an aspect considered when the sequence was designed. However, the procedures of some students to construct the perpendicular bisector and the bisector plane in 3D were not part of our assumption. These students simplified 3D mental operations to build these objects by recovering and adapting to the 3D-DGE construction protocols with a ruler and compass. Proving these constructions led to evoking the corresponding proofs in 2D, a valid strategy for them since the circle and sphere shared the property of equidistance.

The third moment shows a more mature state of transition in which the analogy begins to disappear. The use of geometric objects is observed, endowed with useful properties for constructing and elaborating deductive proofs. The analogy, if used, allows ideas to be extended from one domain to another by recovering construction strategies in the plane that are then adapted to the characteristics of the 3D configuration, thereby advancing the resolution of the problem in the latter domain.

Mario's actions offer a different picture. The use of vertical dragging in the exploration and characterization of sets of points had a greater presence when compared with the actions of his colleagues. Mario transformed vertical dragging into a tool for construction and proof when he worked in 3D, but the inductive nature with which this tool was developed did not allow the level of his proofs to be similar to that of his colleagues. Although the analogy allowed him to solve problems in 3D, his journey to get to this point was slower. This was evident in the episodes in which other students anticipated locus while he resorted to vertical dragging in search of solutions or used the discovered geometric objects without being aware of why they provided a solution to the problem.

Construction-and-proof problems provide a scenario in which the need to construct objects with specific properties leads to the need to articulate the available tools and knowledge. In the context of 3D geometry, there is also a search and connection with what is known about 2D geometry when the tools provided by 3D-DGE do not allow replicating the actions executed in the plane. Analogy can support these processes but requires the design of appropriate problems that promote it. As we have
presented, using a 3D-DGE in a scenario like this produces personal meanings that progressively, and at unequal rates among students, will become mathematical meanings or approximate them.

## Acknowledgements

This paper is part of the R+D+I project PID2020-117395RB-I00, and the predoctoral grant EDU2017-84377-R, both funded by MCIN/AEI/10.13039/501100011033.

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Simon, M., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: an elaboration of the hypothetical learning trajectory. *Mathematical Thinking and Learning, 6*(2), 91–104.
https://doi.org/10.1207/s15327833mtl0602_2

Sinclair, N., & Robutti, O. (2013). Technology and the role of proof: the case of dynamic geometry. In M. A. Clements, A. J. Bishop, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), *Third international handbook of mathematics education* (pp. 571–596). Springer.

Stylianides, A.J., Bieda, K.N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutiérrez, G.C. Leder, & P. Boero (Eds.), *The second handbook of research on the psychology of mathematics education*. Sense. https://doi.org/10.1007/978-94-6300-561-6_9
presented, using a 3D-DGE in a scenario like this produces personal meanings that progressively, and at unequal rates among students, will become mathematical meanings or approximate them.

## Acknowledgements

This paper is part of the R+D+I project PID2020-117395RB-I00, and the predoctoral grant EDU2017-84377-R, both funded by MCIN/AEI/10.13039/501100011033.

## References

Bartolini-Bussi, M., & Mariotti, M. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. D. English, & D. Kirshner (Eds.), *Handbook of international research in mathematics education* (pp. 746–783). Routledge.

Echeverry, A., Camargo, L., Gutiérrez, A., & Samper, C. (2018). Uncertainty and 3d dynamic geometry: a catalyst for the step from 2d to 3d geometry. In D. M. Gómez (Ed.), *Proceedings of the First PME Regional Conference: South America* (pp. 49–56). PME.

English, L. D. (1997). Children’s reasoning processes in classifying and solving computational word problems. In L. D. English (Ed.), *Mathematical reasoning: analogies, metaphors, and images* (pp. 191–220). Lawrence Erlbaum.

Ferrarello, D., Flavia Mammana, M., & Taranto, E. (2020). Dynamic geometry systems in proving 3D geometry properties. *International Journal for Technology in Mathematics Education, 27*(1), 19–27.
https://doi.org/10.1564/tme_v27.1.03

Fischbein, E. (2002). *Intuition in science and mathematics: An educational approach*. Springer.
https://doi.org/10.1007/0-306-47237-6

Gutiérrez, A., & Jaime, A. (2015). Análisis del aprendizaje de geometría espacial en un entorno de geometría dinámica 3-dimensional. *PNA, 9*(2), 53–83. https://doi.org/10.30827/pna.v9i2.6106

Jaime, A., & Gutiérrez, A. (2017). Investigación sobre estudiantes con alta capacidad matemática. In J. M. Muñoz-Escolano, A. Arnal-Bailera, P. Beltrán-Pellicer, M. L. Callejo, & J. Carrillo (Eds.), *Investigación en Educación Matemática XXI* (pp. 71–89). SEIEM.

Mariotti, M. A. (2019). The contribution of information and communication technology to the teaching of proof. In G. Hanna, D. A. Reid, & M. de Villiers (Eds.), *Proof technology in mathematics research and teaching* (pp. 173–195). Springer.

Parzysz, B. (1988). “Knowing” vs “seeing”. Problems of the plane representation of space geometry figures. *Educational Studies in Mathematics, 19*(1), 79–92. https://doi.org/10.1007/BF00428386

Richland, L. E., & Simms, N. (2015). Analogy, higher order thinking, and education. *Wiley Interdisciplinary Reviews: Cognitive Science, 6*(2), 177–192. https://doi.org/10.1002/wcs.1336

Schlimm, D. (2008). Two ways of analogy: extending the study of analogies to mathematical domains. *Philosophy of Science, 75*(2), 178–200. https://doi.org/10.1086/590198

Simon, M., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: an elaboration of the hypothetical learning trajectory. *Mathematical Thinking and Learning, 6*(2), 91–104.
https://doi.org/10.1207/s15327833mtl0602_2

Sinclair, N., & Robutti, O. (2013). Technology and the role of proof: the case of dynamic geometry. In M. A. Clements, A. J. Bishop, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), *Third international handbook of mathematics education* (pp. 571–596). Springer.

Stylianides, A.J., Bieda, K.N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutiérrez, G.C. Leder, & P. Boero (Eds.), *The second handbook of research on the psychology of mathematics education*. Sense. https://doi.org/10.1007/978-94-6300-561-6_9
# CHALLENGES OF INTEGRATING DGS WITH GEOMETRY EDUCATION

Mohamadreza Tavakoli, Zahra Gooya

Shahid Beheshti University

Mathematics teachers in Iran, have many challenges in integrating DGS with their geometry teaching. To overcome these challenges, we conducted research taking qualitative approach to identify the major constraints that mathematics teachers have in using technology in geometry education. The informants were 18 volunteer mathematics teachers from different cities in Iran and the data collected virtually by a semi-structured interview. All recorded interviews were transcribed and the MAXQDA was used for analysis of the data. The findings consisted of four themes as technological infrastructure, policies, mathematics curriculum, and teachers' resistance. The study exposed that teachers developed innovative strategies to reduce these constraints to facilitate the integration of DGS with geometry education. The study concludes that even in highly centralized education system, teachers are main player in promoting technology in mathematics teaching and thus, their professional development is a necessary enterprise.

## INTRODUCTION

Iranian Education system is centralized in which, there is a prescribed curriculum and a single textbook for every school subject at the national level. The tolerance of the highly centralized system to the diversity from different perspectives including culture, language, economic situation and even climate, is limited and instead, the policymakers have more focus on general educational demands (Karimi et al., 2023). However, Gholamazad (2020) criticized this radical look on the degree of centralization prior to that and gave an evidence-based explanation that the highly centralized education system in Iran has undergone unavoidable changes at a slow but overwhelming pace, including technology as becoming one of the infrastructures of educational activities. This forecast was in line with Sinclair et al (2016) report that regarding geometry education, technology has become a global trend. Along with this trend, using “Dynamic Geometry Software” (DGS) for teaching geometry, gained more attention. However, at the end of 2020, the outbreak of Covid-19 pandemic caused the closure of schools around the globe and unexpectedly, all education systems had to switch from physical-face to face- to virtual education and rapidly, technology became the main platform for teaching and learning all school subjects including geometry and the use of DGS. During that era, Iran was not the exception and technology became integrated with all aspects of education rather rapidly and with no well-thought plans as well. Nevertheless, the new situation created a unique opportunity for some young mathematics teachers to use digital technology and DGS for their teaching. We designed a multi-phase study to investigate the role of DGS on geometrical proof and reasoning of senior high school students in Iran. At the first phase, we focused on the teachers' views about using DGS in teaching geometry and found out whether that practice became a legacy to last or just a temporal action during virtual education. So, the first phase of the study focused on teachers' views about the obstacles of using DGS in geometry teaching and this paper is reporting on the preliminary findings.
# CHALLENGES OF INTEGRATING DGS WITH GEOMETRY EDUCATION

Mohamadreza Tavakoli, Zahra Gooya

Shahid Beheshti University

Mathematics teachers in Iran, have many challenges in integrating DGS with their geometry teaching. To overcome these challenges, we conducted research taking qualitative approach to identify the major constraints that mathematics teachers have in using technology in geometry education. The informants were 18 volunteer mathematics teachers from different cities in Iran and the data collected virtually by a semi-structured interview. All recorded interviews were transcribed and the MAXQDA was used for analysis of the data. The findings consisted of four themes as technological infrastructure, policies, mathematics curriculum, and teachers' resistance. The study exposed that teachers developed innovative strategies to reduce these constraints to facilitate the integration of DGS with geometry education. The study concludes that even in highly centralized education system, teachers are main player in promoting technology in mathematics teaching and thus, their professional development is a necessary enterprise.

## INTRODUCTION

Iranian Education system is centralized in which, there is a prescribed curriculum and a single textbook for every school subject at the national level. The tolerance of the highly centralized system to the diversity from different perspectives including culture, language, economic situation and even climate, is limited and instead, the policymakers have more focus on general educational demands (Karimi et al., 2023). However, Gholamazad (2020) criticized this radical look on the degree of centralization prior to that and gave an evidence-based explanation that the highly centralized education system in Iran has undergone unavoidable changes at a slow but overwhelming pace, including technology as becoming one of the infrastructures of educational activities. This forecast was in line with Sinclair et al (2016) report that regarding geometry education, technology has become a global trend. Along with this trend, using “Dynamic Geometry Software” (DGS) for teaching geometry, gained more attention. However, at the end of 2020, the outbreak of Covid-19 pandemic caused the closure of schools around the globe and unexpectedly, all education systems had to switch from physical-face to face- to virtual education and rapidly, technology became the main platform for teaching and learning all school subjects including geometry and the use of DGS. During that era, Iran was not the exception and technology became integrated with all aspects of education rather rapidly and with no well-thought plans as well. Nevertheless, the new situation created a unique opportunity for some young mathematics teachers to use digital technology and DGS for their teaching. We designed a multi-phase study to investigate the role of DGS on geometrical proof and reasoning of senior high school students in Iran. At the first phase, we focused on the teachers' views about using DGS in teaching geometry and found out whether that practice became a legacy to last or just a temporal action during virtual education. So, the first phase of the study focused on teachers' views about the obstacles of using DGS in geometry teaching and this paper is reporting on the preliminary findings.
LITERATURE REVIEW

Straesser (2002) specified that digital tools allow students to construct geometric figures and see relationships that are not limited by the small steps of conventional paper and pencil drawing techniques and so, expand the scope of possible geometric constructions and solutions. On another study, Reyhani et al. (2009) investigated the effect of DGS on improving high school students' reasoning in geometry and found that using DGS enabled them to enhance their problem-solving skills considering geometry. In a project led by Lavicza et al. (2010) in England, a professional development network was created with nine teachers who were enthusiastic in using GeoGebra for their mathematics teaching. They formed a core group to support mathematics teachers via professional development, as well as exploring the ways in which, GeoGebra could be aligned with England national curriculum. The project also involved developing and collecting resources for mathematics teaching and learning with GeoGebra and establishing a local searchable GeoGebra site and an online support structure. However, Stols and Kriek (2011) found that teachers' general technology proficiency is a prerequisite for using DGS effectively in classroom. With the development of digital technology, many mathematics education researchers became interested in attesting the usefulness of DGS in mathematics teaching and learning. Ruthven (2014) alleged that relevant knowledge and expertise of mathematics teachers is the most critical factor for the effective incorporation of digital technologies into the regular teaching practice. To enhance teachers’ knowledge, he introduced a framework with five components of “working environment, resource system, activity structure, curriculum script, and time economy of classroom” to support mathematics teachers for integrating digital technologies into their mathematics teaching. On the other hand, Chan and Leung (2014) systematic review of quasi-experimental studies showed that DGS-based teaching, improved students’ mathematical achievements with a large and positive effect size.

Later, Sinclair et al. (2016) reported that the impact of “Dynamic Geometry Environments” (DGEs) on geometry education from elementary to tertiary level and its various aspects including proof, definition, and visualization, has been widely investigated. However, they stressed that the nature of communication between geometric objects and concepts in this environment is different from the traditional paper-and-pencil methods and it is important to make necessary modifications when using DGS. Aside, Ottenbreit-Leftwich et al. (2018) performed a study with four mathematics teachers who had received extensive training to use technology for teaching. After completing the training, they had various field experiences as well. In all teaching sessions, these teachers demonstrated high levels of self-efficacy, knowledge, beliefs, and intentions to use technology. However, they encountered some external barriers that sometimes, they could not use technology in their real classroom settings. In a recent study, Hillmayr et al. (2020) employed a meta-analysis approach and examined 92 primary studies that investigated the effects of using digital tools for teaching and learning science and mathematics at secondary school. Their analysis showed the significant positive impact of digital tools on students’ attitude and learning outcomes. Furthermore, they concluded that one of the key factors for the successful implementation of digital tools in mathematics and science classes is teacher training in advance. They also suggested that digital tools are more beneficial when are used as complement rather than replacement of other instructional methods. Clark-Wilson et al. (2020) reviewed the current state of research on teaching mathematics with technology at the secondary level. They found that the predominance of studies was on students' learning and barely on teaching. They also highlighted the need for more research on teacher professional development that can
LITERATURE REVIEW

Straesser (2002) specified that digital tools allow students to construct geometric figures and see relationships that are not limited by the small steps of conventional paper and pencil drawing techniques and so, expand the scope of possible geometric constructions and solutions. On another study, Reyhani et al. (2009) investigated the effect of DGS on improving high school students' reasoning in geometry and found that using DGS enabled them to enhance their problem-solving skills considering geometry. In a project led by Lavicza et al. (2010) in England, a professional development network was created with nine teachers who were enthusiastic in using GeoGebra for their mathematics teaching. They formed a core group to support mathematics teachers via professional development, as well as exploring the ways in which, GeoGebra could be aligned with England national curriculum. The project also involved developing and collecting resources for mathematics teaching and learning with GeoGebra and establishing a local searchable GeoGebra site and an online support structure. However, Stols and Kriek (2011) found that teachers' general technology proficiency is a prerequisite for using DGS effectively in classroom. With the development of digital technology, many mathematics education researchers became interested in attesting the usefulness of DGS in mathematics teaching and learning. Ruthven (2014) alleged that relevant knowledge and expertise of mathematics teachers is the most critical factor for the effective incorporation of digital technologies into the regular teaching practice. To enhance teachers’ knowledge, he introduced a framework with five components of “working environment, resource system, activity structure, curriculum script, and time economy of classroom” to support mathematics teachers for integrating digital technologies into their mathematics teaching. On the other hand, Chan and Leung (2014) systematic review of quasi-experimental studies showed that DGS-based teaching, improved students’ mathematical achievements with a large and positive effect size.

Later, Sinclair et al. (2016) reported that the impact of “Dynamic Geometry Environments” (DGEs) on geometry education from elementary to tertiary level and its various aspects including proof, definition, and visualization, has been widely investigated. However, they stressed that the nature of communication between geometric objects and concepts in this environment is different from the traditional paper-and-pencil methods and it is important to make necessary modifications when using DGS. Aside, Ottenbreit-Leftwich et al. (2018) performed a study with four mathematics teachers who had received extensive training to use technology for teaching. After completing the training, they had various field experiences as well. In all teaching sessions, these teachers demonstrated high levels of self-efficacy, knowledge, beliefs, and intentions to use technology. However, they encountered some external barriers that sometimes, they could not use technology in their real classroom settings. In a recent study, Hillmayr et al. (2020) employed a meta-analysis approach and examined 92 primary studies that investigated the effects of using digital tools for teaching and learning science and mathematics at secondary school. Their analysis showed the significant positive impact of digital tools on students’ attitude and learning outcomes. Furthermore, they concluded that one of the key factors for the successful implementation of digital tools in mathematics and science classes is teacher training in advance. They also suggested that digital tools are more beneficial when are used as complement rather than replacement of other instructional methods. Clark-Wilson et al. (2020) reviewed the current state of research on teaching mathematics with technology at the secondary level. They found that the predominance of studies was on students' learning and barely on teaching. They also highlighted the need for more research on teacher professional development that can
support them to use technology in their classrooms. According to Højsted (2020) looked at the
historical development of research regarding DGEs, and he classified his findings into three periods;
first, the main emphasis was on investigating students/learners' cognitive processes to better
understand how they learn geometry in DGEs. Second, the research attention shifted to how to design
effective tasks that align with learning objectives and affordance of DGEs. In the third period,
research focus has moved to how teachers can integrate DGEs into their pedagogical practices and
what kind of professional development is needed to prepare them to teach geometry using DGEs.
Højsted (2020) concluded that by the rapid development of DGEs and their accessibility, it is
necessary to consider learner-task-teacher and their interplay as three main dimensions of research
plans and methods in this field. Ruthven (2014) points to the several factors such as poor resourcing
of schools, limited recognition of DGS in school mathematics curricula, and exclusion of digital tools
in examinations, have hindered the implementation of these tools in mathematics classes. His findings
were in line with Clark-Wilson et al. (2020) results as insufficient time and supportive professional
development, limited access to appropriate digital technology, and poor technical support as
constraints for teachers in using technology in class. In this line, Pittalis (2020) developed and
validated a comprehensive model that could assess secondary school teachers' intention to use DGS
in geometry teaching that depends on their evaluation of how well the software aligns with their
pedagogical and learning goals. However, teachers expressed that sometimes the external barriers
such as structures, policies, culture, and resources of their schools, were quite challenging for them.
Further, Thurm & Barzel (2022) stressed the importance of continues projects that their purpose is to
provide professional development for teachers to become able to incorporate technology into their
teaching and concluded that this is an evolutionary rather than a revolutionary process where teacher
change occurs in incremental steps. They also pointed out that teachers' beliefs toward their own self-
efficacy, potential benefits of technology, and extra time requirement, have visible consequence on
using digital technology for teaching mathematics. Finally, Benning et al. (2023) conducted a study
in Ghana to investigate that how a professional development program using GeoGebra, might change
the dispositions of mathematics teachers concerning technology. The results proved that at the end of
the program, teachers' ability to use technology in their mathematics teaching improved significantly.
They also expressed that three contextual factors comprising administrative support, continuous
professional training, and availability of adequate technology resources are involving teachers' use
of digital technology.

METHOD

The purpose of the present study was to identify the major obstacles for integrating DGS with
geometry teaching in Iran. Thus, the research guided by this research question and according to its
explanatory nature, a qualitative approach was employed. The researchers formulated several
questions according to the research findings that presented in the literature review. After piloting the
questions, the cover letter for interview sent to several mathematics teachers' groups that are formed
in social media including WhatsApp and Telegram and invited those who are interested in using DGS
to voluntarily participate in a semi-structured and online interview. At last, 18 senior high school
teachers were interviewed. The informants' diversity in terms of gender, geographic distribution,
years of teaching experience and economic status of schools that they were teaching. The interviewer
indicated that for the sake of confidentiality, the name and other personal information of participants
support them to use technology in their classrooms. According to Højsted (2020) looked at the
historical development of research regarding DGEs, and he classified his findings into three periods;
first, the main emphasis was on investigating students/learners' cognitive processes to better
understand how they learn geometry in DGEs. Second, the research attention shifted to how to design
effective tasks that align with learning objectives and affordance of DGEs. In the third period,
research focus has moved to how teachers can integrate DGEs into their pedagogical practices and
what kind of professional development is needed to prepare them to teach geometry using DGEs.
Højsted (2020) concluded that by the rapid development of DGEs and their accessibility, it is
necessary to consider learner-task-teacher and their interplay as three main dimensions of research
plans and methods in this field. Ruthven (2014) points to the several factors such as poor resourcing
of schools, limited recognition of DGS in school mathematics curricula, and exclusion of digital tools
in examinations, have hindered the implementation of these tools in mathematics classes. His findings
were in line with Clark-Wilson et al. (2020) results as insufficient time and supportive professional
development, limited access to appropriate digital technology, and poor technical support as
constraints for teachers in using technology in class. In this line, Pittalis (2020) developed and
validated a comprehensive model that could assess secondary school teachers' intention to use DGS
in geometry teaching that depends on their evaluation of how well the software aligns with their
pedagogical and learning goals. However, teachers expressed that sometimes the external barriers
such as structures, policies, culture, and resources of their schools, were quite challenging for them.
Further, Thurm & Barzel (2022) stressed the importance of continues projects that their purpose is to
provide professional development for teachers to become able to incorporate technology into their
teaching and concluded that this is an evolutionary rather than a revolutionary process where teacher
change occurs in incremental steps. They also pointed out that teachers' beliefs toward their own self-
efficacy, potential benefits of technology, and extra time requirement, have visible consequence on
using digital technology for teaching mathematics. Finally, Benning et al. (2023) conducted a study
in Ghana to investigate that how a professional development program using GeoGebra, might change
the dispositions of mathematics teachers concerning technology. The results proved that at the end of
the program, teachers' ability to use technology in their mathematics teaching improved significantly.
They also expressed that three contextual factors comprising administrative support, continuous
professional training, and availability of adequate technology resources are involving teachers' use
of digital technology.

METHOD

The purpose of the present study was to identify the major obstacles for integrating DGS with
geometry teaching in Iran. Thus, the research guided by this research question and according to its
explanatory nature, a qualitative approach was employed. The researchers formulated several
questions according to the research findings that presented in the literature review. After piloting the
questions, the cover letter for interview sent to several mathematics teachers' groups that are formed
in social media including WhatsApp and Telegram and invited those who are interested in using DGS
to voluntarily participate in a semi-structured and online interview. At last, 18 senior high school
teachers were interviewed. The informants' diversity in terms of gender, geographic distribution,
years of teaching experience and economic status of schools that they were teaching. The interviewer
indicated that for the sake of confidentiality, the name and other personal information of participants
is not part of interview protocol. All interviews were recorded by permission and transcribed, and for
the data analysis, MAXQDA software was used.

RESULTS

The data analysis revealed four main themes related to the challenges of using DGS consisted of “technological infrastructures”, “policies”, “mathematics curriculum”, and “teachers’ resistance”. Based on these themes, teachers’ solutions to overcome these challenges, three main themes of “in-classroom strategies”, “out-classroom strategies”, and “promotional activities” were found as well.

CONSTRAINTS OF USING DGS IN GEOMETRY EDUCATION

Mathematics teachers are faced with various challenges in using DGS in geometry education that are grouped into four main themes as "technological infrastructures", "policies", "mathematics curriculum", and "teachers' resistance" (Figure 1.)

<figure><img src="image_180.png" /><figcaption>Figure 10: Constraints of using DGS from mathematics teachers’ perspective.</figcaption></figure>

Technological infrastructures

For teachers, lack of adequate technological infrastructure in most schools, is a major constraint in using the DGS in geometry classes. Although in their views, schools' facilities ranged from technological readiness with a smart board in every classroom and a computer lab in some private schools and special schools for gifted students, to no technological facilities in the classrooms or a single computer and video projector either in every class or entire school. Extremely low internet speed in schools, limits teachers to use online software and tools as well and makes it difficult to share supplementary videos with their students through virtual platforms. Another even more serious challenge is the internet restrictions imposed by external sanctions or internal filtering in Iran that make many online platforms and websites, inaccessible for Iranian users, like some GeoGebra applets and Desmos do not function properly due to filtering is even worse.

Policies

Teachers expressed that in highly centralized educational system in Iran, the Ministry of Education has the authority to form new policies for supporting teachers to use digital technologies. But in contrast, they pointed out that institutionalized procedures and the lack of attention to the preparation
is not part of interview protocol. All interviews were recorded by permission and transcribed, and for
the data analysis, MAXQDA software was used.

RESULTS

The data analysis revealed four main themes related to the challenges of using DGS consisted of “technological infrastructures”, “policies”, “mathematics curriculum”, and “teachers’ resistance”. Based on these themes, teachers’ solutions to overcome these challenges, three main themes of “in-classroom strategies”, “out-classroom strategies”, and “promotional activities” were found as well.

CONSTRAINTS OF USING DGS IN GEOMETRY EDUCATION

Mathematics teachers are faced with various challenges in using DGS in geometry education that are grouped into four main themes as "technological infrastructures", "policies", "mathematics curriculum", and "teachers' resistance" (Figure 1.)

<figure><img src="image_187.png" /><figcaption>Figure 10: Constraints of using DGS from mathematics teachers’ perspective.</figcaption></figure>

Technological infrastructures

For teachers, lack of adequate technological infrastructure in most schools, is a major constraint in using the DGS in geometry classes. Although in their views, schools’ facilities ranged from technological readiness with a smart board in every classroom and a computer lab in some private schools and special schools for gifted students, to no technological facilities in the classrooms or a single computer and video projector either in every class or entire school. Extremely low internet speed in schools, limits teachers to use online software and tools as well and makes it difficult to share supplementary videos with their students through virtual platforms. Another even more serious challenge is the internet restrictions imposed by external sanctions or internal filtering in Iran that make many online platforms and websites, inaccessible for Iranian users, like some GeoGebra applets and Desmos do not function properly due to filtering is even worse.

Policies

Teachers expressed that in highly centralized educational system in Iran, the Ministry of Education has the authority to form new policies for supporting teachers to use digital technologies. But in contrast, they pointed out that institutionalized procedures and the lack of attention to the preparation
of teachers for using DGS is evident. In their view, "many teachers haven't even heard about different software". In this situation, "it is not surprising that they don't feel the need to learn and use DGS". In addition, "some teachers haven't even heard GeoGebra", though "usually, the local education offices offer workshops or in-service courses to introduce GeoGebra to teachers", but "they are not effective and widespread". Most teachers believed that "if mathematics teachers were introduced to DGS and GeoGebra and such, they would definitely use it". Nonetheless, teachers stated that the Ministry of Education is responsible for professional development of teachers and should take this matter seriously by making implementable policies with sufficient monitoring and support. Otherwise, only offering online courses without adequate quality is not responsive to this great need. Some participants tried to promote DGS among other teachers by directing workshops to demonstrate its features. The feedback indicated that "they often had a positive initial impression", but they discontinued its use and did not integrate it into their teaching that might implies that for using DGS, the policy makers should invest on long-term training not only short-term outcomes.

## Mathematics curriculum

Mathematics textbooks are a vital component of school mathematics in Iran and in fact, are curriculum itself and are the main source of teaching. However, teachers argued that "the structure of current mathematics textbooks is not acceptable, and there is rarely any demand for integrating geometry teaching with DGS". In addition, they reasoned that "mathematics textbooks are outdated and don't include technology-based activities and mathematical applications". Teachers also explicated that "time allocated" to geometry is not sufficient to incorporate DGS to its teaching for two reasons; the organization of content that is not based on technology and require them to design their own teaching material suitable for this integration and second, "outdated equipment" and "lack of technical support", consume more time and put more constrains on using DGS<fnref n="23" />. However, for few teachers, "DGS saved time", because "it facilitated the drawing of figures", "attracted students' interests", and "avoided unnecessary repetitions in explaining some topics". Another important constrain for integrating DGS with teaching geometry was national "university entrance examination" that is "multiple choice", and students are not allowed to use technology. Thus, "it is not useful for students to spend their time on technology. They need to do more drills and practice" and even "some families don't want to waste time on technology and rather spend more time on practicing". They also "expected to cover the entire textbook for final exams, and their performance is evaluated by the exam results". So, "using DGS is not a high priority" for many of them.

## Teachers' resistance

Another theme that emerged from the analysis of the data was "teachers' resistance" to learn or adopt DGS in their teaching. For some teachers, using technology is still conceived as "cheating" and a source of "mental laziness."

## TEACHERS' STRATEGIES TO OVERCOME CONSTRAINS

Several active teachers with more positive views towards integrating DGS with geometry teaching, developed some strategies to facilitate this process without expecting any kind of top-down reforms

<footnote n="23">²³ Private schools and schools for gifted, do not follow this policy and usually allocate more time to various subjects. As well, they have modern technology labs with specialized staff.</footnote>
of teachers for using DGS is evident. In their view, "many teachers haven't even heard about different software". In this situation, "it is not surprising that they don't feel the need to learn and use DGS". In addition, "some teachers haven't even heard GeoGebra", though "usually, the local education offices offer workshops or in-service courses to introduce GeoGebra to teachers", but "they are not effective and widespread". Most teachers believed that "if mathematics teachers were introduced to DGS and GeoGebra and such, they would definitely use it". Nonetheless, teachers stated that the Ministry of Education is responsible for professional development of teachers and should take this matter seriously by making implementable policies with sufficient monitoring and support. Otherwise, only offering online courses without adequate quality is not responsive to this great need. Some participants tried to promote DGS among other teachers by directing workshops to demonstrate its features. The feedback indicated that "they often had a positive initial impression", but they discontinued its use and did not integrate it into their teaching that might implies that for using DGS, the policy makers should invest on long-term training not only short-term outcomes.

## Mathematics curriculum

Mathematics textbooks are a vital component of school mathematics in Iran and in fact, are curriculum itself and are the main source of teaching. However, teachers argued that "the structure of current mathematics textbooks is not acceptable, and there is rarely any demand for integrating geometry teaching with DGS". In addition, they reasoned that "mathematics textbooks are outdated and don't include technology-based activities and mathematical applications". Teachers also explicated that "time allocated" to geometry is not sufficient to incorporate DGS to its teaching for two reasons; the organization of content that is not based on technology and require them to design their own teaching material suitable for this integration and second, "outdated equipment" and "lack of technical support", consume more time and put more constrains on using DGS<fnref n="23" />. However, for few teachers, "DGS saved time", because "it facilitated the drawing of figures", "attracted students' interests", and "avoided unnecessary repetitions in explaining some topics". Another important constrain for integrating DGS with teaching geometry was national "university entrance examination" that is "multiple choice", and students are not allowed to use technology. Thus, "it is not useful for students to spend their time on technology. They need to do more drills and practice" and even "some families don't want to waste time on technology and rather spend more time on practicing". They also "expected to cover the entire textbook for final exams, and their performance is evaluated by the exam results". So, "using DGS is not a high priority" for many of them.

## Teachers' resistance

Another theme that emerged from the analysis of the data was "teachers' resistance" to learn or adopt DGS in their teaching. For some teachers, using technology is still conceived as "cheating" and a source of "mental laziness."

## TEACHERS' STRATEGIES TO OVERCOME CONSTRAINS

Several active teachers with more positive views towards integrating DGS with geometry teaching, developed some strategies to facilitate this process without expecting any kind of top-down reforms

<footnote n="23">²³ Private schools and schools for gifted, do not follow this policy and usually allocate more time to various subjects. As well, they have modern technology labs with specialized staff.</footnote>
and shared them with other teachers via their social groups. Their initiatives are grouped into three themes (Figure 2.)

<figure><img src="image_181.png" /><figcaption>Figure 11: Teachers’ strategies to overcome challenges of using DGS in Iran</figcaption></figure>

**In-classroom strategies**

Teachers initiated different strategies to overcome some of the constraints that introduced above. For instance, challenges of inadequate technological resources in schools, many of them used their own personal devices for teaching as they did during the covid-19 pandemic. In one surprising occasion, one teacher in a deprived school, used his laptop to demonstrate a geometry activity in a school that did not have video projector. In his words, “I turned my laptop screen towards 30 students in class and they crowded around to see the activity”. In terms of limited time allocation, some teachers adopt a “less is more” strategy as one of them explained, “instead of covering all the textbook exercises, I did some of them thoroughly and used GeoGebra to demonstrate their different aspects”. She then said with confidence that “they usually could do the rest by themselves”. Other strategies adopted by teachers included “talk less-do more”, “avoid doing repetitive tasks”, “give less exams” and instead, “spend more time using DGS”. At junior high school, geometry is part of mathematics textbook and some teachers “devoted more time to geometry” comparing to other chapters to save time for using DGS. Finally, a promising strategy for overcoming teachers’ resistance was to provide effective professional development programs and offer more support to mathematics teachers.

**Out-classroom strategies**

To deal with teachers’ resistance, those who were enthusiastic to teach geometry using DGS, posted GIFs or video clips created by DGS and were ready to use, to encourage and support other teachers to become knowledgeable and confident to utilize technology in their geometry teaching. Further, many willing teachers initiated virtual environments for their classes and posted extra materials such as video clips and GIFs for their students. For instance, they used DGS to teach “how to sketch different sides of a solid”, and to their experience, “sending several related GIFs was sufficient for students to solve such tasks by themselves”. As well, in schools that did not have video projector, teachers sent video clips and other teaching material via virtual environments to both overcoming lack of technological equipment and insufficient teaching time. In addition, teachers provided tutorials for
and shared them with other teachers via their social groups. Their initiatives are grouped into three themes (Figure 2.)

<figure><img src="image_188.png" /><figcaption>Figure 11: Teachers’ strategies to overcome challenges of using DGS in Iran</figcaption></figure>

**In-classroom strategies**

Teachers initiated different strategies to overcome some of the constraints that introduced above. For instance, challenges of inadequate technological resources in schools, many of them used their own personal devices for teaching as they did during the covid-19 pandemic. In one surprising occasion, one teacher in a deprived school, used his laptop to demonstrate a geometry activity in a school that did not have video projector. In his words, "I turned my laptop screen towards 30 students in class and they crowded around to see the activity". In terms of limited time allocation, some teachers adopt a "less is more" strategy as one of them explained, "instead of covering all the textbook exercises, I did some of them thoroughly and used GeoGebra to demonstrate their different aspects". She then said with confidence that "they usually could do the rest by themselves". Other strategies adopted by teachers included "talk less-do more", "avoid doing repetitive tasks", "give less exams" and instead, "spend more time using DGS". At junior high school, geometry is part of mathematics textbook and some teachers "devoted more time to geometry" comparing to other chapters to save time for using DGS. Finally, a promising strategy for overcoming teachers' resistance was to provide effective professional development programs and offer more support to mathematics teachers.

**Out-classroom strategies**

To deal with teachers' resistance, those who were enthusiastic to teach geometry using DGS, posted GIFs or video clips created by DGS and were ready to use, to encourage and support other teachers to become knowledgeable and confident to utilize technology in their geometry teaching. Further, many willing teachers initiated virtual environments for their classes and posted extra materials such as video clips and GIFs for their students. For instance, they used DGS to teach "how to sketch different sides of a solid", and to their experience, "sending several related GIFs was sufficient for students to solve such tasks by themselves". As well, in schools that did not have video projector, teachers sent video clips and other teaching material via virtual environments to both overcoming lack of technological equipment and insufficient teaching time. In addition, teachers provided tutorials for
students who were instructed to use GeoGebra for solving geometry problems. Some teachers posted interactive activities made by DGS where "students could drag shapes, measure segments and angles, make conjectures, and answer questions". Another teacher described that:

> In my class, I demonstrated the intersection of altitudes of a triangle in different conditions with GeoGebra and then asked students to construct similar dynamic shapes for medians, bisectors, and perpendicular bisectors of a triangle and send them to our virtual group.

In sum, several teachers developed various strategies to face the challenges of integrating technology with geometry education. Even some of them gave supplementary sessions as extra-curricular activity or asked other teachers to give them permission to use their possible spare time. Finally, one of the mathematics teachers asked the school principal to allow her to combine two “Mathematics” and “Information Technology (IT)” subjects to integrate DGS with her teaching.

## Promotional activities

Most teachers indicated that their first exposure to DGS happened in mathematics communities including mathematics houses, conferences, and online communications such as various mathematics teachers’ virtual groups. They used these platforms to disseminate their successful practices with DGS. Even so, some of them initiated numerous free online workshops for their colleagues to decrease their resistance by enhancing their knowledge and skills in using digital technology. They also shared their ready to use material with others via virtual groups to balance teaching time.

## CONCLUSION

The purpose of this study was to identify the main constrains that Iranian mathematics teachers encountered for integrating DGS with teaching geometry. By analysis of the data, the constraints grouped into four themes labeling “technological infrastructure”, “policies”, “mathematics curriculum” and “teachers’ resistance”. In addition, this study presented the initiatives of enthusiastic mathematics teachers who were willing to share their successful practices with other teachers and encourage them to adopt DGS with geometry teaching. The study presented examples of how teachers compensate for the lack of school resources with their great effort and beyond school hours by creating virtual environments for their classes as well as for their colleagues to exchange resources. They also used various educational communities such as mathematics houses and conferences, produced ready to use materials for both students and other teachers, and communicated with the Ministry of Education and offered their willingness to collaborate with them to promote technology in school mathematics from different aspects. These initiatives could inspire mathematics teachers around the world to overcome the constraints in using DGS with geometry teaching. The implication of this research could particularly be useful for highly centralized education systems and disadvantaged schools and districts that “where there is a will, there is a way!”

## References

* Benning, I., Linsell, C., & Ingram, N. (2023). Examining the changes in mathematics teachers’ technology dispositions through GeoGebra-mediated professional development. *Asian Journal for Mathematics Education, 2*, 2752726323116326. https://doi.org/10.1177/27527263231163276.

* Chan, K. K., & Leung, S. (2014). Dynamic geometry software improves mathematical achievement: Systematic review and meta-analysis. *Journal of Educational Computing Research, 51*, 311–325. https://doi.org/10.2190/EC.51.3.c.
students who were instructed to use GeoGebra for solving geometry problems. Some teachers posted interactive activities made by DGS where "students could drag shapes, measure segments and angles, make conjectures, and answer questions". Another teacher described that:

> In my class, I demonstrated the intersection of altitudes of a triangle in different conditions with GeoGebra and then asked students to construct similar dynamic shapes for medians, bisectors, and perpendicular bisectors of a triangle and send them to our virtual group.

In sum, several teachers developed various strategies to face the challenges of integrating technology with geometry education. Even some of them gave supplementary sessions as extra-curricular activity or asked other teachers to give them permission to use their possible spare time. Finally, one of the mathematics teachers asked the school principal to allow her to combine two “Mathematics” and “Information Technology (IT)” subjects to integrate DGS with her teaching.

## Promotional activities

Most teachers indicated that their first exposure to DGS happened in mathematics communities including mathematics houses, conferences, and online communications such as various mathematics teachers' virtual groups. They used these platforms to disseminate their successful practices with DGS. Even so, some of them initiated numerous free online workshops for their colleagues to decrease their resistance by enhancing their knowledge and skills in using digital technology. They also shared their ready to use material with others via virtual groups to balance teaching time.

## CONCLUSION

The purpose of this study was to identify the main constrains that Iranian mathematics teachers encountered for integrating DGS with teaching geometry. By analysis of the data, the constraints grouped into four themes labeling “technological infrastructure”, “policies”, “mathematics curriculum” and “teachers’ resistance”. In addition, this study presented the initiatives of enthusiastic mathematics teachers who were willing to share their successful practices with other teachers and encourage them to adopt DGS with geometry teaching. The study presented examples of how teachers compensate for the lack of school resources with their great effort and beyond school hours by creating virtual environments for their classes as well as for their colleagues to exchange resources. They also used various educational communities such as mathematics houses and conferences, produced ready to use materials for both students and other teachers, and communicated with the Ministry of Education and offered their willingness to collaborate with them to promote technology in school mathematics from different aspects. These initiatives could inspire mathematics teachers around the world to overcome the constraints in using DGS with geometry teaching. The implication of this research could particularly be useful for highly centralized education systems and disadvantaged schools and districts that “where there is a will, there is a way!”

## References

* Benning, I., Linsell, C., & Ingram, N. (2023). Examining the changes in mathematics teachers' technology dispositions through GeoGebra-mediated professional development. *Asian Journal for Mathematics Education, 2*, 2752726323116326. https://doi.org/10.1177/27527263231163276.

* Chan, K. K., & Leung, S. (2014). Dynamic geometry software improves mathematical achievement: Systematic review and meta-analysis. *Journal of Educational Computing Research, 51*, 311–325. https://doi.org/10.2190/EC.51.3.c.
Clark-Wilson, A., Robutti, O., & Thomas, M. (2020). Teaching with digital technology. *ZDM*. 52, 1223–1242.
https://doi.org/10.1007/s11858-020-01196-0.

Gholamazad, S. (2020). School Mathematics Curriculum Change: Challenges and Required Research. *Journal of Curriculum Studies, 15*(57), 107–128. Iranian Curriculum Studies Association: I.C.S.A. (In Farsi.)

Hillmayr, D., Ziernwald, L., Reinhold, F., Hofer, S. I., & Reiss, K. M. (2020). The potential of digital tools to enhance mathematics and science learning in secondary schools: A context-specific meta-analysis. *Computers & Education, 153*, 103897. https://doi.org/10.1016/j.compedu.2020.103897.

Højsted, I. H. (2020). Guidelines for utilizing affordances of dynamic geometry environments to support development of reasoning competency. *Nordic Studies in Mathematics Education, 25*(2), 71–98

Karimi, M. S., Hoveida, R., & Siadat, S. (2023). Paradigm model of decentralization of iranian education system. *Journal of Management and Planning in Educational System, 16*(1), 137–162.
https://doi.org/10.52547/mpes.16.1.137. Shahid Beheshti University. (In Farsi.)

Lavicza, Z., Hohenwarter, M., Jones, K., Lu, A., & Dawes, M. (2010). Establishing a professional development network around dynamic mathematics software in England. *International Journal for Technology in Mathematics Education, 17*, 177–182.

Ottenbreit-Leftwich, A., Liao, Y.-C., Sadik, O., & Ertmer, P. (2018). Evolution of teachers’ technology integration knowledge, beliefs, and practices: How can we support beginning teachers use of technology? *Journal of Research on Technology in Education, 50*, 1–23.
https://doi.org/10.1080/15391523.2018.1487350.

Pittalis, M. (2020). Extending the Technology Acceptance Model to Evaluate Teachers’ Intention to Use Dynamic Geometry Software in Geometry Teaching. *International Journal of Mathematical Education in Science and Technology*. 52, 1–20. https://doi.org/10.1080/0020739X.2020.1766139.

Reyhani, E., Mesgarani, H., & Farmehr, F. (2009). Investigation of the roles of dynamic geometry software in problem solving skills and conjecture making. *Technology of Education Journal (TEJ)*, 3(2), 73–86.
https://doi.org/10.22061/tej.2009.1318. (In Farsi.)

Ruthven, K. (2014). Frameworks for analysing the expertise that underpins successful integration of digital technologies into everyday teaching practice (Vol. 2, pp. 373–394). In A. C. Wilson; O. Robutti & N. Sinclaire. (Eds.), *An international perspective on technology focused professional development: The mathematics teacher in the digital era* (volume 2). https://doi.org/10.1007/978-94-007-4638-1_16.

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. 48(5), 691–719.
https://doi.org/10.1007/s11858-016-0796-6.

Stols, G., & Kriek, J. (2011). Why don’t all maths teachers use dynamic geometry software in their classrooms? *Australasian Journal of Educational Technology, 27*, 137–151. *ASCIIITE: The Australasian Society for Computers in Learning in Tertiary Education*. https://doi.org/10.14742/ajet.988

Straesser, R. (2002). Cabri-géomètre: Does Dynamic Geometry Software (DGS) Change Geometry and its Teaching and Learning? *International Journal of Computers for Mathematical Learning, 6*(3), 319–333.
https://doi.org/10.1023/A:1013361712895.

Thurm, D., & Barzel, B. (2022). Teaching mathematics with technology: A multidimensional analysis of teacher beliefs. *Educational Studies in Mathematics, 109*, 1–23. https://doi.org/10.1007/s10649-021-10072-x.
Clark-Wilson, A., Robutti, O., & Thomas, M. (2020). Teaching with digital technology. *ZDM*. 52, 1223–1242.
https://doi.org/10.1007/s11858-020-01196-0.

Gholamazad, S. (2020). School Mathematics Curriculum Change: Challenges and Required Research. *Journal of Curriculum Studies, 15*(57), 107–128. Iranian Curriculum Studies Association: I.C.S.A. (In Farsi.)

Hillmayr, D., Ziernwald, L., Reinhold, F., Hofer, S. I., & Reiss, K. M. (2020). The potential of digital tools to enhance mathematics and science learning in secondary schools: A context-specific meta-analysis. *Computers & Education, 153*, 103897. https://doi.org/10.1016/j.compedu.2020.103897.

Højsted, I. H. (2020). Guidelines for utilizing affordances of dynamic geometry environments to support development of reasoning competency. *Nordic Studies in Mathematics Education, 25*(2), 71–98

Karimi, M. S., Hoveida, R., & Siadat, S. (2023). Paradigm model of decentralization of iranian education system. *Journal of Management and Planning in Educational System, 16*(1), 137–162.
https://doi.org/10.52547/mpes.16.1.137. Shahid Beheshti University. (In Farsi.)

Lavicza, Z., Hohenwarter, M., Jones, K., Lu, A., & Dawes, M. (2010). Establishing a professional development network around dynamic mathematics software in England. *International Journal for Technology in Mathematics Education, 17*, 177–182.

Ottenbreit-Leftwich, A., Liao, Y.-C., Sadik, O., & Ertmer, P. (2018). Evolution of teachers’ technology integration knowledge, beliefs, and practices: How can we support beginning teachers use of technology? *Journal of Research on Technology in Education, 50*, 1–23.
https://doi.org/10.1080/15391523.2018.1487350.

Pittalis, M. (2020). Extending the Technology Acceptance Model to Evaluate Teachers’ Intention to Use Dynamic Geometry Software in Geometry Teaching. *International Journal of Mathematical Education in Science and Technology*. 52, 1–20. https://doi.org/10.1080/0020739X.2020.1766139.

Reyhani, E., Mesgarani, H., & Farmehr, F. (2009). Investigation of the roles of dynamic geometry software in problem solving skills and conjecture making. *Technology of Education Journal (TEJ)*, 3(2), 73–86.
https://doi.org/10.22061/tej.2009.1318. (In Farsi.)

Ruthven, K. (2014). Frameworks for analysing the expertise that underpins successful integration of digital technologies into everyday teaching practice (Vol. 2, pp. 373–394). In A. C. Wilson; O. Robutti & N. Sinclaire. (Eds.), *An international perspective on technology focused professional development: The mathematics teacher in the digital era* (volume 2). https://doi.org/10.1007/978-94-007-4638-1_16.

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. 48(5), 691–719.
https://doi.org/10.1007/s11858-016-0796-6.

Stols, G., & Kriek, J. (2011). Why don’t all maths teachers use dynamic geometry software in their classrooms? *Australasian Journal of Educational Technology, 27*, 137–151. *ASCIIITE: The Australasian Society for Computers in Learning in Tertiary Education*. https://doi.org/10.14742/ajet.988

Straesser, R. (2002). Cabri-géomètre: Does Dynamic Geometry Software (DGS) Change Geometry and its Teaching and Learning? *International Journal of Computers for Mathematical Learning, 6*(3), 319–333.
https://doi.org/10.1023/A:1013361712895.

Thurm, D., & Barzel, B. (2022). Teaching mathematics with technology: A multidimensional analysis of teacher beliefs. *Educational Studies in Mathematics, 109*, 1–23. https://doi.org/10.1007/s10649-021-10072-x.
# DEVELOPING ANALYTICAL THINKING IN THE RECOGNITION OF UNUSUAL GEOMETRICAL SHAPES WITH YOUNG PUPILS (FROM 4–8-YEAR-OLD)

Céline Vendeira, Sylvia Coutat

Université de Genève

In this research, we present how we conceived a resource for teaching geometrical shapes and their recognition from the age of 4. We show how, through pupils' ways of speaking and acting, in situations, we are able to understand their way of thinking about geometry, in particular by enriching their initial iconic visualization with analytical thinking.

## INTRODUCTION

Several research works in geometry education have pointed out a rupture between primary and secondary school: either due to the transition to deductive geometry (Berthelot and Salin, 1992), to a paradigm shift between natural and axiomatic geometry (Houdement & Kuzniak, 2000). Therefore, we postulate that it is possible to act from kindergarten and contribute to reducing this rupture and smoothening the transition. To this end, we have created a specific artefact (Coutat & Vendeira, 2015) and developed teaching-learning situations in geometry around shapes recognition (Coutat and Vendeira, 2015-2018) for pupils aged 4 to 8. In presenting our manipulative and visual tool and the teaching activities we have designed, we want to contribute to the topic C2: "resources for teaching and learning geometry" of ICMI Study 26 Discussion Document. Indeed, our research tackles the issue of manipulation and visualization as a sufficient mean to allow effective learning processes in geometry in early childhood. In particular, we question the role of language and gestures to identify ways of thinking in geometry with pupils aged from 4 to 8. Duval (2005) distinguishes two possible types of visualization when identifying geometric objects and their relationships: the iconic and noniconic visualization. The first represents a global apprehension of the shape, while the second is more analytical. In the iconic visualization, 2D objects are perceived globally, as pieces of surface of certain shapes arranged in a sort of a catalogue, a bit like in a botanist herbarium (Duval et al., 2004). In a noniconic visualization, one goes beyond the globally perceived shape to deal more analytically with the properties of the object. Duval underlines the necessity to change the way of looking at figures, from an iconic to a noniconic visualization, in order to reach a certain level in geometrical expertise. This change though is not a radical immediate change, but rather a progressive "enrichment" of the way of looking at shapes towards an analytical way of looking at geometric figures. We distinguish shapes describable through their characteristics from figures involving properties and requiring a higher level of abstraction. Objectively, it seems to us that talking about "properties" with pupils aged from 4 to 8 is premature. This is why we prefer to use the term "characteristics" meaning more precisely the state of conceptualization of geometric objects at that age. The nuance between characteristics of shapes and properties of figures is based on the vocabulary used, the different ways of looking at geometric objects and the relationships considered to define an object. In our work (Coutat & Vendeira 2017), we talk about characteristics when pupils still need representations of the
# DEVELOPING ANALYTICAL THINKING IN THE RECOGNITION OF UNUSUAL GEOMETRICAL SHAPES WITH YOUNG PUPILS (FROM 4–8-YEAR-OLD)

Céline Vendeira, Sylvia Coutat

Université de Genève

In this research, we present how we conceived a resource for teaching geometrical shapes and their recognition from the age of 4. We show how, through pupils' ways of speaking and acting, in situations, we are able to understand their way of thinking about geometry, in particular by enriching their initial iconic visualization with analytical thinking.

## INTRODUCTION

Several research works in geometry education have pointed out a rupture between primary and secondary school: either due to the transition to deductive geometry (Berthelot and Salin, 1992), to a paradigm shift between natural and axiomatic geometry (Houdement & Kuzniak, 2000). Therefore, we postulate that it is possible to act from kindergarten and contribute to reducing this rupture and smoothening the transition. To this end, we have created a specific artefact (Coutat & Vendeira, 2015) and developed teaching-learning situations in geometry around shapes recognition (Coutat and Vendeira, 2015-2018) for pupils aged 4 to 8. In presenting our manipulative and visual tool and the teaching activities we have designed, we want to contribute to the topic C2: "resources for teaching and learning geometry" of ICMI Study 26 Discussion Document. Indeed, our research tackles the issue of manipulation and visualization as a sufficient mean to allow effective learning processes in geometry in early childhood. In particular, we question the role of language and gestures to identify ways of thinking in geometry with pupils aged from 4 to 8. Duval (2005) distinguishes two possible types of visualization when identifying geometric objects and their relationships: the iconic and noniconic visualization. The first represents a global apprehension of the shape, while the second is more analytical. In the iconic visualization, 2D objects are perceived globally, as pieces of surface of certain shapes arranged in a sort of a catalogue, a bit like in a botanist herbarium (Duval et al., 2004). In a noniconic visualization, one goes beyond the globally perceived shape to deal more analytically with the properties of the object. Duval underlines the necessity to change the way of looking at figures, from an iconic to a noniconic visualization, in order to reach a certain level in geometrical expertise. This change though is not a radical immediate change, but rather a progressive "enrichment" of the way of looking at shapes towards an analytical way of looking at geometric figures. We distinguish shapes describable through their characteristics from figures involving properties and requiring a higher level of abstraction. Objectively, it seems to us that talking about "properties" with pupils aged from 4 to 8 is premature. This is why we prefer to use the term "characteristics" meaning more precisely the state of conceptualization of geometric objects at that age. The nuance between characteristics of shapes and properties of figures is based on the vocabulary used, the different ways of looking at geometric objects and the relationships considered to define an object. In our work (Coutat & Vendeira 2017), we talk about characteristics when pupils still need representations of the
objects such as diagrams or drawings even if these are partial. This is however no longer necessary
for the properties of figures because they exist independently of any representation. At the stage of
the characteristics, the pupils have only limited access to the comprehension of geometric shapes: 1)
all properties are not considered and 2) these are not necessarily put in relation. Our research began
in 2015. We built some equipment made of various shapes cut in wood or synthetic resin with which
we have experimented many activities in classes, in the French-speaking part of Switzerland, mainly
in Geneva with pupils aged from 4 to 8. Overall, the whole experiment has been very positive. Pupils
manage to juggle between the two different types of visualization of geometrical objects, highlighting
the enrichment of the way of looking at them. Beyond this enrichment, we also understood that there
is a need of a certain flexibility in these changes. Indeed, the visualization must be adapted to the
situation: an expert procedure in a certain situation could, for example, no longer be adequate in
another, depending on the modifications of the didactic variables (Brousseau, 1982). In this
contribution, we propose to first describe the developed equipment and explain our didactic choices
to enrich pupils' way of looking at shapes. We then give some elements on our analysis framework.
Finally, we pursue by presenting succinctly some examples of concrete classroom analyses before
concluding.

THE MATERIAL AND DIDACTIC CHOICES

As in kindergarten we cannot use neither instruments, nor construction tasks we have developed a
specific equipment adapted to work around shapes recognition at this level. Our idea was to work on
a special collection of shapes, which force to change the way of looking at them and to gradually
work around the characteristics of shapes from the age of 4 up to higher levels in primary education.
At four already, pupils are quite aware of the characteristic of the number of sides of a shape. Yet,
pupils of this age are still learning the concept of number, which therefore remains fragile when it
comes to counting sides. In addition, other skills may interfere, such as enumeration (Briand et al.,
2004), which further complicates the task. For these different reasons, it is important not to focus
exclusively or preferentially on this characteristic. It is also necessary to deal with the presence of
straight or curved side, symmetry, parallel opposite sides or even the fact that the shape is convex or
not. Correct mathematical terms are, however, not expected. At this early age, it is important to let
the identification of such characteristics be made with a personal vocabulary as long as it is
understandable and shared by the other pupils. Our pieces are not the usual shapes like squares,
rectangles, triangles and circles. We made this choice because of some obstacles we observed in
classrooms (Coutat & Vendeira, 2015). Indeed, dealing with usual shapes, the pupils do not refer to
their characteristics but only use the name of the shape, which seems to be its only characteristic.
Moreover, non-prototypical shapes (such as the rightangled triangle or the rectangle not having
lengths twice as long as the widths) are not recognized as belonging to their class (triangle/rectangle),
because they are perceptually different from their model. This phenomenon is also confirmed by
research in psychology, for example, Gentaz (2013) questions how children during their early
development manage to overcome specificities in favor of generality. Based on the above, we
developed a collection of unusual shapes that do not have names (Figure 1).
objects such as diagrams or drawings even if these are partial. This is however no longer necessary
for the properties of figures because they exist independently of any representation. At the stage of
the characteristics, the pupils have only limited access to the comprehension of geometric shapes: 1)
all properties are not considered and 2) these are not necessarily put in relation. Our research began
in 2015. We built some equipment made of various shapes cut in wood or synthetic resin with which
we have experimented many activities in classes, in the French-speaking part of Switzerland, mainly
in Geneva with pupils aged from 4 to 8. Overall, the whole experiment has been very positive. Pupils
manage to juggle between the two different types of visualization of geometrical objects, highlighting
the enrichment of the way of looking at them. Beyond this enrichment, we also understood that there
is a need of a certain flexibility in these changes. Indeed, the visualization must be adapted to the
situation: an expert procedure in a certain situation could, for example, no longer be adequate in
another, depending on the modifications of the didactic variables (Brousseau, 1982). In this
contribution, we propose to first describe the developed equipment and explain our didactic choices
to enrich pupils' way of looking at shapes. We then give some elements on our analysis framework.
Finally, we pursue by presenting succinctly some examples of concrete classroom analyses before
concluding.

THE MATERIAL AND DIDACTIC CHOICES

As in kindergarten we cannot use neither instruments, nor construction tasks we have developed a
specific equipment adapted to work around shapes recognition at this level. Our idea was to work on
a special collection of shapes, which force to change the way of looking at them and to gradually
work around the characteristics of shapes from the age of 4 up to higher levels in primary education.
At four already, pupils are quite aware of the characteristic of the number of sides of a shape. Yet,
pupils of this age are still learning the concept of number, which therefore remains fragile when it
comes to counting sides. In addition, other skills may interfere, such as enumeration (Briand et al.,
2004), which further complicates the task. For these different reasons, it is important not to focus
exclusively or preferentially on this characteristic. It is also necessary to deal with the presence of
straight or curved side, symmetry, parallel opposite sides or even the fact that the shape is convex or
not. Correct mathematical terms are, however, not expected. At this early age, it is important to let
the identification of such characteristics be made with a personal vocabulary as long as it is
understandable and shared by the other pupils. Our pieces are not the usual shapes like squares,
rectangles, triangles and circles. We made this choice because of some obstacles we observed in
classrooms (Coutat & Vendeira, 2015). Indeed, dealing with usual shapes, the pupils do not refer to
their characteristics but only use the name of the shape, which seems to be its only characteristic.
Moreover, non-prototypical shapes (such as the rightangled triangle or the rectangle not having
lengths twice as long as the widths) are not recognized as belonging to their class (triangle/rectangle),
because they are perceptually different from their model. This phenomenon is also confirmed by
research in psychology, for example, Gentaz (2013) questions how children during their early
development manage to overcome specificities in favor of generality. Based on the above, we
developed a collection of unusual shapes that do not have names (Figure 1).
<figure><img src="image_182.png" /><figcaption>Fig. 1: The developed material consisting of 47 pieces of different unusual and unnamable shapes</figcaption></figure>

The choice of this collection does not prevent the pupils from focusing on the overall aspect of the shapes. A resemblance to a known everyday object is possible, such as a fish, a vase, or a mountain. However, depending on the set of the pieces and their iconic proximity, a treatment by characteristics only will be effective (case of the selection made in fig. 2). In fact, a pupil would not be able to use the resemblance of one of the pieces below to a fish, since they all resemble to a fish. According to the objectives to reach, an adequate selection of shapes by the teachers is therefore essential.

<figure><img src="image_183.png" /><figcaption>Fig. 2: Example of iconically close shapes</figcaption></figure>

Research in cognitive science (Gentaz et al., 2009) and mathematics education (Duval & Godin, 2005; Duval, 2005), pointed out the importance for young pupils to be able to manipulate objects. This is why we chose to have physical manipulable objects involving haptic exploration rather than images on paper (even if we use these sometimes too) (Coutat & Vendeira, 2019a). Gentaz et al. (2009) showed that the haptic modality enables pupils to represent shapes better than when they are only represented as images on paper. Our equipment (Figure 3) is therefore composed of two interlocking parts (templates and stencils) allowing two distinct types of manipulation (the inside of the stencil and the perimeter of the template). Thus, when touching the pieces, where points are detectable on a template, holes are associated on the corresponding stencil.

<figure><img src="image_184.png" /><figcaption>Fig. 3: The stencil (hollowed-out part) – The template (full part) – The two-part puzzle</figcaption></figure>

These different configurations give rise to the perception (both visual and haptic) of four distinct “shapes”. Those of the stencil (the empty part and the solid part), that of the template and that of the puzzle of the template and the stencil. In order to avoid favoring any orientation, the stencils were cut
<figure><img src="image_189.png" /><figcaption>Fig. 1: The developed material consisting of 47 pieces of different unusual and unnamable shapes</figcaption></figure>

The choice of this collection does not prevent the pupils from focusing on the overall aspect of the shapes. A resemblance to a known everyday object is possible, such as a fish, a vase, or a mountain. However, depending on the set of the pieces and their iconic proximity, a treatment by characteristics only will be effective (case of the selection made in fig. 2). In fact, a pupil would not be able to use the resemblance of one of the pieces below to a fish, since they all resemble to a fish. According to the objectives to reach, an adequate selection of shapes by the teachers is therefore essential.

<figure><img src="image_190.png" /><figcaption>Fig. 2: Example of iconically close shapes</figcaption></figure>

Research in cognitive science (Gentaz et al., 2009) and mathematics education (Duval & Godin, 2005; Duval, 2005), pointed out the importance for young pupils to be able to manipulate objects. This is why we chose to have physical manipulable objects involving haptic exploration rather than images on paper (even if we use these sometimes too) (Coutat & Vendeira, 2019a). Gentaz et al. (2009) showed that the haptic modality enables pupils to represent shapes better than when they are only represented as images on paper. Our equipment (Figure 3) is therefore composed of two interlocking parts (templates and stencils) allowing two distinct types of manipulation (the inside of the stencil and the perimeter of the template). Thus, when touching the pieces, where points are detectable on a template, holes are associated on the corresponding stencil.

<figure><img src="image_191.png" /><figcaption>Fig. 3: The stencil (hollowed-out part) – The template (full part) – The two-part puzzle</figcaption></figure>

These different configurations give rise to the perception (both visual and haptic) of four distinct “shapes”. Those of the stencil (the empty part and the solid part), that of the template and that of the puzzle of the template and the stencil. In order to avoid favoring any orientation, the stencils were cut
out in a form of a disc. The embedding of the templates in the stencils provides generally an appropriate and reliable feedback for the pupils (Brousseau, 1997). A priori, if the pupil does not manage to embed her or his template in the stencil, it is that it is not the right one (or possibly that it would have to be rotated (turn the piece in the plane) or returned (in the case of a non-symmetrical shape). Once the material had been produced, an accompanying handbook was developed to support its use. This handbook offers a few explanations of the learning challenges and objectives and present some possible tasks for achieving them. The two main activities are 1) recognizing shapes from a set and embedding the template and stencil for validation, and 2) classifying shapes according to several criteria. We then play with the didactic variables to generate a variety set of tasks (Vendeira, 2019). In the following section we develop the analytical framework, which enabled us to observe pupils in class in order to determine their way of looking at shapes.

## ANALYTICAL FRAMEWORK

In order to analyze the classroom sessions, we use the tools from Bernié (2002) such as the ways of acting, speaking and thinking necessary for the notion of a discursive community. We enrich it with the approach of iconic and non iconic visualizations from Duval (2005). As each of these ways of thinking is associated with distinct ways of speaking and acting, we can use this tool in order to analyze activity in class. When pupils think globally with an iconic visualization, they usually try to name geometric objects based on their resemblance to corresponding familiar objects. On the other hand, a pupil who thinks about objects analytically with a non iconic visualization, will refer to their characteristics or properties. Depending on the age of the pupils, the geometric vocabulary used is either conventional and formal or more spontaneous and referring to everyday life. For instance, “non-convex figures” can be designated as: “shapes with holes”. However, in both cases it is indeed a property or a characteristic that is referred to. Thus, the difference in language provides information on the pupil's state of knowledge. Below is the summary table of the elements taken into account for our analyses. We take as an example the following pieces of our collection:

<table><caption>Fig. 4: Summary table of the elements taken into account for the analyses</caption><thead><tr><th rowspan="2">Visualization</th><th>Iconic</th><th colspan="2">non- iconic</th></tr><tr><th></th><th>Characteristics</th><th>Properties</th></tr></thead><tbody><tr><td>think</td><td>Global</td><td colspan="2">Analytical</td></tr><tr><td>talk</td><td>Common and spontaneous vocabulary related to the global aspect of the piece<br>"It looks like a fish"</td><td>Common and spontaneous vocabulary related to characteristics<br>"It is a shape with holes and with straight and curved sides etc. »</td><td>Conventional and formal geometric vocabulary<br>"parallel lines, an axial symmetry, ..."</td></tr><tr><td>act</td><td>Frequent handling<br>Many insets are attempted (all pieces that look like fish).<br>Gestures such as: - put the pieces side by side to compare them; - return the pieces in an orientation that favors comparison; - move around the work table, etc.</td><td>Precise/focused handling<br>Pieces are touched non-randomly.<br>Gestures such as: - make the outline of the piece with the finger - touch a specific part such as a vertex</td><td>Almost no manipulation<br>The pieces are scanned.</td></tr></tbody></table>

Regarding ways of acting, it is very likely that pupils who think more globally manipulate more than others who think more analytically. Some gestures can be identified with one way of thinking rather
out in a form of a disc. The embedding of the templates in the stencils provides generally an appropriate and reliable feedback for the pupils (Brousseau, 1997). A priori, if the pupil does not manage to embed her or his template in the stencil, it is that it is not the right one (or possibly that it would have to be rotated (turn the piece in the plane) or returned (in the case of a non-symmetrical shape). Once the material had been produced, an accompanying handbook was developed to support its use. This handbook offers a few explanations of the learning challenges and objectives and present some possible tasks for achieving them. The two main activities are 1) recognizing shapes from a set and embedding the template and stencil for validation, and 2) classifying shapes according to several criteria. We then play with the didactic variables to generate a variety set of tasks (Vendeira, 2019). In the following section we develop the analytical framework, which enabled us to observe pupils in class in order to determine their way of looking at shapes.

## ANALYTICAL FRAMEWORK

In order to analyze the classroom sessions, we use the tools from Bernié (2002) such as the ways of acting, speaking and thinking necessary for the notion of a discursive community. We enrich it with the approach of iconic and non iconic visualizations from Duval (2005). As each of these ways of thinking is associated with distinct ways of speaking and acting, we can use this tool in order to analyze activity in class. When pupils think globally with an iconic visualization, they usually try to name geometric objects based on their resemblance to corresponding familiar objects. On the other hand, a pupil who thinks about objects analytically with a non iconic visualization, will refer to their characteristics or properties. Depending on the age of the pupils, the geometric vocabulary used is either conventional and formal or more spontaneous and referring to everyday life. For instance, “non-convex figures” can be designated as: “shapes with holes”. However, in both cases it is indeed a property or a characteristic that is referred to. Thus, the difference in language provides information on the pupil's state of knowledge. Below is the summary table of the elements taken into account for our analyses. We take as an example the following pieces of our collection:

<table><caption>Fig. 4: Summary table of the elements taken into account for the analyses</caption><thead><tr><th rowspan="2">Visualization</th><th>Iconic</th><th colspan="2">non iconic</th></tr><tr><th></th><th>Characteristics</th><th>Properties</th></tr></thead><tbody><tr><td>think</td><td>Global</td><td colspan="2">Analytical</td></tr><tr><td>talk</td><td>Common and spontaneous vocabulary related to the global aspect of the piece<br>"It looks like a fish"</td><td>Common and spontaneous vocabulary related to characteristics<br>"It is a shape with holes and with straight and curved sides etc. »</td><td>Conventional and formal geometric vocabulary<br>"parallel lines, an axial symmetry, ..."</td></tr><tr><td>act</td><td>Frequent handling<br>Many insets are attempted (all pieces that look like fish).<br>Gestures such as: - put the pieces side by side to compare them; - return the pieces in an orientation that favors comparison; - move around the work table, etc.</td><td>Precise/focused handling<br>Pieces are touched non-randomly.<br>Gestures such as: - make the outline of the piece with the finger - touch a specific part such as a vertex</td><td>Almost no manipulation<br>The pieces are scanned.</td></tr></tbody></table>

Regarding ways of acting, it is very likely that pupils who think more globally manipulate more than others who think more analytically. Some gestures can be identified with one way of thinking rather
than another. For example, a pupil who turns over a piece to orient it in the same direction as another one (to compare them visually) is very likely to mobilize an iconic visualization. On the other hand, a pupil who identifies a specific feature of a part of shape with his finger is more likely in an analytical thought process. Nevertheless, we shall remind that it is not necessarily always the analytical way of thinking that is the most effective, but that it depends on the situation. For instance, if the task given to the pupils is to decide on a way of separating a group of shapes (only the templates without the stencil) into two sub-groups, a pupil can use the properties of convexity (even if he or she talks about shapes with and without holes) to make the two sub-group. However, if in the group all shapes are non-convex, this analytical way of seeing will be totally inefficient and maybe separating the shapes between these which looks like fishes and the others will be very efficient. Therefore, we believe that the two ways of thinking must coexist and that a certain flexibility is necessary in order to adapt to the proposed situations. The class sessions with the developed equipment are thus analyzed according to two levels: the operationalization of the enrichment of ways of thinking and their flexibility according to a given situation. In the following section, we show how we used this analytical framework to analyze pupils' productions.

## SOME EXAMPLES OF ANALYSES

In this section, we do not seek to prove the effectiveness of the developed material. Instead, we give examples that highlight, through the ways of speaking, the enrichment of the way of looking at shapes and how a flexible way of looking is developed. The examples are based on data collected over several years with pupils aged between 4 and 8. In this paper, because we focus on ways of talking, which are easier to describe than ways of acting, our examples are with 6-8-year-olds where tasks involving language are mainly proposed. Our data are based on a variety of tasks. In order not to complicate the reader's understanding, we have deliberately chosen to focus our examples only on two of them (classification and recognition of a shape by its written description).

### Global or analytical ways of thinking

Here are a few examples of how the pupils we observed think. In a task involving the classification of a group of given shapes into 2 to 4 subgroups, pupils regularly group together figures without being able to justify their choices other than "because they look a bit alike". It's an iconic visualization. We also find more substantiated arguments relating to this same visualization. For example, for figures 1 and 2 below, one pupil justifies his choice by saying "this is the round family". It's the resemblance of the two shapes to a circle that counts.

<figure><img src="image_185.png" /><figcaption>Fig. 5: Iconic visualization "the round family"</figcaption></figure>

In another task when asking to divide a group of six shapes into two subgroups, one pupil justifies putting A, C and F together because they all contain at least one "half-circle". This is also a use of a type of iconic visualization.