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_192.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.
<figure><img src="image_186.png" /><figcaption>Fig. 6: a Iconic visualization with “half-circle” - b Summary table of the elements taken into account for the analyses</figcaption></figure>

In many cases, however, we see a more analytical classification of these same three shapes. Some pupils refer to waves (for curved sides) or to freehand tracings as opposed to those made with a ruler (fig.6b). In this case, we see a common and spontaneous vocabulary related to the characteristics. Finally, in some rare occasions, we find examples where the vocabulary is more conventional and formal. For shapes B, D and E, pupils say: “they are polygons and have almost the same number of sides”, “they have straight sides” or “they all have right angles”. These few examples show how it is possible to obtain information about pupil's way of thinking.

**Flexibility: Visualization must be adapted to the situation**

For pupils aged from 6 to 8, it seems difficult to be exhaustive in enumerating the characteristics needed to solve the task described below (Figure 8). In fact, each of these figures can be distinguished from the one marked with the star by a single geometric criterion. These are, in order, the number of sides, convexity, symmetry and the presence of only straight or some curved sides. When only characteristics are used, those are rarely exhaustive and do not allow to succeed. Pupils must therefore be flexible, mobilizing both iconic and non-iconic visualization. This situation is an opportunity to test this skill.

<figure><img src="image_187.png" /><figcaption>Fig. 7: Task to assess visualization flexibility</figcaption></figure>

Here are two examples where only the global aspect is exploited which is not enough to find the right shape: “the shape I’ve chosen looks like a hexagon”, “I’d like a shape with one piece of a hexagon and one piece of a triangle”. Since many of the shapes are similar, an exclusive use of the iconic visualization is not operational in this situation. Pupils of this age often use the term “hexagon” to designate any polygon with more than 4 sides, which makes it difficult to ascertain its meaning. On the other hand, relying on the characteristics of the shapes only requires a degree of comprehensiveness that is difficult to achieve for pupils of this age. Indeed, between 6 and 8, they generally haven't yet worked on all these characteristics in class. Here are some typical examples: “It has 6 straight sides. One side is bigger than the other”, “It's drawn with a ruler. It has 6 sides and 6 vertices”. In these descriptions, the iconic visualization is not mobilized and the proposed characteristics are insufficient in order to determine the right shape. In addition to the mastery of the characteristics, it is also necessary to know the geometry vocabulary and its signification. From the following example, we can see that the pupils are not yet familiar with the specific vocabulary: “It has no right angle. It has a vertex. It has straight lines. Lots of sides”. We can imagine that this pupil, when he uses the term vertex, refers to something sharp. If the pupil knew the geometric definition
<figure><img src="image_193.png" /><figcaption>Fig. 6: a Iconic visualization with “half-circle” - b Summary table of the elements taken into account for the analyses</figcaption></figure>

In many cases, however, we see a more analytical classification of these same three shapes. Some pupils refer to waves (for curved sides) or to freehand tracings as opposed to those made with a ruler (fig.6b). In this case, we see a common and spontaneous vocabulary related to the characteristics. Finally, in some rare occasions, we find examples where the vocabulary is more conventional and formal. For shapes B, D and E, pupils say: “they are polygons and have almost the same number of sides”, “they have straight sides” or “they all have right angles”. These few examples show how it is possible to obtain information about pupil's way of thinking.

## Flexibility: Visualization must be adapted to the situation

For pupils aged from 6 to 8, it seems difficult to be exhaustive in enumerating the characteristics needed to solve the task described below (Figure 8). In fact, each of these figures can be distinguished from the one marked with the star by a single geometric criterion. These are, in order, the number of sides, convexity, symmetry and the presence of only straight or some curved sides. When only characteristics are used, those are rarely exhaustive and do not allow to succeed. Pupils must therefore be flexible, mobilizing both iconic and non-iconic visualization. This situation is an opportunity to test this skill.

<figure><img src="image_194.png" /><figcaption>Fig. 7: Task to assess visualization flexibility</figcaption></figure>

Here are two examples where only the global aspect is exploited which is not enough to find the right shape: “the shape I’ve chosen looks like a hexagon”, “I’d like a shape with one piece of a hexagon and one piece of a triangle”. Since many of the shapes are similar, an exclusive use of the iconic visualization is not operational in this situation. Pupils of this age often use the term “hexagon” to designate any polygon with more than 4 sides, which makes it difficult to ascertain its meaning. On the other hand, relying on the characteristics of the shapes only requires a degree of comprehensiveness that is difficult to achieve for pupils of this age. Indeed, between 6 and 8, they generally haven't yet worked on all these characteristics in class. Here are some typical examples: “It has 6 straight sides. One side is bigger than the other”, “It's drawn with a ruler. It has 6 sides and 6 vertices”. In these descriptions, the iconic visualization is not mobilized and the proposed characteristics are insufficient in order to determine the right shape. In addition to the mastery of the characteristics, it is also necessary to know the geometry vocabulary and its signification. From the following example, we can see that the pupils are not yet familiar with the specific vocabulary: “It has no right angle. It has a vertex. It has straight lines. Lots of sides”. We can imagine that this pupil, when he uses the term vertex, refers to something sharp. If the pupil knew the geometric definition
of this term, he would probably understand that all the proposed shapes contain several vertices. We
might also wonder what the pupil means when using the term “right angle” because this notion has
not been yet worked on in class. We now present some examples where pupils jointly mobilize both
ways of thinking as intended. However, this doesn't always lead to success, as in this case: “it looks
like a pointed flint, the sides are straight”. In this case, there's not enough information to determine a
single shape. Finally, here are some examples of successful productions, such as: “The shape has 6
sides, it's almost a hexagon and the part on the right is deformed”. In this production, however, we
can see that the pupil uses spatial cues, even though the instruction forbids it. In many productions,
pupils use spatial vocabulary such as “on top”, “on the right” and so on. A certain number of the
pupils’ productions are also based on other aspects such as measurement, or the decomposition of
shapes into known subshapes. Here's an example of what one could expect: “It doesn't have curves
sides. It has 6 angles. It has 6 lines. There's one place that looks like a hexagon and another place that
looks like a triangle.” These few examples show the diversity of what pupils can produce, depending
on whether they use iconic, non iconic or mixed visualizations.

CONCLUSION

Our research proposes to work on geometry in early school in order to reduce the gaps highlighted
by several research studies in mathematics education. To do this, we have developed material based
on manipulation and visualization. In order to see how the pupils learn, we observed the ways in
which they act and speak, to have tray to determine in their way of thinking about geometrical shapes,
whether they use either iconic or non iconic visualization. Our research shows that the use of the
equipment leads pupils from the age of 4 to enrich their initial iconic visualization with some elements
of non iconic visualization. So, they are able to take into account some of the characteristics of shapes.
The choice of tasks and the group of shapes proposed by the teachers are essential in training the
flexibility of their look. Pupils had to be able to mobilize the most effective visualization according
to the environment they were confronted with. However, the results on this point are more nuanced.
It is more difficult for the pupils to know when to use a global rather than an analytical way of thinking
depending on the situation. Similarly, for teachers, understanding this more cross-curricular objective
also seems more complex. At the moment, we're looking at ways of disseminating this material
ensuring that teachers make the most of it. We therefore continue conducting our investigations in
two directions: looking at issues concerning in-service teacher training and considering evaluation
and institutionalization.

References

Bernié, J.-P. (2002). L’approche des pratiques langagières scolaires à travers la notion de «communauté discursive»: un apport à la didactique comparée ? *Revue Française de Pédagogie*, 141, 77-88.

Briand, J., Loubet, M., & Salin, M.-H. (2004). *Apprentissages mathématiques en maternelle: situations et analyses*. Hatier.

Brousseau, G. (1982). D'un problème à l'étude a priori d'une situation didactique, *Actes de la 2ème école Ecole d'été de didactique des mathématiques* (pp. 9 – 60). IREM d'Orléans.

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

Coutat, S., & Vendeira, C. (2015). Des pics, des pointes et des arrondis en 1P-2P. *Math-Ecole, 223*, 14-19.
of this term, he would probably understand that all the proposed shapes contain several vertices. We
might also wonder what the pupil means when using the term “right angle” because this notion has
not been yet worked on in class. We now present some examples where pupils jointly mobilize both
ways of thinking as intended. However, this doesn't always lead to success, as in this case: “it looks
like a pointed flint, the sides are straight”. In this case, there's not enough information to determine a
single shape. Finally, here are some examples of successful productions, such as: “The shape has 6
sides, it's almost a hexagon and the part on the right is deformed”. In this production, however, we
can see that the pupil uses spatial cues, even though the instruction forbids it. In many productions,
pupils use spatial vocabulary such as “on top”, “on the right” and so on. A certain number of the
pupils’ productions are also based on other aspects such as measurement, or the decomposition of
shapes into known subshapes. Here's an example of what one could expect: “It doesn't have curves
sides. It has 6 angles. It has 6 lines. There's one place that looks like a hexagon and another place that
looks like a triangle.” These few examples show the diversity of what pupils can produce, depending
on whether they use iconic, non iconic or mixed visualizations.

CONCLUSION

Our research proposes to work on geometry in early school in order to reduce the gaps highlighted
by several research studies in mathematics education. To do this, we have developed material based
on manipulation and visualization. In order to see how the pupils learn, we observed the ways in
which they act and speak, to have tray to determine in their way of thinking about geometrical shapes,
whether they use either iconic or non iconic visualization. Our research shows that the use of the
equipment leads pupils from the age of 4 to enrich their initial iconic visualization with some elements
of non iconic visualization. So, they are able to take into account some of the characteristics of shapes.
The choice of tasks and the group of shapes proposed by the teachers are essential in training the
flexibility of their look. Pupils had to be able to mobilize the most effective visualization according
to the environment they were confronted with. However, the results on this point are more nuanced.
It is more difficult for the pupils to know when to use a global rather than an analytical way of thinking
depending on the situation. Similarly, for teachers, understanding this more cross-curricular objective
also seems more complex. At the moment, we're looking at ways of disseminating this material
ensuring that teachers make the most of it. We therefore continue conducting our investigations in
two directions: looking at issues concerning in-service teacher training and considering evaluation
and institutionalization.

References

Bernié, J.-P. (2002). L’approche des pratiques langagières scolaires à travers la notion de «communauté discursive»: un apport à la didactique comparée ? *Revue Française de Pédagogie*, 141, 77-88.

Briand, J., Loubet, M., & Salin, M.-H. (2004). *Apprentissages mathématiques en maternelle: situations et analyses*. Hatier.

Brousseau, G. (1982). D'un problème à l'étude a priori d'une situation didactique, *Actes de la 2ème école Ecole d'été de didactique des mathématiques* (pp. 9 – 60). IREM d'Orléans.

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

Coutat, S., & Vendeira, C. (2015). Des pics, des pointes et des arrondis en 1P-2P. *Math-Ecole, 223*, 14-19.
Coutat, S., & Vendeira, C. (2015-2018). *Activités pour les classes de 1P-2P, figures géométriques, travailler autrement les formes au cycle 1*. https://www.unige.ch/fapse/dimage/files/3915/4953/5358/CV-Activites-1P-2P.pdf

Coutat-Gousseau, S., & Vendeira-Maréchal, C. (2016). Shape recognition in early school. *13th International Congress on Mathematical Education*.

Celi, V., Coutat, S., & Vendeira-Maréchal, C. (2019a). Travailler avec les formes en maternelle: premiers pas vers des connaissances géométriques ? *Actes du 45e colloque de la Copirelem* (pp. 35–55). ARPEME.

Duval, R. (2005). Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements. *Annales de Didactique et de Sciences Cognitives, 10*, 5–53.

Duval, R., & Godin, M. (2005). Les changements de regards nécessaires sur les figures, *Grand N*, **76**, 7-27.

Gentaz, E. (2013). Comment aider les enfants de 5-6 ans à connaître les figures géométriques planes ? Un point de vue des sciences cognitives de l'éducation. *Actes du XXXXème Colloque Copirelem*, (pp. 81-116).

Gentaz, É., Bara, F., Palluel-Germain, R., Pinet, L., Hillairet, & De Boisferon, A. (2009). Apports de la modalité haptique manuelle dans les apprentissages scolaires. *Cognito*, **3**(3), 1–38.

Houdement, C., & Kuzniak, A. (2000). Paradigmes géométriques et enseignement de la géométrie. *Annales de Didactiques des Sciences Cognitives*, **11**, 175-193.

Vendeira, C., & Coutat, S. (2017). « C'est une montagne ou une trompette ? » entre perception globale et caractéristiques des formes au cycle 1 et 2. *Grand N*, **100**, 79-103.

Vendeira, C. (2019). Quelle transférabilité d'un matériel de géométrie d'un contexte d'enseignement à un autre? *RMé*, **233**, 47-57.
Coutat, S., & Vendeira, C. (2015-2018). *Activités pour les classes de 1P-2P, figures géométriques, travailler autrement les formes au cycle 1*. https://www.unige.ch/fapse/dimage/files/3915/4953/5358/CV-Activites-1P-2P.pdf

Coutat-Gousseau, S., & Vendeira-Maréchal, C. (2016). Shape recognition in early school. *13th International Congress on Mathematical Education*.

Celi, V., Coutat, S., & Vendeira-Maréchal, C. (2019a). Travailler avec les formes en maternelle: premiers pas vers des connaissances géométriques ? *Actes du 45e colloque de la Copirelem* (pp. 35–55). ARPEME.

Duval, R. (2005). Les conditions cognitives de l'apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements. *Annales de Didactique et de Sciences Cognitives, 10*, 5–53.

Duval, R., & Godin, M. (2005). Les changements de regards nécessaires sur les figures, *Grand N*, **76**, 7-27.

Gentaz, E. (2013). Comment aider les enfants de 5-6 ans à connaître les figures géométriques planes ? Un point de vue des sciences cognitives de l'éducation. *Actes du XXXXème Colloque Copirelem*, (pp. 81-116).

Gentaz, É., Bara, F., Palluel-Germain, R., Pinet, L., Hillairet, & De Boisferon, A. (2009). Apports de la modalité haptique manuelle dans les apprentissages scolaires. *Cognito*, **3**(3), 1–38.

Houdement, C., & Kuzniak, A. (2000). Paradigmes géométriques et enseignement de la géométrie. *Annales de Didactiques des Sciences Cognitives*, **11**, 175-193.

Vendeira, C., & Coutat, S. (2017). « C'est une montagne ou une trompette ? » entre perception globale et caractéristiques des formes au cycle 1 et 2. *Grand N*, **100**, 79-103.

Vendeira, C. (2019). Quelle transférabilité d'un matériel de géométrie d'un contexte d'enseignement à un autre? *RMé*, **233**, 47-57.
# DESCRIPTIVE AUTOMATED ASSESSMENT: FACILITATING INQUIRY IN GEOMETRY

Michal Yerushalmy, Shai Olsher

Department of Mathematics Education, University of Haifa, Israel

As designers of technology to support learning, we are interested in supporting feedback processes in the context of guided inquiry instruction of geometry. We explore the potential of automatically associating mathematical descriptions as resources for reflection and inquiry learning. We study the potential of what might be thought of as a mirror that speaks: it reflects student's actions and describes the mathematics it sees in a student's submission constructed using interactive diagrams. We illustrate the inquiry process by a thought experiment around a task related to inscribed triangles. We discuss the design of descriptive assessment where students reflect on and communicate their own learning, utilizing individually-reported multi-dimensional automatic analysis of their submissions in response to example-elicitig tasks. Based on Hershkowitz's theory of fundamental concepts (1990) and on theories of exemplification and variation we outline a research based framework for automated descriptive assessment that is designed to facilitate inquiry in geometry.

## INTRODUCTION: INQUIRY LEARNING AND TECH-BASED FEEDBACK PROCESSES

In this paper, we suggest that automated descriptions of student work are a new strategy for feedback that can be designed into technology for providing students with support for learning geometry. We illustrate this strategy with an example-elicitig-task (EET) designed in the STEP platform (Olsher et al., 2016) that is intended to support students in developing conjectures about areas’ relations among inscribed triangles. We argue that such tasks have the potential to initiate and support autonomous inquiry situations. We are interested in conceptualizing feedback as an ongoing process and study the use of automated information to support technology-enhanced didactical situations that involve inquiry processes with non-judgmental personal feedback. The tasks include interactive diagrams (applets) in Geogebra (Hohenwarter et al., 2009). The focus of our design effort is providing students with mathematical verbal descriptions of their submitted examples. A main challenge is to design an environment (tasks and tools) that can involve students in exploration of examples and submission of answers to tasks that go beyond the specific examples. Personal feedback provided automatically to students, which allows them to reflect on their work, can serve as a fundamental tool to define pedagogical and mathematical goals amenable to automatic assessment. Our overarching research question is how automatic characterization of students’ examples serves as feedback which involves students’ reflections on their own understandings of the examples they have created.

## THEORETICAL BACKGROUND

### Exemplification

We view inquiry learning in mathematics as centrally involving the coordination of examples and of concepts, where the definition of a concept identifies criteria for classifying instances as examples or non-examples of that concept. DGE is a natural tool for designing inquiry tasks and inquiry situations
# DESCRIPTIVE AUTOMATED ASSESSMENT: FACILITATING INQUIRY IN GEOMETRY

Michal Yerushalmy, Shai Olsher

Department of Mathematics Education, University of Haifa, Israel

As designers of technology to support learning, we are interested in supporting feedback processes in the context of guided inquiry instruction of geometry. We explore the potential of automatically associating mathematical descriptions as resources for reflection and inquiry learning. We study the potential of what might be thought of as a mirror that speaks: it reflects student's actions and describes the mathematics it sees in a student's submission constructed using interactive diagrams. We illustrate the inquiry process by a thought experiment around a task related to inscribed triangles. We discuss the design of descriptive assessment where students reflect on and communicate their own learning, utilizing individually-reported multi-dimensional automatic analysis of their submissions in response to example-eliciting tasks. Based on Hershkowitz's theory of fundamental concepts (1990) and on theories of exemplification and variation we outline a research based framework for automated descriptive assessment that is designed to facilitate inquiry in geometry.

## INTRODUCTION: INQUIRY LEARNING AND TECH-BASED FEEDBACK PROCESSES

In this paper, we suggest that automated descriptions of student work are a new strategy for feedback that can be designed into technology for providing students with support for learning geometry. We illustrate this strategy with an example-eliciting-task (EET) designed in the STEP platform (Olsher et al., 2016) that is intended to support students in developing conjectures about areas' relations among inscribed triangles. We argue that such tasks have the potential to initiate and support autonomous inquiry situations. We are interested in conceptualizing feedback as an ongoing process and study the use of automated information to support technology-enhanced didactical situations that involve inquiry processes with non-judgmental personal feedback. The tasks include interactive diagrams (applets) in Geogebra (Hohenwarter et al., 2009). The focus of our design effort is providing students with mathematical verbal descriptions of their submitted examples. A main challenge is to design an environment (tasks and tools) that can involve students in exploration of examples and submission of answers to tasks that go beyond the specific examples. Personal feedback provided automatically to students, which allows them to reflect on their work, can serve as a fundamental tool to define pedagogical and mathematical goals amenable to automatic assessment. Our overarching research question is how automatic characterization of students' examples serves as feedback which involves students' reflections on their own understandings of the examples they have created.

## THEORETICAL BACKGROUND

### Exemplification

We view inquiry learning in mathematics as centrally involving the coordination of examples and of concepts, where the definition of a concept identifies criteria for classifying instances as examples or non-examples of that concept. DGE is a natural tool for designing inquiry tasks and inquiry situations
for learning geometry. Studies suggest how dragging allows students to experience multiple examples of multiple representations, visualize mathematical concepts, and explore dependencies among properties of the diagram. This can help students learn to categorize diagrams according to their characteristics, discern what could be true, and experience opportunities to overcome difficulties in conjecturing (Arzarello et al., 2002; Luz & Yerushalmy 2023). Participating in such learning, students need to learn to note and communicate about patterns, structures, or regularities in mathematical examples and to question reasons for these phenomena. Although an example cannot confirm a universal statement they can be “supporting” in the context of universal statements (Buchbinder et al. 2017) and having students construct examples of a concept is an important activity that can be a catalyst for enhancing students’ conceptual understanding and expanding students’ concept image (Zaslavsky & Zodik, 2014). Given this perspective, example-elicitig-tasks (EETs) are an important component of our approach to guided inquiry with a digital environment.

Interactions with DGEs are known as helping students to generate examples, identify patterns and categories of constructions in order to formulate conjectures. Yet, research has followed learning episodes where the feedback did not lead to concept-formation and the achievement of curricular goals. Thus, a major challenge is to create a feedback process that would support students who are working on EETs and guide conceptualization of mathematical objects and actions that are central to the curricular goals. To overcome the obstacle of the particularity of an example and to gain insight into the breadth of students’ concept image, we describe two theoretical views that helped us to design the STEP platform and outlay the principal aspects for automated descriptive assessment.

## Variation

To discern a concept or an object of learning, learners must be familiar with a large example space: a collection of examples of the concept, together with the methods that make it possible for them to construct their own examples (Mason & Watson, 2008). Learners can engage with new ideas through making their own examples. Within STEP, students’ submissions to EETs consist mainly of examples, which are instances created with the interactive diagram of the task and stored on the platform. EETs prompt students to generate their own examples but also to produce multiple-examples’ submissions that are as different as possible from one another. The collection of outcomes, including the personal example spaces (PESs) of individual students (or of the collective example space of the entire group), are analyzed online and filtered according to specified characteristics.

Generating examples is rarely sufficient for meaningful learning in autonomous inquiry situations. Most learners need to have explicit stimuli to form a class they can denote as a collection of similar or related examples. However, what is required for learning through exemplification should exceed observations of patterns; it has to involve developing awareness of the variants and invariants of the PES. And it has to become part of a feedback process that supports discerning and formalizing patterns symbolically or verbally. Usually, such awareness is triggered by external guidance, most often by teacher’s probes. The theory of variation (Marton & Tsui, 2004) analyzes the critical aspects of how learners can learn concepts from examples (for a variation-based automated analysis in geometry see Luz & Yerushalmy, 2023). Watson and Mason (2005) relates to variation as the features of an example that learners recognize as eligible for change, without losing examplehood.
for learning geometry. Studies suggest how dragging allows students to experience multiple examples of multiple representations, visualize mathematical concepts, and explore dependencies among properties of the diagram. This can help students learn to categorize diagrams according to their characteristics, discern what could be true, and experience opportunities to overcome difficulties in conjecturing (Arzarello et al., 2002; Luz & Yerushalmy 2023). Participating in such learning, students need to learn to note and communicate about patterns, structures, or regularities in mathematical examples and to question reasons for these phenomena. Although an example cannot confirm a universal statement they can be “supporting” in the context of universal statements (Buchbinder et al. 2017) and having students construct examples of a concept is an important activity that can be a catalyst for enhancing students’ conceptual understanding and expanding students’ concept image (Zaslavsky & Zodik, 2014). Given this perspective, example-elicitig-tasks (EETs) are an important component of our approach to guided inquiry with a digital environment.

Interactions with DGEs are known as helping students to generate examples, identify patterns and categories of constructions in order to formulate conjectures. Yet, research has followed learning episodes where the feedback did not lead to concept-formation and the achievement of curricular goals. Thus, a major challenge is to create a feedback process that would support students who are working on EETs and guide conceptualization of mathematical objects and actions that are central to the curricular goals. To overcome the obstacle of the particularity of an example and to gain insight into the breadth of students’ concept image, we describe two theoretical views that helped us to design the STEP platform and outlay the principal aspects for automated descriptive assessment.

## Variation

To discern a concept or an object of learning, learners must be familiar with a large example space: a collection of examples of the concept, together with the methods that make it possible for them to construct their own examples (Mason & Watson, 2008). Learners can engage with new ideas through making their own examples. Within STEP, students’ submissions to EETs consist mainly of examples, which are instances created with the interactive diagram of the task and stored on the platform. EETs prompt students to generate their own examples but also to produce multiple-examples’ submissions that are as different as possible from one another. The collection of outcomes, including the personal example spaces (PESs) of individual students (or of the collective example space of the entire group), are analyzed online and filtered according to specified characteristics.

Generating examples is rarely sufficient for meaningful learning in autonomous inquiry situations. Most learners need to have explicit stimuli to form a class they can denote as a collection of similar or related examples. However, what is required for learning through exemplification should exceed observations of patterns; it has to involve developing awareness of the variants and invariants of the PES. And it has to become part of a feedback process that supports discerning and formalizing patterns symbolically or verbally. Usually, such awareness is triggered by external guidance, most often by teacher’s probes. The theory of variation (Marton & Tsui, 2004) analyzes the critical aspects of how learners can learn concepts from examples (for a variation-based automated analysis in geometry see Luz & Yerushalmy, 2023). Watson and Mason (2005) relates to variation as the features of an example that learners recognize as eligible for change, without losing examplehood.
## Characterizing examples with words

To further enhance conceptualization using mathematical characteristics as stimuli we follow Hershkowitz's theory of fundamental concepts with its focus on definitions and examples (Hershkowitz, 1990). By concept, this theory refers to a combination of critical and non-critical characteristics; by concept definition, to the minimal combination of critical characteristics (necessary and sufficient) to define the concept; and, by concept image, to the collection of examples and the derived properties reflected in students' work. From this perspective, characterizing students' examples in words might be thought of as providing a mirror that speaks, in some ways like the mirror in the Snow White tale. The sort of "talking mirror" we aim to create "speaks" to the student by offering a description of each example and of the generated PES. Yerushalmy et al. (2022) further explain the theoretical foundations and roles of the mirror that speaks. The kind of mirroring that STEP provides is led by the designers of the tasks or the teachers, to enable students to think in terms of categories rather than of an example; compare and distinguish between their examples, suggesting mathematical descriptions or contextual descriptions of mathematical phenomena. As Watson & Shipman (2009) who quote Vigotsky, we appreciate autonomous learning that is "structured by expert others" (1978, p.57) where these experts speak to the student by automatically characterizing PESs.

## ILLUSTRATING DESCRIPTIVE AUTOMATED ASSESSMENT IN GEOMETRY

In this section, we build on the terms and theories described above to illustrate the descriptive assessment of a geometry inquiry process: The section starts by presenting a STEP geometry task (Figure 1). Then it presents the characteristics and the mathematical and pedagogical considerations for choosing them. Thirdly we demonstrate an automated personal task report; the section ends with a description of a learning situation with the resources of STEP.

<figure><img src="image_188.png" /><figcaption>Figure 1: The inscribed triangles task</figcaption></figure>

### An example-eliciting-task in geometry:

This task requires that students construct a triangle DEF inside triangle ABC where the area is ¼ of the original ABC and only one vertex is on a red dot. The red dots subdivide each of the three sides
## Characterizing examples with words

To further enhance conceptualization using mathematical characteristics as stimuli we follow Hershkowitz's theory of fundamental concepts with its focus on definitions and examples (Hershkowitz, 1990). By concept, this theory refers to a combination of critical and non-critical characteristics; by concept definition, to the minimal combination of critical characteristics (necessary and sufficient) to define the concept; and, by concept image, to the collection of examples and the derived properties reflected in students' work. From this perspective, characterizing students' examples in words might be thought of as providing a mirror that speaks, in some ways like the mirror in the Snow White tale. The sort of "talking mirror" we aim to create "speaks" to the student by offering a description of each example and of the generated PES. Yerushalmy et al. (2022) further explain the theoretical foundations and roles of the mirror that speaks. The kind of mirroring that STEP provides is led by the designers of the tasks or the teachers, to enable students to think in terms of categories rather than of an example; compare and distinguish between their examples, suggesting mathematical descriptions or contextual descriptions of mathematical phenomena. As Watson & Shipman (2009) who quote Vigotsky, we appreciate autonomous learning that is "structured by expert others" (1978, p.57) where these experts speak to the student by automatically characterizing PESs.

## ILLUSTRATING DESCRIPTIVE AUTOMATED ASSESSMENT IN GEOMETRY

In this section, we build on the terms and theories described above to illustrate the descriptive assessment of a geometry inquiry process: The section starts by presenting a STEP geometry task (Figure 1). Then it presents the characteristics and the mathematical and pedagogical considerations for choosing them. Thirdly we demonstrate an automated personal task report; the section ends with a description of a learning situation with the resources of STEP.

<figure><img src="image_195.png" /><figcaption>Figure 1: The inscribed triangles task</figcaption></figure>

### An example-eliciting-task in geometry:

This task requires that students construct a triangle DEF inside triangle ABC where the area is ¼ of the original ABC and only one vertex is on a red dot. The red dots subdivide each of the three sides
of ABC into four equal segments. The shaded triangle is triangle DEF. Points A, B, C, D, E, F are
drageable (the positions of points D, E, F are limited to the perimeter of triangle ABC).

The task's aim is to prompt interaction with the area of a triangle formula while using proportional reasoning to define a set of examples that would preserve a final ratio between two results of multiplication: segment * altitude to the segment (or it's extension). The low entry point for this task enables the students to rely on the division of the big triangle into 4 equal-area sub-triangles by dividing one of its sides into 4 equal-length parts. This method enables the students to initially succeed when working on the task while setting the stage for prompting them to further interact with the problem.

<table><caption>Table 1: Critical and non-critical characteristics</caption><thead><tr><th>Critical characteristics</th><th>Non-critical characteristics</th></tr></thead><tbody><tr><td><ul><li>Exactly one vertex of DEF coincides with a red dot</li><li>The area of DEF is 1/4 of the area of ABC</li></ul></td><td><ol><li>One common vertex to triangles ABC and DEF</li><li>Two common vertices to triangles ABC and DEF</li><li>Three common vertices to triangles ABC and DEF</li><li>Triangles ABC and DEF have no common sides</li><li>One side of DEF lies on a side of ABC</li><li>Two sides of DEF lies on a side of ABC</li><li>ABC has a right angle</li><li>DEF has a right angle</li><li>Common altitude to triangles ABC and DEF</li><li>In triangle DEF one side is a midsegment of triangle ABC</li><li>In triangle DEF two sides are midsegments of triangle ABC</li><li>In triangle DEF no sides are midsegments of triangle ABC</li></ol></td></tr></tbody></table>

## Characterizing variations

The key for the variation of the answers lies within the understanding that the result of the multiplication of the segment and its corresponding altitude of triangle DEF is a quarter of the multiplication result of the altitude and segment of triangle ABC. Below we describe ways in which the non-critical characteristics can offer students ways to either reflect on their submitted work, or plan for different examples that might fit or not fit the marked characteristics. We group the different characteristics to categories and explain the rationale of the choice:

*Reducing the complexity of the problem* (7,8,9): ABC has a right angle; DEF has a right angle; Common altitude to triangles ABC and DEF.

Students might find it easier to drag ABC to be a right triangle. In which the altitude coincides with a side of the triangle. The same could be said for the construction of triangle DEF. Another way to deal with only one side's ratio is to use a common altitude. While this might not be the way that students perceive their examples, reflecting this characteristic in their answers could help them communicate their strategy in a more mathematically robust fashion.

*Limitations of the example space* (1,2,3): One common vertex to triangles ABC and DEF; Two common vertices to triangles ABC and DEF; Three common vertices to triangles ABC and DEF.
of ABC into four equal segments. The shaded triangle is triangle DEF. Points A, B, C, D, E, F are
draggable (the positions of points D, E, F are limited to the perimeter of triangle ABC).

The task's aim is to prompt interaction with the area of a triangle formula while using proportional reasoning to define a set of examples that would preserve a final ratio between two results of multiplication: segment * altitude to the segment (or it's extension). The low entry point for this task enables the students to rely on the division of the big triangle into 4 equal-area sub-triangles by dividing one of its sides into 4 equal-length parts. This method enables the students to initially succeed when working on the task while setting the stage for prompting them to further interact with the problem.

<table><caption>Table 1: Critical and non-critical characteristics</caption><thead><tr><th>Critical characteristics</th><th>Non-critical characteristics</th></tr></thead><tbody><tr><td><ul><li>Exactly one vertex of DEF coincides with a red dot</li><li>The area of DEF is 1/4 of the area of ABC</li></ul></td><td><ol><li>One common vertex to triangles ABC and DEF</li><li>Two common vertices to triangles ABC and DEF</li><li>Three common vertices to triangles ABC and DEF</li><li>Triangles ABC and DEF have no common sides</li><li>One side of DEF lies on a side of ABC</li><li>Two sides of DEF lies on a side of ABC</li><li>ABC has a right angle</li><li>DEF has a right angle</li><li>Common altitude to triangles ABC and DEF</li><li>In triangle DEF one side is a midsegment of triangle ABC</li><li>In triangle DEF two sides are midsegments of triangle ABC</li><li>In triangle DEF no sides are midsegments of triangle ABC</li></ol></td></tr></tbody></table>

## Characterizing variations

The key for the variation of the answers lies within the understanding that the result of the multiplication of the segment and its corresponding altitude of triangle DEF is a quarter of the multiplication result of the altitude and segment of triangle ABC. Below we describe ways in which the non-critical characteristics can offer students ways to either reflect on their submitted work, or plan for different examples that might fit or not fit the marked characteristics. We group the different characteristics to categories and explain the rationale of the choice:

*Reducing the complexity of the problem* (7,8,9): ABC has a right angle; DEF has a right angle; Common altitude to triangles ABC and DEF.

Students might find it easier to drag ABC to be a right triangle. In which the altitude coincides with a side of the triangle. The same could be said for the construction of triangle DEF. Another way to deal with only one side's ratio is to use a common altitude. While this might not be the way that students perceive their examples, reflecting this characteristic in their answers could help them communicate their strategy in a more mathematically robust fashion.

*Limitations of the example space* (1,2,3): One common vertex to triangles ABC and DEF; Two common vertices to triangles ABC and DEF; Three common vertices to triangles ABC and DEF.
Students might look into these characteristics in pursuit to meet all of the listed criteria. This would
eventually lead them to see that some characteristics contradict critical characteristics. In this case,
there cannot be a correct answer with three common vertices, while one and two are possible.

*Variation of non-critical characteristics* (4,5,6): Triangles ABC and DEF have no common sides;
One side of DEF lies on a side of ABC; Two sides of DEF lies on a side of ABC.

All of these non-critical characteristics can be demonstrated in correct answers.

*Contradicting conditions* (10,11,12): In triangle DEF one side is a midsegment of triangle ABC; In triangle DEF two sides are midsegments of triangle ABC; In triangle DEF no sides are midsegments of triangle ABC.

As the midsegments connect two red dots, 10 and 11 are impossible to achieve. This could also be
hinted by the fact that any examples meeting the critical characteristics also meet 12.

**Personal descriptive assessment**

In this section, we describe a thought experiment based on our experience with student's feedback
processes using STEP. We demonstrate how interactions with the personal characterization of each
example are manifested in the feedback process. Below we mark these actions as (a) discerning the
PES: Students analyze the examples, observing critical characteristics described by STEP, thus
distinguishing between examples and non-examples. To comply with the diversity's requirement,
they would (b) use the characterizations for categorizing examples according to non-critical
characteristics - turning the non-critical to "critical" within a special case of the solution. This, in
turn, leads to (c) formulating conjectures through generalizing distinctions between examples and
non-examples using descriptive statements or symbolic expressions. Sometimes they will generate
new examples to argue for truth through supporting examples, refutation by new examples and by
following relations, contradictions, and dependencies among non-critical characteristics. It could
provide the initial stepping stone for proof construction. Often STEP tasks ask for textual or symbolic
explanations that are manually assessed.

In Figure 2, we see that the student submitted three different examples; each example is an instance
describing construction of an inscribed triangle DEF that its three vertices are on the sides or vertices
of ABC and its area is 1/4 of ABC. Example 1 demonstrates a triangle DEF having altitude that is 1/4
of the altitude to a common side AC (which is also FD); Example 2 demonstrates an DEF that its
altitude to FD is 1/2 of the altitude to AC and FD is 1/2 of AC (AC and ED are overlapping); In Example
3 triangle DEF complements the area of the triangle constructed in Example 2 to 1/2 of ABC, probably
imagining folding ABC upon the median BD to get triangle ABD which is half area of ABC and
recognizing DF to be a median, thus DEF is 1/2 of 1/2 of ABC.

Following the submission of the answer and the examples, the student starts analyzing the information
presented online in the personal report. If the student answered "No" the report would present the
student's explanation. Otherwise, the personal report consists of the list of characteristics that the
algorithm assess and highlights automatically per submission. Firstly, it identified critical
characteristics (action (a)): The report identifies example 1 as fulfilling both requirements while
Examples 2 & 3 are identified by only one requirement. While both - example 1 and 2 - could have
been a result of a similar strategy for producing the required 1/4 area, DEF in example 2 is located on
Students might look into these characteristics in pursuit to meet all of the listed criteria. This would
eventually lead them to see that some characteristics contradict critical characteristics. In this case,
there cannot be a correct answer with three common vertices, while one and two are possible.

*Variation of non-critical characteristics (4,5,6):* Triangles ABC and DEF have no common sides;
One side of DEF lies on a side of ABC; Two sides of DEF lies on a side of ABC.

All of these non-critical characteristics can be demonstrated in correct answers.

*Contradicting conditions (10,11,12):* In triangle DEF one side is a midsegment of triangle ABC; In triangle DEF two sides are midsegments of triangle ABC; In triangle DEF no sides are midsegments of triangle ABC.

As the midsegments connect two red dots, 10 and 11 are impossible to achieve. This could also be
hinted by the fact that any examples meeting the critical characteristics also meet 12.

**Personal descriptive assessment**

In this section, we describe a thought experiment based on our experience with student's feedback
processes using STEP. We demonstrate how interactions with the personal characterization of each
example are manifested in the feedback process. Below we mark these actions as (a) discerning the
PES: Students analyze the examples, observing critical characteristics described by STEP, thus
distinguishing between examples and non-examples. To comply with the diversity's requirement,
they would (b) use the characterizations for categorizing examples according to non-critical
characteristics - turning the non-critical to "critical" within a special case of the solution. This, in
turn, leads to (c) formulating conjectures through generalizing distinctions between examples and
non-examples using descriptive statements or symbolic expressions. Sometimes they will generate
new examples to argue for truth through supporting examples, refutation by new examples and by
following relations, contradictions, and dependencies among non-critical characteristics. It could
provide the initial stepping stone for proof construction. Often STEP tasks ask for textual or symbolic
explanations that are manually assessed.

In Figure 2, we see that the student submitted three different examples; each example is an instance
describing construction of an inscribed triangle DEF that its three vertices are on the sides or vertices
of ABC and its area is 1/4 of ABC. Example 1 demonstrates a triangle DEF having altitude that is 1/4
of the altitude to a common side AC (which is also FD); Example 2 demonstrates an DEF that its
altitude to FD is 1/2 of the altitude to AC and FD is 1/2 of AC (AC and ED are overlapping); In Example
3 triangle DEF complements the area of the triangle constructed in Example 2 to 1/2 of ABC, probably
imagining folding ABC upon the median BD to get triangle ABD which is half area of ABC and
recognizing DF to be a median, thus DEF is 1/2 of 1/2 of ABC.

Following the submission of the answer and the examples, the student starts analyzing the information
presented online in the personal report. If the student answered "No" the report would present the
student's explanation. Otherwise, the personal report consists of the list of characteristics that the
algorithm assess and highlights automatically per submission. Firstly, it identified critical
characteristics (action (a)): The report identifies example 1 as fulfilling both requirements while
Examples 2 & 3 are identified by only one requirement. While both - example 1 and 2 - could have
been a result of a similar strategy for producing the required 1/4 area, DEF in example 2 is located on
two red dots. Example 3 as well has two red dots as vertices. At this point the student considers more
general inscribed triangle than the limiting requirement of the given task.

<figure><img src="image_189.png" /><figcaption>Figure 2: An online personal post-task report, listing and highlighting characteristics for three submitted examples (From left to right)</figcaption></figure>

The inquiry continues by action (b): Comparatively reflecting upon the non-critical characteristics of an example and a non-example.

"I notice that characteristics 3,4,5 identify the number of common vertices. In Example1 ABC and DEF share two vertices and in Example2 they share a single vertex. Mmm! so it must be that the other two vertices are located on the sides of ABC. And, Example3 also demonstrates ABC and DEF that share one vertex."

The comparative analysis of the first three noncritical characteristics then led to a question:

"Should I always construct DEF to share two vertices with ABC?"

Seeking further information that can support an answer to this question the student continues to scan
to non-critical characteristics list (action (c)). The next characteristic on the list (6) relates to common
two red dots. Example 3 as well has two red dots as vertices. At this point the student considers more
general inscribed triangle than the limiting requirement of the given task.

<figure><img src="image_196.png" /><figcaption>Figure 2: An online personal post-task report, listing and highlighting characteristics for three submitted examples (From left to right)</figcaption></figure>

The inquiry continues by action (b): Comparatively reflecting upon the non-critical characteristics of an example and a non-example.

"I notice that characteristics 3,4,5 identify the number of common vertices. In Example1 ABC and DEF share two vertices and in Example2 they share a single vertex. Mmm! so it must be that the other two vertices are located on the sides of ABC. And, Example3 also demonstrates ABC and DEF that share one vertex."

The comparative analysis of the first three noncritical characteristics then led to a question:

"Should I always construct DEF to share two vertices with ABC?"

Seeking further information that can support an answer to this question the student continues to scan
to non-critical characteristics list (action (c)). The next characteristic on the list (6) relates to common
sides. Both non-examples are marked as not having a common side. Are the two aspects of sharing sides and sharing one common vertex provide consistent feedback?

A positive answer helps to conclude that: *Two triangles share a side if and only if they share two vertices*. But this quite trivial conclusion does not provide further explanation and the student returns to comparatively analyze the remaining list of the non-critical characteristics of Example1 and Example2:

> "...there must be another non-critical characteristic that distinguishes between an example and a non-example of triangles that share a side and ¼ area."

Idyllically, one can come closer to an answer by returning to the interactive diagram and generating new constructions such as in Figure 3.

<figure><img src="image_190.png" /><figcaption>Figure 3: Possible correct constructions for the inscribed triangle task</figcaption></figure>

> "Ah! Blue dots are not necessarily vertices! the vertices of DEF might be any point on the ABC sides which isn't red."

In the case of this task, developing symbolic arguments most probably triggered by external teacher prompts. Generalizing the answers, for $\frac{DF \cdot h_{DF}}{2} = \frac{1}{4} \cdot \frac{AC \cdot h_{AC}}{2}$ (DF and AC can be replaced by any segment of the triangle, or in terms of the original triangle: $DF = aAC, h_{DF} = bh_{AC} \Rightarrow \frac{aAC \cdot bh_{AC}}{2} = \frac{1}{4}$). We just need to keep this relation: $a \cdot b = \frac{1}{4}$.

While this generalization is a goal within itself in terms of student's understanding of ratios between areas in geometry, there are many meaningful interactions with different mathematical characteristics of this specific setting that could promote inquiry that can be supported by an interactive feedback process that suggests different perspectives, as described in the list of non-critical characteristics.

## TAKING DESCRIPTIVE ASSESSMENT IN GEOMETRY FURTHER

As this illustration suggests, the strategy of characterizing student submissions with mathematical descriptions points to less common directions in the field of learning analytics (LA): Mostly used LA uses measures like time on task, correctness, misconceptions, and more such information that could suggest important aspects of student's learning in general. We consider automatic characterization of student examples in words as a way for students to interact with points of view that may be different from their point of view while engaging in inquiry. We argue that through reports constructed for individual learners, such communication could be useful, enabling students to further their learning by reflecting on their understanding.
sides. Both non-examples are marked as not having a common side. Are the two aspects of sharing sides and sharing one common vertex provide consistent feedback?

A positive answer helps to conclude that: *Two triangles share a side if and only if they share two vertices*. But this quite trivial conclusion does not provide further explanation and the student returns to comparatively analyze the remaining list of the non-critical characteristics of Example 1 and Example 2:

> "...there must be another non-critical characteristic that distinguishes between an example and a non-example of triangles that share a side and ¼ area."

Idyllically, one can come closer to an answer by returning to the interactive diagram and generating new constructions such as in Figure 3.

<figure><img src="image_197.png" /><figcaption>Figure 3: Possible correct constructions for the inscribed triangle task</figcaption></figure>

> "Ah! Blue dots are not necessarily vertices! the vertices of DEF might be any point on the ABC sides which isn't red."

In the case of this task, developing symbolic arguments most probably triggered by external teacher prompts. Generalizing the answers, for $\frac{DF \cdot h_{DF}}{2} = \frac{1}{4} \cdot \frac{AC \cdot h_{AC}}{2}$ (DF and AC can be replaced by any segment of the triangle, or in terms of the original triangle: $DF = aAC, h_{DF} = bh_{AC} \Rightarrow \frac{aAC \cdot bh_{AC}}{2} = \frac{1}{4}$). We just need to keep this relation: $a \cdot b = \frac{1}{4}$.

While this generalization is a goal within itself in terms of student's understanding of ratios between areas in geometry, there are many meaningful interactions with different mathematical characteristics of this specific setting that could promote inquiry that can be supported by an interactive feedback process that suggests different perspectives, as described in the list of non-critical characteristics.

## TAKING DESCRIPTIVE ASSESSMENT IN GEOMETRY FURTHER

As this illustration suggests, the strategy of characterizing student submissions with mathematical descriptions points to less common directions in the field of learning analytics (LA): Mostly used LA uses measures like time on task, correctness, misconceptions, and more such information that could suggest important aspects of student's learning in general. We consider automatic characterization of student examples in words as a way for students to interact with points of view that may be different from their point of view while engaging in inquiry. We argue that through reports constructed for individual learners, such communication could be useful, enabling students to further their learning by reflecting on their understanding.
Furthermore, the digital nature of these tasks, and the rich descriptions in a digitally analyzable format, enables us to envision mechanisms that would benefit and use this information as a “training ground” for artificial intelligence agents that could then automatically suggest relevant characterizations of mathematical tasks, as well as sequencing of tasks to create a relevant learning trajectory. In addition, we end with a thought about the possible interactions with large language models, to be trained with the relevant mathematical content, as part of personal learning with a “speaking mirror”.

## References

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

Buchbinder, O., & Zaslavsky, O. (2009). A framework for understanding the status of examples in establishing the validity of mathematical statements. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), *Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 225–232). PME.

Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P., Nesher, & J., Kilpatrick. (Eds.), *Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education*, (pp. 70–95). Cambridge University Press.

Hohenwarter, M., Jarvis, D., & Lavicza, Z. (2009). Linking geometry, algebra, and mathematics teachers: GeoGebra software and the establishment of the International GeoGebra Institute. *International Journal for Technology in Mathematics Education*, 16(2), 83–87.

Luz, Y., & Yerushalmy, M. (2023). Computer-supported assessment of geometric exploration using variation theory. *Journal for Research in Mathematics Education, 54*(2), 141–174.

Marton, F., & Tsui, A. B. M. (2004). *Classroom discourse and the space of learning*. Routledge.
https://doi.org/10.4324/9781410609762

Mason, J., & Watson, A. (2008). Mathematics as a constructive activity: Exploiting dimensions of possible variation. In M. P. Carlson & C. Rasmussen (Eds.), *Making the connection: Research and teaching in undergraduate mathematics education* (pp. 191–204). Mathematical Association of America.
https://doi.org/10.5948/UPO9780883859759.016

Olsher, S., Yerushalmy, M., & Chazan, D. (2016). How might the use of technology in formative assessment support changes in mathematics teaching? *For the Learning of Mathematics*, 36(3), 11–18.

Popper, P., & Yerushalmy, M. (2021). Online example-based assessment as a resource for teaching about quadrilaterals. *Educational Studies in Mathematics*, 110, 83–100. https://doi.org/10.1007/s10649-021-10109-1

Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples.
Routledge.

Watson, A., & Shipman, S. (2008). Using learner generated examples to introduce new concepts. *Educational Studies in Mathematics*, 69, 97–109. https://doi.org/10.1007/s10649-008-9142-4

Yerushalmy, M., Olsher, S., Harel, R. & Chazan, D. (2022). Supporting inquiry learning: an intellectual mirror that describes what it “sees”. *Digital Experiences in Mathematics Education*.
https://doi.org/10.1007/s40751-022-00120-3

Zaslavsky, O., & Zodik, I. (2014). Example-generation as indicator and catalyst of mathematical and pedagogical understandings. In Y. Li, A. E. Silver & S. Li (Eds.), *Transforming mathematics instruction: Multiple approaches and practices* (pp. 525–546). Springer.
Furthermore, the digital nature of these tasks, and the rich descriptions in a digitally analyzable format, enables us to envision mechanisms that would benefit and use this information as a “training ground” for artificial intelligence agents that could then automatically suggest relevant characterizations of mathematical tasks, as well as sequencing of tasks to create a relevant learning trajectory. In addition, we end with a thought about the possible interactions with large language models, to be trained with the relevant mathematical content, as part of personal learning with a “speaking mirror”.

## References

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

Buchbinder, O., & Zaslavsky, O. (2009). A framework for understand the status of examples in establishing the validity of mathematical statements. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), *Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 225–232). PME.

Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P., Nesher, & J., Kilpatrick. (Eds.), *Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education*, (pp. 70–95). Cambridge University Press.

Hohenwarter, M., Jarvis, D., & Lavicza, Z. (2009). Linking geometry, algebra, and mathematics teachers: GeoGebra software and the establishment of the International GeoGebra Institute. *International Journal for Technology in Mathematics Education*, 16(2), 83–87.

Luz, Y., & Yerushalmy, M. (2023). Computer-supported assessment of geometric exploration using variation theory. *Journal for Research in Mathematics Education, 54*(2), 141–174.

Marton, F., & Tsui, A. B. M. (2004). *Classroom discourse and the space of learning*. Routledge.
https://doi.org/10.4324/9781410609762

Mason, J., & Watson, A. (2008). Mathematics as a constructive activity: Exploiting dimensions of possible variation. In M. P. Carlson & C. Rasmussen (Eds.), *Making the connection: Research and teaching in undergraduate mathematics education* (pp. 191–204). Mathematical Association of America.
https://doi.org/10.5948/UPO9780883859759.016

Olsher, S., Yerushalmy, M., & Chazan, D. (2016). How might the use of technology in formative assessment support changes in mathematics teaching? *For the Learning of Mathematics*, 36(3), 11–18.

Popper, P., & Yerushalmy, M. (2021). Online example-based assessment as a resource for teaching about quadrilaterals. *Educational Studies in Mathematics, 110*, 83–100. https://doi.org/10.1007/s10649-021-10109-1

Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples.
Routledge.

Watson, A., & Shipman, S. (2008). Using learner generated examples to introduce new concepts. *Educational Studies in Mathematics*, 69, 97–109. https://doi.org/10.1007/s10649-008-9142-4

Yerushalmy, M., Olsher, S., Harel, R. & Chazan, D. (2022). Supporting inquiry learning: an intellectual mirror that describes what it “sees”. *Digital Experiences in Mathematics Education*.
https://doi.org/10.1007/s40751-022-00120-3

Zaslavsky, O., & Zodik, I. (2014). Example-generation as indicator and catalyst of mathematical and pedagogical understandings. In Y. Li, A. E. Silver & S. Li (Eds.), *Transforming mathematics instruction: Multiple approaches and practices* (pp. 525–546). Springer.
TOPIC D

Multidisciplinary perspectives
TOPIC D

Multidisciplinary perspectives
<figure><img src="image_191.png" /></figure>
<figure><img src="image_198.png" /></figure>
# EXPLORING CREATIVE PROBLEM-SOLVING WITH EYE-TRACKING METHODOLOGY

Adi Eraky, Bat-Sheva Hadad, Hagit Hel-Or, Elie Abboud, Roza Leikin

University of Haifa

We present an empirical investigation of creativity-directed problem-solving linked to Visual Geometric Multiple Solution Tasks (VG-MST). The problem-solving strategies used by the subjects were analysed for fluency, flexibility, and originality of the problem-solving process, using a thinking-aloud procedure. The corresponding eye-tracking patterns reordered along the solution process were examined using scan paths, heat maps and entropy. Differences between the solution strategies employed by two university students who are both expert in mathematics suggest that (a) producing similar problem-solving strategies can be linked to different cognitive processes, and (b) mathematical insight is linked to more efficient eye-movement patterns.

## BACKGROUND

Studies have indicated that students with creative thinking abilities are more adept at tackling the intricate problems of the modern era (Pellegrino & Hilton, 2012). *Creativity in mathematics* promotes the development of mathematical proficiency and, in turn, depends on the level of mathematical proficiency (Elgrably & Leikin, 202; Schoevers et al., 2022; Leikin & Lev, 2013; Levav-Waynberg & Leikin, 2012). The present study is aimed to shed light on creative problem solving in geometry problems, with special attention to creative mathematical processing, using both behavioral (problem-solving ways used and their accuracy) and neurophysiological (eye tracking) measures.

We explore creativity using Multiple Solution Tasks (MSTs), which explicitly require solving a problem using multiple strategies (Leikin, 2009, 2013). Over the past decade, MSTs have been found among the primary tools for assessing mathematical creativity (Leikin & Sriraman, 2022), with creativity assessed based on the analysis of the solver's fluency, flexibility and originality (Leikin, 2009). In addition, due to the centrality of insight to creative endeavors (Haavold & Sriraman, 2022; Leikin et al, 2016; Leikin & Guberman, 2023), we use insight-allowing MSTs for which one solution is insight-based.

This paper presents a study that focuses on solving Visual Geometric Multiple Solution Tasks (VG-MSTs), with a visual insightful solution. These tasks require implementation of visual skills, deductive, abstract and logical reasoning (Hanna, 1996; Herbst & Brach, 2006; Clements & Battista, 1992). VG-MSTs provide a rich basis for inquiry, implementation of proving skills and generalization. While similarly to most geometry problems, solutions of VG-MSTs are not algorithmic and, therefore, involve heuristic reasoning (Polya, 1981; Schoenfeld, 1985), insightful solutions additionally require seeing the underlying structure of the problem.

Over the past decade, neuroscience has been applied to enhance understanding of mathematical processing (e.g., Grabner & De Smedt, 2016). Among other neuroscience methodologies, the eye-tracking methodology has also entered the terrain of mathematics education research and is shown to
EXPLORING CREATIVE PROBLEM-SOLVING
WITH EYE-TRACKING METHODOLOGY

Adi Eraky, Bat-Sheva Hadad, Hagit Hel-Or, Elie Abboud, Roza Leikin

University of Haifa

We present an empirical investigation of creativity-directed problem-solving linked to Visual Geometric Multiple Solution Tasks (VG-MST). The problem-solving strategies used by the subjects were analysed for fluency, flexibility, and originality of the problem-solving process, using a thinking-aloud procedure. The corresponding eye-tracking patterns reordered along the solution process were examined using scan paths, heat maps and entropy. Differences between the solution strategies employed by two university students who are both expert in mathematics suggest that (a) producing similar problem-solving strategies can be linked to different cognitive processes, and (b) mathematical insight is linked to more efficient eye-movement patterns.

BACKGROUND

Studies have indicated that students with creative thinking abilities are more adept at tackling the intricate problems of the modern era (Pellegrino & Hilton, 2012). *Creativity in mathematics* promotes the development of mathematical proficiency and, in turn, depends on the level of mathematical proficiency (Elgrably & Leikin, 202; Schoevers et al., 2022; Leikin & Lev, 2013; Levav-Waynberg & Leikin, 2012). The present study is aimed to shed light on creative problem solving in geometry problems, with special attention to creative mathematical processing, using both behavioral (problem-solving ways used and their accuracy) and neurophysiological (eye tracking) measures.

We explore creativity using Multiple Solution Tasks (MSTs), which explicitly require solving a problem using multiple strategies (Leikin, 2009, 2013). Over the past decade, MSTs have been found among the primary tools for assessing mathematical creativity (Leikin & Sriraman, 2022), with creativity assessed based on the analysis of the solver's fluency, flexibility and originality (Leikin, 2009). In addition, due to the centrality of insight to creative endeavors (Haavold & Sriraman, 2022; Leikin et al, 2016; Leikin & Guberman, 2023), we use insight-allowing MSTs for which one solution is insight-based.

This paper presents a study that focuses on solving Visual Geometric Multiple Solution Tasks (VG-MSTs), with a visual insightful solution. These tasks require implementation of visual skills, deductive, abstract and logical reasoning (Hanna, 1996; Herbst & Brach, 2006; Clements & Battista, 1992). VG-MSTs provide a rich basis for inquiry, implementation of proving skills and generalization. While similarly to most geometry problems, solutions of VG-MSTs are not algorithmic and, therefore, involve heuristic reasoning (Polya, 1981; Schoenfeld, 1985), insightful solutions additionally require seeing the underlying structure of the problem.

Over the past decade, neuroscience has been applied to enhance understanding of mathematical processing (e.g., Grabner & De Smedt, 2016). Among other neuroscience methodologies, the eye-tracking methodology has also entered the terrain of mathematics education research and is shown to
be beneficial. It was demonstrated that eye fixations and eye movements correspond to mental operations (Grant & Spivey, 2003). Learning about mental processing during problem-solving using eye trackers is increasingly key to educational research (e.g., Scheiter & van Gog, 2009), particularly in mathematics education (e.g., Andrá et al., 2015). For example, Dewolf, Van Dooren, Hermens, and Verschaffel (2013) used eye movements to validate students' strategies related to solving mathematical word problems and found that students scarcely looked at the representational illustrations and that the illustrations did not affect the realistic nature of their solutions.

Strohmaier (2020) found that eye tracking is useful for studying processes, revealing mental representations, and evaluating subconscious aspects of mathematical thinking (Strohmaier et al., 2020). For example, Norqvist (2019) utilized eye tracking to examine algorithmic and creative reasoning strategies in mathematics tasks and discovered that students who practiced creative tasks performed better than those who practiced algorithmic tasks in subsequent evaluations (Norqvist et al., 2019). While eye-tracking technology shows potential when it comes to understanding students' mathematical strategies, its application in graph interpretation is still limited. In this study we employed eye-tracking methodology, keeping in mind the challenges outlined above, with the aim of bridging the gap between behavioral data and learners' cognitive processes.

## THE RESEARCH GOAL

The study's main goal was to characterize creativity-directed problem-solving processing in order to deepen our understanding of the flexibility associated with the production of multiple solution strategies and the originality associated with mathematical insight. We evaluated the creativity of the problem-solving process using Leikin's model (2009, 2013). The research methodology integrated a thinking-aloud procedure with eye-tracking to obtain a deeper understanding of the behavioral and neuro-physiological patterns associated with solving MSTs.

## PARTICIPANTS AND DATA COLLECTION

As mentioned before, we used a VG-MST, depicted in Figure 1. The problem allowed algorithmic computational solution strategies and insightful solution strategies without computations (see Results section).

<figure><img src="image_192.png" /><figcaption>Figure 1: The Visual Geometric Multiple Solution Task (VG-MST) used in the study</figcaption></figure>

We report on the problem-solving performance on the VG-MST of two BSc students studying Computer Sciences (S1, S2). These students both have a high level of mathematical competencies. The problem chosen for the study was easily approachable for these students; the requirement to find multiple solutions allowed us to examine the participants' creativity rather than their knowledge.

We employed individual interviews with a thinking-aloud procedure accompanied by Camtasia software recording and the Pupil Labs GmbH software. The subjects solved the task while wearing a pupil eye-tracker [Pupil Labs GmbH] and using a DELL 24" touch screen on which the tasks were displayed and on which the participants could write while tackling the tasks. Camtasia allowed
be beneficial. It was demonstrated that eye fixations and eye movements correspond to mental operations (Grant & Spivey, 2003). Learning about mental processing during problem-solving using eye trackers is increasingly key to educational research (e.g., Scheiter & van Gog, 2009), particularly in mathematics education (e.g., Andrá et al., 2015). For example, Dewolf, Van Dooren, Hermens, and Verschaffel (2013) used eye movements to validate students' strategies related to solving mathematical word problems and found that students scarcely looked at the representational illustrations and that the illustrations did not affect the realistic nature of their solutions.

Strohmaier (2020) found that eye tracking is useful for studying processes, revealing mental representations, and evaluating subconscious aspects of mathematical thinking (Strohmaier et al., 2020). For example, Norqvist (2019) utilized eye tracking to examine algorithmic and creative reasoning strategies in mathematics tasks and discovered that students who practiced creative tasks performed better than those who practiced algorithmic tasks in subsequent evaluations (Norqvist et al., 2019). While eye-tracking technology shows potential when it comes to understanding students' mathematical strategies, its application in graph interpretation is still limited. In this study we employed eye-tracking methodology, keeping in mind the challenges outlined above, with the aim of bridging the gap between behavioral data and learners' cognitive processes.

## THE RESEARCH GOAL

The study's main goal was to characterize creativity-directed problem-solving processing in order to deepen our understanding of the flexibility associated with the production of multiple solution strategies and the originality associated with mathematical insight. We evaluated the creativity of the problem-solving process using Leikin's model (2009, 2013). The research methodology integrated a thinking-aloud procedure with eye-tracking to obtain a deeper understanding of the behavioral and neuro-physiological patterns associated with solving MSTs.

## PARTICIPANTS AND DATA COLLECTION

As mentioned before, we used a VG-MST, depicted in Figure 1. The problem allowed algorithmic computational solution strategies and insightful solution strategies without computations (see Results section).

<figure><img src="image_199.png" /></figure>

Given two congruent squares with side 4.
One square subscribes 4 small tangent circles while the other subscribes one big circle.
Compare the area of the big circle and the area of 4 small circles?

<caption>Figure 1: The Visual Geometric Multiple Solution Task (VG-MST) used in the study</caption>

We report on the problem-solving performance on the VG-MST of two BSc students studying Computer Sciences (S1, S2). These students both have a high level of mathematical competencies. The problem chosen for the study was easily approachable for these students; the requirement to find multiple solutions allowed us to examine the participants' creativity rather than their knowledge.

We employed individual interviews with a thinking-aloud procedure accompanied by Camtasia software recording and the Pupil Labs GmbH software. The subjects solved the task while wearing a pupil eye-tracker [Pupil Labs GmbH] and using a DELL 24" touch screen on which the tasks were displayed and on which the participants could write while tackling the tasks. Camtasia allowed
synchronized recording of the thinking-aloud problem-solving process with solution writing on the computer screen, the shifts between different solution strategies, and eye movements.

The study was conducted in a small and quiet room, while the participant was sitting approximately 35 cm away from the computer screen. The eye tracking data was translated into eye movement characteristics, including the number, frequency, and length of the saccades, their directions, and the location of fixations.

## DATA ANALYSES

The solution strategies employed by the participants were categorized into distinct categories, and the evaluation of the creativity linked to the problem-solving strategies was aligned with these categories (Leikin, 2009). Subsequently, the student's performance was evaluated based on creativity components - fluency, flexibility, and originality - using the framework and scoring system developed by Leikin (2009). This model for evaluating creativity has been consistently utilized in prior research investigations (Leikin, 2009, 2013; Levav-Waynberg, 2014).

The eye tracker reordered eye movements and screen coordinates at approximately 200Hz. This generated a continuous stream of (x, y) coordinates and screen images every 5 milliseconds. The correspondence between the solution strategies and eye movement patterns was classified based on this analysis using techniques such as heatmaps, scan paths, and entropy. The strategies were then classified into distinct qualitative categories, each characterized by distinct movement patterns, including scan paths and the duration of fixations. The final phase encompassed a validation and refinement of these identified categories.

Our calibration procedure for the Pupil Eye tracker embraced both internal and external methods, as outlined by Schweizer et al. (2021). The external calibration technique utilized barcodes at the screen periphery to establish the screen as a two-dimensional surface within a three-dimensional context (Elmadjian et al., 2018). This calibration allowed us to generate a gaze direction graph during task completion. The screen was divided into a 9x9 grid of bins, wherein fixation points were quantified for each bin and depicted as varying-sized green circles. Saccades, or rapid eye movements between fixations, were visually represented by connecting lines, as illustrated in Figure 3. In our analytical framework, we harnessed an array of quantitative metrics derived from eye-tracking data. Beyond basic statistical parameters like mean and variance, we gauged the dispersion of fixation points across the screen using Shannon's Entropy (Shannon, 1948). To achieve this, we assigned P(i) to signify the proportion of fixations occurring within bin *i* relative to the entire set of fixations. High entropy indicated significant dispersion, whereas low entropy corresponded to clustered fixations. The entropy calculation followed this formula—$\sum_i p_i \log(p_i)$. Furthermore, our analysis encompassed the assessment of saccades based on their length, variability, and angles between successive saccades. Larger angles denoted smoother transitions between fixations, while smaller angles indicated more erratic movements between different locations.

The problem-solving process was meticulously recorded, including participants' handwritten notations and their concurrent verbalized thoughts through the think-aloud procedure. Each instance of a solution strategy was temporally bounded, stretching from the initiation of the solution process to its culmination.
synchronized recording of the thinking-aloud problem-solving process with solution writing on the computer screen, the shifts between different solution strategies, and eye movements.

The study was conducted in a small and quiet room, while the participant was sitting approximately 35 cm away from the computer screen. The eye tracking data was translated into eye movement characteristics, including the number, frequency, and length of the saccades, their directions, and the location of fixations.

## DATA ANALYSES

The solution strategies employed by the participants were categorized into distinct categories, and the evaluation of the creativity linked to the problem-solving strategies was aligned with these categories (Leikin, 2009). Subsequently, the student's performance was evaluated based on creativity components - fluency, flexibility, and originality - using the framework and scoring system developed by Leikin (2009). This model for evaluating creativity has been consistently utilized in prior research investigations (Leikin, 2009, 2013; Levav-Waynberg, 2014).

The eye tracker reordered eye movements and screen coordinates at approximately 200Hz. This generated a continuous stream of (x, y) coordinates and screen images every 5 milliseconds. The correspondence between the solution strategies and eye movement patterns was classified based on this analysis using techniques such as heatmaps, scan paths, and entropy. The strategies were then classified into distinct qualitative categories, each characterized by distinct movement patterns, including scan paths and the duration of fixations. The final phase encompassed a validation and refinement of these identified categories.

Our calibration procedure for the Pupil Eye tracker embraced both internal and external methods, as outlined by Schweizer et al. (2021). The external calibration technique utilized barcodes at the screen periphery to establish the screen as a two-dimensional surface within a three-dimensional context (Elmadjian et al., 2018). This calibration allowed us to generate a gaze direction graph during task completion. The screen was divided into a 9x9 grid of bins, wherein fixation points were quantified for each bin and depicted as varying-sized green circles. Saccades, or rapid eye movements between fixations, were visually represented by connecting lines, as illustrated in Figure 3. In our analytical framework, we harnessed an array of quantitative metrics derived from eye-tracking data. Beyond basic statistical parameters like mean and variance, we gauged the dispersion of fixation points across the screen using Shannon's Entropy (Shannon, 1948). To achieve this, we assigned P(i) to signify the proportion of fixations occurring within bin *i* relative to the entire set of fixations. High entropy indicated significant dispersion, whereas low entropy corresponded to clustered fixations. The entropy calculation followed this formula—$\sum_i p_i \log(p_i)$. Furthermore, our analysis encompassed the assessment of saccades based on their length, variability, and angles between successive saccades. Larger angles denoted smoother transitions between fixations, while smaller angles indicated more erratic movements between different locations.

The problem-solving process was meticulously recorded, including participants' handwritten notations and their concurrent verbalized thoughts through the think-aloud procedure. Each instance of a solution strategy was temporally bounded, stretching from the initiation of the solution process to its culmination.
# FINDINGS

## Problem-solving strategies used by the participants

<figure><img src="image_193.png" /><figcaption>Figure 2: The task and solution strategies produced by the participants.</figcaption></figure>

<figure><img src="image_194.png" /><figcaption>Figure 3: Appearance of the different strategies when solving the tasks</figcaption></figure>

Figures 2 and 3 depict the different solution strategies and their sequences and durations as employed by the two study participants. Four solution strategies were identified (see Figure 2). Figure 3 shows that S1 implemented four solution strategies, and S2 implemented three solutions. Both S1 and S2 started by solving the task with an insight-based solution. They each solved the problem using calculations. The two additional solutions performed by S1 and S2 were similar (denoted as A1 and A2 in Figure 3). S2 provided an additional solution using the area of a circle formula.

## Eye movements related to the creativity of the problem-solving performance

In this section, we present an analysis of the eye movements linked to the creativity-emerging problem-solving process for the task. We focus on the differences in the thinking processes of the two subjects when performing similar solutions, and on the different ways of solving performed by the same subjects. We examine the differences displayed in the scan paths of specific solutions for
# FINDINGS

## Problem-solving strategies used by the participants

<table><caption>Figure 2 presents different solution strategies for the VG-MST used in this study.</caption><thead><tr><th></th><th>Solution A1: Calculate the area of a large circle and one small circle</th><th>Solution A2: Calculate the area of a large circle and 4 small circles</th><th>Solution B: Insight-based – similarity of the small and big squares</th><th>Solution C: Calculate: Check and multiply by 4</th></tr></thead><tbody><tr><td>Figure 2: The task and solution strategies produced by the participants.</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<figure><img src="image_200.png" /></figure>

<figure><img src="image_201.png" /></figure>

<figure><img src="image_202.png" /></figure>

<figure><img src="image_203.png" /><figcaption></figcaption></figure>

<figure><img src="image_204.png" /><figcaption>Figure 3: Appearance of the different strategies when solving the tasks</figcaption></figure>

Figures 2 and 3 depict the different solution strategies and their sequences and durations as employed by the two study participants. Four solution strategies were identified (see Figure 2). Figure 3 shows that S1 implemented four solution strategies, and S2 implemented three solutions. Both S1 and S2 started by solving the task with an insight-based solution. They each solved the problem using calculations. The two additional solutions performed by S1 and S2 were similar (denoted as A1 and A2 in Figure 3). S2 provided an additional solution using the area of a circle formula.

## Eye movements related to the creativity of the problem-solving performance

In this section, we present an analysis of the eye movements linked to the creativity-emerging problem-solving process for the task. We focus on the differences in the thinking processes of the two subjects when performing similar solutions, and on the different ways of solving performed by the same subjects. We examine the differences displayed in the scan paths of specific solutions for
the task performed by S1 and S2 (Figure 4).

<figure><img src="image_195.png" /><figcaption>Figure 4: Eye tracking analysis of specific solutions for the task by S1 and S2</figcaption></figure>

Figure 4 demonstrates that different solution strategies are (naturally) associated with different thinking processes reflected in different loci of attention and different scan paths. This difference can be seen in the difference between the scan paths corresponding to strategies A1, A2, and B for each one of the participants. Moreover, we find that similar solution strategies in different participants are associated with different thinking processes and foci of attentions. For example, solution B, performed both by S1 and S2 as the first solution strategy, was a similar insight-based strategy:

* S1-B: The same diameter here, i.e., the ratio between them is equal, i.e., the radius of 2 circles equals the radius of one circle.

* S2-B: It is possible to refer to the ratios between the radii. They are equal.

The corresponding scan paths demonstrate that S1 and S2 focused on different locations on the figure as well as following different connections between the figure and the text. While S1's utterance related to this solution is longer, the scan path demonstrates that he was more efficient, with less time spent on the solution, fewer fixations and a smaller number of saccades. At the same time, while S2 was less verbal he spent more time producing the solution and his scan path differed in location, number of fixations and saccades. S1 primarily focused on the square's sides, whereas S2 focused on radiuses. These differences may indicate different thinking processes that led to the use of similar solution strategies.
the task performed by S1 and S2 (Figure 4).

<figure><img src="image_205.png" /><figcaption>Figure 4: Eye tracking analysis of specific solutions for the task by S1 and S2</figcaption></figure>

Figure 4 demonstrates that different solution strategies are (naturally) associated with different thinking processes reflected in different loci of attention and different scan paths. This difference can be seen in the difference between the scan paths corresponding to strategies A1, A2, and B for each one of the participants. Moreover, we find that similar solution strategies in different participants are associated with different thinking processes and foci of attentions. For example, solution B, performed both by S1 and S2 as the first solution strategy, was a similar insight-based strategy:

* S1-B: The same diameter here, i.e., the ratio between them is equal, i.e., the radius of 2 circles equals the radius of one circle.

* S2-B: It is possible to refer to the ratios between the radii. They are equal.

The corresponding scan paths demonstrate that S1 and S2 focused on different locations on the figure as well as following different connections between the figure and the text. While S1's utterance related to this solution is longer, the scan path demonstrates that he was more efficient, with less time spent on the solution, fewer fixations and a smaller number of saccades. At the same time, while S2 was less verbal he spent more time producing the solution and his scan path differed in location, number of fixations and saccades. S1 primarily focused on the square's sides, whereas S2 focused on radiuses. These differences may indicate different thinking processes that led to the use of similar solution strategies.
Similarly, the scan paths of S1 and S2 corresponding to solution strategy A1 differ. The analysis
shows that they focused on different parts of the figure, but these different approaches led to similar
calculation strategies. In solution A1, S1 focused on the circles and writing below the illustration, and
S2 focused mainly on the top side of the rectangle and returned to reading the text.

S1-A1: [...] The diameter is 4, the radius is 2 cm, and the area of the circle is $4\pi$ [...] It is half a square, 2 cm minus 1, is a radius, $\pi$ in a square, one square is a $\pi$ and a big circle $4\pi$.

S1-A2: Suppose it is four and divide by that to calculate the area of a tiny circle and then double four and one big circle. Moreover, if there are four here, then $4\pi$.

S2-A1: Check the area of the large circle and each area of the small circles.

S2-A2: It is also possible to take the area of a tiny circle multiplied by four circles, and it is the same area, check and multiply by 4.

As mentioned above, comparison between the scan paths of different solutions - for example, A1 and A2 - demonstrates the differences in corresponding eye movements of each participant in the same strategy: when performing solution A2, S1 focuses on the radius and inner areas, compared to S2, who concentrates on the top left side of the square, and thinks of division into four small circles. As seen in the scan path, eye movements differed for S1 and S2, each of whom focused on specific regions of interest.

This analysis assumes that the scan path identified areas of interest during mental processing, and identified both *subject-dependency* and *strategy-dependency* aspects of problem-solving processes. In this way, eye-tracking analysis deepens our understanding of the different thinking processes involved in different subjects' performance of similar solutions.

An additional eye-tracking variable that allows analysis of the thinking process associated with problem solving is *entropy*. Entropy refers to the degree of dispersion of fixation points on a display. High entropy indicates a large dispersion, while low entropy indicates a tight clustering of fixations. Figure 5 presents the analyses of each solution's entropy and total fixations.

<figure><img src="image_196.png" /><figcaption>Figure 5: Eye tracking analysis of specific solutions for the task by S1 and S2</figcaption></figure>

From the entropy analysis, which demonstrates the dispersion of the fixations, we can understand the
differences between S1 and S2 in solving the tasks. In solution B, S1 was lower entropy than S2,
which means the fixations are tightly clustered for S1 and more dispersed for S2. Another exciting
example was in solution A1 (similar for both subjects), which indicated that S1 and S2 had high
entropy and large dispersion. The last example for the same solution is demonstrated in solution A2;
S1 had high entropy, his dispersion was large, and S2 had very low entropy, which means the
dispersion of the fixations was tightly clustered, implying the same strategy may still involve a
Similarly, the scan paths of S1 and S2 corresponding to solution strategy A1 differ. The analysis
shows that they focused on different parts of the figure, but these different approaches led to similar
calculation strategies. In solution A1, S1 focused on the circles and writing below the illustration, and
S2 focused mainly on the top side of the rectangle and returned to reading the text.

S1-A1: [...] The diameter is 4, the radius is 2 cm, and the area of the circle is $4\pi$ [...] It is half a square, 2 cm minus 1, is a radius, $\pi$ in a square, one square is a $\pi$ and a big circle $4\pi$.

S1-A2: Suppose it is four and divide by that to calculate the area of a tiny circle and then double four and one big circle. Moreover, if there are four here, then $4\pi$.

S2-A1: Check the area of the large circle and each area of the small circles.

S2-A2: It is also possible to take the area of a tiny circle multiplied by four circles, and it is the same area, check and multiply by 4.

As mentioned above, comparison between the scan paths of different solutions - for example, A1 and A2 - demonstrates the differences in corresponding eye movements of each participant in the same strategy: when performing solution A2, S1 focuses on the radius and inner areas, compared to S2, who concentrates on the top left side of the square, and thinks of division into four small circles. As seen in the scan path, eye movements differed for S1 and S2, each of whom focused on specific regions of interest.

This analysis assumes that the scan path identified areas of interest during mental processing, and identified both *subject-dependency* and *strategy-dependency* aspects of problem-solving processes. In this way, eye-tracking analysis deepens our understanding of the different thinking processes involved in different subjects' performance of similar solutions.

An additional eye-tracking variable that allows analysis of the thinking process associated with problem solving is *entropy*. Entropy refers to the degree of dispersion of fixation points on a display. High entropy indicates a large dispersion, while low entropy indicates a tight clustering of fixations. Figure 5 presents the analyses of each solution's entropy and total fixations.

<figure><img src="image_206.png" /><figcaption>Figure 5: Eye tracking analysis of specific solutions for the task by S1 and S2</figcaption></figure>

From the entropy analysis, which demonstrates the dispersion of the fixations, we can understand the
differences between S1 and S2 in solving the tasks. In solution B, S1 was lower entropy than S2,
which means the fixations are tightly clustered for S1 and more dispersed for S2. Another exciting
example was in solution A1 (similar for both subjects), which indicated that S1 and S2 had high
entropy and large dispersion. The last example for the same solution is demonstrated in solution A2;
S1 had high entropy, his dispersion was large, and S2 had very low entropy, which means the
dispersion of the fixations was tightly clustered, implying the same strategy may still involve a
different thinking process. The total fixations showed that S2 performed more eye movement in each solution than S1. For example, in solution A1, S2 solved more quickly than S1, but used more eye movement in this strategy.

Based on the above, we argue that eye-tracking analysis is adequate for understanding differences in mathematical thinking that lead to similar solution strategies and outcomes. Such understanding could not be attained from the interview transcript analysis alone.

## DISCUSSION

This paper presents an empirical investigation of creativity-emerging problem-solving by employing a VG-MST that explicitly requires participants to solve a mathematical problem in multiple ways. Our goal was to characterize creativity-emerging problem-solving processing by participants with the same level of creativity through integrating behavioral and neuro-physiological perspectives that focus on problem-solving strategies and corresponding eye-tracking patterns. We call the process of solving VG-MST creativity-directed since we do not require participants to perform creative solutions but fosters creative thinking.

We integrated eye-tracking methodologies with individual clinical interviews to characterize patterns of processing associated with flexible mathematical thinking and insight-based processing in solvers with different strategies. Due to the small sample size (the major limitation of our study), we performed qualitative data analysis. The data analysis included behavioral measures – fluency, flexibility, and originality and the corresponding eye-tracking tools: scan paths and entropy. The findings described in this paper shed light on several processes and phenomena.

Eye tracking analysis further reveals differences in thinking processes associated with employing seemingly similar solution strategies. This finding is based on observing differences in scan paths of identical solution strategies. The differences are related to regions of interest, entropy values, number of saccades, and total fixations. The number of saccades and the longer duration of fixations, combined with a higher level of cognitive load, can indicate either a higher level of problem-solving creativity or problem complexity. Uniquely, our study demonstrated that insight-based solutions have different and identifiable eye-tracking patterns related to more focused regions of interest with smaller numbers of saccades and fixations. In summary, based on the findings of our study, we argue that eye-tracking methodology can make a meaningful contribution to research in mathematics education since it reveals underlying problem-solving processes that cannot be discovered by behavior analysis alone.

## References

* Andrá, C., Lindström, P., Arzarello, F., Holmqvist, K., Robutti, O., & Sabena, C. (2015). Reading mathematics representations: an eye-tracking study. *International Journal of Science and Mathematics Education, 13*(2), 237–259.

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

* Dewolf, T., Van Dooren, W., Hermens, F., & Verschaffel, L. (2013). Do students attend to and profit from representational illustrations of non-standard mathematical word problems? In A. M. Lindmeier & A. Heinze (Eds.), *Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 217-224). PME.
different thinking process. The total fixations showed that S2 performed more eye movement in each solution than S1. For example, in solution A1, S2 solved more quickly than S1, but used more eye movement in this strategy.

Based on the above, we argue that eye-tracking analysis is adequate for understanding differences in
mathematical thinking that lead to similar solution strategies and outcomes. Such understanding could
not be attained from the interview transcript analysis alone.

DISCUSSION

This paper presents an empirical investigation of creativity-emerging problem-solving by employing a VG-MST that explicitly requires participants to solve a mathematical problem in multiple ways. Our goal was to characterize creativity-emerging problem-solving processing by participants with the same level of creativity through integrating behavioral and neuro-physiological perspectives that focus on problem-solving strategies and corresponding eye-tracking patterns. We call the process of solving VG-MST creativity-directed since we do not require participants to perform creative solutions but fosters creative thinking.

We integrated eye-tracking methodologies with individual clinical interviews to characterize patterns of processing associated with flexible mathematical thinking and insight-based processing in solvers with different strategies. Due to the small sample size (the major limitation of our study), we performed qualitative data analysis. The data analysis included behavioral measures – fluency, flexibility, and originality and the corresponding eye-tracking tools: scan paths and entropy. The findings described in this paper shed light on several processes and phenomena.

Eye tracking analysis further reveals differences in thinking processes associated with employing seemingly similar solution strategies. This finding is based on observing differences in scan paths of identical solution strategies. The differences are related to regions of interest, entropy values, number of saccades, and total fixations. The number of saccades and the longer duration of fixations, combined with a higher level of cognitive load, can indicate either a higher level of problem-solving creativity or problem complexity. Uniquely, our study demonstrated that insight-based solutions have different and identifiable eye-tracking patterns related to more focused regions of interest with smaller numbers of saccades and fixations. In summary, based on the findings of our study, we argue that eye-tracking methodology can make a meaningful contribution to research in mathematics education since it reveals underlying problem-solving processes that cannot be discovered by behavior analysis alone.

References

Andrá, C., Lindström, P., Arzarello, F., Holmqvist, K., Robutti, O., & Sabena, C. (2015). Reading mathematics representations: an eye-tracking study. *International Journal of Science and Mathematics Education, 13*(2), 237–259.

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

Dewolf, T., Van Dooren, W., Hermens, F., & Verschaffel, L. (2013). Do students attend to and profit from representational illustrations of non-standard mathematical word problems? In A. M. Lindmeier & A. Heinze (Eds.), *Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 217-224). PME.
Elgrably, H., & Leikin, R. (2021). Creativity as a function of problem-solving expertise: posing new problems through investigations. *ZDM - Mathematics Education*.

Grabner, R. & De Smedt, B. (2016). Special issue “Cognitive neuroscience and mathematics learning – revisited after five years.” *ZDM - Mathematics Education, 48*(3).

Grant, E. R., & Spivey, M. J. (2003). Eye movements and problem solving: guiding attention guides thought. *Psychological Science, 14*(5), 462–466.

Haavold, P.Ø., & Sriraman, B. (2022). Creativity in problem solving: integrating two different views of insight. *ZDM - Mathematics Education, 54*, 83–96.

Hanna, G. (1996). The ongoing value of proof. In L. Puig, & A. Gutiérrez (Eds.), *Proceedings of the PME 20. PME*.

Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: what is it that is going on for students? *Cognition and Instruction, 24*(1), 73-122.

Leikin, R. & Guberman, R. (2023). Creativity and challenge: task complexity as a function of insight and multiplicity of solutions. In R. Leikin (Ed.). *Mathematical challenges for all* (pp. 325-342). Springer.

Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman & B. Koichu (Eds.), *Creativity in mathematics and the education of gifted students*. (pp. 129-145). Sense Publisher.

Leikin, R. (2013). Evaluating mathematical creativity: the interplay between multiplicity and insight. *Psychological Test and Assessment Modeling, 55*(4), 385–400.

Leikin, R., & Lev, M. (2013). Mathematical creativity in generally gifted and mathematically excelling adolescents: What makes the difference? *ZDM Mathematics Education, 45*(2), 183–197.

Leikin, R., & Sriraman, B. (2022). Empirical research on creativity in mathematics (education): from the wastelands of psychology to the current state of the art. *ZDM, Mathematics Education, 54*, 1–17.

Leikin, R., Pitta-Pantazi, D. (2013). Creativity and mathematics education: the state of the art. *ZDM-Mathematics Education, 45*, 159–166.

Leikin, R., Waisman, I. & Leikin, M. (2016). Does solving insight-based problems differ from solving learning-based problems? Some evidence from an ERP study. *ZDM Mathematics Education, 48*, 305–319.

Levav-Waynberg, A. & Leikin, R. (2012). The role of multiple solution tasks in developing knowledge and creativity in geometry. *Journal of Mathematical Behavior, 31*, 73–90.

Norqvist, M., Jonsson, B., Lithner, J., Qwillbard, T., & Holm, L. (2019). Investigating algorithmic and creative reasoning strategies by eye tracking. *The Journal of Mathematical Behavior, 55*, 100701.

Polya, G. (1981). Mathematics discovery. Willey & Son.

Scheiter, K., & van Gog, T. (2009). Using eye-tracking in applied research to study and stimulate the process of information from multi-representational sources. *Applied Cognitive Psychology, 23*, 1209–1214.

Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.

Schoevers, E.M., Kroesbergen, E.H., Moerbeek, M.(2022). The relation between creativity and students' performance on different types of geometrical problems in elementary education. *ZDM Mathematics Education, 54*, 133–147.

Strohmaier, A.R., MacKay, K.J., Obersteiner, A., & Reiss, K. (2020). Eye-tracking methodology in mathematics education research: a systematic literature review. *Educational Studies in Mathematics, 104*, 147–200.
Elgrably, H., & Leikin, R. (2021). Creativity as a function of problem-solving expertise: posing new problems through investigations. *ZDM - Mathematics Education*.

Grabner, R. & De Smedt, B. (2016). Special issue “Cognitive neuroscience and mathematics learning – revisited after five years.” *ZDM - Mathematics Education, 48*(3).

Grant, E. R., & Spivey, M. J. (2003). Eye movements and problem solving: guiding attention guides thought. *Psychological Science, 14*(5), 462–466.

Haavold, P.Ø., & Sriraman, B. (2022). Creativity in problem solving: integrating two different views of insight. *ZDM - Mathematics Education, 54*, 83–96.

Hanna, G. (1996). The ongoing value of proof. In L. Puig, & A. Gutiérrez (Eds.), *Proceedings of the PME 20. PME*.

Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: what is it that is going on for students? *Cognition and Instruction, 24*(1), 73-122.

Leikin, R. & Guberman, R. (2023). Creativity and challenge: task complexity as a function of insight and multiplicity of solutions. In R. Leikin (Ed.). *Mathematical challenges for all* (pp. 325-342). Springer.

Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman & B. Koichu (Eds.), *Creativity in mathematics and the education of gifted students*. (pp. 129-145). Sense Publisher.

Leikin, R. (2013). Evaluating mathematical creativity: the interplay between multiplicity and insight. *Psychological Test and Assessment Modeling, 55*(4), 385–400.

Leikin, R., & Lev, M. (2013). Mathematical creativity in generally gifted and mathematically excelling adolescents: What makes the difference? *ZDM Mathematics Education, 45*(2), 183–197.

Leikin, R., & Sriraman, B. (2022). Empirical research on creativity in mathematics (education): from the wastelands of psychology to the current state of the art. *ZDM, Mathematics Education, 54*, 1–17.

Leikin, R., Pitta-Pantazi, D. (2013). Creativity and mathematics education: the state of the art. *ZDM-Mathematics Education, 45*, 159–166.

Leikin, R., Waisman, I. & Leikin, M. (2016). Does solving insight-based problems differ from solving learning-based problems? Some evidence from an ERP study. *ZDM Mathematics Education, 48*, 305–319.

Levav-Waynberg, A. & Leikin, R. (2012). The role of multiple solution tasks in developing knowledge and creativity in geometry. *Journal of Mathematical Behavior, 31*, 73–90.

Norqvist, M., Jonsson, B., Lithner, J., Qwillbard, T., & Holm, L. (2019). Investigating algorithmic and creative reasoning strategies by eye tracking. *The Journal of Mathematical Behavior, 55*, 100701.

Polya, G. (1981). Mathematics discovery. Willey & Son.

Scheiter, K., & van Gog, T. (2009). Using eye-tracking in applied research to study and stimulate the process of information from multi-representational sources. *Applied Cognitive Psychology, 23*, 1209–1214.

Schoenfeld, A. H. (1985). *Mathematical problem solving*. Academic Press.

Schoevers, E.M., Kroesbergen, E.H., Moerbeek, M.(2022). The relation between creativity and students’ performance on different types of geometrical problems in elementary education. *ZDM Mathematics Education, 54*, 133–147.

Strohmaier, A.R., MacKay, K.J., Obersteiner, A., & Reiss, K. (2020). Eye-tracking methodology in mathematics education research: a systematic literature review. *Educational Studies in Mathematics, 104*, 147–200.
# TEACHING GEOMETRY TO ADVANCE DESIGN JUSTICE

Gloriana González, Saadeddine Shehab, Edward Powers

University of Illinois Urbana-Champaign

The study of geometry provides unique opportunities for students to appreciate geometric patterns in the world and apply geometry to solve authentic problems. These opportunities are well aligned with design thinking approaches to problem solving. The geometry curriculum can incorporate design challenges that require geometric thinking. We apply a design justice perspective to geometry instruction and provide examples of lessons that can allow students to advance social justice in their communities. This perspective can help to rehumanize mathematics by engaging students in creating new math, broadening what constitutes math, and developing students’ sense of ownership. We posit that by embracing a multidisciplinary approach to the teaching and learning of geometry, we can position students as designers, thus preparing them for future jobs where design thinking is at the core. At the same time, by drawing on a design justice perspective, students can develop empathy and empower their communities through the study of geometry, thus making their math instruction more meaningful. Overall, design thinking can model the historical development of geometry knowledge.

## INTRODUCTION

The U.S. Geometry curriculum has tried to attend to various goals which are represented by four arguments justifying the high school geometry course (González & Herbst, 2006). Proponents of the *intuitive argument* established that geometry provides unique opportunities to read the world. Consequently, students’ ability to appreciate geometry in their surrounding such as seeing patterns in geometric designs is important. Proponents of the *mathematical argument* stressed that the study of geometry offers opportunities for students to think like mathematicians. From this perspective, students’ development of conjectures as they explore geometric problems is a key activity. Proponents of the *formal argument* consider that geometry can allow students to apply logical reasoning that they can transfer to other situations. In the U.S., the use of the two-column proof format is an example of how school geometry incorporated resources for teaching students how to make logical arguments (Herbst, 2002). Proponents of the *utilitarian argument* considered that geometry is especially important in preparing students for the workforce. Geometry problems that position students as professionals such as navigators or architects are helpful to achieve the goals of the utilitarian arguments. While these four arguments stem from the study of the U.S. geometry curriculum in the 20th century, they continue to co-exist in textbooks such as in problems pertaining to geometry and visual arts (González, 2020). Initial evidence from U.S. geometry teachers suggests that the intuitive and utilitarian arguments are most valued in contrast with the formal argument, which seems to be less valued (González & Rinkenberger, 2023). Additionally, research suggests that the emphasis on proofs characteristic of the formal argument is less relevant in the geometry curriculum and geometry instruction (Otten et al., 2014; Sears & Chávez, 2014).
# TEACHING GEOMETRY TO ADVANCE DESIGN JUSTICE

Gloriana González, Saadeddine Shehab, Edward Powers

University of Illinois Urbana-Champaign

The study of geometry provides unique opportunities for students to appreciate geometric patterns in the world and apply geometry to solve authentic problems. These opportunities are well aligned with design thinking approaches to problem solving. The geometry curriculum can incorporate design challenges that require geometric thinking. We apply a design justice perspective to geometry instruction and provide examples of lessons that can allow students to advance social justice in their communities. This perspective can help to rehumanize mathematics by engaging students in creating new math, broadening what constitutes math, and developing students' sense of ownership. We posit that by embracing a multidisciplinary approach to the teaching and learning of geometry, we can position students as designers, thus preparing them for future jobs where design thinking is at the core. At the same time, by drawing on a design justice perspective, students can develop empathy and empower their communities through the study of geometry, thus making their math instruction more meaningful. Overall, design thinking can model the historical development of geometry knowledge.

## INTRODUCTION

The U.S. Geometry curriculum has tried to attend to various goals which are represented by four arguments justifying the high school geometry course (González & Herbst, 2006). Proponents of the *intuitive argument* established that geometry provides unique opportunities to read the world. Consequently, students' ability to appreciate geometry in their surrounding such as seeing patterns in geometric designs is important. Proponents of the *mathematical argument* stressed that the study of geometry offers opportunities for students to think like mathematicians. From this perspective, students' development of conjectures as they explore geometric problems is a key activity. Proponents of the *formal argument* consider that geometry can allow students to apply logical reasoning that they can transfer to other situations. In the U.S., the use of the two-column proof format is an example of how school geometry incorporated resources for teaching students how to make logical arguments (Herbst, 2002). Proponents of the *utilitarian argument* considered that geometry is especially important in preparing students for the workforce. Geometry problems that position students as professionals such as navigators or architects are helpful to achieve the goals of the utilitarian arguments. While these four arguments stem from the study of the U.S. geometry curriculum in the 20th century, they continue to co-exist in textbooks such as in problems pertaining to geometry and visual arts (González, 2020). Initial evidence from U.S. geometry teachers suggests that the intuitive and utilitarian arguments are most valued in contrast with the formal argument, which seems to be less valued (González & Rinkenberger, 2023). Additionally, research suggests that the emphasis on proofs characteristic of the formal argument is less relevant in the geometry curriculum and geometry instruction (Otten et al., 2014; Sears & Chávez, 2014).
At the same time, questions about the role of school mathematics in supporting students to adopt a
critical stance towards society have become increasingly relevant. The pandemic has made explicit
societal inequities for which mathematics can be an important tool to examine the world (Bakker et
al., 2021). Mathematics educators' discussions about whether geometry instruction will meet the
current needs to teach students mathematics to read the world are important as we consider the
justifications for teaching geometry in schools. Moreover, the study of geometry can prompt students
to dream of a better future and conceive of new ways of living in the world. Gutiérrez's (2018)
"Rehumanizing Mathematics Framework" exemplifies the call for new ways of engaging students in
mathematical activity by centering mathematics around social justice goals.

FOCUS

We are investigating how to use design-based pedagogy for geometry instruction. Design-based pedagogy is a learning environment that uses instructional scaffolds to support students as they solve a problem by implementing design practices (Royalty, 2018). We are particularly interested in applying a design justice approach so that students can inquire into problems that are relevant to them and their communities (Costanza-Chock, 2020). In doing so, we align the justification for why students need to learn geometry with the goals of the intuitive and utilitarian arguments. A design-based perspective can allow students to appreciate geometry in their surroundings and address challenges affecting them, their communities, or people in the world. We strive for students to develop empathy by understanding the needs of those affected by the challenges (including themselves). At the same time, we are in alignment with the utilitarian argument by positioning students as designers. The opportunity to learn skills and procedures that designers use can strengthen their desire to study geometry and find connections with the ways in which designers apply geometry in their profession. Moreover, with a design justice approach, we seek that students will be able to examine their agency in proposing solutions that are just and equitable.

Main questions

The main questions guiding our study are:

1) In what ways does geometry allow for students to study the world by using design?

2) How does a design-based approach to geometry instruction enable students to investigate authentic problems as designers?

3) What are examples of geometry lessons that use a design justice approach?

With the first question, we are interested in connections between geometry and design by considering
the ways in which geometry knowledge is fundamental in addressing design challenges. With the
second question, we are interested in identifying characteristics of design challenges that position
students as designers. The third question advances the combination of design and geometry with a
focus on design justice. Altogether, we are interested in answering the question of how we can best
position students to use geometry to solve design challenges addressing social justice issues and
leveraging on their cultural and community-based knowledge.
At the same time, questions about the role of school mathematics in supporting students to adopt a
critical stance towards society have become increasingly relevant. The pandemic has made explicit
societal inequities for which mathematics can be an important tool to examine the world (Bakker et
al., 2021). Mathematics educators' discussions about whether geometry instruction will meet the
current needs to teach students mathematics to read the world are important as we consider the
justifications for teaching geometry in schools. Moreover, the study of geometry can prompt students
to dream of a better future and conceive of new ways of living in the world. Gutiérrez's (2018)
"Rehumanizing Mathematics Framework" exemplifies the call for new ways of engaging students in
mathematical activity by centering mathematics around social justice goals.

FOCUS

We are investigating how to use design-based pedagogy for geometry instruction. Design-based pedagogy is a learning environment that uses instructional scaffolds to support students as they solve a problem by implementing design practices (Royalty, 2018). We are particularly interested in applying a design justice approach so that students can inquire into problems that are relevant to them and their communities (Costanza-Chock, 2020). In doing so, we align the justification for why students need to learn geometry with the goals of the intuitive and utilitarian arguments. A design-based perspective can allow students to appreciate geometry in their surroundings and address challenges affecting them, their communities, or people in the world. We strive for students to develop empathy by understanding the needs of those affected by the challenges (including themselves). At the same time, we are in alignment with the utilitarian argument by positioning students as designers. The opportunity to learn skills and procedures that designers use can strengthen their desire to study geometry and find connections with the ways in which designers apply geometry in their profession. Moreover, with a design justice approach, we seek that students will be able to examine their agency in proposing solutions that are just and equitable.

Main questions

The main questions guiding our study are:

1) In what ways does geometry allow for students to study the world by using design?

2) How does a design-based approach to geometry instruction enable students to investigate authentic problems as designers?

3) What are examples of geometry lessons that use a design justice approach?

With the first question, we are interested in connections between geometry and design by considering
the ways in which geometry knowledge is fundamental in addressing design challenges. With the
second question, we are interested in identifying characteristics of design challenges that position
students as designers. The third question advances the combination of design and geometry with a
focus on design justice. Altogether, we are interested in answering the question of how we can best
position students to use geometry to solve design challenges addressing social justice issues and
leveraging on their cultural and community-based knowledge.
# THEORETICAL UNDERPINNINGS

## Human-Centered Design

Human-Centered Design (HCD) is a problem-solving approach where designers rely on tools and methods to address the needs of a population and develop meaningful and creative solutions through iteration (Brown, 2008). When designers use HCD they implement different processes that are guided by empathy and iteration (Brown, 2008). These processes include examining current designs related to the new design challenge, conducting user research, brainstorming design solutions, and creating prototypes (Zhang & Dong, 2008). Lawrence et al. (in press) developed an HCD taxonomy that incorporates and defines the HCD processes and practices in a way that can foster the teaching and learning of HCD in educational settings specifically for non-designers. The taxonomy includes 5 spaces: understand, synthesize, ideate, prototype, and implement. Each of the spaces include various processes such as observing, defining, proposing, evaluating, and executing. We integrate the HCD processes listed in the HCD taxonomy in geometry problems that focus on addressing social justice issues. The taxonomy allows us to apply design as an interdisciplinary perspective that can enrich the teaching and learning of geometry.

## Design justice

One critique to design thinking perspectives is that the innovation cycle could continue to reinforce inequities in the world, perpetuate oppression, and erase the role of people who come up with ideas and solutions but are not named as designers (Benjamin, 2019). People who embrace a design justice approach are continuously working to overcome the barriers that designers may have in perpetuating inequities and seek to increase the agency and engagement of marginalized communities to achieve social justice through design (Costanza-Chock, 2020). New perspectives such as “design for the pluriverse” (Escobar, 2017) and “design justice” (Costanza-Chock, 2020) propose new accounts of the design process. Works like “The Black experience in design,” (Berry et al., 2022) center Black designers in the design process to reclaim their contributions and extend opportunities that amplify what design is about. In our work we have used the ten principles by the Design Justice Network (n.d.) to conceive of ways in which geometry can integrate social justice issues.

## OUR MODEL

Figure 1 shows our model for creating geometry lessons anchored in a design justice approach.

<figure><img src="image_197.png" /><figcaption>Figure 1: Model for Geometry Design Justice Lessons</figcaption></figure>
# THEORETICAL UNDERPINNINGS

## Human-Centered Design

Human-Centered Design (HCD) is a problem-solving approach where designers rely on tools and methods to address the needs of a population and develop meaningful and creative solutions through iteration (Brown, 2008). When designers use HCD they implement different processes that are guided by empathy and iteration (Brown, 2008). These processes include examining current designs related to the new design challenge, conducting user research, brainstorming design solutions, and creating prototypes (Zhang & Dong, 2008). Lawrence et al. (in press) developed an HCD taxonomy that incorporates and defines the HCD processes and practices in a way that can foster the teaching and learning of HCD in educational settings specifically for non-designers. The taxonomy includes 5 spaces: understand, synthesize, ideate, prototype, and implement. Each of the spaces include various processes such as observing, defining, proposing, evaluating, and executing. We integrate the HCD processes listed in the HCD taxonomy in geometry problems that focus on addressing social justice issues. The taxonomy allows us to apply design as an interdisciplinary perspective that can enrich the teaching and learning of geometry.

## Design justice

One critique to design thinking perspectives is that the innovation cycle could continue to reinforce inequities in the world, perpetuate oppression, and erase the role of people who come up with ideas and solutions but are not named as designers (Benjamin, 2019). People who embrace a design justice approach are continuously working to overcome the barriers that designers may have in perpetuating inequities and seek to increase the agency and engagement of marginalized communities to achieve social justice through design (Costanza-Chock, 2020). New perspectives such as “design for the pluriverse” (Escobar, 2017) and “design justice” (Costanza-Chock, 2020) propose new accounts of the design process. Works like “The Black experience in design,” (Berry et al., 2022) center Black designers in the design process to reclaim their contributions and extend opportunities that amplify what design is about. In our work we have used the ten principles by the Design Justice Network (n.d.) to conceive of ways in which geometry can integrate social justice issues.

## OUR MODEL

Figure 1 shows our model for creating geometry lessons anchored in a design justice approach.

<figure><img src="image_207.png" /><figcaption>Figure 1: Model for Geometry Design Justice Lessons</figcaption></figure>
In the model we consider the three fundamental perspectives in our curricular innovation. The geometry lessons must be anchored in curricular standards. In our case, we use the Common Core Standards for Mathematics, attending to the geometry standards (National Governors Association Center for Best Practices, 2010). We draw on recommendations by Goldman and Zielezinski (2022) who have adopted a four-stage process when applying design thinking to classrooms: exploring the problem space, empathizing, brainstorming, and prototyping cycles. In contrast with a problem-based approach which is often used in mathematics classrooms (see Boaler 1998; Chazan, 2000; Lampert, 2001), a design-thinking approach is more open-ended and involves students' investigation of a problem space and empathizing with those who will benefit from the design. As a result of their brainstorming, students propose possible solutions by creating prototypes that showcase a way to address the challenge. The prototypes are tangible deliverables typically the result of a collaborative process. For example, students may create a sketch, a display, or a diorama showing a prototype of their design challenge solution. In doing so, the students communicate with designers about how they addressed the challenge and use the feedback to improve their design.

Finally, the design justice approach allows us to identify problem spaces that are relevant to the students and their communities. This is important because in contrast with a typical problem-based lesson, the starting point for us in designing a lesson is not the geometry content, but the problem space. In identifying the problem space, we use the design justice principles as a starting point to brainstorm possible ways in which geometry can be applied to solve a design challenge.

## PROTOTYPES OF LESSONS

Here we show examples of three prototypes of lessons that we have created. We will use the prototypes for teachers to create new lessons in collaboration with peers (González & Deal, 2019). There are some examples of design-based thinking lessons in mathematics (Bush et al., 2018), typically for middle grades. In our case, we aim that teachers construct lessons with substantial geometry content, typical of high school geometry instruction. Therefore, the prototypes can spur the teachers' creativity for seeking further connections with issues pertaining to their students and their communities.

### Designing environmentally friendly and affordable house

**Problem:** Affordable housing is crucial to address important issues such as child poverty and homelessness. Event such as the 2008 mortgage crisis continue to affect affordable housing, and particularly affect underserved communities. Organizations such as Habitat for Humanity have advocated for safe and affordable housing. At the same time, there is a need for designing environmentally friendly buildings.

**Design challenge:** The state has launched a competition to design a 3-story environmentally friendly housing complex. The building should house 12 apartments and include features to be a carbon-negative home. That is, it should produce more energy than the one it consumes.

**Learning goals:** Students will produce a 3D scale model of the 2-story housing complex. Using geometric properties of solids and an understanding of cross-sections of 3D figures, students will apply their creativity to design affordable housing that suits the needs of their community. For example, the design must allocate spaces for play for youth as well as spaces for elderly folks to
In the model we consider the three fundamental perspectives in our curricular innovation. The geometry lessons must be anchored in curricular standards. In our case, we use the Common Core Standards for Mathematics, attending to the geometry standards (National Governors Association Center for Best Practices, 2010). We draw on recommendations by Goldman and Zielezinski (2022) who have adopted a four-stage process when applying design thinking to classrooms: exploring the problem space, empathizing, brainstorming, and prototyping cycles. In contrast with a problem-based approach which is often used in mathematics classrooms (see Boaler 1998; Chazan, 2000; Lampert, 2001), a design-thinking approach is more open-ended and involves students' investigation of a problem space and empathizing with those who will benefit from the design. As a result of their brainstorming, students propose possible solutions by creating prototypes that showcase a way to address the challenge. The prototypes are tangible deliverables typically the result of a collaborative process. For example, students may create a sketch, a display, or a diorama showing a prototype of their design challenge solution. In doing so, the students communicate with designers about how they addressed the challenge and use the feedback to improve their design.

Finally, the design justice approach allows us to identify problem spaces that are relevant to the students and their communities. This is important because in contrast with a typical problem-based lesson, the starting point for us in designing a lesson is not the geometry content, but the problem space. In identifying the problem space, we use the design justice principles as a starting point to brainstorm possible ways in which geometry can be applied to solve a design challenge.

## PROTOTYPES OF LESSONS

Here we show examples of three prototypes of lessons that we have created. We will use the prototypes for teachers to create new lessons in collaboration with peers (González & Deal, 2019). There are some examples of design-based thinking lessons in mathematics (Bush et al., 2018), typically for middle grades. In our case, we aim that teachers construct lessons with substantial geometry content, typical of high school geometry instruction. Therefore, the prototypes can spur the teachers' creativity for seeking further connections with issues pertaining to their students and their communities.

### Designing environmentally friendly and affordable house

**Problem:** Affordable housing is crucial to address important issues such as child poverty and homelessness. Event such as the 2008 mortgage crisis continue to affect affordable housing, and particularly affect underserved communities. Organizations such as Habitat for Humanity have advocated for safe and affordable housing. At the same time, there is a need for designing environmentally friendly buildings.

**Design challenge:** The state has launched a competition to design a 3-story environmentally friendly housing complex. The building should house 12 apartments and include features to be a carbon-negative home. That is, it should produce more energy than the one it consumes.

**Learning goals:** Students will produce a 3D scale model of the 2-story housing complex. Using geometric properties of solids and an understanding of cross-sections of 3D figures, students will apply their creativity to design affordable housing that suits the needs of their community. For example, the design must allocate spaces for play for youth as well as spaces for elderly folks to
interact. The design must also consider ways in which to minimize the effects on the environment such as including green spaces and solar panels.

## Circular patterns and cultural experiences and identities

**Problem:** Geometric patterns have been used in different cultures for centuries and remain popular today. In many cultures, geometric patterns are used to decorate religious buildings, clothes, and other objects. Geometric patterns are also used primarily as decoration. For example, in art, geometric patterns are often used to express a certain emotion or mood. These patterns are also used to create illusions of movement and color in artworks. Geometric patterns can bring balance to a design by helping to create harmony between different elements. Moreover, geometric patterns can be used to express identity and values in relation to people's cultural wealth and experiences. Geometric patterns usually include many geometric figures such as circles, triangles, and trapezoids.

**Design challenge:** The Spurlock Museum in Urbana is excited to announce a community exhibit focusing on stories of immigration and identity. The Museum is seeking to borrow objects to display in this exhibit and help create a deeper, richer understanding of the cultural experiences and identities of community members. Objects, documents, and photographs can connect us to other people, no matter how far away in time or place. The Museum is interested in displaying objects that include circular patterns.

**Learning goals:** Students will design a low-fidelity prototype of an object that can be displayed in the exhibit. The object needs to have some circular patterns for visitors to empathize with the cultural experiences and identities of community members. The object needs to target a specific cultural experience or identity that is relevant to a specific community. To support students in completing these tasks, students will have access to supplementary materials about various cultural experiences and identities in their community. Students will sketch circular patterns with appropriate dimensions for the exhibit to be visible.

## Designing a school garden

**Problem:** Designing a sustainable garden is an art. Gardens can be a form of expression, provide food security, and supply valuable nutrients to people's diets. Designing a school garden can be a viable solution to providing students with a fresh source of fruits and vegetables for their daily lunches. Designers' considerations about what is to be planted and how to arrange the plants in the garden can result in an incredible resource for students.

**Design challenge:** The Student Council has launched a competition to design a sustainable and nutritious school garden. The garden will increase access to food and gardening opportunities in your school community. The design must include quadrilaterals and circles for representing various garden plots. The design teams must present a scale model of the garden in the next meeting.

**Learning goals:** Students will collaborate in teams to explore spaces within their school for a viable garden location, research sustainable plant options, and create a prototype garden using geometry software. Students will use ratios and proportions to create a scale model of their garden, apply coordinate geometry, and model with geometry throughout the prototyping process. Students will be empowered to bring their knowledge home to design their own gardens in their communities.
interact. The design must also consider ways in which to minimize the effects on the environment such as including green spaces and solar panels.

## Circular patterns and cultural experiences and identities

**Problem:** Geometric patterns have been used in different cultures for centuries and remain popular today. In many cultures, geometric patterns are used to decorate religious buildings, clothes, and other objects. Geometric patterns are also used primarily as decoration. For example, in art, geometric patterns are often used to express a certain emotion or mood. These patterns are also used to create illusions of movement and color in artworks. Geometric patterns can bring balance to a design by helping to create harmony between different elements. Moreover, geometric patterns can be used to express identity and values in relation to people's cultural wealth and experiences. Geometric patterns usually include many geometric figures such as circles, triangles, and trapezoids.

**Design challenge:** The Spurlock Museum in Urbana is excited to announce a community exhibit focusing on stories of immigration and identity. The Museum is seeking to borrow objects to display in this exhibit and help create a deeper, richer understanding of the cultural experiences and identities of community members. Objects, documents, and photographs can connect us to other people, no matter how far away in time or place. The Museum is interested in displaying objects that include circular patterns.

**Learning goals:** Students will design a low-fidelity prototype of an object that can be displayed in the exhibit. The object needs to have some circular patterns for visitors to empathize with the cultural experiences and identities of community members. The object needs to target a specific cultural experience or identity that is relevant to a specific community. To support students in completing these tasks, students will have access to supplementary materials about various cultural experiences and identities in their community. Students will sketch circular patterns with appropriate dimensions for the exhibit to be visible.

## Designing a school garden

**Problem:** Designing a sustainable garden is an art. Gardens can be a form of expression, provide food security, and supply valuable nutrients to people's diets. Designing a school garden can be a viable solution to providing students with a fresh source of fruits and vegetables for their daily lunches. Designers' considerations about what is to be planted and how to arrange the plants in the garden can result in an incredible resource for students.

**Design challenge:** The Student Council has launched a competition to design a sustainable and nutritious school garden. The garden will increase access to food and gardening opportunities in your school community. The design must include quadrilaterals and circles for representing various garden plots. The design teams must present a scale model of the garden in the next meeting.

**Learning goals:** Students will collaborate in teams to explore spaces within their school for a viable garden location, research sustainable plant options, and create a prototype garden using geometry software. Students will use ratios and proportions to create a scale model of their garden, apply coordinate geometry, and model with geometry throughout the prototyping process. Students will be empowered to bring their knowledge home to design their own gardens in their communities.
## Characteristics of the lesson prototypes

The three lessons show various characteristics in our model. First, each lesson is anchored around one of the design justice principles. For example, Principle 1 states, “We Use Design to Sustain, Heal, and Empower our Communities, as Well as to Seek Liberation from Exploitative and Oppressive Systems” (Costanza-Chock, 2020, p. 190). This principle is important in the school garden problem as the food garden will result in providing new opportunities for students to connect with the Earth through faming and develop sustainable farming skills. Second, the problems intend to engage students in design-thinking processes such as building a prototype while also integrating activities where they empathize with the stakeholders who will be affected by the design. For example, in the affordable housing problem, students can interview people in their communities and learn about their needs to address their needs in their designs. Finally, the lessons require a thorough study of geometry, at times extending the geometric content in the standards. For example, in the circular patterns problems, students’ examination of how circles have been used in various cultures can result in their deep examination of cultural practices (see Lawlor, 1982). We expect that students’ motivation to propose solutions to the design challenges will drive their study of geometry as they will need to learn about geometric concepts to complete their designs. The more examples of lessons that we can create following this model, the better resources geometry teachers will have to engage in design justice challenges with their students.

## FUTURE DIRECTIONS

The problems that we introduce here are meant to be prototypes for geometry teachers to create design challenges for their students. We are not seeking to create a geometry curriculum. Instead, we seek to empower geometry teachers to work with their students in becoming designers. Through this process, we expect students to become agents of change in their communities through applying geometry in identifying and addressing design challenges. Geometry provides unique opportunities for students to become designers as they can use geometry to achieve multiple goals such as solving authentic problems, creating beautiful designs, and appreciating how math is embedded in nature and culture. Our development of a framework for teachers to create design challenges for their students is meant to be a first step in addressing ways in which geometry can uniquely propel students’ engagement in mathematics that would transform their educational experience. We believe that a multidisciplinary approach could help in making geometry relevant to address current social issues. Students who apply a design justice approach to geometry problem solving can be creators and develop a sense of ownership towards mathematics, two dimensions in the Rehumanizing Mathematics Framework (Gutiérrez, 2018). More importantly, students will be creators of new mathematics that will have a direct impact on their communities. Escobar (2017) uses the notion of “disoñar,” or dream with design. We expect that when students are engaged in disoñar by using geometry they start imagining ways in which math is relevant for them and their communities to build a better future. Students’ engagement in solving design problems with geometry can follow the ways in which geometry knowledge has been developed in the history of mathematics. Geometry knowledge was fundamental in real-world problems (Henderson, 2000; Joseph, 2010). Returning to the goal of situating geometry within human activities can re-engage students in appreciating and using geometry.
## Characteristics of the lesson prototypes

The three lessons show various characteristics in our model. First, each lesson is anchored around one of the design justice principles. For example, Principle 1 states, “We Use Design to Sustain, Heal, and Empower our Communities, as Well as to Seek Liberation from Exploitative and Oppressive Systems” (Costanza-Chock, 2020, p. 190). This principle is important in the school garden problem as the food garden will result in providing new opportunities for students to connect with the Earth through faming and develop sustainable farming skills. Second, the problems intend to engage students in design-thinking processes such as building a prototype while also integrating activities where they empathize with the stakeholders who will be affected by the design. For example, in the affordable housing problem, students can interview people in their communities and learn about their needs to address their needs in their designs. Finally, the lessons require a thorough study of geometry, at times extending the geometric content in the standards. For example, in the circular patterns problems, students’ examination of how circles have been used in various cultures can result in their deep examination of cultural practices (see Lawlor, 1982). We expect that students’ motivation to propose solutions to the design challenges will drive their study of geometry as they will need to learn about geometric concepts to complete their designs. The more examples of lessons that we can create following this model, the better resources geometry teachers will have to engage in design justice challenges with their students.

## FUTURE DIRECTIONS

The problems that we introduce here are meant to be prototypes for geometry teachers to create design challenges for their students. We are not seeking to create a geometry curriculum. Instead, we seek to empower geometry teachers to work with their students in becoming designers. Through this process, we expect students to become agents of change in their communities through applying geometry in identifying and addressing design challenges. Geometry provides unique opportunities for students to become designers as they can use geometry to achieve multiple goals such as solving authentic problems, creating beautiful designs, and appreciating how math is embedded in nature and culture. Our development of a framework for teachers to create design challenges for their students is meant to be a first step in addressing ways in which geometry can uniquely propel students’ engagement in mathematics that would transform their educational experience. We believe that a multidisciplinary approach could help in making geometry relevant to address current social issues. Students who apply a design justice approach to geometry problem solving can be creators and develop a sense of ownership towards mathematics, two dimensions in the Rehumanizing Mathematics Framework (Gutiérrez, 2018). More importantly, students will be creators of new mathematics that will have a direct impact on their communities. Escobar (2017) uses the notion of “disoñar,” or dream with design. We expect that when students are engaged in disoñar by using geometry they start imagining ways in which math is relevant for them and their communities to build a better future. Students’ engagement in solving design problems with geometry can follow the ways in which geometry knowledge has been developed in the history of mathematics. Geometry knowledge was fundamental in real-world problems (Henderson, 2000; Joseph, 2010). Returning to the goal of situating geometry within human activities can re-engage students in appreciating and using geometry.
## Acknowledgements

The research described in this paper was supported by a National Science Foundation's grant to Gloriana González (Principal Investigator) and Saadeddine Shehab (Co-Principal Investigator) for the project, "Engaging Teachers in Integrating Human-Centered Design for Geometry Problem-based Instruction," Grant No. DRL-2300689. Opinions, findings, conclusions, or recommendations are those of the authors and do not necessarily reflect the views of the National Science Foundation.

## References

Bakker, A., Cai, J., & Zenger, L. (2021). Future themes of mathematics education research: An international survey before and during the pandemic. *Educational Studies in Mathematics*, *107*(1), 1-24.
https://doi.org/10.1007/s10649-021-10049-w

Benjamin, R. (2020). *Race after technology: Abolitionist tools for the new Jim code*. Polity.

Berry, A. H., Collie, K., Laker, P. A., Noel, L. A., Rittner, J., & Walters, K. (Eds.). (2022). *The Black Experience in Design: Identity, Expression & Reflection*. Simon and Schuster.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. *Journal for Research in Mathematics Education*, 29, 41-62.

Brown, T. (2008). Design thinking. *Harvard Business Review*, 86(6), 84-94.

Bush, S. B., Karp, K. S., Cox, R., Cook, K. L., Albanese, J., & Karp, M. (2018). Design thinking framework: Shaping powerful mathematics. *Mathematics Teaching in the Middle School*, 23(4), 1-5.

Chazan, D. (2000). *Beyond formulas in mathematics and teaching: Dynamics of the high school algebra classroom*. Teachers College.

Costanza-Chock, S. (2020). Design justice: Community-led practices to build the worlds we need. The MIT Press.

Design Justice Network. (n.d.). https://designjustice.org/

Escobar, A. (2017). *Designs for the pluriverse*. Duke University Press.

Goldman, S. & Zielezinski, M. B. (2022). *Design thinking for every classroom*. Routledge.

González, G. (2020). Visual art contexts in U.S. geometry textbooks aligned with the Common Core Standards. *Annales de Didactique et de Sciences Cognitives*, 25, 183-210. https://doi.org/10.4000/adsc.560

González, G., & Deal, J. T. (2019). Using a creativity framework to promote teacher learning in Lesson Study. *Thinking Skills and Creativity*, 32, 114-128.

González, G., & Herbst, P. (2006). Competing arguments for the geometry course: Why were American high school students to study geometry in the twentieth century? *International Journal for the History of Mathematics Education*, 1(1), 7-33.

González, G., & Rinkenberger, C. (2023). Teachers' evaluation of geometry problems situated in visual arts contexts. *Electronic Journal for Research in Science & Mathematics Education*, 27(3), 1-19.

Gutiérrez, R. (2018). Introduction: The need to rehumanize mathematics. In I. Goffney & Gutiérrez, R. (Eds.) *Rehumanizing mathematics for Black, Indigenous, and Latinx students* (pp. 1-10). National Council of Teachers of Mathematics.

Henderson, D. W. (2000). Square Roots in the Sulba Sūtras. *Geometry at Work*, 53, 39.

Herbst, P. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in early twentieth century. *Educational Studies in Mathematics*, 49, 283-312.

Joseph, G. G. (2010). *The crest of the peacock: Non-European roots of mathematics*. Princeton University Press.
## Acknowledgements

The research described in this paper was supported by a National Science Foundation's grant to Gloriana González (Principal Investigator) and Saadeddine Shehab (Co-Principal Investigator) for the project, "Engaging Teachers in Integrating Human-Centered Design for Geometry Problem-based Instruction," Grant No. DRL-2300689. Opinions, findings, conclusions, or recommendations are those of the authors and do not necessarily reflect the views of the National Science Foundation.

## References

Bakker, A., Cai, J., & Zenger, L. (2021). Future themes of mathematics education research: An international survey before and during the pandemic. *Educational Studies in Mathematics*, *107*(1), 1-24.
https://doi.org/10.1007/s10649-021-10049-w

Benjamin, R. (2020). *Race after technology: Abolitionist tools for the new Jim code*. Polity.

Berry, A. H., Collie, K., Laker, P. A., Noel, L. A., Rittner, J., & Walters, K. (Eds.). (2022). *The Black Experience in Design: Identity, Expression & Reflection*. Simon and Schuster.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. *Journal for Research in Mathematics Education*, 29, 41-62.

Brown, T. (2008). Design thinking. *Harvard Business Review*, 86(6), 84-94.

Bush, S. B., Karp, K. S., Cox, R., Cook, K. L., Albanese, J., & Karp, M. (2018). Design thinking framework: Shaping powerful mathematics. *Mathematics Teaching in the Middle School*, 23(4), 1-5.

Chazan, D. (2000). *Beyond formulas in mathematics and teaching: Dynamics of the high school algebra classroom*. Teachers College.

Costanza-Chock, S. (2020). Design justice: Community-led practices to build the worlds we need. The MIT Press.

Design Justice Network. (n.d.). https://designjustice.org/

Escobar, A. (2017). *Designs for the pluriverse*. Duke University Press.

Goldman, S. & Zielezinski, M. B. (2022). *Design thinking for every classroom*. Routledge.

González, G. (2020). Visual art contexts in U.S. geometry textbooks aligned with the Common Core Standards. *Annales de Didactique et de Sciences Cognitives*, 25, 183-210. https://doi.org/10.4000/adsc.560

González, G., & Deal, J. T. (2019). Using a creativity framework to promote teacher learning in Lesson Study. *Thinking Skills and Creativity*, 32, 114-128.

González, G., & Herbst, P. (2006). Competing arguments for the geometry course: Why were American high school students to study geometry in the twentieth century? *International Journal for the History of Mathematics Education*, 1(1), 7-33.

González, G., & Rinkenberger, C. (2023). Teachers' evaluation of geometry problems situated in visual arts contexts. *Electronic Journal for Research in Science & Mathematics Education*, 27(3), 1-19.

Gutiérrez, R. (2018). Introduction: The need to rehumanize mathematics. In I. Goffney & Gutiérrez, R. (Eds.) *Rehumanizing mathematics for Black, Indigenous, and Latinx students* (pp. 1-10). National Council of Teachers of Mathematics.

Henderson, D. W. (2000). Square Roots in the Sulba Sūtras. *Geometry at Work*, 53, 39.

Herbst, P. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in early twentieth century. *Educational Studies in Mathematics*, 49, 283-312.

Joseph, G. G. (2010). *The crest of the peacock: Non-European roots of mathematics*. Princeton University Press.
Lampert, M. (2001). *Teaching problems and the problems of teaching*. Yale.

Lawlor, R. (1982). *Sacred geometry: Philosophy and practice*. Crossroad.

Lawrence, L., Shehab, S., & Tissenbaum, M. (in press). Understanding non-designers' practices and processes in a human-centered design course. *International Journal of Innovation in Education*.

National Governors Association Center for Best Practices, Council of Chief State School Officers [NGAC]. (2010). *Common core state standards for mathematics*. Author.

Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. *Mathematical Thinking and Learning, 16*(1), 51-79.

Royalty, A. (2018). Design-based pedagogy: Investigating an emerging approach to teaching design to non-designers. *Mechanism and Machine Theory, 125*, 137-145.

Sears, R., & Chávez, O. (2014). Opportunities to engage with proof: the nature of proof tasks in two geometry textbooks and its influence on enacted lessons. *ZDM, 46*(5), 767-780.

Zhang, T., & Dong, H. (2008). *Human-Centered Design: An Emergent Conceptual Model*. 1-7.
Lampert, M. (2001). *Teaching problems and the problems of teaching*. Yale.

Lawlor, R. (1982). *Sacred geometry: Philosophy and practice*. Crossroad.

Lawrence, L., Shehab, S., & Tissenbaum, M. (in press). Understanding non-designers' practices and processes in a human-centered design course. *International Journal of Innovation in Education*.

National Governors Association Center for Best Practices, Council of Chief State School Officers [NGAC]. (2010). *Common core state standards for mathematics*. Author.

Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. *Mathematical Thinking and Learning, 16*(1), 51-79.

Royalty, A. (2018). Design-based pedagogy: Investigating an emerging approach to teaching design to non-designers. *Mechanism and Machine Theory, 125*, 137-145.

Sears, R., & Chávez, O. (2014). Opportunities to engage with proof: the nature of proof tasks in two geometry textbooks and its influence on enacted lessons. *ZDM, 46*(5), 767-780.

Zhang, T., & Dong, H. (2008). *Human-Centered Design: An Emergent Conceptual Model*. 1–7.
# VISUALIZATION AND SPATIAL VISUALIZATION IN GEOMETRY

Danielle Harris, Tracy Logan, Tom Lowrie

University of Canberra

*The disciplines of mathematics education and psychology have various definitions for visualization and spatial visualization with important implications for geometry. In this study we explored the influence of spatial visualization, as measured in psychology, and visualization on primary pre-proof geometry tasks. Results suggest that spatial visualization and visualization are related but distinct. Implications are discussed for incorporating psychological findings into geometry education.*

## INTRODUCTION

Ask mathematics educators what links geometry, visualization, and spatial reasoning and they might conclude that they are part of the same idea; that visualization is integral to geometry, geometry is inherently spatial, and spatial reasoning is geometry (Battista et al., 2018; Gutiérrez, 1996; Woolcott et al., 2020). Posing the same question to cognitive psychologists may initiate a discussion around the mechanisms that link geometry with spatial reasoning and visualization. Each of these discussions are underpinned by the definitions adopted in the respective fields (Resnick & Stieff, 2024).

In this paper, we conceptualize geometry in the context of primary-school curricula. From this perspective, space and geometry are woven together in curriculum guidelines (ACARA, 2023) and teaching practice (Lowrie & Logan, 2018). Primary, pre-proof geometry is fundamentally about exploring shape and space (Battista et al., 2018). These foundations may lead to the ability to visualize abstract geometric concepts when solving proofs (Arcavi, 2003) but primary school geometry largely involves imagining transformations, exploring representations, and engaging with concrete materials, all of which form the basis of *visualization* and are supported by spatial skills (Gutiérrez et al., 2018).

In parallel, cognitive psychologists find the strongest links to spatial skills within mathematics in geometry (Xie et al., 2020). However, geometry is rarely discussed in psychological literature because of the difficulty untangling geometric conventions from pure spatial skills (Newcombe, 2018). Spatial skills are clearly defined in psychology, yet they are examined in a vacuum and removed from their applications within classrooms (Resnick & Stieff, 2024). Although psychological research identifies specific spatial skills beneficial for geometry, there is minimal support for how those skills might be enacted in practice (Hawes et al., 2023). In this study we explore visualization and spatial visualization as they relate to geometry problem-solving, with the aim to understand the affordances of psychological spatial constructs for primary-school geometry.

## Visualization

Visualization incorporates mental processes and the use of representational tools (such as diagrams) and has been considered synonymous with spatial reasoning (Gutiérrez, 1996). Psychological views of visualization are traditionally positioned more narrowly, for example in the process of restructuring, manipulation or transformation of objects (Ekstrom et al., 1976). Such definitions are
# VISUALIZATION AND SPATIAL VISUALIZATION IN GEOMETRY

Danielle Harris, Tracy Logan, Tom Lowrie

University of Canberra

*The disciplines of mathematics education and psychology have various definitions for visualization and spatial visualization with important implications for geometry. In this study we explored the influence of spatial visualization, as measured in psychology, and visualization on primary pre-proof geometry tasks. Results suggest that spatial visualization and visualization are related but distinct. Implications are discussed for incorporating psychological findings into geometry education.*

## INTRODUCTION

Ask mathematics educators what links geometry, visualization, and spatial reasoning and they might conclude that they are part of the same idea; that visualization is integral to geometry, geometry is inherently spatial, and spatial reasoning is geometry (Battista et al., 2018; Gutiérrez, 1996; Woolcott et al., 2020). Posing the same question to cognitive psychologists may initiate a discussion around the mechanisms that link geometry with spatial reasoning and visualization. Each of these discussions are underpinned by the definitions adopted in the respective fields (Resnick & Stieff, 2024).

In this paper, we conceptualize geometry in the context of primary-school curricula. From this perspective, space and geometry are woven together in curriculum guidelines (ACARA, 2023) and teaching practice (Lowrie & Logan, 2018). Primary, pre-proof geometry is fundamentally about exploring shape and space (Battista et al., 2018). These foundations may lead to the ability to visualize abstract geometric concepts when solving proofs (Arcavi, 2003) but primary school geometry largely involves imagining transformations, exploring representations, and engaging with concrete materials, all of which form the basis of *visualization* and are supported by spatial skills (Gutiérrez et al., 2018).

In parallel, cognitive psychologists find the strongest links to spatial skills within mathematics in geometry (Xie et al., 2020). However, geometry is rarely discussed in psychological literature because of the difficulty untangling geometric conventions from pure spatial skills (Newcombe, 2018). Spatial skills are clearly defined in psychology, yet they are examined in a vacuum and removed from their applications within classrooms (Resnick & Stieff, 2024). Although psychological research identifies specific spatial skills beneficial for geometry, there is minimal support for how those skills might be enacted in practice (Hawes et al., 2023). In this study we explore visualization and spatial visualization as they relate to geometry problem-solving, with the aim to understand the affordances of psychological spatial constructs for primary-school geometry.

## Visualization

Visualization incorporates mental processes and the use of representational tools (such as diagrams) and has been considered synonymous with spatial reasoning (Gutiérrez, 1996). Psychological views of visualization are traditionally positioned more narrowly, for example in the process of restructuring, manipulation or transformation of objects (Ekstrom et al., 1976). Such definitions are
by nature connected with mathematics content (e.g., imagining transformations on a co-ordinate grid; Battista et al., 2018). Yet often learners struggle to connect the notions of space and geometry in their mathematical practice that are intuitive to mathematics educators (Presmeg, 2014).

Seminal works in mathematics education conceptualize visualization as noun and verb, product and process; the generation of mental images and representations, and the integration of these in solving problems (Arcavi, 2003; Bishop, 1980). Underpinning these functions are the spatial skills that evolved out of psychology (Gutiérrez, 1996; Woolcott et al., 2020). However, the way psychologists conceptualize and measure spatial skills remains abstract (Resnick & Stieff, 2024). Although training spatial skills can improve mathematics achievement, it may be that these skills are so closely aligned to geometry that there is a transfer effect (Hawes et al., 2022, 2023) or that an intersection exists between the skills and their implementation within visualization that can be leveraged.

## The intersection of visualization and spatial visualization

Spatial visualization, the ability to imagine and transform objects and relations (Battista et al., 2018), forms one component of Gutiérrez’s (1996) visualization framework where visual or spatial elements are used to reason and problem-solve (Gutiérrez et al., 2018). Decades of research suggest that visualization is most effective when problem-solvers are supported by spatial skills and the ability to produce functional representations (Larkin & Simon, 1987; Presmeg, 2014).

Presmeg (1986) categorized visualization into types of imagery with different affordances for problem-solving and diverse solution pathways. The efficacy of these types of imagery would depend on the spatial skills of the problem-solver. For example, for pictorial imagery, strong spatial thinkers tend to construct mental images that facilitate problem-solving, while poor spatial thinkers tend to produce detailed mental images that do little to support problem-solving (Presmeg, 2006).

There is a pervasive discussion in the psychology community about the mechanisms linking spatial skills and mathematics (Resnick & Stieff, 2024). That is, whether a mechanistic or neural link exists between spatial and mathematics skills (Hawes et al., 2023) or is the link in the use of tools such as imagery, embodiment, and representation (Newcombe, 2018)? The former remains the focus of psychologists (Mix & Levine, 2018), the latter connects to the mathematical notion of visualization (Gutiérrez, 1996). Psychological research suggests that training isolated spatial skills is sufficient to improve geometry performance but proposes no reasons why or how to implement experimental findings into practice (Hawes et al., 2022, 2023). In an education framework we seek to understand how the skills or tools can be enacted in practice to develop content knowledge (Battista et al., 2018).

## Context of this study

Our early scenario neglected to consider the expertise of mathematics educators. The hypothetical educators may “see through symbolic forms” (Arcavi, 2003, p. 223) when they describe all geometry as spatial. Although possessing strong spatial skills, many experts do not default to spatial strategies (Uttal & Cohen, 2012). Despite the inherent link between geometry and space, we cannot assume that students are intuitively making these connections in their mathematical practice (Presmeg, 2014). Psychological research has isolated spatial skills that support geometry understanding. For example, Hawes et al., (2022) found spatial visualization to be the most trained spatial skill with the greatest link to geometry performance. However, to date, there are few studies that explore the links between spatial visualization and implementation in geometry problem-solving using tools of visualization.
by nature connected with mathematics content (e.g., imagining transformations on a co-ordinate grid; Battista et al., 2018). Yet often learners struggle to connect the notions of space and geometry in their mathematical practice that are intuitive to mathematics educators (Presmeg, 2014).

Seminal works in mathematics education conceptualize visualization as noun and verb, product and process; the generation of mental images and representations, and the integration of these in solving problems (Arcavi, 2003; Bishop, 1980). Underpinning these functions are the spatial skills that evolved out of psychology (Gutiérrez, 1996; Woolcott et al., 2020). However, the way psychologists conceptualize and measure spatial skills remains abstract (Resnick & Stieff, 2024). Although training spatial skills can improve mathematics achievement, it may be that these skills are so closely aligned to geometry that there is a transfer effect (Hawes et al., 2022, 2023) or that an intersection exists between the skills and their implementation within visualization that can be leveraged.

## The intersection of visualization and spatial visualization

Spatial visualization, the ability to imagine and transform objects and relations (Battista et al., 2018), forms one component of Gutiérrez’s (1996) visualization framework where visual or spatial elements are used to reason and problem-solve (Gutiérrez et al., 2018). Decades of research suggest that visualization is most effective when problem-solvers are supported by spatial skills and the ability to produce functional representations (Larkin & Simon, 1987; Presmeg, 2014).

Presmeg (1986) categorized visualization into types of imagery with different affordances for problem-solving and diverse solution pathways. The efficacy of these types of imagery would depend on the spatial skills of the problem-solver. For example, for pictorial imagery, strong spatial thinkers tend to construct mental images that facilitate problem-solving, while poor spatial thinkers tend to produce detailed mental images that do little to support problem-solving (Presmeg, 2006).

There is a pervasive discussion in the psychology community about the mechanisms linking spatial skills and mathematics (Resnick & Stieff, 2024). That is, whether a mechanistic or neural link exists between spatial and mathematics skills (Hawes et al., 2023) or is the link in the use of tools such as imagery, embodiment, and representation (Newcombe, 2018)? The former remains the focus of psychologists (Mix & Levine, 2018), the latter connects to the mathematical notion of visualization (Gutiérrez, 1996). Psychological research suggests that training isolated spatial skills is sufficient to improve geometry performance but proposes no reasons why or how to implement experimental findings into practice (Hawes et al., 2022, 2023). In an education framework we seek to understand how the skills or tools can be enacted in practice to develop content knowledge (Battista et al., 2018).

## Context of this study

Our early scenario neglected to consider the expertise of mathematics educators. The hypothetical educators may “see through symbolic forms” (Arcavi, 2003, p. 223) when they describe all geometry as spatial. Although possessing strong spatial skills, many experts do not default to spatial strategies (Uttal & Cohen, 2012). Despite the inherent link between geometry and space, we cannot assume that students are intuitively making these connections in their mathematical practice (Presmeg, 2014). Psychological research has isolated spatial skills that support geometry understanding. For example, Hawes et al., (2022) found spatial visualization to be the most trained spatial skill with the greatest link to geometry performance. However, to date, there are few studies that explore the links between spatial visualization and implementation in geometry problem-solving using tools of visualization.
*Research question:* Does the function of visualization strategies differ depending on spatial visualization skills in primary, pre-proof, geometry problem-solving?

# METHOD

## Participants

Grades 3, 4, and 5 students (N = 282, age range 8 – 11 years) from two (non-government) schools in Canberra participated<fnref n="24" />. One school was girls-only, the other was a co-educational primary school. Consequently, the sample consisted of 81.9% females. All students obtained parental consent and gave consent to have their data included in the study.

## Measures and procedure

Students completed a geometry test, a Mathematics Processing Instrument, and a spatial visualization test. All measures were administered on paper in the same order to class groups over three lessons.

*Geometry Test (GeoT):* The geometry items were sourced from Australia's National Assessment Program: Literacy and Numeracy (NAPLAN) for Grade 3 and 5 students. As there is no NAPLAN assessment for Grade 4, a combination of items from Grades 3 and 5 were used to provide a balanced assessment. Each grade had a unique, developmentally appropriate test (8 items for Grade 3, 10 items for Grade 4, and 13 items for Grade 5). Percentages were used for comparison purposes.

*Mathematics Processing Instrument (MPI):* After the GeoT, students completed an MPI (Suwarsono, 1982) which displayed the items and multiple-choice solution strategies based on previous work (Lowrie et al., 2016). Students were instructed to select the first strategy they used or use their own words. The strategies were categorized into analytical, mental imagery, and embodied. Analytical strategies involved calculation, counting or component matching. Mental imagery occurred when students reported imagining transformations in their mind. Embodied strategies included drawings or manipulating the question booklet. There were often multiple options within categories for each item. Initial inter-rater reliability was .80 and disputed ratings were discussed to reach agreement.

*Spatial Visualization (SV):* The items in the Paper Folding Test (16 items, 10 minutes; Lohman, 2012) displayed a series of paper folds. The folded paper was shown with holes and students had to pick the unfolded paper from multiple-choice options. This task is analogous to a psychological test (Ekstrom et al., 1976) but developed and normed on a primary school population (Cronbach's Alpha = .69).

# RESULTS

Figure 1 contains descriptive statistics. Students were deemed Low (n = 81), Mid (n = 105), or High (n = 80) Spatial using frequencies on the SV measure to determine three relatively equal groupings. Students completed all measures above chance, with no violations to normality. Analysis of Variance revealed significant differences between the spatial groups on the GeoT, F(2, 263) = 39.40, p < .001 with High Spatial students outperforming Mid and Low Spatial students and Mid outperforming Low Spatial students. There were significant differences in mental imagery based on SV, F(2, 265) = 5.15, p = .006 but not analytical or embodied strategies, F(2, 265) = .45, p = .64 and F(2, 265) = 2.40, p = .09 respectively. High Spatial students used more mental imagery than Low and Mid students.

<footnote n="24">²⁴ This sample was taken from a larger Australian Research Council Discovery project [DP230100328]</footnote>
*Research question:* Does the function of visualization strategies differ depending on spatial visualization skills in primary, pre-proof, geometry problem-solving?

# METHOD

## Participants

Grades 3, 4, and 5 students (N = 282, age range 8 – 11 years) from two (non-government) schools in Canberra participated<fnref n="24" />. One school was girls-only, the other was a co-educational primary school. Consequently, the sample consisted of 81.9% females. All students obtained parental consent and gave consent to have their data included in the study.

## Measures and procedure

Students completed a geometry test, a Mathematics Processing Instrument, and a spatial visualization test. All measures were administered on paper in the same order to class groups over three lessons.

*Geometry Test (GeoT):* The geometry items were sourced from Australia's National Assessment Program: Literacy and Numeracy (NAPLAN) for Grade 3 and 5 students. As there is no NAPLAN assessment for Grade 4, a combination of items from Grades 3 and 5 were used to provide a balanced assessment. Each grade had a unique, developmentally appropriate test (8 items for Grade 3, 10 items for Grade 4, and 13 items for Grade 5). Percentages were used for comparison purposes.

*Mathematics Processing Instrument (MPI):* After the GeoT, students completed an MPI (Suwarsono, 1982) which displayed the items and multiple-choice solution strategies based on previous work (Lowrie et al., 2016). Students were instructed to select the first strategy they used or use their own words. The strategies were categorized into analytical, mental imagery, and embodied. Analytical strategies involved calculation, counting or component matching. Mental imagery occurred when students reported imagining transformations in their mind. Embodied strategies included drawings or manipulating the question booklet. There were often multiple options within categories for each item. Initial inter-rater reliability was .80 and disputed ratings were discussed to reach agreement.

*Spatial Visualization (SV):* The items in the Paper Folding Test (16 items, 10 minutes; Lohman, 2012) displayed a series of paper folds. The folded paper was shown with holes and students had to pick the unfolded paper from multiple-choice options. This task is analogous to a psychological test (Ekstrom et al., 1976) but developed and normed on a primary school population (Cronbach's Alpha = .69).

# RESULTS

Figure 1 contains descriptive statistics. Students were deemed Low (n = 81), Mid (n = 105), or High (n = 80) Spatial using frequencies on the SV measure to determine three relatively equal groupings. Students completed all measures above chance, with no violations to normality. Analysis of Variance revealed significant differences between the spatial groups on the GeoT, F(2, 263) = 39.40, p < .001 with High Spatial students outperforming Mid and Low Spatial students and Mid outperforming Low Spatial students. There were significant differences in mental imagery based on SV, F(2, 265) = 5.15, p = .006 but not analytical or embodied strategies, F(2, 265) = .45, p = .64 and F(2, 265) = 2.40, p = .09 respectively. High Spatial students used more mental imagery than Low and Mid students.

<footnote n="24">²⁴ This sample was taken from a larger Australian Research Council Discovery project [DP230100328]</footnote>
<figure><img src="image_198.png" /><figcaption>Figure 1: (a) Descriptive statistics</figcaption></figure>

<figure><img src="image_199.png" /><figcaption>(b) Frequency of strategy use</figcaption></figure>

To determine the extent of association between GeoT, SV, and strategy use, correlations were calculated, including a strategy discrepancy variable (based on Battista, 1990) where higher scores relate to more mental imagery and lower scores relate to more analytical strategies (see Table 2).

<table><caption>Table 2: Correlation table *p < .05, **p < .01</caption><thead><tr><th></th><th>SV</th><th>Disc.</th><th>M.I.</th><th>E.</th><th>Analytical</th></tr></thead><tbody><tr><td>Geometry (GeoT)</td><td>.52**</td><td>.19**</td><td>.31**</td><td>-.16**</td><td>0.04</td></tr><tr><td>Spatial Visualization (SV)</td><td>-</td><td>0.12</td><td>.21**</td><td>-.15*</td><td>0.06</td></tr><tr><td>Discrepancy</td><td></td><td>-</td><td>.88**</td><td>-.25**</td><td>-.77**</td></tr><tr><td>Mental imagery (M.I.) strategies</td><td></td><td></td><td>-</td><td>-.44**</td><td>-.39**</td></tr><tr><td>Embodied (E.) strategies</td><td></td><td></td><td></td><td>-</td><td>-0.09</td></tr></tbody></table>

Performance on the GeoT had a strong, positive association with SV and the use of mental imagery but negative association with the use of embodied strategies. Spatial visualization was positively associated with mental imagery strategies and negatively associated with embodied strategies.

Figure 2 illustrates the impact of strategy dominance on GeoT as a function of SV. Low Spatial students who used mostly mental imagery tended to perform better on the GeoT compared with Low Spatial students who were more analytical. For Mid Spatial students, GeoT performance was fairly consistent, regardless of strategy dominance. For High Spatial students, those with a tendency for one dominant strategy performed better than those who used equal parts analytical and mental imagery.

<figure><img src="image_200.png" /><figcaption>Figure 2: Geometry performance by spatial level and strategy dominance</figcaption></figure>
<figure><img src="image_208.png" /><figcaption>Figure 1: (a) Descriptive statistics</figcaption></figure>

<figure><img src="image_209.png" /><figcaption>(b) Frequency of strategy use</figcaption></figure>

To determine the extent of association between GeoT, SV, and strategy use, correlations were calculated, including a strategy discrepancy variable (based on Battista, 1990) where higher scores relate to more mental imagery and lower scores relate to more analytical strategies (see Table 2).

<table><caption>Table 2: Correlation table *p < .05, **p < .01</caption><thead><tr><th></th><th>SV</th><th>Disc.</th><th>M.I.</th><th>E.</th><th>Analytical</th></tr></thead><tbody><tr><td>Geometry (GeoT)</td><td>.52**</td><td>.19**</td><td>.31**</td><td>-.16**</td><td>0.04</td></tr><tr><td>Spatial Visualization (SV)</td><td>-</td><td>0.12</td><td>.21**</td><td>-.15*</td><td>0.06</td></tr><tr><td>Discrepancy</td><td></td><td>-</td><td>.88**</td><td>-.25**</td><td>-.77**</td></tr><tr><td>Mental imagery (M.I.) strategies</td><td></td><td></td><td>-</td><td>-.44**</td><td>-.39**</td></tr><tr><td>Embodied (E.) strategies</td><td></td><td></td><td></td><td>-</td><td>-0.09</td></tr></tbody></table>

Performance on the GeoT had a strong, positive association with SV and the use of mental imagery but negative association with the use of embodied strategies. Spatial visualization was positively associated with mental imagery strategies and negatively associated with embodied strategies.

Figure 2 illustrates the impact of strategy dominance on GeoT as a function of SV. Low Spatial students who used mostly mental imagery tended to perform better on the GeoT compared with Low Spatial students who were more analytical. For Mid Spatial students, GeoT performance was fairly consistent, regardless of strategy dominance. For High Spatial students, those with a tendency for one dominant strategy performed better than those who used equal parts analytical and mental imagery.

<figure><img src="image_210.png" /><figcaption>Figure 2: Geometry performance by spatial level and strategy dominance</figcaption></figure>
Performance, visualization, and spatial visualization

Qualitative data was used to analyze strategy and performance on pairs of tasks (see Tables 3 and 4). Percentages relate to the proportion of students in each category<fnref n="25" /> who correctly solved the problem.

<table><thead><tr><th></th><th>Jack folded this piece of paper along the dotted lines to make a model.</th><th>Which one of these is the net of a cube?</th><th></th><th></th><th></th><th></th></tr><tr><th></th><th>Which of these models did Jack make?</th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>Correct</th><th>Imagery</th><th>Embodied</th><th>Analytical</th><th>Imagery</th><th>Embodied</th><th>Analytical</th></tr></thead><tbody><tr><th>Overall</th><td>77.8%</td><td>85.7%</td><td>93.6%</td><td>41.0%</td><td>16.7%</td><td>-</td></tr><tr><th>Low</th><td>84.6%</td><td>66.7%</td><td>71.4%</td><td>23.1%</td><td>7.1%</td><td>-</td></tr><tr><th>Mid</th><td>93.2%</td><td>-</td><td>80.8%</td><td>43.9%</td><td>36.4%</td><td>-</td></tr><tr><th>High</th><td>98.4%</td><td>-</td><td>87.5%</td><td>58.8%</td><td>-</td><td>-</td></tr></tbody></table>

<figure><img src="image_201.png" /></figure>

<figure><img src="image_202.png" /></figure>

<figure><img src="image_203.png" /></figure>

<figure><img src="image_204.png" /></figure>

<figure><img src="image_205.png" /><figcaption>Table 3: Net items with different difficulty</figcaption></figure>

Table 3 includes two net tasks but the first did not use any mathematical terminology while the second included the terms *net* and *cube*. The task with the added language convention was more difficult for all students. Spatial skills benefited students who used mental imagery. Even when students drew and cut out the shapes, they were not successful without knowledge of the mathematical language. In the task without the language convention, mental imagery was advantageous regardless of spatial skill. Success rates for analytical were still high, these involved matching components which, although methodical, still relied on visual processing, hence the greater success amongst High Spatial students.

<table><thead><tr><th></th><th>Jack folded this piece of paper along the dotted lines to make a model.</th><th>Which one of these is the net of a cube?</th><th></th><th></th><th></th><th></th></tr><tr><th></th><th>Which of these models did Jack make?</th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>Correct</th><th>Imagery</th><th>Embodied</th><th>Analytical</th><th>Imagery</th><th>Embodied</th><th>Analytical</th></tr></thead><tbody><tr><th>Overall</th><td>60.0%</td><td>46.2%</td><td>55.9%</td><td>50.6%</td><td>-</td><td>50.6%</td></tr><tr><th>Low</th><td>29.4%</td><td>-</td><td>29.4%</td><td>43.8%</td><td>-</td><td>54.2%</td></tr><tr><th>Mid</th><td>61%</td><td>33.3%</td><td>60.8%</td><td>40.7%</td><td>-</td><td>51.4%</td></tr><tr><th>High</th><td>86.1%</td><td>-</td><td>78.4%</td><td>70%</td><td>-</td><td>95.7%</td></tr></tbody></table>

<figure><img src="image_206.png" /></figure>

<figure><img src="image_207.png" /></figure>

Mike had a circular piece of paper. He folded it in half twice and cut a piece out as shown.

<figure><img src="image_208.png" /><figcaption>Table 4: SV tasks with similar difficulty, different intent</figcaption></figure>

Table 4 presented two items that required skills associated with spatial visualization (i.e., reflection and paper folding) and had similar difficulty (based on success) but the task objectives were different.

<footnote n="25">²⁵ Percentages were not reported when less than 5 students in a category used that strategy.</footnote>
Performance, visualization, and spatial visualization

Qualitative data was used to analyze strategy and performance on pairs of tasks (see Tables 3 and 4). Percentages relate to the proportion of students in each category<fnref n="25" /> who correctly solved the problem.

<table><thead><tr><th></th><th>Jack folded this piece of paper along the dotted lines to make a model.</th><th>Which one of these is the net of a cube?</th><th></th><th></th><th></th><th></th></tr><tr><th></th><th>Which of these models did Jack make?</th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>Correct</th><th>Imagery</th><th>Embodied</th><th>Analytical</th><th>Imagery</th><th>Embodied</th><th>Analytical</th></tr></thead><tbody><tr><th>Overall</th><td>77.8%</td><td>85.7%</td><td>93.6%</td><td>41.0%</td><td>16.7%</td><td>-</td></tr><tr><th>Low</th><td>84.6%</td><td>66.7%</td><td>71.4%</td><td>23.1%</td><td>7.1%</td><td>-</td></tr><tr><th>Mid</th><td>93.2%</td><td>-</td><td>80.8%</td><td>43.9%</td><td>36.4%</td><td>-</td></tr><tr><th>High</th><td>98.4%</td><td>-</td><td>87.5%</td><td>58.8%</td><td>-</td><td>-</td></tr></tbody></table>

<figure><img src="image_211.png" /></figure>

<figure><img src="image_212.png" /></figure>

<figure><img src="image_213.png" /></figure>

<figure><img src="image_214.png" /></figure>

<figure><img src="image_215.png" /></figure>

<figure><img src="image_216.png" /><figcaption>Table 3: Net items with different difficulty</figcaption></figure>

Table 3 includes two net tasks but the first did not use any mathematical terminology while the second included the terms *net* and *cube*. The task with the added language convention was more difficult for all students. Spatial skills benefited students who used mental imagery. Even when students drew and cut out the shapes, they were not successful without knowledge of the mathematical language. In the task without the language convention, mental imagery was advantageous regardless of spatial skill. Success rates for analytical were still high, these involved matching components which, although methodical, still relied on visual processing, hence the greater success amongst High Spatial students.

<table><thead><tr><th></th><th>Jack folded this piece of paper along the dotted lines to make a model.</th><th>Which one of these is the net of a cube?</th><th></th><th></th><th></th><th></th></tr><tr><th></th><th>Which of these models did Jack make?</th><th></th><th></th><th></th><th></th><th></th></tr><tr><th>Correct</th><th>Imagery</th><th>Embodied</th><th>Analytical</th><th>Imagery</th><th>Embodied</th><th>Analytical</th></tr></thead><tbody><tr><th>Overall</th><td>60.0%</td><td>46.2%</td><td>55.9%</td><td>50.6%</td><td>-</td><td>50.6%</td></tr><tr><th>Low</th><td>29.4%</td><td>-</td><td>29.4%</td><td>43.8%</td><td>-</td><td>54.2%</td></tr><tr><th>Mid</th><td>61%</td><td>33.3%</td><td>60.8%</td><td>40.7%</td><td>-</td><td>51.4%</td></tr><tr><th>High</th><td>86.1%</td><td>-</td><td>78.4%</td><td>70%</td><td>-</td><td>95.7%</td></tr></tbody></table>

<figure><img src="image_217.png" /></figure>

<figure><img src="image_218.png" /></figure>

Mike had a circular piece of paper. He folded it in half twice and cut a piece out as shown.

<figure><img src="image_219.png" /></figure>

<figure><img src="image_220.png" /><figcaption>Table 4: SV tasks with similar difficulty, different intent</figcaption></figure>

Table 4 presented two items that required skills associated with spatial visualization (i.e., reflection and paper folding) and had similar difficulty (based on success) but the task objectives were different.

<footnote n="25">²⁵ Percentages were not reported when less than 5 students in a category used that strategy.</footnote>