ICMI Study 26 document where it is emphasized how much “it is important to consider how students
can be offered a much wider variety of experiences (and inputs)” (p. 4).

Regarding the second research question, it became evident that the choice of visual mediators during
students' performance significantly influenced their ability to provide correct answers. This
phenomenon likely arises from the fact that the use of graphical representation facilitated students in
visually discerning the number of straight lines intersecting with the parabola. Conversely, when
employing the algebraic approach, students attempted to determine the angular coefficient, a method
that did not enable them to identify all the requisite straight lines. This study underscores the efficacy
of incorporating and utilizing graphical representation in successfully tackling the task at hand.
According to Duval’s (2006) research, this study showed the importance of management of different
representations, particularly the graphical representation of a mathematical object. The correct use of
graphical representation to solve the task showed students’ ability to handle the representations of a
parabola. Moreover, those who used the GG strategy correctly in the Type 1 task revealed an ability
to switch between algebraic and graphical representations. This grasp and management of different
representations failed when routines took over.

When tackling a new mathematical topic, the student is involved in mathematical activity pushed by
social needs such as acceptance, expectations and consensus of the teacher or schoolmates. The
mathematical routine has no link with the objective of the task situation and instead of being outcome-
oriented (as it should be in nature), it becomes process-oriented. Students who adopted a process that
belongs to the GR category (probably because they preferred to use a familiar practice that meets the
expectations of the teacher) instead focus attention on the request made by the task.

The use of the GG or GR categories strategies affected not only the percentage of correct answers,
but also the approach to the problem and the aim of this approach; in one case, it is geared toward
solving the task, while in the case of routines it is focused on the procedure employed. Further studies,
combined with interviews, will provide further insight into the causes and features of the choice and
implementation of respective resolution strategies.

References

Dell'Agnello, R., Gambini, A., Maffia, A., & Viola, G. (2022). University students' discourse about irreducible polynomials. In: Fernández, Ceneida, et al. (Eds.). *Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education*. Alicante, Spain, 18-23 of July 2022. Alicante: PME, 2022. Vol. 2. pp. 219-226.

Duval R. (1993). Registres de représentations sémiotiques et fonctionnement cognitif de la pensée. *Annales de Didactique et de Science Cognitives*. ULP, IREM Strasbourg. 5, 37-65.

Duval R. (2006), The cognitive analysis of problems of comprehension in the learning of mathematics. *Educational Studies in Mathematics*, 61(1), 103-131.

Gambini, A., Viola, G. & Ferretti, F. (2023). The influence of visual mediators in university students' resolution processes. In: *Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education*, in press.

Kuckartz, U. (2019). Qualitative text analysis: A systematic approach. In *Compendium for early career researchers in mathematics education* (pp. 181-197). Springer.

Lavie, I., Steiner, A., & Sfard, A. (2019). Routines we live by: From ritual to exploration. *Educational Studies in Mathematics*, 101(2), 153-176.
Mayring, P. (2015). Qualitative content analysis: Theoretical background and procedures. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), *Approaches to qualitative research in mathematics education. Examples of methodology and methods* (pp. 365–380). Springer.

Presmeg, N. (2006). Research on visualization in learning and teaching mathematics. In A. Gutierrez & P. Boero (Eds.), *Handbook of research on the psychology of mathematics education: Past, present and future* (pp. 205–236). Sense.

Presmeg, N. (2020). Visualization and learning in mathematics education. *Encyclopedia of mathematics education*, 900-904.

Ryve, A., Nilsson, P., & Pettersson, K. (2013). Analyzing effective communication in mathematics group work: The role of visual mediators and technical terms. *Educational Studies in Mathematics*, 82(3), 497-514.

Sfard, A. (2008). *Thinking as communicating. Human development, the growth of discourse, and mathematizing*. New York, NY: Cambridge University Press.
Mayring, P. (2015). Qualitative content analysis: Theoretical background and procedures. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), *Approaches to qualitative research in mathematics education. Examples of methodology and methods* (pp. 365–380). Springer.

Presmeg, N. (2006). Research on visualization in learning and teaching mathematics. In A. Gutierrez & P. Boero (Eds.), *Handbook of research on the psychology of mathematics education: Past, present and future* (pp. 205–236). Sense.

Presmeg, N. (2020). Visualization and learning in mathematics education. *Encyclopedia of mathematics education*, 900-904.

Ryve, A., Nilsson, P., & Pettersson, K. (2013). Analyzing effective communication in mathematics group work: The role of visual mediators and technical terms. *Educational Studies in Mathematics*, 82(3), 497–514.

Sfard, A. (2008). *Thinking as communicating. Human development, the growth of discourse, and mathematizing*. New York, NY: Cambridge University Press.
# FROM GEOMETRIC THINKING TO GEOMETRIC PRACTICE: THE POTENTIAL OF REPRESENTATIONS OF PRACTICE FOR TEACHING AND LEARNING GEOMETRY IN SECONDARY SCHOOLS

Patricio Herbst¹, Daniel Chazan²

¹ University of Michigan, USA ² University of Maryland, USA

*This contribution to the theoretical strand and the visualization substrand provides a commentary on possibilities for research and development in secondary geometry. We argue that visualization tools need to include tools for representing the human practices of doing geometry, particularly in the service of problematizing and supporting the evolution of concrete experiences with shape and space into mathematizing those experiences in social interaction and reflecting on those experiences.*

## TWO DIFFERENT POSITIONS IN RESEARCH IN GEOMETRY EDUCATION

Poincaré classically described geometry as “the art of reasoning well from badly-drawn figures.” In their graduate textbook on the subject, Herbst et al. (2017) build a definition of secondary geometry on that quote, assuming that, after elementary school, youth have had formative experiences with shape and space that knowledgeable adults might recognize as geometrical. They propose that

the development of geometric knowledge at the secondary level consists of the progressive sophistication of students’ intellectual means to model, predict, and control geometric representations — that is, the progressive sophistication in students’ ways of organizing those artifacts (words, diagrams, and others) so that they can be reliably used in making and transacting meanings (p. 3).

In proposing that definition, the authors complicate our field’s relationship to two positions that have currency in the education research community. The first of them can be called the *study of geometry* and takes for granted that geometry is a scholarly body of knowledge produced by mathematicians over centuries, converted into content of studies for students to acquire. The second can be called *experiencing geometry* and takes for granted that human experiences with shape and space across the lifespan confront them with challenges that they undertake by acting in ways that mathematically educated observers might describe as differentially knowledgeable of geometric concepts and properties. In the latter, it is worth noting that *space* alludes to any of the three spaces described by Berthelot and Salin (1995): The microspace of the page or screen (with objects much smaller than the human handling them), the macrospace of sea and landscapes (which much smaller human beings traverse), and the mesospace of objects of size commensurate with the human body.

Each of those positions, the study of geometry and experiencing geometry, has warranted distinct research agendas and educational proposals. In the area we call the study of geometry, the research questions concern understanding how students learn and teachers teach the academic content derived from that store of scholarly knowledge (Du & Zhang, 2019; Lawson & Chinappan, 2000; Levav-Waynberg & Leikin, 2012). This research has also included studies of the cognition involved in reading and doing geometric proofs (e.g., Braithwaite, 2022; Cirillo & Hummer, 2021; Koedinger & Anderson, 1990; Heinze et al., 2008; Lin & Yang, 2007). This research has tended to focus on modeling the cognitive operations students make use of in tasks designed to study geometric
# FROM GEOMETRIC THINKING TO GEOMETRIC PRACTICE: THE POTENTIAL OF REPRESENTATIONS OF PRACTICE FOR TEACHING AND LEARNING GEOMETRY IN SECONDARY SCHOOLS

Patricio Herbst¹, Daniel Chazan²

¹ University of Michigan, USA ² University of Maryland, USA

*This contribution to the theoretical strand and the visualization substrand provides a commentary on possibilities for research and development in secondary geometry. We argue that visualization tools need to include tools for representing the human practices of doing geometry, particularly in the service of problematizing and supporting the evolution of concrete experiences with shape and space into mathematizing those experiences in social interaction and reflecting on those experiences.*

## TWO DIFFERENT POSITIONS IN RESEARCH IN GEOMETRY EDUCATION

Poincaré classically described geometry as “the art of reasoning well from badly-drawn figures.” In their graduate textbook on the subject, Herbst et al. (2017) build a definition of secondary geometry on that quote, assuming that, after elementary school, youth have had formative experiences with shape and space that knowledgeable adults might recognize as geometrical. They propose that

the development of geometric knowledge at the secondary level consists of the progressive sophistication of students’ intellectual means to model, predict, and control geometric representations — that is, the progressive sophistication in students’ ways of organizing those artifacts (words, diagrams, and others) so that they can be reliably used in making and transacting meanings (p. 3).

In proposing that definition, the authors complicate our field’s relationship to two positions that have currency in the education research community. The first of them can be called the *study of geometry* and takes for granted that geometry is a scholarly body of knowledge produced by mathematicians over centuries, converted into content of studies for students to acquire. The second can be called *experiencing geometry* and takes for granted that human experiences with shape and space across the lifespan confront them with challenges that they undertake by acting in ways that mathematically educated observers might describe as differentially knowledgeable of geometric concepts and properties. In the latter, it is worth noting that *space* alludes to any of the three spaces described by Berthelot and Salin (1995): The microspace of the page or screen (with objects much smaller than the human handling them), the macrospace of sea and landscapes (which much smaller human beings traverse), and the mesospace of objects of size commensurate with the human body.

Each of those positions, the study of geometry and experiencing geometry, has warranted distinct research agendas and educational proposals. In the area we call the study of geometry, the research questions concern understanding how students learn and teachers teach the academic content derived from that store of scholarly knowledge (Du & Zhang, 2019; Lawson & Chinappan, 2000; Levav-Waynberg & Leikin, 2012). This research has also included studies of the cognition involved in reading and doing geometric proofs (e.g., Braithwaite, 2022; Cirillo & Hummer, 2021; Koedinger & Anderson, 1990; Heinze et al., 2008; Lin & Yang, 2007). This research has tended to focus on modeling the cognitive operations students make use of in tasks designed to study geometric
knowledge and investigate the effectiveness of teaching and learning approaches to promote this study. This research on the study of geometry has led to the development of interventions (e.g., cognitive tutors) whose effectiveness might be measured in terms of students' successful completion of school tasks. In contrast, the current we call experiencing geometry has warranted research on how children and youth's activity in the physical (or in virtual) world(s) calls for them to act and develop more sophisticated ways of acting geometrically (Hall et al., 2014, Herbst & Boileau, 2018; Nathan & Walkington, 2017; Dimmel, et al., 2021; Soto, 2022). This research done by educational psychologists and mathematics educators has concentrated on interpreting human activity using geometry concepts and casting human actions as attesting to geometric meanings even if those actions are not valued in school. In this line of research, the extent to which students can consciously relate their activity to geometric knowledge has been less of a focus. This line of research has also led to the design of experiences, spaces, and technologies in which students can experience geometry and act in increasingly sophisticated ways (e.g., Ludwig & Jesberg, 2015).

<figure><img src="image_29.png" /><figcaption>Figure 1. Experiencing geometry and the study of geometry</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>Figure 2. A modeling perspective (from Herbst et al., 2017, p. 4)</figcaption></figure>

The diagram in Figure 1 can help represent these two research perspectives in their differences and their similarities. While research on the study of geometry is concerned with understanding the interactions between school students and school geometry content (bottom of Figure 1), research on experiencing geometry is concerned with interpreting the interactions between children and youth and the real world (top of figure 1). In both cases, geometry is a resource for the observer: In the first, the discipline of geometry legitimizes the content of studies as well as informs researchers' interpretation of what students do in those tasks; in the second, the discipline of geometry informs the researchers' selection of the real world objects and activities in which they observe youth and the researchers' interpretation of what the youth do therein. Geometry is wielded by the observer to sanction youth's meanings either by certifying matches and mismatches (in the study of geometry) or by providing the key for projecting correspondences (in the case of experiencing geometry). In both cases, one still has to ask on what basis may one assert that the subjective experience of the youth can be described as geometric.

Both approaches seem to ignore that the geometries that serve such sanctioning are the products of mathematicians practice and that mathematical practice is what endows the eventual mathematical knowledge with this mathematical status. Poincaré's notion of "thinking correctly about poorly drawn figures" is an epigram of this mathematical practice and alludes to how mathematicians' work with
knowledge and investigate the effectiveness of teaching and learning approaches to promote this study. This research on the study of geometry has led to the development of interventions (e.g., cognitive tutors) whose effectiveness might be measured in terms of students' successful completion of school tasks. In contrast, the current we call experiencing geometry has warranted research on how children and youth's activity in the physical (or in virtual) world(s) calls for them to act and develop more sophisticated ways of acting geometrically (Hall et al., 2014, Herbst & Boileau, 2018; Nathan & Walkington, 2017; Dimmel, et al., 2021; Soto, 2022). This research done by educational psychologists and mathematics educators has concentrated on interpreting human activity using geometry concepts and casting human actions as attesting to geometric meanings even if those actions are not valued in school. In this line of research, the extent to which students can consciously relate their activity to geometric knowledge has been less of a focus. This line of research has also led to the design of experiences, spaces, and technologies in which students can experience geometry and act in increasingly sophisticated ways (e.g., Ludwig & Jesberg, 2015).

<figure><img src="image_34.png" /><figcaption>Figure 1. Experiencing geometry and the study of geometry</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>Figure 2. A modeling perspective (from Herbst et al., 2017, p. 4)</figcaption></figure>

The diagram in Figure 1 can help represent these two research perspectives in their differences and their similarities. While research on the study of geometry is concerned with understanding the interactions between school students and school geometry content (bottom of Figure 1), research on experiencing geometry is concerned with interpreting the interactions between children and youth and the real world (top of figure 1). In both cases, geometry is a resource for the observer: In the first, the discipline of geometry legitimizes the content of studies as well as informs researchers' interpretation of what students do in those tasks; in the second, the discipline of geometry informs the researchers' selection of the real world objects and activities in which they observe youth and the researchers' interpretation of what the youth do therein. Geometry is wielded by the observer to sanction youth's meanings either by certifying matches and mismatches (in the study of geometry) or by providing the key for projecting correspondences (in the case of experiencing geometry). In both cases, one still has to ask on what basis may one assert that the subjective experience of the youth can be described as geometric.

Both approaches seem to ignore that the geometries that serve such sanctioning are the products of mathematicians practice and that mathematical practice is what endows the eventual mathematical knowledge with this mathematical status. Poincaré's notion of "thinking correctly about poorly drawn figures" is an epigram of this mathematical practice and alludes to how mathematicians' work with
representations (e.g., diagrams, inscriptions) seeks to overcome the limitations of representations
through disciplined thinking. The communications about that thinking that have served to build
geometries as bodies of knowledge (using publications and letters) over more than 5000 years (Scriba
& Schreiber, 2015) in which concrete problems (e.g., architectural design) interacted with theoretical
problems (e.g., the possibility of a construction) have been key for constituting geometry as a body
of knowledge. While the approach we call the study of geometry seems to extract the product of such
evolution to use it as an academic text of studies, the approach we call experiencing geometry takes
for granted how mathematical practice has historically connected human experiences with shape and
space and imposes on these experiences a geometric status that is in fact contingent on mathematical
practice. The modeling perspective Herbst et al. (2017) propose seeks to create in classrooms the
practices that would allow students to turn their experiences with shape and space into discourses that
might be comparable to the geometries produced through mathematical practice.

A MODELING PERSPECTIVE ON SECONDARY GEOMETRY EDUCATION

Figure 2 represents that mathematical activity as a process of modeling: The construction of representations of real world objects and activities (e.g., drawing perspective diagrams of buildings) and the visualization of real world phenomena (e.g., buildings) in terms of such diagrams is at once a first concrete modeling activity (what Kuzniak, 2018, would call the first geometric working space) and the source of a second modeling activity where the key features of those representations are proposed (e.g., in the form of definitions and assumptions; such as Piero della Francesca’s theory of perspective; Peterson, 1997; or what Kuzniak, 2018, would call the second geometric working space). This second, geometric modeling, as it accumulates and gets organized by logical principles (e.g., axiomatics), may give rise to the geometries that eventually legitimate the school geometry content (what Kuzniak, 2018, calls the third geometric working space). A key feature of the mathematical modeling described is an epistemological shift in the relationship with representations (e.g., diagrams): While at one time representations portray the real and hence they can be read as disclosing (e.g., through visualization) other features of the real, at another time, these representations are constrained by stipulations (e.g., definitions, norms for visualization) that allow some visualizations as reasonable but discourage others, and furthermore necessitate some visualizations as reasonable even if they are not perceived. This is what Herbst (2004) called a generative interaction with diagrams; other representations (e.g., configurations of objects taken as representations of real-world objects and activities, or media representing geometric objects over time) are open to the same considerations.

The conceptualization of secondary school geometry offered by Herbst et al. (2017) seeks to organize
the teaching and learning of geometry so that students can engage in a similar mathematical modeling
of their relationships with diagrams and other representations as Poincaré and other geometers have
done. They represent this engagement with the diagram in Figure 2. In this conception of secondary
geometry, the geometric experience in the world (e.g., the human creation of artifacts that others
might endow of geometric meaning) is the beginning of a modeling process that aims at producing
those meanings as ways of controlling inferences about the represented world. If the research and
development agendas that motivated geometers to create geometric knowledge in the first place are
not available to teachers and students at present, the general question one needs to ask is in what
representations (e.g., diagrams, inscriptions) seeks to overcome the limitations of representations
through disciplined thinking. The communications about that thinking that have served to build
geometries as bodies of knowledge (using publications and letters) over more than 5000 years (Scriba
& Schreiber, 2015) in which concrete problems (e.g., architectural design) interacted with theoretical
problems (e.g., the possibility of a construction) have been key for constituting geometry as a body
of knowledge. While the approach we call the study of geometry seems to extract the product of such
evolution to use it as an academic text of studies, the approach we call experiencing geometry takes
for granted how mathematical practice has historically connected human experiences with shape and
space and imposes on these experiences a geometric status that is in fact contingent on mathematical
practice. The modeling perspective Herbst et al. (2017) propose seeks to create in classrooms the
practices that would allow students to turn their experiences with shape and space into discourses that
might be comparable to the geometries produced through mathematical practice.

A MODELING PERSPECTIVE ON SECONDARY GEOMETRY EDUCATION

Figure 2 represents that mathematical activity as a process of modeling: The construction of representations of real world objects and activities (e.g., drawing perspective diagrams of buildings) and the visualization of real world phenomena (e.g., buildings) in terms of such diagrams is at once a first concrete modeling activity (what Kuzniak, 2018, would call the first geometric working space) and the source of a second modeling activity where the key features of those representations are proposed (e.g., in the form of definitions and assumptions; such as Piero della Francesca’s theory of perspective; Peterson, 1997; or what Kuzniak, 2018, would call the second geometric working space). This second, geometric modeling, as it accumulates and gets organized by logical principles (e.g., axiomatics), may give rise to the geometries that eventually legitimate the school geometry content (what Kuzniak, 2018, calls the third geometric working space). A key feature of the mathematical modeling described is an epistemological shift in the relationship with representations (e.g., diagrams): While at one time representations portray the real and hence they can be read as disclosing (e.g., through visualization) other features of the real, at another time, these representations are constrained by stipulations (e.g., definitions, norms for visualization) that allow some visualizations as reasonable but discourage others, and furthermore necessitate some visualizations as reasonable even if they are not perceived. This is what Herbst (2004) called a generative interaction with diagrams; other representations (e.g., configurations of objects taken as representations of real-world objects and activities, or media representing geometric objects over time) are open to the same considerations.

The conceptualization of secondary school geometry offered by Herbst et al. (2017) seeks to organize
the teaching and learning of geometry so that students can engage in a similar mathematical modeling
of their relationships with diagrams and other representations as Poincaré and other geometers have
done. They represent this engagement with the diagram in Figure 2. In this conception of secondary
geometry, the geometric experience in the world (e.g., the human creation of artifacts that others
might endow of geometric meaning) is the beginning of a modeling process that aims at producing
those meanings as ways of controlling inferences about the represented world. If the research and
development agendas that motivated geometers to create geometric knowledge in the first place are
not available to teachers and students at present, the general question one needs to ask is in what
experiential context can specific geometric concepts become sensible ways of giving students control
over the meanings they make of those experiences.

The general question raised above is not new and indeed it represents a whole field of research in geometry education—the didactique of geometry (Houdement & Kuzniak, 2006). An important question within that approach concerns students’ development of the need to represent (including the need to characterize discursively) the properties of concrete representations of their geometry experiences. Poincaré’s notion of “reasoning well” does not stress enough the importance that actual statements (i.e., semiotic productions about those poorly drawn diagrams) have in the production of such reasoning (Duval, 1995). If the study of geometry may take those statements as canonical elements of the content of studies, experiencing geometry may take those statements as unnecessary to constitute the geometric meaning of experience. In contrast, the didactique of geometry asks in what conditions might the statement of properties about geometric representations be instrumental for students to control how those representations are made and interpreted.

Didacticians of mathematics are mathematics education researchers concerned with the meaningful construction of mathematical knowledge in schools. Didacticians of geometry can see value in both types of research described above (research on the study of geometry and on experiencing geometry) while still noting that there is a gap that practitioners are bound to recognize: Students at times might produce the right answers to school tasks without providing any evidence that their actions in the real world are informed by geometric understanding; students might also encounter school tasks for which their actions in the world could provide valuable prior knowledge to use but fail to recognize those connections. That is, it is possible, even common, to see that knowledge fails to inform action and that skilled action fails to inform knowing. A compelling illustration of this gap in practice can be read in Schoenfeld’s (1989) observation of students who had proved theorems about lines tangents to circles but failed to use them to control the ways in which they constructed a circle tangent to intersecting lines. Equally interesting are observations of adults who know geometry but fail to use this knowledge to understand practical problems in housekeeping such as moving furniture or fixing exercising equipment. For sure, we are not saying that those practical activities require an explicit use of geometry, but that a way to make geometric knowledge and its meanings valuable to people is to create the conditions in which they recognize this knowledge as enabling them to act differently.

The separation between problem types that Schoenfeld (1989) alluded to suggests not only an undesirable outcome in geometry education, but also a gap in education design and research. The definition of secondary geometry proposed by Herbst et al (2017) seeks to bridge the gap illustrated by those examples by proposing a modeling approach to geometry. By this they mean

the development of the capacity to intellectually organize, predict, and control the world of representations
of physical objects and experiences. We describe this as a *modeling perspective* insofar as it sees as central
the process of creating models of diagrams and other concrete representations (p. 3)

Fleshing out this conception of secondary geometry requires the development of a kind of instruction that connects meaningful experiences in the spaces in which students (could) live with the mathematical ontologies, epistemologies, and axiologies produced by the discipline of mathematics. The reasons one might want to do that have been articulated over more than a century of geometry instruction (González & Herbst, 2006). They boil down to the belief that geometric knowledge can make lives better in a variety of ways (where lives include those of the mathematics enthusiast, the
experiential context can specific geometric concepts become sensible ways of giving students control
over the meanings they make of those experiences.

The general question raised above is not new and indeed it represents a whole field of research in geometry education—the didactique of geometry (Houdement & Kuzniak, 2006). An important question within that approach concerns students’ development of the need to represent (including the need to characterize discursively) the properties of concrete representations of their geometry experiences. Poincaré’s notion of “reasoning well” does not stress enough the importance that actual statements (i.e., semiotic productions about those poorly drawn diagrams) have in the production of such reasoning (Duval, 1995). If the study of geometry may take those statements as canonical elements of the content of studies, experiencing geometry may take those statements as unnecessary to constitute the geometric meaning of experience. In contrast, the didactique of geometry asks in what conditions might the statement of properties about geometric representations be instrumental for students to control how those representations are made and interpreted.

Didacticians of mathematics are mathematics education researchers concerned with the meaningful construction of mathematical knowledge in schools. Didacticians of geometry can see value in both types of research described above (research on the study of geometry and on experiencing geometry) while still noting that there is a gap that practitioners are bound to recognize: Students at times might produce the right answers to school tasks without providing any evidence that their actions in the real world are informed by geometric understanding; students might also encounter school tasks for which their actions in the world could provide valuable prior knowledge to use but fail to recognize those connections. That is, it is possible, even common, to see that knowledge fails to inform action and that skilled action fails to inform knowing. A compelling illustration of this gap in practice can be read in Schoenfeld’s (1989) observation of students who had proved theorems about lines tangents to circles but failed to use them to control the ways in which they constructed a circle tangent to intersecting lines. Equally interesting are observations of adults who know geometry but fail to use this knowledge to understand practical problems in housekeeping such as moving furniture or fixing exercising equipment. For sure, we are not saying that those practical activities require an explicit use of geometry, but that a way to make geometric knowledge and its meanings valuable to people is to create the conditions in which they recognize this knowledge as enabling them to act differently.

The separation between problem types that Schoenfeld (1989) alluded to suggests not only an undesirable outcome in geometry education, but also a gap in education design and research. The definition of secondary geometry proposed by Herbst et al (2017) seeks to bridge the gap illustrated by those examples by proposing a modeling approach to geometry. By this they mean

the development of the capacity to intellectually organize, predict, and control the world of representations
of physical objects and experiences. We describe this as a *modeling perspective* insofar as it sees as central
the process of creating models of diagrams and other concrete representations (p. 3)

Fleshing out this conception of secondary geometry requires the development of a kind of instruction that connects meaningful experiences in the spaces in which students (could) live with the mathematical ontologies, epistemologies, and axiologies produced by the discipline of mathematics. The reasons one might want to do that have been articulated over more than a century of geometry instruction (González & Herbst, 2006). They boil down to the belief that geometric knowledge can make lives better in a variety of ways (where lives include those of the mathematics enthusiast, the
manual worker, the city dweller, the computer gamer, etc.). Inasmuch as secondary schools can be
places where students explore various possible lives, the design of such instruction does not need to
be limited to any given conception of the real (material or social) world but can make different worlds
(including virtual worlds in immersive digital environments) available to students’ exploration.

The added complexity of secondary geometry instruction is how to make experience in real worlds an object of mathematical modeling in which geometric ontologies, epistemologies, and axiologies, can emerge. By geometric ontologies we refer to the notion that geometric objects and concepts as construed in the discipline of mathematics are different kinds of beings than the beings students encounter in their daily, real or virtual, experiences with space and shape; and that one role of secondary geometry is for students to create models of the beings they encounter in their experiential world that are comparable with the concepts and objects of geometry. By geometric epistemologies we refer to the notion that coming to know in the discipline of mathematics (including calculating, constructing, exploring, proving) involves a different relationship to knowledge than coming to know in concrete experiences with shape and space: While the latter may seek timely effectiveness or efficiency in context, the former seeks enduring truth and reproducibility across contexts. By geometric axiologies we mean that the production of geometric knowledge in the discipline is sustained by ethical and aesthetical values that are not necessarily present in the experiences with shape and space in which youth are involved daily: While the latter may privilege the self’s well-being (succeeding, feeling good) and their preferred relationship with their temporal and physical surroundings (seeking usefulness or comfort), the ethical and aesthetical values of the discipline seek to add an outer orientation (to develop knowledge that enlightens the world) and depersonalizable ways of appreciating the world (seeking intersubjective truth and beauty, for example). Clearly, that secondary geometry may pursue the expansion of students ontological, epistemological, and axiological relations with the world, does not denigrate earlier experiences with shape and space or suggest that the mathematics discipline provides the only way of expanding those experiences—the visual and performative arts clearly are other such disciplines that expand similar earlier experiences toward alternative relationships with the world. Just as the recognition of a drawing as art requires an artistic processing of the drawing, the recognition of experiences with shape and space as geometry require a geometric modeling of those experiences. If outside observers may bestow the labels of art or geometry to such drawing or experiences, the possibility that the producer of such drawing or the protagonist of such experience might be able to construct meaning for such labels, use them productively in later practice, and be able to allocate them appropriately are still different, compelling possibilities. And we are saying that secondary geometry is the place where learning to do such geometric modeling needs to happen—a geometric practice may unfold there: A reflective discourse of action with shapes and spaces in social circumstances that require communication progressively commensurate with that of mathematicians. In what follows we are much more concrete, focusing on construction as part of geometric practice and for the sake of illustrating the intricacies of a pedagogy (or, should we say, a didactique) that seeks to induct students into a geometric practice.

USING REPRESENTATIONS OF EXPERIENCES WITH SHAPE AND SPACE TO TEACH

Students’ induction into a geometric practice can be supported by a different kind of curriculum
development, one that engages students with representations of practice. We propose that for
manual worker, the city dweller, the computer gamer, etc.). Inasmuch as secondary schools can be
places where students explore various possible lives, the design of such instruction does not need to
be limited to any given conception of the real (material or social) world but can make different worlds
(including virtual worlds in immersive digital environments) available to students’ exploration.

The added complexity of secondary geometry instruction is how to make experience in real worlds an object of mathematical modeling in which geometric ontologies, epistemologies, and axiologies, can emerge. By geometric ontologies we refer to the notion that geometric objects and concepts as construed in the discipline of mathematics are different kinds of beings than the beings students encounter in their daily, real or virtual, experiences with space and shape; and that one role of secondary geometry is for students to create models of the beings they encounter in their experiential world that are comparable with the concepts and objects of geometry. By geometric epistemologies we refer to the notion that coming to know in the discipline of mathematics (including calculating, constructing, exploring, proving) involves a different relationship to knowledge than coming to know in concrete experiences with shape and space: While the latter may seek timely effectiveness or efficiency in context, the former seeks enduring truth and reproducibility across contexts. By geometric axiologies we mean that the production of geometric knowledge in the discipline is sustained by ethical and aesthetical values that are not necessarily present in the experiences with shape and space in which youth are involved daily: While the latter may privilege the self’s well-being (succeeding, feeling good) and their preferred relationship with their temporal and physical surroundings (seeking usefulness or comfort), the ethical and aesthetical values of the discipline seek to add an outer orientation (to develop knowledge that enlightens the world) and depersonalizable ways of appreciating the world (seeking intersubjective truth and beauty, for example). Clearly, that secondary geometry may pursue the expansion of students ontological, epistemological, and axiological relations with the world, does not denigrate earlier experiences with shape and space or suggest that the mathematics discipline provides the only way of expanding those experiences—the visual and performative arts clearly are other such disciplines that expand similar earlier experiences toward alternative relationships with the world. Just as the recognition of a drawing as art requires an artistic processing of the drawing, the recognition of experiences with shape and space as geometry require a geometric modeling of those experiences. If outside observers may bestow the labels of art or geometry to such drawing or experiences, the possibility that the producer of such drawing or the protagonist of such experience might be able to construct meaning for such labels, use them productively in later practice, and be able to allocate them appropriately are still different, compelling possibilities. And we are saying that secondary geometry is the place where learning to do such geometric modeling needs to happen—a geometric practice may unfold there: A reflective discourse of action with shapes and spaces in social circumstances that require communication progressively commensurate with that of mathematicians. In what follows we are much more concrete, focusing on construction as part of geometric practice and for the sake of illustrating the intricacies of a pedagogy (or, should we say, a didactique) that seeks to induct students into a geometric practice.

USING REPRESENTATIONS OF EXPERIENCES WITH SHAPE AND SPACE TO TEACH

Students’ induction into a geometric practice can be supported by a different kind of curriculum
development, one that engages students with representations of practice. We propose that for
experiences with shape and space to become objects of geometric practice in which students may produce knowledge compatible with what is typically aimed for in the study of geometry, students needs to engage in work that problematizes those experiences, making them subject of inquiry. Ontological, epistemological, and axiological transitions can be promoted through multimodal argumentation via a semiotic infrastructure of practice.

Consider as an example the theorem that characterizes an angle bisector as the locus of the points equidistant from the sides of an angle. Earlier in their studies, students have likely constructed angle bisectors using a protractor. How can that experience be turned into an object of reflection and argument in which the theorem listed above can be conjectured and become a useful control? Arguably, this requires appropriating the goal to describe the drawing as a set of points with some properties. Figure 3 shows a storyboard that can support discussions promoting students’ conjecturing of the theorem and their use to engage in arguments in which points are singled out and described in terms of being or not being on the angle bisector as well as in terms of their distance to the sides.

<figure><img src="image_31.png" /><figcaption>Figure 3. Toward seeing the angle bisector as made of points with particular properties.</figcaption></figure>

Consider high school students being asked to annotate this storyboard, including answering questions such as “What is the point of posing these questions at the board as opposed to giving them to each student on a worksheet?”, “How is the teacher choosing which points to erase and to keep in Frames 4 and 6?”, “Would there be a way to know whether a segment determined by two points will cross the angle bisector without drawing the angle bisector?”, “What could you do to find a point that would turn out to be on the angle bisector?”, or, “If a segment is such that you think it would intersect the angle bisector, could you find out where they would intersect before drawing the angle bisector?” These questions would likely get students to make a conjecture along the lines that for a segment to intersect the bisector, one of its extremes would need to be closer to one side than the other and the other extreme closer to the other side and then that for a point to be on the angle bisector, it should be at the same distance from both sides of the angle. The task itself could be used to promote the conjecture as a theorem-in-action; it might be sensible to start the lesson by having students try themselves the question in Frame 1.
experiences with shape and space to become objects of geometric practice in which students may produce knowledge compatible with what is typically aimed for in the study of geometry, students needs to engage in work that problematizes those experiences, making them subject of inquiry. Ontological, epistemological, and axiological transitions can be promoted through multimodal argumentation via a semiotic infrastructure of practice.

Consider as an example the theorem that characterizes an angle bisector as the locus of the points equidistant from the sides of an angle. Earlier in their studies, students have likely constructed angle bisectors using a protractor. How can that experience be turned into an object of reflection and argument in which the theorem listed above can be conjectured and become a useful control? Arguably, this requires appropriating the goal to describe the drawing as a set of points with some properties. Figure 3 shows a storyboard that can support discussions promoting students’ conjecturing of the theorem and their use to engage in arguments in which points are singled out and described in terms of being or not being on the angle bisector as well as in terms of their distance to the sides.

<figure><img src="image_36.png" /><figcaption>Figure 3. Toward seeing the angle bisector as made of points with particular properties.</figcaption></figure>

Consider high school students being asked to annotate this storyboard, including answering questions such as “What is the point of posing these questions at the board as opposed to giving them to each student on a worksheet?”, “How is the teacher choosing which points to erase and to keep in Frames 4 and 6?”, “Would there be a way to know whether a segment determined by two points will cross the angle bisector without drawing the angle bisector?”, “What could you do to find a point that would turn out to be on the angle bisector?,” or, “If a segment is such that you think it would intersect the angle bisector, could you find out where they would intersect before drawing the angle bisector?” These questions would likely get students to make a conjecture along the lines that for a segment to intersect the bisector, one of its extremes would need to be closer to one side than the other and the other extreme closer to the other side and then that for a point to be on the angle bisector, it should be at the same distance from both sides of the angle. The task itself could be used to promote the conjecture as a theorem-in-action; it might be sensible to start the lesson by having students try themselves the question in Frame 1.
Offering the storyboard and posing questions that involve the interpretation of the storyboard may help initiate the above-named transitions. The storyboard instantiates a semiotic infrastructure (icons, indices, and symbols organized in a multimodal grammar; see Herbst et al., 2023) that supports talking about points as both actual and potential depending on events in the storyline (e.g., between Frames 4 and 5) and of constructions as anticipated and performed (e.g., between Frames 1 and 3). In the context of a discussion of that semiotic environment (e.g., the narration or critique of the story), there is a possible transition in the ontological status of points: Points are no longer only specific objects in the diagram, but also can be described (i.e., modeled) as possible contingencies among properties (i.e., points can be characterized as closer or farther from one or the other side). Further, the storyboard supports an epistemological transition away from drawing to see whether something is the case to anticipating what could or would happen if one were to draw. The storyboard medium provides also access to the storyline not only as a vicarious experience but also as a replayable experience that can support an applied rationalist (as opposed to empirical) construction of reality along the lines of Bachelard’s “Le réel n’est jamais ce qu’on pourrait croire, mais il est toujours ce qu’on aurait dû penser.”<fnref n="4" /> The surveyability of experiences with shape and space that a storyboard provides allows for someone who would initially only know from observing what happened to one who might come to realize they should have anticipated the outcome. Finally, and perhaps this is the most important aspect of the storyboard representation of geometric experience, this semiotic mediation supports decentering knowledge from the function of serving a selfish goal (e.g., solving one’s problems) to solving problems that anyone could have, without eliding the human need for knowing. Thus, while the reader of the storyboard is not herself on the spot in Frame 6, they might feel empathy with the student at the board that could compel them to offer a characterization of the points as a solution to a human need. The ethical option of helping the student at the board may further support the aesthetical value of obtaining a good definition of the distance between a point and a line.

**Acknowledgment**

The writing of this paper was supported by National Science Foundation grant DUE- 1725837. All opinions are those of the authors and do not necessarily represent the views of the Foundation.

**References**

Berthelot, R., & Salin, M. H. (1998). The role of pupils' spatial knowledge in the elementary teaching of geometry. In C. Mammana & V. Villani, (Eds.) *Perspectives on the teaching of geometry for the 21st century: An ICMI study* (pp. 71-77). Kluwer.

Braithwaite, D. W. (2022). Relations between geometric proof justification and probabilistic reasoning. *Learning and Individual Differences, 98*, 102201.

Cirillo, M., & Hummer, J. (2021). Competencies and behaviors observed when students solve geometry proof problems: An interview study with smartpen technology. *ZDM-Mathematics Education, 53*, 861-875.

Dimmel, J., Pandiscio, E., & Bock, C. (2021). The geometry of movement: Encounters with spatial inscriptions for making and exploring mathematical figures. *Digital Experiences in Mathematics Education, 7*, 122-148.

Du, X., & Zhang, Q. (2019). Tracing worked examples: Effects on learning in geometry. *Educational Psychology, 39*(2), 169-187.

<footnote n="4">⁴ The real is never what one could come to believe but rather that which one should have thought about.</footnote>
Offering the storyboard and posing questions that involve the interpretation of the storyboard may help initiate the above-named transitions. The storyboard instantiates a semiotic infrastructure (icons, indices, and symbols organized in a multimodal grammar; see Herbst et al., 2023) that supports talking about points as both actual and potential depending on events in the storyline (e.g., between Frames 4 and 5) and of constructions as anticipated and performed (e.g., between Frames 1 and 3). In the context of a discussion of that semiotic environment (e.g., the narration or critique of the story), there is a possible transition in the ontological status of points: Points are no longer only specific objects in the diagram, but also can be described (i.e., modeled) as possible contingencies among properties (i.e., points can be characterized as closer or farther from one or the other side). Further, the storyboard supports an epistemological transition away from drawing to see whether something is the case to anticipating what could or would happen if one were to draw. The storyboard medium provides also access to the storyline not only as a vicarious experience but also as a replayable experience that can support an applied rationalist (as opposed to empirical) construction of reality along the lines of Bachelard’s “Le réel n’est jamais ce qu’on pourrait croire, mais il est toujours ce qu’on aurait dû penser.”<fnref n="4" /> The surveyability of experiences with shape and space that a storyboard provides allows for someone who would initially only know from observing what happened to one who might come to realize they should have anticipated the outcome. Finally, and perhaps this is the most important aspect of the storyboard representation of geometric experience, this semiotic mediation supports decentering knowledge from the function of serving a selfish goal (e.g., solving one’s problems) to solving problems that anyone could have, without eliding the human need for knowing. Thus, while the reader of the storyboard is not herself on the spot in Frame 6, they might feel empathy with the student at the board that could compel them to offer a characterization of the points as a solution to a human need. The ethical option of helping the student at the board may further support the aesthetical value of obtaining a good definition of the distance between a point and a line.

**Acknowledgment**

The writing of this paper was supported by National Science Foundation grant DUE- 1725837. All opinions are those of the authors and do not necessarily represent the views of the Foundation.

**References**

Berthelot, R., & Salin, M. H. (1998). The role of pupils' spatial knowledge in the elementary teaching of geometry. In C. Mammana & V. Villani, (Eds.) *Perspectives on the teaching of geometry for the 21st century: An ICMI study* (pp. 71-77). Kluwer.

Braithwaite, D. W. (2022). Relations between geometric proof justification and probabilistic reasoning. *Learning and Individual Differences, 98*, 102201.

Cirillo, M., & Hummer, J. (2021). Competencies and behaviors observed when students solve geometry proof problems: An interview study with smartpen technology. *ZDM-Mathematics Education, 53*, 861-875.

Dimmel, J., Pandiscio, E., & Bock, C. (2021). The geometry of movement: Encounters with spatial inscriptions for making and exploring mathematical figures. *Digital Experiences in Mathematics Education, 7*, 122-148.

Du, X., & Zhang, Q. (2019). Tracing worked examples: Effects on learning in geometry. *Educational Psychology, 39*(2), 169-187.

<footnote n="4">⁴ The real is never what one could come to believe but rather that which one should have thought about.</footnote>
Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.) *Exploiting mental imagery with computers in mathematics education* (pp. 142-157). Springer.

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

Hall, R., Ma, J. Y., & Nemirovsky, R. (2014). Rescaling bodies in/as representational instruments in GPS drawing. In *Learning technologies and the body* (pp. 112-131). Routledge.

Heinze, A., Cheng, Y. H., Ufer, S., Lin, F. L., & Reiss, K. (2008). Strategies to foster students’ competencies in constructing multi-steps geometric proofs: Teaching experiments in Taiwan and Germany. *ZDM—Mathematics Education, 40*, 443-453.

Herbst, P. (2004). Interactions with diagrams and the making of reasoned conjectures in geometry. *ZDM—Mathematics Education, 36*(5), 129-139. https://doi.org/10.1007/bf02655665

Herbst, P. & Boileau, N. (2018). Geometric modeling of mesospace objects: A task, its didactical variables, and the mathematics at stake. In K. Mix & M. Battista (Eds.). *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (p. 277-308). Springer.

Herbst, P., Chazan, D., & Schleppegrell, M. (2023). How digital storyboards support the transaction of practice: The semiotic infrastructure of representations of practice. In: Pepin, B., Gueudet, G., Choppin, J. (Eds), *Handbook of Digital Resources in Mathematics Education*. Springer.

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

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

Koedinger, K. R., & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. *Cognitive Science, 14*(4), 511-550.

Kuzniak, A. (2018). Thinking about the teaching of geometry through the lens of the theory of geometric working spaces. In P. Herbst, U. Cheah, K. Jones, & P. Richard (Eds.), *International Perspectives on the Teaching and Learning of Geometry in Secondary Schools* (pp. 5-21). Springer.

Lawson, M. & Chinnappan, M. (2000). Knowledge connectedness in geometry problem solving. *Journal for Research in Mathematics Education, 31* (1), 26-43

Levav-Waynberg, A., & Leikin, R. (2012). The role of multiple solution tasks in developing knowledge and creativity in geometry. *The Journal of Mathematical Behavior, 31*(1), 73-90.

Lin, F. L., & Yang, K. L. (2007). The reading comprehension of geometric proofs: The contribution of knowledge and reasoning. *International Journal of Science and Mathematics Education, 5*, 729-754.

Ludwig, M., & Jesberg, J. (2015). Using mobile technology to provide outdoor modelling tasks-The MathCityMap-Project. *Procedia-Social and Behavioral Sciences, 191*, 2776-2781.

Nathan, M. J., & Walkington, C. (2017). Grounded and embodied mathematical cognition: Promoting mathematical insight and proof using action and language. *Cognitive Research: Principles and Implications, 2*(9).

Peterson, M. A. (1997). The geometry of Piero della Francesca. *The Mathematical Intelligencer, 19*, 33-40.

Schoenfeld, A. H. (1989). Explorations of students' mathematical beliefs and behavior. *Journal for Research in Mathematics Education, 20*(4), 338-355.

Scriba, C. J., & Schreiber, P. (2015). *5000 years of geometry: Mathematics in history and culture*. Birkhäuser.

Soto, H. (2022). Acting Out Euclidean Transformations. *PRIMUS, 32*(8), 902-916.
Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.) *Exploiting mental imagery with computers in mathematics education* (pp. 142-157). Springer.

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

Hall, R., Ma, J. Y., & Nemirovsky, R. (2014). Rescaling bodies in/as representational instruments in GPS drawing. In *Learning technologies and the body* (pp. 112-131). Routledge.

Heinze, A., Cheng, Y. H., Ufer, S., Lin, F. L., & Reiss, K. (2008). Strategies to foster students' competencies in constructing multi-steps geometric proofs: Teaching experiments in Taiwan and Germany. *ZDM—Mathematics Education, 40*, 443-453.

Herbst, P. (2004). Interactions with diagrams and the making of reasoned conjectures in geometry. *ZDM—Mathematics Education, 36*(5), 129-139. https://doi.org/10.1007/bf02655665

Herbst, P. & Boileau, N. (2018). Geometric modeling of mesospace objects: A task, its didactical variables, and the mathematics at stake. In K. Mix & M. Battista (Eds.). *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (p. 277-308). Springer.

Herbst, P., Chazan, D., & Schleppegrell, M. (2023). How digital storyboards support the transaction of practice: The semiotic infrastructure of representations of practice. In: Pepin, B., Gueudet, G., Choppin, J. (Eds), *Handbook of Digital Resources in Mathematics Education*. Springer.

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

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

Koedinger, K. R., & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. *Cognitive Science, 14*(4), 511-550.

Kuzniak, A. (2018). Thinking about the teaching of geometry through the lens of the theory of geometric working spaces. In P. Herbst, U. Cheah, K. Jones, & P. Richard (Eds.), *International Perspectives on the Teaching and Learning of Geometry in Secondary Schools* (pp. 5-21). Springer.

Lawson, M. & Chinnappan, M. (2000). Knowledge connectedness in geometry problem solving. *Journal for Research in Mathematics Education, 31* (1), 26-43

Levav-Waynberg, A., & Leikin, R. (2012). The role of multiple solution tasks in developing knowledge and creativity in geometry. *The Journal of Mathematical Behavior, 31*(1), 73-90.

Lin, F. L., & Yang, K. L. (2007). The reading comprehension of geometric proofs: The contribution of knowledge and reasoning. *International Journal of Science and Mathematics Education, 5*, 729-754.

Ludwig, M., & Jesberg, J. (2015). Using mobile technology to provide outdoor modelling tasks-The MathCityMap-Project. *Procedia-Social and Behavioral Sciences, 191*, 2776-2781.

Nathan, M. J., & Walkington, C. (2017). Grounded and embodied mathematical cognition: Promoting mathematical insight and proof using action and language. *Cognitive Research: Principles and Implications, 2*(9).

Peterson, M. A. (1997). The geometry of Piero della Francesca. *The Mathematical Intelligencer, 19*, 33-40.

Schoenfeld, A. H. (1989). Explorations of students' mathematical beliefs and behavior. *Journal for Research in Mathematics Education, 20*(4), 338-355.

Scriba, C. J., & Schreiber, P. (2015). *5000 years of geometry: Mathematics in history and culture*. Birkhäuser.

Soto, H. (2022). Acting Out Euclidean Transformations. *PRIMUS, 32*(8), 902-916.
PROOFS IN GEOMETRY TEACHING IN THE BRAZILIAN CONTEXT: A REPRESENTATION OF YESTERDAY AND TODAY

Ana Paula Jahn¹, Maria Célia Leme da Silva²

¹ University of São Paulo, ² Federal University of São Paulo

The topic regarding advances in geometry teaching leads us to reflect on the past of this teaching, mainly about changes in curricular proposals and how they were reflected in textbooks, aiming to identify progress or setbacks. This study focuses on investigating, from a cultural historical perspective, how the teaching of geometric proofs has been proposed and discussed in Brazilian regulations, as well as in textbooks. We analyzed three educational reforms from the first half of the 20th century, the 1998 National Curricular Parameters and the 2018 National Common Curricular Base, and we examined four textbooks representing different moments of changing approaches to teaching geometry, from 1930 to the present day. One result throughout the various regulations and pedagogical approaches found in textbooks is the persistent tension between intuitive and deductive geometry. The importance of the transition between pragmatic proofs and conceptual proofs in geometry teaching is a topic that remains central and relevant in the Brazilian reality.

INTRODUCTION

The call for ICMI 26 (Advances in Geometry Education) encourages us to think about the past of geometry teaching, changes in pedagogical proposals and how they were translated into didactic-pedagogical tasks in textbooks, to identify progress and setbacks. It is in this context that the present work is inserted, by choosing the theme of proofs in geometry teaching and analyzing, from a cultural historical perspective (Chartier, 2009), how this theme has been proposed and discussed in official curricular regulations in Brazil, as well as in textbooks. Thus, we took a period of almost one hundred years, from 1930 to the present day, beginning our analysis at the time of the unification of the fields of Algebra, Arithmetic and Geometry into a single school subject called Mathematics for secondary school programs (aimed at students aged 11 to 14, after primary school). The research sources were the regulations of three educational reforms from the first half of the 20th century (1931, 1942 and 1951), the National Curricular Parameters (PCN) at the end of the same century, and the last regulation, approved in 2018, the Base National Common Curricular (BNCC). We also selected four textbooks as representative of important moments in changing perspectives, the books by Osvaldo Sangiorgi from the 1950s and 1960s, to examine the approach of the Modern Mathematics Movement and the books by Edwaldo Bianchini from the 2010s and 2020s, to investigate possible changes between the last two regulations, the PCN (Brasil, 1998) and the BNCC (Brasil, 2018).

It is necessary to highlight that the study considers and examines national regulations, however Brazil is a country of continental dimensions, with 27 states distributed in 5 regions, with important social, economic, cultural, and educational differences, each state having the autonomy to organize its school programs under national regulations. Thus, the choice of textbooks, which will be explained below, represents a sample that we consider significant, but without any intention of representing the entire
# PROOFS IN GEOMETRY TEACHING IN THE BRAZILIAN CONTEXT: A REPRESENTATION OF YESTERDAY AND TODAY

Ana Paula Jahn¹, Maria Célia Leme da Silva²

¹ University of São Paulo, ² Federal University of São Paulo

The topic regarding advances in geometry teaching leads us to reflect on the past of this teaching, mainly about changes in curricular proposals and how they were reflected in textbooks, aiming to identify progress or setbacks. This study focuses on investigating, from a cultural historical perspective, how the teaching of geometric proofs has been proposed and discussed in Brazilian regulations, as well as in textbooks. We analyzed three educational reforms from the first half of the 20th century, the 1998 National Curricular Parameters and the 2018 National Common Curricular Base, and we examined four textbooks representing different moments of changing approaches to teaching geometry, from 1930 to the present day. One result throughout the various regulations and pedagogical approaches found in textbooks is the persistent tension between intuitive and deductive geometry. The importance of the transition between pragmatic proofs and conceptual proofs in geometry teaching is a topic that remains central and relevant in the Brazilian reality.

## INTRODUCTION

The call for ICMI 26 (Advances in Geometry Education) encourages us to think about the past of geometry teaching, changes in pedagogical proposals and how they were translated into didactic-pedagogical tasks in textbooks, to identify progress and setbacks. It is in this context that the present work is inserted, by choosing the theme of proofs in geometry teaching and analyzing, from a cultural historical perspective (Chartier, 2009), how this theme has been proposed and discussed in official curricular regulations in Brazil, as well as in textbooks. Thus, we took a period of almost one hundred years, from 1930 to the present day, beginning our analysis at the time of the unification of the fields of Algebra, Arithmetic and Geometry into a single school subject called Mathematics for secondary school programs (aimed at students aged 11 to 14, after primary school). The research sources were the regulations of three educational reforms from the first half of the 20th century (1931, 1942 and 1951), the National Curricular Parameters (PCN) at the end of the same century, and the last regulation, approved in 2018, the Base National Common Curricular (BNCC). We also selected four textbooks as representative of important moments in changing perspectives, the books by Osvaldo Sangiorgi from the 1950s and 1960s, to examine the approach of the Modern Mathematics Movement and the books by Edwaldo Bianchini from the 2010s and 2020s, to investigate possible changes between the last two regulations, the PCN (Brasil, 1998) and the BNCC (Brasil, 2018).

It is necessary to highlight that the study considers and examines national regulations, however Brazil is a country of continental dimensions, with 27 states distributed in 5 regions, with important social, economic, cultural, and educational differences, each state having the autonomy to organize its school programs under national regulations. Thus, the choice of textbooks, which will be explained below, represents a sample that we consider significant, but without any intention of representing the entire
national reality. The focus of the investigation is not on an in-depth study of specific moments, but rather on analyzing long-term processes with the necessary methodological care, to support the historical narrative of the issue at hand. The objective is to analyze how the proofs were and are proposed in this selection to discuss the different conceptions of proofs for teaching geometry to students aged 11 to 14, in the period.

## Educational Reforms in Brazil in the first half of the 20th century

The reflections of the international movement to reform mathematics teaching, discussed at the IV International Mathematics Congress, held in Rome in 1908, under the leadership of Félix Klein (1849-1925), were appropriated in Brazilian regulations only in the 1930s. Euclides Roxo (1890-1950), engineer and teacher at Colégio Pedro II (a reference in Brazilian secondary education) was the leader of the renovation in the Francisco Campos Reform in 1931, which instituted, for the first time, the subject Mathematics for the secondary course.

According to Roxo, this reform brought together the trends of the international movement, both from a methodological point of view, as in the contents, by merging the different fields of mathematics: arithmetic, algebra, and geometry into a single discipline. The instructions for teaching mathematics recommended that:

> Starting from the living and concrete intuition, the logical feature will grow, little by little, until it gradually reaches formal exposition; [...] knowledge will be acquired, at first through experimentation and sensory perception and then, slowly, through analytical reasoning. Thus, regarding geometry, the formal demonstrative study must be preceded by a propaedeutic course, aimed at intuitive teaching, of an experimental and constructive nature (Bicudo, 1942, p. 157).

The detailed analysis of the regulations emphasizes the importance of intuitive and experimental teaching preceding geometric proofs. The 1931 Reform lasted a short time and was reorganized in 1942, by the Capanema Reform. In 1951, the Simões Filho Reform kept the 1942 structure. For geometry teaching, the three reforms designated deductive geometry, that is, the introduction to the study of geometry proofs for the final grades (students aged 13 to 14) as we can see in Table 1.

<table><caption>Table 1: Geometry Teaching in prescribed programs (1931-1951). Source: The authors based on the regulations.</caption><thead><tr><td>Reform</td><td>Geometry organization</td><td>Instructions</td></tr></thead><tbody><tr><td>Campos (1931)</td><td>Geometric Initiation Geometry</td><td>- Geometry propaedeutic course<br/>- Starting from intuition to <strong>gradually reach</strong> formal exposure: from experimentation and sensory perception to analytical reasoning</td></tr><tr><td>Capanema (1942)</td><td>Intuitive Geometry Deductive Geometry</td><td>- Intuitive geometry as a <strong>smooth transaction</strong> between experiences with shapes and deductive conception of Geometry</td></tr><tr><td>Simões Filho (1951)</td><td>Geometry (Only to students 13 and 14 years old)</td><td>- Do not ignore appeal to intuition<br/>- <strong>Gradually awaken</strong> the feeling of the need for justification and proof</td></tr></tbody></table>

The three reforms emphasized the need for an intuitive geometry, with experimentation, and sensory perception, as preparation for understanding a deductive geometry, but the 1951 reform excluded the space allocated to introductory and intuitive part of previous reforms, maintaining only the deductive approach. Chervel (1990) warns us about the difficulties in distinguishing objective finalities and real
national reality. The focus of the investigation is not on an in-depth study of specific moments, but rather on analyzing long-term processes with the necessary methodological care, to support the historical narrative of the issue at hand. The objective is to analyze how the proofs were and are proposed in this selection to discuss the different conceptions of proofs for teaching geometry to students aged 11 to 14, in the period.

## Educational Reforms in Brazil in the first half of the 20th century

The reflections of the international movement to reform mathematics teaching, discussed at the IV International Mathematics Congress, held in Rome in 1908, under the leadership of Félix Klein (1849-1925), were appropriated in Brazilian regulations only in the 1930s. Euclides Roxo (1890-1950), engineer and teacher at Colégio Pedro II (a reference in Brazilian secondary education) was the leader of the renovation in the Francisco Campos Reform in 1931, which instituted, for the first time, the subject Mathematics for the secondary course.

According to Roxo, this reform brought together the trends of the international movement, both from a methodological point of view, as in the contents, by merging the different fields of mathematics: arithmetic, algebra, and geometry into a single discipline. The instructions for teaching mathematics recommended that:

> Starting from the living and concrete intuition, the logical feature will grow, little by little, until it gradually reaches formal exposition; [...] knowledge will be acquired, at first through experimentation and sensory perception and then, slowly, through analytical reasoning. Thus, regarding geometry, the formal demonstrative study must be preceded by a propaedeutic course, aimed at intuitive teaching, of an experimental and constructive nature (Bicudo, 1942, p. 157).

The detailed analysis of the regulations emphasizes the importance of intuitive and experimental teaching preceding geometric proofs. The 1931 Reform lasted a short time and was reorganized in 1942, by the Capanema Reform. In 1951, the Simões Filho Reform kept the 1942 structure. For geometry teaching, the three reforms designated deductive geometry, that is, the introduction to the study of geometry proofs for the final grades (students aged 13 to 14) as we can see in Table 1.

<table><caption>Table 1: Geometry Teaching in prescribed programs (1931-1951). Source: The authors based on the regulations.</caption><thead><tr><td>Reform</td><td>Geometry organization</td><td>Instructions</td></tr></thead><tbody><tr><td>Campos (1931)</td><td>Geometric Initiation Geometry</td><td>- Geometry propaedeutic course<br/>- Starting from intuition to <strong>gradually reach</strong> formal exposure: from experimentation and sensory perception to analytical reasoning</td></tr><tr><td>Capanema (1942)</td><td>Intuitive Geometry Deductive Geometry</td><td>- Intuitive geometry as a <strong>smooth transaction</strong> between experiences with shapes and deductive conception of Geometry</td></tr><tr><td>Simões Filho (1951)</td><td>Geometry (Only to students 13 and 14 years old)</td><td>- Do not ignore appeal to intuition<br/>- <strong>Gradually awaken</strong> the feeling of the need for justification and proof</td></tr></tbody></table>

The three reforms emphasized the need for an intuitive geometry, with experimentation, and sensory perception, as preparation for understanding a deductive geometry, but the 1951 reform excluded the space allocated to introductory and intuitive part of previous reforms, maintaining only the deductive approach. Chervel (1990) warns us about the difficulties in distinguishing objective finalities and real
finalities and, in this sense, the study of regulations indicates the objective finalities of teaching, however, pedagogical practices are those that reveal the real finalities, and they do not always converge. For instance, in the 1950s, despite the regulations signaling attention for intuitive study and gradually awakening the need for justifications, the reports on practices in classrooms that took place at the II National Congress of Mathematics Teaching, in 1957, constitute traces of what the approach to deductive geometry was like, associated with memorizing the proofs of theorems:

> The lack of initial logical concatenation of the theorems and the intuitive character of most of them produces the harmful impression, in the student's mind, that the proofs constitute juggling on the part of the teacher. [...] It also does not form a clear idea of what a theory is or does not have a theory at all. [...] The ill-fated theorems have been the lifeline of mediocre students who, in exams, achieve the minimum grade thanks to a memorized formal proofs (Congresso ..., 1959 *apud* Búrigo, 2015, p. 8).

Despite the attempt to insert the principles of the international movement of the beginning of the 20th century into Brazilian regulations, the innovations suffered a lot of resistance from a traditional school culture (Julia, 2001), based on the rigor and abstraction that was expected from a study of deductive geometry. We reached the middle of the 20th century with the teaching of deductive geometry characterized by the exposition of a set of postulates, followed by numerous formal proofs of a significant set of theorems, to be memorized and taken in exams, as we will see below.

## Textbooks by Osvaldo Sangiorgi in the 1950s and 1960s

Osvaldo Sangiorgi (1921-2017), mathematician and teacher, author of best-selling textbooks in several editions since the 1950s, was a relevant name in the process of appropriation and dissemination of the ideas of the Modern Mathematics Movement (MMM) in Brazil. He did an internship at the University of Kansas, in the USA, in the mid-60s and when he returned to Brazil, even without changing the regulations, he created a new collection of textbooks with modern proposals and expressive editions.

The study of Jahn and Leme da Silva (2023) compared the first textbooks by Osvaldo Sangiorgi (OS) that introduced students to deductive geometry (13-year-old students) in the 1950s, considered pre-modern and, in 1960, modern. An initial quantitative analysis indicated that practically the number of pages dedicated to the study of geometry remained stable, however the approach between the two textbooks proved to be completely different. When comparing the topics related to the teaching of geometry - in common in the two textbooks - it was identified that the same concepts started to occupy 28% more space in the modern collection. On the other hand, the number of theorems proved in the two textbooks went from 54 to 28, that is, a reduction of around 50% in the number of theorems. For the study of proofs, the same have indicated changes in the selection of theorems and in the approach to presenting them. The qualitative analysis made it possible to understand the numerical differences: 120 preparatory pages were inserted before starting the presentation of postulates and theorems, and the author explains the change:

> At this stage, the student is already “accustomed” to seeing that some properties are consequences of other more elementary ones. It is the preparation for the beginning of a simple axiomatization, where the situations encountered in the **exploratory exercises** will be brought together in the form of postulates (axioms). A theory (recommended by Hilbert) is thus constructed from primitive concepts, postulates, and theorems, easily proved, through a logical chain of reasoning. [...] The current treatment used to begin deductive geometry is very different from what was traditionally done. **Only after much experience** can the student be initiated into the **proofs** themselves (Sangiorgi, 1966, p. 51, emphasis added).
finalities and, in this sense, the study of regulations indicates the objective finalities of teaching, however, pedagogical practices are those that reveal the real finalities, and they do not always converge. For instance, in the 1950s, despite the regulations signaling attention for intuitive study and gradually awakening the need for justifications, the reports on practices in classrooms that took place at the II National Congress of Mathematics Teaching, in 1957, constitute traces of what the approach to deductive geometry was like, associated with memorizing the proofs of theorems:

> The lack of initial logical concatenation of the theorems and the intuitive character of most of them produces the harmful impression, in the student's mind, that the proofs constitute juggling on the part of the teacher. [...] It also does not form a clear idea of what a theory is or does not have a theory at all. [...] The ill-fated theorems have been the lifeline of mediocre students who, in exams, achieve the minimum grade thanks to a memorized formal proofs (Congresso ..., 1959 *apud* Búrigo, 2015, p. 8).

Despite the attempt to insert the principles of the international movement of the beginning of the 20th century into Brazilian regulations, the innovations suffered a lot of resistance from a traditional school culture (Julia, 2001), based on the rigor and abstraction that was expected from a study of deductive geometry. We reached the middle of the 20th century with the teaching of deductive geometry characterized by the exposition of a set of postulates, followed by numerous formal proofs of a significant set of theorems, to be memorized and taken in exams, as we will see below.

## Textbooks by Osvaldo Sangiorgi in the 1950s and 1960s

Osvaldo Sangiorgi (1921-2017), mathematician and teacher, author of best-selling textbooks in several editions since the 1950s, was a relevant name in the process of appropriation and dissemination of the ideas of the Modern Mathematics Movement (MMM) in Brazil. He did an internship at the University of Kansas, in the USA, in the mid-60s and when he returned to Brazil, even without changing the regulations, he created a new collection of textbooks with modern proposals and expressive editions.

The study of Jahn and Leme da Silva (2023) compared the first textbooks by Osvaldo Sangiorgi (OS) that introduced students to deductive geometry (13-year-old students) in the 1950s, considered pre-modern and, in 1960, modern. An initial quantitative analysis indicated that practically the number of pages dedicated to the study of geometry remained stable, however the approach between the two textbooks proved to be completely different. When comparing the topics related to the teaching of geometry - in common in the two textbooks - it was identified that the same concepts started to occupy 28% more space in the modern collection. On the other hand, the number of theorems proved in the two textbooks went from 54 to 28, that is, a reduction of around 50% in the number of theorems. For the study of proofs, the same have indicated changes in the selection of theorems and in the approach to presenting them. The qualitative analysis made it possible to understand the numerical differences: 120 preparatory pages were inserted before starting the presentation of postulates and theorems, and the author explains the change:

> At this stage, the student is already “accustomed” to seeing that some properties are consequences of other more elementary ones. It is the preparation for the beginning of a simple axiomatization, where the situations encountered in the **exploratory exercises** will be brought together in the form of postulates (axioms). A theory (recommended by Hilbert) is thus constructed from primitive concepts, postulates, and theorems, easily proved, through a logical chain of reasoning. [...] The current treatment used to begin deductive geometry is very different from what was traditionally done. **Only after much experience** can the student be initiated into the **proofs** themselves (Sangiorgi, 1966, p. 51, emphasis added).
OS clarifies that the exploratory exercises were introduced so that the student can experiment with verifying some properties as a result of others, an attempt to insert preparatory experimental geometry before the deductive study. The author also indicates the construction of a new axiomatization for proofs in which the exploratory exercises carried out by students become postulates when proving the theorems. Everything indicates that the main point is the understanding of a deductive process, which incorporates “new postulates”, to allow a more adequate understanding of logical-deductive reasoning.

An example of exploratory exercises proposed by OS in the 1960 textbook were the geometric constructions of triangles involving cases of congruence, in which students were invited to carry out the constructions of triangles with the respective data, for example, the case side, angle and side (LAL) and then compare their triangle with their classmates in order to identify congruence. In the 1950 textbook, the four cases of triangle congruences were stated as theorems and proved; while in the 1960 textbook, after the exploratory study carried out with geometric constructions, they were then taken as postulates. The list of postulates between the two textbooks was significantly altered, both in quantity and choice (Jahn & Leme da Silva, 2023). Another example that helps us understand the change in the proposal made by OS is the way the theorems are presented. The theorem on the property of the base angles of an isosceles triangle, in the 1950 textbook, was done in a classic way: the theorem is stated, with the figure on the side, the hypothesis and thesis are identified, and the proof is carried out. In the modern approach, students were first invited to carry out exploratory exercises, which correspond to constructing an isosceles triangle, measuring the angles, comparing with their colleagues whether the result obtained was also verified by the group and stating the identified property. Other exercises complement this stage, such as attention tests, which ask students to identify the corresponding vertices in different triangle congruence situations. OS presents three distinct proofs: the first as an “action plan”, which corresponds to drawing the bisector of the angle between the congruent sides; the second, as “another way” for the proof, tracing the height relative to the base of the isosceles triangle and the third, through “drawn diagrams”, colored preferably, where a series of deductions appear, through constructions (→), of equivalences (↔) and implications (⇒). The emphasis of different resolutions and different representations to proof the same theorem, so that the student can understand the diversity of paths in the deductive process, is an innovative aspect in Brazilian textbooks from the 1960s. OS encourages students to develop their own proofs, warning: “Don't 'memorize' proof of theorem! Value yourself, using any of the methods presented. Do it your way by using these methods and you will be accomplishing in Mathematics!” (Sangiorgi, 1967, p. 258).

It is necessary to consider that the appropriation made by OS for the study of modern deductive geometry must have been influenced by the North American trend based on Birkhoff; however, other trends were present in Brazilian textbooks, such as the proposal to teach geometry through geometric transformations linked to Klein. Anyhow, OS's modern approach to deductive geometry significantly changed the representation of geometry teaching in the 1950s of a formal deductive geometry, of memorization, both within the scope of Euclidean geometry and in the methodological didactic aspects, by proposing the insertion of exploratory exercises, in addition to different records of representations as a way of make understandable the deductive geometry.
OS clarifies that the exploratory exercises were introduced so that the student can experiment with verifying some properties as a result of others, an attempt to insert preparatory experimental geometry before the deductive study. The author also indicates the construction of a new axiomatization for proofs in which the exploratory exercises carried out by students become postulates when proving the theorems. Everything indicates that the main point is the understanding of a deductive process, which incorporates “new postulates”, to allow a more adequate understanding of logical-deductive reasoning.

An example of exploratory exercises proposed by OS in the 1960 textbook were the geometric constructions of triangles involving cases of congruence, in which students were invited to carry out the constructions of triangles with the respective data, for example, the case side, angle and side (LAL) and then compare their triangle with their classmates in order to identify congruence. In the 1950 textbook, the four cases of triangle congruences were stated as theorems and proved; while in the 1960 textbook, after the exploratory study carried out with geometric constructions, they were then taken as postulates. The list of postulates between the two textbooks was significantly altered, both in quantity and choice (Jahn & Leme da Silva, 2023). Another example that helps us understand the change in the proposal made by OS is the way the theorems are presented. The theorem on the property of the base angles of an isosceles triangle, in the 1950 textbook, was done in a classic way: the theorem is stated, with the figure on the side, the hypothesis and thesis are identified, and the proof is carried out. In the modern approach, students were first invited to carry out exploratory exercises, which correspond to constructing an isosceles triangle, measuring the angles, comparing with their colleagues whether the result obtained was also verified by the group and stating the identified property. Other exercises complement this stage, such as attention tests, which ask students to identify the corresponding vertices in different triangle congruence situations. OS presents three distinct proofs: the first as an “action plan”, which corresponds to drawing the bisector of the angle between the congruent sides; the second, as “another way” for the proof, tracing the height relative to the base of the isosceles triangle and the third, through “drawn diagrams”, colored preferably, where a series of deductions appear, through constructions (→), of equivalences (↔) and implications (⇒). The emphasis of different resolutions and different representations to proof the same theorem, so that the student can understand the diversity of paths in the deductive process, is an innovative aspect in Brazilian textbooks from the 1960s. OS encourages students to develop their own proofs, warning: “Don't 'memorize' proof of theorem! Value yourself, using any of the methods presented. Do it your way by using these methods and you will be accomplishing in Mathematics!” (Sangiorgi, 1967, p. 258).

It is necessary to consider that the appropriation made by OS for the study of modern deductive geometry must have been influenced by the North American trend based on Birkhoff; however, other trends were present in Brazilian textbooks, such as the proposal to teach geometry through geometric transformations linked to Klein. Anyhow, OS's modern approach to deductive geometry significantly changed the representation of geometry teaching in the 1950s of a formal deductive geometry, of memorization, both within the scope of Euclidean geometry and in the methodological didactic aspects, by proposing the insertion of exploratory exercises, in addition to different records of representations as a way of make understandable the deductive geometry.
In summary, in the period examined, from 1930 to 1960, it can be said that deductive geometry was present, both in Brazilian regulations and in textbooks, from the perspective of *conceptual proofs*<fnref n="5" />. However, the moment described above, of the incorporation of modernist ideals by OS, seems representative to us, which, even without abandoning the conceptual proof, the exposition of the postulates and the many theorems to 13-year-old students, innovates in the textbooks, through a differentiated proposal to prepare the study of deductive geometry, by inserting *pragmatic proofs*, a call to the student to build their own proofs, to argue, to justify. Everything indicates that the insertion of this new approach was not incorporated into school culture, however, it seems to have been the moment to decrease the formal approach, that of many theorems to be memorized.

## Educational reforms in Brazil at the end of the 20th century and beginning of the 21st century

From the 1970s onwards, criticism of the MMM became increasingly accentuated and, in parallel, a movement of reflection on mathematics teaching began, bringing together study groups, teachers, mathematicians, psychologists, educators, without, however, result in new curricular programs at a national level. At the end of the 20th century, and with the presence of mathematics educators participating in the development of curricular programs, the National Curricular Parameters (PCN) were implemented. Recently, in 2018, with controversy and criticism from educators, the National Common Curricular Base (BNCC) was implemented, which we began to examine comparatively, on how deductive geometry has been proposed.

The PCN (Brasil, 1998) continue to advise that the study of deductive geometry should be started for students aged 13 and emphasize that the skills of arguing and proving in Mathematics are very important for the formation of a critical spirit:

> Therefore, it is desirable to work on developing argumentation, so that students are not satisfied with just producing responses to statements but **have the attitude of always trying to justify them**. [...] Geometry activities are very suitable for the teacher to build, with his students, a path that, **based on concrete experiences, leads them to understand the importance and need for proof** to legitimize the hypotheses raised (Brasil, 1998, p. 71 and 126, emphasis added).

The document explains the relevance of tasks that favor the path from concrete experiences to the need for evidence, in this case, in the sense of pragmatic evidence. The formal proofs are not explained in the regulations. It is understood that the focus must be on arguing, justifying, and proving with usual language.

Twenty years later, the BNCC (Brasil, 2018) removes any discussion about proofs processes, the need for proof and the transition between intuitive and experimental geometry to deductive geometry. The regulations emphasize tasks involving concepts of congruence and similarity so that students “know how to apply this concept to perform simple proofs, contributing to the formation of an important type of reasoning for mathematics, hypothetical-deductive reasoning” (Brasil, 2018, p 272, emphasis added). However, we do not discuss or comment on what is meant by simple proofs. In this way, we can infer that the BNCC, compared to the PCN, presents a setback for the study of proofs,

<footnote n="5">⁵ We mobilize the concepts of pragmatic proofs as proofs that include resources of action and empirical observation on mathematical objects, aiming to verify or prove the mathematical property and conceptual or intellectual proofs are based on the formulation of relevant properties and connections are established between them in accordance with Balacheff (1988).</footnote>
In summary, in the period examined, from 1930 to 1960, it can be said that deductive geometry was present, both in Brazilian regulations and in textbooks, from the perspective of *conceptual proofs*<fnref n="5" />. However, the moment described above, of the incorporation of modernist ideals by OS, seems representative to us, which, even without abandoning the conceptual proof, the exposition of the postulates and the many theorems to 13-year-old students, innovates in the textbooks, through a differentiated proposal to prepare the study of deductive geometry, by inserting *pragmatic proofs*, a call to the student to build their own proofs, to argue, to justify. Everything indicates that the insertion of this new approach was not incorporated into school culture, however, it seems to have been the moment to decrease the formal approach, that of many theorems to be memorized.

## Educational reforms in Brazil at the end of the 20th century and beginning of the 21st century

From the 1970s onwards, criticism of the MMM became increasingly accentuated and, in parallel, a movement of reflection on mathematics teaching began, bringing together study groups, teachers, mathematicians, psychologists, educators, without, however, result in new curricular programs at a national level. At the end of the 20th century, and with the presence of mathematics educators participating in the development of curricular programs, the National Curricular Parameters (PCN) were implemented. Recently, in 2018, with controversy and criticism from educators, the National Common Curricular Base (BNCC) was implemented, which we began to examine comparatively, on how deductive geometry has been proposed.

The PCN (Brasil, 1998) continue to advise that the study of deductive geometry should be started for students aged 13 and emphasize that the skills of arguing and proving in Mathematics are very important for the formation of a critical spirit:

> Therefore, it is desirable to work on developing argumentation, so that students are not satisfied with just producing responses to statements but **have the attitude of always trying to justify them**. [...] Geometry activities are very suitable for the teacher to build, with his students, a path that, **based on concrete experiences, leads them to understand the importance and need for proof** to legitimize the hypotheses raised (Brasil, 1998, p. 71 and 126, emphasis added).

The document explains the relevance of tasks that favor the path from concrete experiences to the need for evidence, in this case, in the sense of pragmatic evidence. The formal proofs are not explained in the regulations. It is understood that the focus must be on arguing, justifying, and proving with usual language.

Twenty years later, the BNCC (Brasil, 2018) removes any discussion about proofs processes, the need for proof and the transition between intuitive and experimental geometry to deductive geometry. The regulations emphasize tasks involving concepts of congruence and similarity so that students “know how to apply this concept to perform simple proofs, contributing to the formation of an important type of reasoning for mathematics, hypothetical-deductive reasoning” (Brasil, 2018, p 272, emphasis added). However, we do not discuss or comment on what is meant by simple proofs. In this way, we can infer that the BNCC, compared to the PCN, presents a setback for the study of proofs,

<footnote n="5">⁵ We mobilize the concepts of pragmatic proofs as proofs that include resources of action and empirical observation on mathematical objects, aiming to verify or prove the mathematical property and conceptual or intellectual proofs are based on the formulation of relevant properties and connections are established between them in accordance with Balacheff (1988).</footnote>
which ceases to occupy a prominent place and becomes specific applications for the study of
congruences and similarities of triangles.

Textbooks by Edwaldo Bianchini in the 2010s and 2020s

Edwaldo Bianchini (1935-2018), science graduate, mathematics teacher and textbook author since the end of the 20th century. Bianchini's collection was selected for its positive emphasis on the approach to geometry at PNLD<fnref n="6" /> 2011, which proposes the study of geometry in a rigorous and extensive manner, with frequent formal proofs. We seek to comparatively analyze the introduction of deductive geometry in two textbooks by Edwaldo Bianchini (EB), aimed at 13-year-old students, one from the 2010s (book 1), in which the PCN were in force, and the other, from the 2020s (book 2), shortly after the implementation of the BNCC.

An initial quantitative analysis showed that the number of pages dedicated to teaching geometry decreased, from 46% of the total in book 1 to around 27% in book 2. However, the same does not occur when comparing the common themes that deal with of deductive geometry, in which the number of pages in both textbooks was practically the same, with a very similar approach. For example, regarding postulates, the same list appears in both volumes, with no mention of students' familiarity or prior experience with such propositions. This choice may represent the stability of the Euclidean Geometry model at this level of education.

Regarding the theorems proved, in both textbooks, EB begins with a topic on geometric proofs, in which he defines what a theorem is, but he no longer uses this term in the sequence, adopting the designation of “property” as a synonym. The number of properties proved in each textbook remains the same, with 16 proofs. In both textbooks, the author explains the need to prove without resorting to measurements or observations on the figure: “measurement or simple observation are not always sufficient to confirm whether a geometric property is true or false. As evident as it may seem, it can only be considered true after being proven” (Bianchini, 2015, p. 156). Complementarily, in book 2, the author drew attention to the fact that “Verification is not proof” (Bianchini, 2022, p. 181), referring to verification through measurements with instruments (ruler and protractor). We can interpret it as a way for EB to differentiate pragmatic proofs from conceptual proofs and to highlight the need for mathematical proofs, that is, validation using a hypothetical-deductive method. On the other hand, in both textbooks, pragmatic proofs are not explicitly identified in the exercises proposed to students. For example, in the two volumes, after the introduction of the concept of congruence of triangles, the cases of congruence were presented in an expository way, stated, accepted as true without proof and exemplified. In other words, such cases are not explored, discovered, or verified experimentally by students, they must be accepted as truths, and are used in identification exercises as an assumed result, similar to applying a formula. Only when he properly begins the proofs of the properties, EB states that: "In the following situations, we will consider that cases of triangle congruence are already proved truths<fnref n="7" />" (Bianchini, 2015, p. 158; 2022, p. 180). We also note that in the book 1 Teacher's Manual there are no suggestions for exploratory tasks or experimental proposals

<footnote n="6">⁶ From the 1990s onwards, Brazilian textbooks started to be periodically evaluated by a team of experts designated by the Brazilian Ministry of Education, in the National Textbook Program (PNLD) and distributed at no charge to public schools.</footnote>

<footnote n="7">⁷ This choice was not justified, EB bluntly states that: “we chose not to present the proofs of congruences of triangles cases as we consider them didactically inappropriate for this moment” (Bianchini, 2015, p. 158).</footnote>
which ceases to occupy a prominent place and becomes specific applications for the study of
congruences and similarities of triangles.

Textbooks by Edwaldo Bianchini in the 2010s and 2020s

Edwaldo Bianchini (1935-2018), science graduate, mathematics teacher and textbook author since the end of the 20th century. Bianchini's collection was selected for its positive emphasis on the approach to geometry at PNLD<fnref n="6" /> 2011, which proposes the study of geometry in a rigorous and extensive manner, with frequent formal proofs. We seek to comparatively analyze the introduction of deductive geometry in two textbooks by Edwaldo Bianchini (EB), aimed at 13-year-old students, one from the 2010s (book 1), in which the PCN were in force, and the other, from the 2020s (book 2), shortly after the implementation of the BNCC.

An initial quantitative analysis showed that the number of pages dedicated to teaching geometry decreased, from 46% of the total in book 1 to around 27% in book 2. However, the same does not occur when comparing the common themes that deal with of deductive geometry, in which the number of pages in both textbooks was practically the same, with a very similar approach. For example, regarding postulates, the same list appears in both volumes, with no mention of students' familiarity or prior experience with such propositions. This choice may represent the stability of the Euclidean Geometry model at this level of education.

Regarding the theorems proved, in both textbooks, EB begins with a topic on geometric proofs, in which he defines what a theorem is, but he no longer uses this term in the sequence, adopting the designation of “property” as a synonym. The number of properties proved in each textbook remains the same, with 16 proofs. In both textbooks, the author explains the need to prove without resorting to measurements or observations on the figure: “measurement or simple observation are not always sufficient to confirm whether a geometric property is true or false. As evident as it may seem, it can only be considered true after being proven” (Bianchini, 2015, p. 156). Complementarily, in book 2, the author drew attention to the fact that “Verification is not proof” (Bianchini, 2022, p. 181), referring to verification through measurements with instruments (ruler and protractor). We can interpret it as a way for EB to differentiate pragmatic proofs from conceptual proofs and to highlight the need for mathematical proofs, that is, validation using a hypothetical-deductive method. On the other hand, in both textbooks, pragmatic proofs are not explicitly identified in the exercises proposed to students. For example, in the two volumes, after the introduction of the concept of congruence of triangles, the cases of congruence were presented in an expository way, stated, accepted as true without proof and exemplified. In other words, such cases are not explored, discovered, or verified experimentally by students, they must be accepted as truths, and are used in identification exercises as an assumed result, similar to applying a formula. Only when he properly begins the proofs of the properties, EB states that: "In the following situations, we will consider that cases of triangle congruence are already proved truths"<fnref n="7" /> (Bianchini, 2015, p. 158; 2022, p. 180). We also note that in the book 1 Teacher's Manual there are no suggestions for exploratory tasks or experimental proposals

<footnote n="6">⁶ From the 1990s onwards, Brazilian textbooks started to be periodically evaluated by a team of experts designated by the Brazilian Ministry of Education, in the National Textbook Program (PNLD) and distributed at no charge to public schools.</footnote>

<footnote n="7">⁷ This choice was not justified, EB bluntly states that: “we chose not to present the proofs of congruences of triangles cases as we consider them didactically inappropriate for this moment” (Bianchini, 2015, p. 158).</footnote>
that could represent pragmatic tests, as was done in the book 2 Teacher's Manual, which contains
suggestions for students to represent triangles with a ruler and protractor, cut out these triangles and
compare them with the productions of other colleagues, in order to verify that all the triangles
produced are congruent (Bianchini, 2022, p. 166).

The theorem on congruence of the angles of the base of an isosceles triangle, in both textbooks, was made through the classic presentation: statement, identification of hypothesis and thesis, an illustrative figure and the proof steps for which the student is called to “to follow”. In this sense, we can recognize the same approach as the pre-modern OS textbook. Furthermore, EB exposes only one way of proving properties, both in its deductive path and in its way of representing, which can reiterate the idea of a single solution to be memorized. This way, students, in general, are not called upon to reflect or carry out any deduction steps, much less to try to produce their own proof.

In summary, the examination of the textbooks revealed that the tone of the proposal reinforces the
conclusion of Martins and Mandarino (2013): in most textbooks, formal proofs (or *conceptual proofs*)
are presented ready and finished, not giving space for the student to develop their own conjectures
and conclusions. The exercises, in most cases, approach the content under analysis with a procedural
focus, “where calculations and formulas are the basis of the questions with no apparent intention of
developing the spirit of argumentation in the student” (Martins & Mandarino, 2013, p. 112).

CONCLUSIONS

The objective of this study, in creating a long-term retrospective is to construct a broad trajectory on
proofs in geometry, emphasizing significant historical moments and seeking to identify tensions,
ruptures, and continuities. We reinforce it is important not to generalize the notes obtained in the
analytical exercise but rather to discuss the complexity of the teaching of proofs in geometry.

The tension between intuitive and deductive geometry in geometry teaching is evident in the regulations and pedagogical proposals of textbooks. Curriculum programs state the importance of intuitive approaches for students aged 11 to 14 but also highlight the need to introduce deductive practices from the age of 13. Striking a balance between these approaches is challenging, requiring strategies for the transition between pragmatic and conceptual proofs. The reforms of 1931 and 1942, the modern ideas of the MMM appropriated by Sangiorgi, along with the PCN, constitute convergences in valuing intuitive geometry. They also indicate pertinent perspectives for the current moment. The results reinforce the relevance of spaces such as the 26th ICMI to internationally debate advances in geometry teaching and collectively discuss indicators of continuity from different realities and cultures in the search for balance. On the other hand, we identified moments of ruptures and setbacks, such as the 1951 reform, which removed the space allocated to the study of intuitive geometry, as well as the current BNCC, which in normative terms, retreats by giving little emphasis to the transition between tense approaches – intuitive and deductive. However, according to Chervel (1990), objective finalities do not always follow teaching finalities; the production of OS is an example of the insertion of new approaches, even if they are not explained in the regulations. Regarding EB production, few differences were identified, despite regulations indicating detachment.

The study invites us to reflect on how the past can teach us by examining old proposals that evoke
current issues, such as the expressions “intuitive geometry as a smooth transition between experiences
and deductive conceptions” from 1942, or the OS's option in 1966 to postulate cases of triangle
that could represent pragmatic tests, as was done in the book 2 Teacher's Manual, which contains
suggestions for students to represent triangles with a ruler and protractor, cut out these triangles and
compare them with the productions of other colleagues, in order to verify that all the triangles
produced are congruent (Bianchini, 2022, p. 166).

The theorem on congruence of the angles of the base of an isosceles triangle, in both textbooks, was made through the classic presentation: statement, identification of hypothesis and thesis, an illustrative figure and the proof steps for which the student is called to “to follow”. In this sense, we can recognize the same approach as the pre-modern OS textbook. Furthermore, EB exposes only one way of proving properties, both in its deductive path and in its way of representing, which can reiterate the idea of a single solution to be memorized. This way, students, in general, are not called upon to reflect or carry out any deduction steps, much less to try to produce their own proof.

In summary, the examination of the textbooks revealed that the tone of the proposal reinforces the conclusion of Martins and Mandarino (2013): in most textbooks, formal proofs (or *conceptual proofs*) are presented ready and finished, not giving space for the student to develop their own conjectures and conclusions. The exercises, in most cases, approach the content under analysis with a procedural focus, “where calculations and formulas are the basis of the questions with no apparent intention of developing the spirit of argumentation in the student” (Martins & Mandarino, 2013, p. 112).

CONCLUSIONS

The objective of this study, in creating a long-term retrospective is to construct a broad trajectory on proofs in geometry, emphasizing significant historical moments and seeking to identify tensions, ruptures, and continuities. We reinforce it is important not to generalize the notes obtained in the analytical exercise but rather to discuss the complexity of the teaching of proofs in geometry.

The tension between intuitive and deductive geometry in geometry teaching is evident in the regulations and pedagogical proposals of textbooks. Curriculum programs state the importance of intuitive approaches for students aged 11 to 14 but also highlight the need to introduce deductive practices from the age of 13. Striking a balance between these approaches is challenging, requiring strategies for the transition between pragmatic and conceptual proofs. The reforms of 1931 and 1942, the modern ideas of the MMM appropriated by Sangiorgi, along with the PCN, constitute convergences in valuing intuitive geometry. They also indicate pertinent perspectives for the current moment. The results reinforce the relevance of spaces such as the 26th ICMI to internationally debate advances in geometry teaching and collectively discuss indicators of continuity from different realities and cultures in the search for balance. On the other hand, we identified moments of ruptures and setbacks, such as the 1951 reform, which removed the space allocated to the study of intuitive geometry, as well as the current BNCC, which in normative terms, retreats by giving little emphasis to the transition between tense approaches – intuitive and deductive. However, according to Chervel (1990), objective finalities do not always follow teaching finalities; the production of OS is an example of the insertion of new approaches, even if they are not explained in the regulations. Regarding EB production, few differences were identified, despite regulations indicating detachment.

The study invites us to reflect on how the past can teach us by examining old proposals that evoke current issues, such as the expressions “intuitive geometry as a smooth transition between experiences and deductive conceptions” from 1942, or the OS's option in 1966 to postulate cases of triangle
congruence after experimental conjectures. All these debates precede the carefully developed study by Balacheff, who, since the 1980s, highlighted the complexity of teaching mathematical proof. In summary, it is crucial to reinforce the importance of the transition between pragmatic proofs and conceptual proofs in geometry teaching, at least in Brazilian reality.

The study also allows for identifying connections, interactions, and appropriations of international movements (particularly Klein and Birkoff) by Brazilian educators. However, such appropriations were not fully integrated into the school culture, and what becomes relevant is the complexity of changing pedagogical practices, especially, breaking with the tradition of conceptual proofs, balancing them appropriately with pragmatic proofs enables a mathematical practice supported by perspectives from the field of Mathematics Education. We can conjecture that factors such as teacher training, cultural differences, and educational policies may not contribute as significantly to advances in geometry teaching as desired.

## Acknowledgements

This study was supported by the agency FAPESP/Brazil (proc. n. 2023/04639-8) and by the university UNESP-Rio Claro/SP where the first author carries out a post-doctoral internship under the supervision of the second.

## References

- Balacheff, N. (1988). Aspects of proof in pupil's practice of school mathematics. In D. Pimm (Ed.), *Mathematics teachers and children* (pp. 216-235). Hodder and Stoughton.

- Bianchini, E. (2015). *Matemática Bianchini*. Manual do Professor, 8ª ed. Editora Moderna.

- Bianchini, E. (2022). *Matemática Bianchini*. Manual do Professor, 10ª ed. Editora Moderna.

- Bicudo, J. C. (1942). *O ensino secundário no Brasil e sua atual legislação* (de 1931 a 1941 inclusive). Ministério da Educação e Saúde. Divisão de ensino secundário. Rio de Janeiro.

- Brasil (1998). Ministério da Educação e Cultura. *Parâmetros Curriculares Nacionais (5ª a 8ª Séries): Matemática*. MEC – Secretaria de Educação Fundamental.

- Brasil (2018). Ministério da Educação. *Base Nacional Comum Curricular – Educação é a base*. MEC – Secretaria de Educação Básica.

- Búrigo, E. Z. (2015). Professores de Matemática deveriam estudar história? In *Anais do XII Encontro Gaúcho de Educação Matemática*, (pp. 1-14). SBEM/RS.

- Chartier, R. (2009). *A história ou a leitura do tempo*. Trad. Cristina Antunes. Autêntica Editora.

- Chervel, A. (1990). História das disciplinas escolares: reflexões sobre um campo de pesquisa. *Teoria & Educação, 2*, 177-229.

- Jahn, A. P. & Leme da Silva, M. C. (2023). Don't memorize theorem proofs! – The modern geometry of Osvaldo Sangiorgi. *International Journal for Research in Mathematics Education, 13*(4), 1-19.

- Julia, D. (2001). A cultura escolar como objeto histórico. *Revista Brasileira de História da Educação, 1*, 9-43.

- Martins, R. B. & Freire Mandarino, M. C. (2013). Argumentação, prova e demonstração em geometria: análise de coleções de livros didáticos dos anos finais do Ensino Fundamental. *Boletim GEPEM, 62*, 101-115.

- Sangiorgi, O. (1966). *GUIA para uso dos PROFESSORES. 3º Volume – Matemática Curso Moderno*. Companhia Editora Nacional.

- Sangiorgi, O. (1967). *Matemática Curso Moderno. 3º Volume para os ginásios*. Companhia Editora Nacional.
congruence after experimental conjectures. All these debates precede the carefully developed study by Balacheff, who, since the 1980s, highlighted the complexity of teaching mathematical proof. In summary, it is crucial to reinforce the importance of the transition between pragmatic proofs and conceptual proofs in geometry teaching, at least in Brazilian reality.

The study also allows for identifying connections, interactions, and appropriations of international movements (particularly Klein and Birkoff) by Brazilian educators. However, such appropriations were not fully integrated into the school culture, and what becomes relevant is the complexity of changing pedagogical practices, especially, breaking with the tradition of conceptual proofs, balancing them appropriately with pragmatic proofs enables a mathematical practice supported by perspectives from the field of Mathematics Education. We can conjecture that factors such as teacher training, cultural differences, and educational policies may not contribute as significantly to advances in geometry teaching as desired.

## Acknowledgements

This study was supported by the agency FAPESP/Brazil (proc. n. 2023/04639-8) and by the university UNESP-Rio Claro/SP where the first author carries out a post-doctoral internship under the supervision of the second.

## References

Balacheff, N. (1988). Aspects of proof in pupil's practice of school mathematics. In D. Pimm (Ed.), *Mathematics teachers and children* (pp. 216-235). Hodder and Stoughton.

Bianchini, E. (2015). *Matemática Bianchini*. Manual do Professor, 8ª ed. Editora Moderna.

Bianchini, E. (2022). *Matemática Bianchini*. Manual do Professor, 10ª ed. Editora Moderna.

Bicudo, J. C. (1942). *O ensino secundário no Brasil e sua atual legislação* (de 1931 a 1941 inclusive). Ministério da Educação e Saúde. Divisão de ensino secundário. Rio de Janeiro.

Brasil (1998). Ministério da Educação e Cultura. *Parâmetros Curriculares Nacionais (5ª a 8ª Séries): Matemática*. MEC – Secretaria de Educação Fundamental.

Brasil (2018). Ministério da Educação. *Base Nacional Comum Curricular – Educação é a base*. MEC – Secretaria de Educação Básica.

Búrigo, E. Z. (2015). Professores de Matemática deveriam estudar história? In *Anais do XII Encontro Gaúcho de Educação Matemática*, (pp. 1-14). SBEM/RS.

Chartier, R. (2009). *A história ou a leitura do tempo*. Trad. Cristina Antunes. Autêntica Editora.

Chervel, A. (1990). História das disciplinas escolares: reflexões sobre um campo de pesquisa. *Teoria & Educação, 2*, 177-229.

Jahn, A. P. & Leme da Silva, M. C. (2023). Don't memorize theorem proofs! – The modern geometry of Osvaldo Sangiorgi. *International Journal for Research in Mathematics Education, 13*(4), 1-19.

Julia, D. (2001). A cultura escolar como objeto histórico. *Revista Brasileira de História da Educação, 1*, 9-43.

Martins, R. B. & Freire Mandarino, M. C. (2013). Argumentação, prova e demonstração em geometria: análise de coleções de livros didáticos dos anos finais do Ensino Fundamental. *Boletim GEPEM, 62*, 101-115.

Sangiorgi, O. (1966). *GUIA para uso dos PROFESSORES. 3º Volume – Matemática Curso Moderno*. Companhia Editora Nacional.

Sangiorgi, O. (1967). *Matemática Curso Moderno. 3º Volume para os ginásios*. Companhia Editora Nacional.
# A DYNAMIC AND SPATIAL APPROACH TO ENRICH THE TEACHING AND LEARNING OF GEOMETRY IN PRIMARY SCHOOL

Patricia Marchand¹, Nathalie Sinclair², Nelly Julien³

¹ Université de Sherbrooke, ² Simon Fraser University, ³ Université de Montréal

We propose a more dynamic and spatial way of teaching and learning geometry in primary school. Based on the NCTM's "Essential Understanding Series", which sets out four main ideas and ten essential understandings for the teaching of geometry and the report on recent research in this field (see Sinclair et al., 2012), we propose strategies to enrich classical geometry tasks. This contribution is based on research showing the central role of spatial reasoning in learning geometry (Battista, 2020) as well as the need to further value the dynamic aspects of geometric objects (Sinclair et al., 2016). Our reflection on new ways of implementing a geometry lesson is anchored in the theory of grounded cognition (Barsalou, 2008) and more specifically on the "Activity Generating Structure" (Marchand, 2020) which takes into account, among other things, the model of Uttal et al. (2013) which highlights the static/dynamic duality at play in the development of spatial reasoning. Our study exemplifies how this framework can lead to more dynamic and spatial experiences of geometry.

## INTRODUCTION

The teaching of geometry has always been a part of the mathematic elementary curriculum, but its importance may vary according to the different reforms in recent decades. In Canada, geometry has received little attention in the curriculum and in classroom teaching (e.g., Bruce et al., 2012). However, there has been a recent push for spatialized the curriculum (Hawes et al., 2023), which is part of the more general trend in research focusing on "advances in the understandings of visuospatial reasoning", as identified in Sinclair et al. (2016). This push is based on growing research that shows that spatial reasoning<fnref n="8" /> (SR) is correlated with mathematical success and that it is malleable (see Davis et al., 2015; Lowrie et al., 2017; Mulligan et al., 2020). Our work fits within this spatializing trend but focusses specifically on geometry. Although the role of spatial reasoning should be an obvious part of geometry, if teachers focus only on naming shapes and measuring them, little spatial reasoning is actually needed. We are particularly interested in dynamic forms of spatial reasoning, given their importance in mathematics, and the availability of digital technologies to support dynamic imagery and interactions. In our work, we have been focussed on how to transform static and quantitative geometric activities and tasks into dynamic and spatial ones.

The role of the SR is central to develop geometric reasoning (GR), but it is also distinct yet inextricable from it. Behind the understanding of a geometric object is also a spatial understanding of the object in question. Students must create a mental image of the geometry object, which they may do by simulating the experience of seeing and/or manipulating an object (Barsalou, 2020), connect this image with their geometric understanding and transform them according to the situation

<footnote n="8">⁸ Globally, SR referred to “the reasoning [...] that helps each individual better anticipate the effects of their actions on space, to control these effects, and to communicate spatial information” (Berthelot & Salin, 1992, p. 9, freely translated).</footnote>
# A DYNAMIC AND SPATIAL APPROACH TO ENRICH THE TEACHING AND LEARNING OF GEOMETRY IN PRIMARY SCHOOL

Patricia Marchand¹, Nathalie Sinclair², Nelly Julien³, Mylène Forest⁴

¹ Univ. de Sherbrooke, ² Simon Fraser Univ., ³ Univ. de Montréal, ⁴ Univ. du Quebec à Montréal

We propose a more dynamic and spatial way of teaching and learning geometry in primary school. Based on the NCTM's "Essential Understanding Series", which sets out four main ideas and ten essential understandings for the teaching of geometry and the report on recent research in this field (see Sinclair et al., 2012), we propose strategies to enrich classical geometry tasks. This contribution is based on research showing the central role of spatial reasoning in learning geometry (Battista, 2020) as well as the need to further value the dynamic aspects of geometric objects (Sinclair et al., 2016). Our reflection on new ways of implementing a geometry lesson is anchored in the theory of grounded cognition (Barsalou, 2008) and more specifically on the “Activity Generating Structure” (Marchand, 2020) which takes into account, among other things, the model of Uttal et al. (2013) which highlights the static/dynamic duality at play in the development of spatial reasoning. Our study exemplifies how this framework can lead to more dynamic and spatial experiences of geometry.

## INTRODUCTION

The teaching of geometry has always been a part of the mathematic elementary curriculum, but its importance may vary according to the different reforms in recent decades. In Canada, geometry has received little attention in the curriculum and in classroom teaching (e.g., Bruce et al., 2012). However, there has been a recent push for spatialized the curriculum (Hawes et al., 2023), which is part of the more general trend in research focusing on "advances in the understandings of visuospatial reasoning", as identified in Sinclair et al. (2016). This push is based on growing research that shows that spatial reasoning<fnref n="8" /> (SR) is correlated with mathematical success and that it is malleable (see Davis et al., 2015; Lowrie et al., 2017; Mulligan et al., 2020). Our work fits within this spatializing trend but focusses specifically on geometry. Although the role of spatial reasoning should be an obvious part of geometry, if teachers focus only on naming shapes and measuring them, little spatial reasoning is actually needed. We are particularly interested in dynamic forms of spatial reasoning, given their importance in mathematics, and the availability of digital technologies to support dynamic imagery and interactions. In our work, we have been focussed on how to transform static and quantitative geometric activities and tasks into dynamic and spatial ones.

The role of the SR is central to develop geometric reasoning (GR), but it is also distinct yet inextricable from it. Behind the understanding of a geometric object is also a spatial understanding of the object in question. Students must create a mental image of the geometry object, which they may do by simulating the experience of seeing and/or manipulating an object (Barsalou, 2020), connect this image with their geometric understanding and transform them according to the situation

<footnote n="8">⁸ Globally, SR referred to “the reasoning [...] that helps each individual better anticipate the effects of their actions on space, to control these effects, and to communicate spatial information” (Berthelot & Salin, 1992, p. 9, freely translated).</footnote>
to conceptualize them. SR can represent not only a gateway to the study of geometry, but it constitutes
also a powerful cognitive tool for the study of formal geometry (Moss et al., 2016; Clements &
Battista, 1992). For example, it is used to anticipate the figure-image of a rotation or establish the
relevant properties to find a missing measure of a figure from geometric statements. Research
conducted in Italy by Giofrè et al. (2013) cited in Sinclair et al. (2016) shows that SR is not only
central, but it is essential for geometric learning: “According to these researchers, the importance of
visuospatial working memory is critical in learning geometry” (p. 697). Our research is rooted in this
work and has as its starting point the conclusion that Sinclair et al. (2016) had following their review
of the research conducted so far and recommendations for future research to be carried out:

Since geometry is a school mathematics topic that explicitly engages visuo-spatial reasoning, several authors have argued that an increased focus on geometry in the curriculum would be well advised (Sinclair & Bruce 2015). In particular, given the importance of dynamic forms of spatial reasoning, highlighted in the work of Newcombe and others, and the relevance to motion and time across ages and cultures (humans live in a world that is constantly moving, changing), there is reason to conjecture that developing dynamic and haptic forms of visuospatial reasoning would have a positive impact on children’s learning. There are interesting and long-standing mathematical challenges in moving away from paper-based static representations, taking inspiration perhaps from non-Western geometric ways of making sense of the world, but as shown in the Sect. 5, digital technologies can provide alternatives to such representations that offer learners more opportunities to create and reason with dynamic imagery. (pp. 698-699)

The research question asked in this proposal is: How can we conduct a classic geometry activity in class to enhance a more spatial and dynamic approach? To respond to this question, we present the frame of reference on which we based our analysis and examples of experiments we have carried out.

FRAMEWORK

In recent years, frameworks for studying SR have been developed in different fields of research (psychology, cognition, didactics, etc.). The framework of this project fits globally into embodied approaches that ground cognition in the body and the environment (see Barsalou, 2020) and more specifically on the exploitation of a dynamic approach focused on the development of SR to enrich the teaching and learning of geometry. Grounded cognition guides several choices in the design and analysis of the activities proposed in classrooms for both the mathematical teaching components. This framework is not based on the study of information processing, but rather on the study of the sensory experiences of learners, including their actions and perceptions. In grounded cognition, all learning is based on a perceptual and physical interaction of the student within their affective, material and social environment. Actions and perception shape and are shaped by sociocultural forces such as language, tools and values (de Freitas & Sinclair, 2013).

And more specifically, the activity generating structure (AGS) guides us in our choices to enhance
the development of SR in the proposed activities (Marchand, 2020). This framework provides
different benchmarks for generating and analyzing SR development in the classroom. Thus, the
benchmarks chosen for this project will be detailed in the methodology since they constitute the
analysis variables of our *Design-based research*. The Uttal et al. (2013) model offers a typology of
SR according to four categories, as in Figure 1 (the hammer could be any object, such as a cube):
to conceptualize them. SR can represent not only a gateway to the study of geometry, but it constitutes
also a powerful cognitive tool for the study of formal geometry (Moss et al., 2016; Clements &
Battista, 1992). For example, it is used to anticipate the figure-image of a rotation or establish the
relevant properties to find a missing measure of a figure from geometric statements. Research
conducted in Italy by Giofrè et al. (2013) cited in Sinclair et al. (2016) shows that SR is not only
central, but it is essential for geometric learning: “According to these researchers, the importance of
visuospatial working memory is critical in learning geometry” (p. 697). Our research is rooted in this
work and has as its starting point the conclusion that Sinclair et al. (2016) had following their review
of the research conducted so far and recommendations for future research to be carried out:

Since geometry is a school mathematics topic that explicitly engages visuo-spatial reasoning, several authors have argued that an increased focus on geometry in the curriculum would be well advised (Sinclair & Bruce 2015). In particular, given the importance of dynamic forms of spatial reasoning, highlighted in the work of Newcombe and others, and the relevance to motion and time across ages and cultures (humans live in a world that is constantly moving, changing), there is reason to conjecture that developing dynamic and haptic forms of visuospatial reasoning would have a positive impact on children’s learning. There are interesting and long-standing mathematical challenges in moving away from paper-based static representations, taking inspiration perhaps from non-Western geometric ways of making sense of the world, but as shown in the Sect. 5, digital technologies can provide alternatives to such representations that offer learners more opportunities to create and reason with dynamic imagery. (pp. 698-699)

The research question asked in this proposal is: How can we conduct a classic geometry activity in class to enhance a more spatial and dynamic approach? To respond to this question, we present the frame of reference on which we based our analysis and examples of experiments we have carried out.

FRAMEWORK

In recent years, frameworks for studying SR have been developed in different fields of research (psychology, cognition, didactics, etc.). The framework of this project fits globally into embodied approaches that ground cognition in the body and the environment (see Barsalou, 2020) and more specifically on the exploitation of a dynamic approach focused on the development of SR to enrich the teaching and learning of geometry. Grounded cognition guides several choices in the design and analysis of the activities proposed in classrooms for both the mathematical teaching components. This framework is not based on the study of information processing, but rather on the study of the sensory experiences of learners, including their actions and perceptions. In grounded cognition, all learning is based on a perceptual and physical interaction of the student within their affective, material and social environment. Actions and perception shape and are shaped by sociocultural forces such as language, tools and values (de Freitas & Sinclair, 2013).

And more specifically, the activity generating structure (AGS) guides us in our choices to enhance
the development of SR in the proposed activities (Marchand, 2020). This framework provides
different benchmarks for generating and analyzing SR development in the classroom. Thus, the
benchmarks chosen for this project will be detailed in the methodology since they constitute the
analysis variables of our *Design-based research*. The Uttal et al. (2013) model offers a typology of
SR according to four categories, as in Figure 1 (the hammer could be any object, such as a cube):
<table><thead><tr><th></th><th>Intrinsic</th><th>Extrinsic</th></tr></thead><tbody><tr><th>Static</th><td>Relationships between parts of an object; static geometric characteristics.</td><td>Coding the location of objects relative to other objects or a reference frame.</td></tr><tr><th>Dynamic</th><td>Folding, bending, rotating, scaling, cross-sectioning, comparing 2-d and 3-d views.</td><td>How one's perception of the relations among objects changes as one moves through the environment.</td></tr></tbody></table>

<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /><figcaption>Figure 1. A 2x2 classification of spatial skills (adapted from Uttal et al., 2013, p. 354)</figcaption></figure>

The development of SR is based as much on activities involving objects that remain **static** as on activities involving **dynamic** objects by moving or transforming them or by observing them from different perspectives. As mentioned above, the geometric activities usually offered to students are static. This absence of dynamic consideration in activities is also rooted in a number of education programs, including in the mathematic education programs of Québec (Ministère de l'Éducation, du Loisir et du Sport, 2003). Dynamic and spatial approaches to teaching and learning geometry at school are rare, despite the fact that psychology research has for many years shown a correlation between, for example, the development of SR related to mental object rotations, which falls under the dynamic category, and student performance in elementary and high school (Hawes et al., 2015).

The other angle used to characterize SR is the type of spatial relationships at play. These relationships can be between a whole and one or more of its parts (intrinsic) or between different wholes (extrinsic). The emphasis for us here is to consider the management of these relationships as a central element in the design and implementation of geometric activities in the classroom.

By articulating this framework with our overall approach based on grounded cognition theory, it is possible to highlight another element of Marchand (2020). For each of these four categories, the three type of action will necessarily interact with each another. The development of SR and GR is an iterative process between action (physical and mental) on the one hand, and reflection and abstraction on the other (Battista, 2007). In the activities that we propose, students will be in action or be called upon to manipulate physical or virtual objects; they will therefore be in what we have called an archaeological level of action/reflection (Marchand, 2020): they will have to move, observe, manipulate and deduce spatial and geometric regularities, relationships or properties. This action must be accompanied by spatial and geometric mental work. To ensure that students can have access to this internalization, the activities we propose must create this bridge. It is therefore necessary to value activities that require students to create mental images, photographic level of action/reflection (Marchand, 2020) of the manipulated objects. A good example of an activity that enhances this bridge is the Tangram activity of Yackel and Wheatley (1990) in which students are asked to observe an image composed of Tangram pieces for three seconds, and then reproduce the image in his absence. Students can observe the image two or three times; emphasis is placed on the spatial and geometric strategies they use to reproduce it (What do you see in your head? How do you take into account the position of the figures and their articulation? What were your actions between each observation?). This activity is located in the static/intrinsic category and it allows students to connect, several times
<table><thead><tr><th></th><th>Intrinsic</th><th>Extrinsic</th></tr></thead><tbody><tr><th>Static</th><td>Relationships between parts of an object; static geometric characteristics.</td><td>Coding the location of objects relative to other objects or a reference frame.</td></tr><tr><th>Dynamic</th><td>Folding, bending, rotating, scaling, cross-sectioning, comparing 2-d and 3-d views.</td><td>How one's perception of the relations among objects changes as one moves through the environment.</td></tr></tbody></table>

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /><figcaption>Figure 1. A 2x2 classification of spatial skills (adapted from Uttal et al., 2013, p. 354)</figcaption></figure>

The development of SR is based as much on activities involving objects that remain **static** as on activities involving **dynamic** objects by moving or transforming them or by observing them from different perspectives. As mentioned above, the geometric activities usually offered to students are static. This absence of dynamic consideration in activities is also rooted in a number of education programs, including in the mathematic education programs of Québec (Ministère de l'Éducation, du Loisir et du Sport, 2003). Dynamic and spatial approaches to teaching and learning geometry at school are rare, despite the fact that psychology research has for many years shown a correlation between, for example, the development of SR related to mental object rotations, which falls under the dynamic category, and student performance in elementary and high school (Hawes et al., 2015).

The other angle used to characterize SR is the type of spatial relationships at play. These relationships can be between a whole and one or more of its parts (intrinsic) or between different wholes (extrinsic). The emphasis for us here is to consider the management of these relationships as a central element in the design and implementation of geometric activities in the classroom.

By articulating this framework with our overall approach based on grounded cognition theory, it is possible to highlight another element of Marchand (2020). For each of these four categories, the three type of action will necessarily interact with each another. The development of SR and GR is an iterative process between action (physical and mental) on the one hand, and reflection and abstraction on the other (Battista, 2007). In the activities that we propose, students will be in action or be called upon to manipulate physical or virtual objects; they will therefore be in what we have called an archaeological level of action/reflection (Marchand, 2020): they will have to move, observe, manipulate and deduce spatial and geometric regularities, relationships or properties. This action must be accompanied by spatial and geometric mental work. To ensure that students can have access to this internalization, the activities we propose must create this bridge. It is therefore necessary to value activities that require students to create mental images, photographic level of action/reflection (Marchand, 2020) of the manipulated objects. A good example of an activity that enhances this bridge is the Tangram activity of Yackel and Wheatley (1990) in which students are asked to observe an image composed of Tangram pieces for three seconds, and then reproduce the image in his absence. Students can observe the image two or three times; emphasis is placed on the spatial and geometric strategies they use to reproduce it (What do you see in your head? How do you take into account the position of the figures and their articulation? What were your actions between each observation?). This activity is located in the static/intrinsic category and it allows students to connect, several times
through a back and forth between action and reflection (observation, construction and verbalization), through archaeological and photographic tasks. For both dynamic categories, it will be necessary to extend the action/reflection to tasks that stage the geometric objects studied in motion (by moving, decomposing, transforming, folding them, etc.) physically and mentally. This requires a dynamic and articulated vision of the objects in play (the scenographic level of action/reflection). This study falls into both dynamic categories and therefore we will focus on these aspects henceforth.

## METHODOLOGY

Broadly speaking, we draw on the *Design-based Research* methodology for the lesson design (Anderson & Shattuck, 2012). However, in this paper, we present already-designed lessons and analyze them based on the two dynamic categories of activities proposed by Uttal et al. (2013) and the three types of action/reflection proposed by Marchand (2020). We considered three variables that appear to be crucial, given that they are recurring in our work and that of other researchers. We chose the two lessons analyzed in this paper both because their static versions will be familiar to most mathematics educators researchers and because they have been used numerous times in our research.

The **first** variable is the conception of space (micro, meso and macro) (Berthelot & Salin, 1992; Brousseau, 1983). Briefly, micro-space is the space of small objects that can be manipulated, or the space of a sheet of paper. This is the space usually exploited in mathematics classrooms. Meso-space is the space encompassing the movements of a student or object and is fully accessible via the student's field of view. The Tangram activity mentioned before was in the micro-space, but it can be carried out in the meso-space using figures constructed from sheets.

<figure><img src="image_36.png" /><figcaption>Figure 2. Example of a Tangram constructed from sheets</figcaption></figure>

Macro-space is too large for a student “to be able to take in at once” (Brousseau, 2001, p. 6, our translation). Only a local, partial view of the space is possible. The connection between micro- and meso-space seems to be a promising avenue for conceptualizing both SR (Braconne-Michoux & Marchand, 2021; Brousseau, 1983) and GR and this variable is consistent with grounded cognition theory. The **second** variable is the nature and the chronology of tasks that can be given to students. Davis et al. (2015) have identified 29 possible tasks (ex.: observe or touch, coordinate perspectives, move, break down, fold or reorganize). One of the main findings of our previous research is that the chronology of tasks is a decisive factor in the level of difficulty of the activity. Specifically, the following two chronologies seem promising to enrich the teaching/learning of geometry from a dynamic and spatial point of view: 1) anticipate the sufficient and necessary materials required before proceeding to any construction, build, and then describe the solid obtained (Marchand, 2006, 2009) and 2) observes to anticipate geometric properties and spatial relations to be retained for construction, construct without reference to the object to be made and describe the Tangram (Yackel & Wheatley, 1990). In these two cases, the anticipation task is carried out differently, but always takes place before the manipulation or resolution (Michoux-Braconne & Marchand, 2021). Finally, the **third** variable is the nature and dimension of the objects in play, which can be 0D (points), 1D (lines), 2D (figures) or 3D (solids) and can be prototypical or non-prototypical and positioned in a typical or non-typical
through a back and forth between action and reflection (observation, construction and verbalization), through archaeological and photographic tasks. For both dynamic categories, it will be necessary to extend the action/reflection to tasks that stage the geometric objects studied in motion (by moving, decomposing, transforming, folding them, etc.) physically and mentally. This requires a dynamic and articulated vision of the objects in play (the scenographic level of action/reflection). This study falls into both dynamic categories and therefore we will focus on these aspects henceforth.

## METHODOLOGY

Broadly speaking, we draw on the *Design-based Research* methodology for the lesson design (Anderson & Shattuck, 2012). However, in this paper, we present already-designed lessons and analyze them based on the two dynamic categories of activities proposed by Uttal et al. (2013) and the three types of action/reflection proposed by Marchand (2020). We considered three variables that appear to be crucial, given that they are recurring in our work and that of other researchers. We chose the two lessons analyzed in this paper both because their static versions will be familiar to most mathematics educators researchers and because they have been used numerous times in our research.

The **first** variable is the conception of space (micro, meso and macro) (Berthelot & Salin, 1992; Brousseau, 1983). Briefly, micro-space is the space of small objects that can be manipulated, or the space of a sheet of paper. This is the space usually exploited in mathematics classrooms. Meso-space is the space encompassing the movements of a student or object and is fully accessible via the student's field of view. The Tangram activity mentioned before was in the micro-space, but it can be carried out in the meso-space using figures constructed from sheets.

<figure><img src="image_41.png" /><figcaption>Figure 2. Example of a Tangram constructed from sheets</figcaption></figure>

Macro-space is too large for a student “to be able to take in at once” (Brousseau, 2001, p. 6, our translation). Only a local, partial view of the space is possible. The connection between micro- and meso-space seems to be a promising avenue for conceptualizing both SR (Braconne-Michoux & Marchand, 2021; Brousseau, 1983) and GR and this variable is consistent with grounded cognition theory. The **second** variable is the nature and the chronology of tasks that can be given to students. Davis et al. (2015) have identified 29 possible tasks (ex.: observe or touch, coordinate perspectives, move, break down, fold or reorganize). One of the main findings of our previous research is that the chronology of tasks is a decisive factor in the level of difficulty of the activity. Specifically, the following two chronologies seem promising to enrich the teaching/learning of geometry from a dynamic and spatial point of view: 1) anticipate the sufficient and necessary materials required before proceeding to any construction, build, and then describe the solid obtained (Marchand, 2006, 2009) and 2) observes to anticipate geometric properties and spatial relations to be retained for construction, construct without reference to the object to be made and describe the Tangram (Yackel & Wheatley, 1990). In these two cases, the anticipation task is carried out differently, but always takes place before the manipulation or resolution (Michoux-Braconne & Marchand, 2021). Finally, the **third** variable is the nature and dimension of the objects in play, which can be 0D (points), 1D (lines), 2D (figures) or 3D (solids) and can be prototypical or non-prototypical and positioned in a typical or non-typical
orientation. This variable plays a key role in geometric conceptualization. The fact that we are in a dynamic approach necessarily implies that students will be confronted with geometric objects in motion and therefore in non-typical positions and spatial relationships.

A FIRST STEP TOWARDS A DYNAMIC AND SPATIAL APPROACH TO ENRICH THE TEACHING AND LEARNING OF GEOMETRY IN PRIMARY SCHOOL

We explore this avenue through the analysis of two classical activities: the reproduction of an image formed by geometric figures (as in the Tangram activity) and the representation of the possible developments of a solid, such as those of the cube.

The Tangram activity discussed earlier is a classic in the SR literature and involves students identifying and manipulating 2D shapes<fnref n="9" /> to create certain designs. Changing the orientation and chirality of the shapes allows students to experience transformation directly, which might help them mentally rotate these shapes. Geometrically, the main pedagogical benefits might be (1) to identify the names of shapes, such as triangles, squares and parallelograms) and (2) to decompose and recompose shapes in such a way that preserves area. Given the research showing that students tend to develop prototypical images of shapes (squares that are sitting on their sides, triangles that are equilateral, etc.), and the role that dynamic geometry can play in which students experience a wider variety of these target shapes, as well as a sense of their properties (Sinclair & Yurita, 2008), this sequence of DGE tasks<fnref n="10" /> was designed to support both SR and GR.

In this activity, students are asked to use a set of arbitrary triangles (triangles that can be dragged into any shape) and fit them into a given shape (such as a square, a pentagon, a start, etc.) (see Figure 2a). Instead of manipulating a fixed figure, they are producing triangles of various shapes and sizes. While we often see the flat figures statically at school, here they are dynamic as much in relation to the intrinsic spatial relationships, between the triangles and the reference figure (square, pentagon, star), as extrinsic between each of the triangles that make up this figure since they can take any shape and they must fill, without leaving a hole, the reference figure (see Figure 2b). Students use archaeological, photographic and scenographic actions/reflections here. They transform the triangles, they position them in the reference figure (archaeologic), anticipate how to position it in relation to the reference figure (photographic) and in relation to the next triangle to be placed (scenographic). The main tasks required are deformation, anticipation, decomposition and recomposition. The activity takes place in micro-space and students drag the vertices (0D), segments (1D) and figures (2D) to fill in the reference figure (2D). Finally, since students will necessarily deform the triangles to fill the figure, their positions and appearance will not be typical. A conceptualization of the figures involved (triangle, square, pentagon) and the notion of area will thus be enriched.

<figure><img src="image_37.png" /><figcaption>Figure 3a: Triangles to fill the square; b: one solution; c: fitting the last triangle into a pentagon</figcaption></figure>

<footnote n="9">⁹ They are meant to be treated as figures, as they are called squares and triangles, and so on, but are in fact solids, which introduces a complexity that is rarely discussed in the mathematics education research.</footnote>

<footnote n="10">¹⁰ http://www.sfu.ca/content/dam/sfu/geometry4yl/sketchpadfiles/Triangle%20Designs/index.html</footnote>
orientation. This variable plays a key role in geometric conceptualization. The fact that we are in a dynamic approach necessarily implies that students will be confronted with geometric objects in motion and therefore in non-typical positions and spatial relationships.

## A FIRST STEP TOWARDS A DYNAMIC AND SPATIAL APPROACH TO ENRICH THE TEACHING AND LEARNING OF GEOMETRY IN PRIMARY SCHOOL

We explore this avenue through the analysis of two classical activities: the reproduction of an image formed by geometric figures (as in the Tangram activity) and the representation of the possible developments of a solid, such as those of the cube.

The Tangram activity discussed earlier is a classic in the SR literature and involves students identifying and manipulating 2D shapes<fnref n="9" /> to create certain designs. Changing the orientation and chirality of the shapes allows students to experience transformation directly, which might help them mentally rotate these shapes. Geometrically, the main pedagogical benefits might be (1) to identify the names of shapes, such as triangles, squares and parallelograms) and (2) to decompose and recompose shapes in such a way that preserves area. Given the research showing that students tend to develop prototypical images of shapes (squares that are sitting on their sides, triangles that are equilateral, etc.), and the role that dynamic geometry can play in which students experience a wider variety of these target shapes, as well as a sense of their properties (Sinclair & Yurita, 2008), this sequence of DGE tasks<fnref n="10" /> was designed to support both SR and GR.

In this activity, students are asked to use a set of arbitrary triangles (triangles that can be dragged into any shape) and fit them into a given shape (such as a square, a pentagon, a start, etc.) (see Figure 2a). Instead of manipulating a fixed figure, they are producing triangles of various shapes and sizes. While we often see the flat figures statically at school, here they are dynamic as much in relation to the intrinsic spatial relationships, between the triangles and the reference figure (square, pentagon, star), as extrinsic between each of the triangles that make up this figure since they can take any shape and they must fill, without leaving a hole, the reference figure (see Figure 2b). Students use archaeological, photographic and scenographic actions/reflections here. They transform the triangles, they position them in the reference figure (archaeologic), anticipate how to position it in relation to the reference figure (photographic) and in relation to the next triangle to be placed (scenographic). The main tasks required are deformation, anticipation, decomposition and recomposition. The activity takes place in micro-space and students drag the vertices (0D), segments (1D) and figures (2D) to fill in the reference figure (2D). Finally, since students will necessarily deform the triangles to fill the figure, their positions and appearance will not be typical. A conceptualization of the figures involved (triangle, square, pentagon) and the notion of area will thus be enriched.

<figure><img src="image_42.png" /><figcaption>Figure 3a: Triangles to fill the square; b: one solution; c: fitting the last triangle into a pentagon</figcaption></figure>

<footnote n="9">⁹ They are meant to be treated as figures, as they are called squares and triangles, and so on, but are in fact solids, which introduces a complexity that is rarely discussed in the mathematics education research.</footnote>

<footnote n="10">¹⁰ http://www.sfu.ca/content/dam/sfu/geometry4yl/sketchpadfiles/Triangle%20Designs/index.html</footnote>
The concept of folding a net into a solid is in the curriculum at the primary and secondary levels. And, in this sense, several textbooks present the classic static task: Observe these nets and say which can be folded into a cube. We propose a more dynamic and spatial approach, as follows. Provide students with eleven nets of the cube, each on a different page, all having semicircles on the centre of each side of the six squares of the net (see below) and inform students they must remain on their desks and cannot be cut or folded. The eleven nets are also available at the front of the classroom and can be folded, but they cannot be written on (the semicircles are also not coloured). In teams, the students must colour each net, each semicircle of twelve distinct colours so that they form a circle of the same colour when the cube is formed (one side with a blue semicircle joins another side with a blue semicircle to form an edge of the cube). The students can manipulate the net, are challenged to go do to at the front of the classroom as few times as possible.

<figure><img src="image_38.png" /><figcaption>Figure 4a: One page from the student leaflet; b: the nets available in front of the classroom</figcaption></figure>

The didactic choices of not being able to cut the net to colour the semicircles and not being able to manipulate the net so easily function to force students to resort to other strategies that are not only rooted in archaeological action/reflection, but also in the photographic and scenographic. They have to imagine the cube formed, to take into account each of the rotations made by each of the six squares that constitute the cube and retain the relationships that each of the segments of the squares maintain once folded to form the twelve edges of the cube. To do this, students can perform these movements mentally, but also with the help of gestures. These gestures express their starting point (the square on which the cube is placed), the transformations they anticipate, one rotation at a time, the relationship between the different segments to identify the colour to be inscribed and the end point of the folding envisaged. These gestures are very telling about how each of the students folds and designs the cube, as are their verbalization to the other members of the team.

In this activity, the actions/reflections also represent a back and forth between archaeological (the net in front of the class), photographic (the image of the cube to obtain) and especially scenographic (anticipation of how to fold the cube). Students are in the micro-space; they mainly deal with the 2D-to-3D passage, but to identify the corresponding semicircles, they must also resort to the study of segments (1D). In addition, activity lies within the dynamic/intrinsic quadrant since students analyze the relationships that each of the segments has with the others forming the whole, the cube. Usually, of the eleven developments in the cube, students see the one shown in Figure 3a, but as they complete this activity, they will explore the eleven nets and understand how each one forms the cube. This more dynamic and spatial way of approaching this mathematical concept seems to be conducive to a richer understanding of nets.
The concept of folding a net into a solid is in the curriculum at the primary and secondary levels. And, in this sense, several textbooks present the classic static task: Observe these nets and say which can be folded into a cube. We propose a more dynamic and spatial approach, as follows. Provide students with eleven nets of the cube, each on a different page, all having semicircles on the centre of each side of the six squares of the net (see below) and inform students they must remain on their desks and cannot be cut or folded. The eleven nets are also available at the front of the classroom and can be folded, but they cannot be written on (the semicircles are also not coloured). In teams, the students must colour each net, each semicircle of twelve distinct colours so that they form a circle of the same colour when the cube is formed (one side with a blue semicircle joins another side with a blue semicircle to form an edge of the cube). The students can manipulate the net, are challenged to go do to at the front of the classroom as few times as possible.

<figure><img src="image_43.png" /><figcaption>Figure 4a: One page from the student leaflet; b: the nets available in front of the classroom</figcaption></figure>

The didactic choices of not being able to cut the net to colour the semicircles and not being able to manipulate the net so easily function to force students to resort to other strategies that are not only rooted in archaeological action/reflection, but also in the photographic and scenographic. They have to imagine the cube formed, to take into account each of the rotations made by each of the six squares that constitute the cube and retain the relationships that each of the segments of the squares maintain once folded to form the twelve edges of the cube. To do this, students can perform these movements mentally, but also with the help of gestures. These gestures express their starting point (the square on which the cube is placed), the transformations they anticipate, one rotation at a time, the relationship between the different segments to identify the colour to be inscribed and the end point of the folding envisaged. These gestures are very telling about how each of the students folds and designs the cube, as are their verbalization to the other members of the team.

In this activity, the actions/reflections also represent a back and forth between archaeological (the net in front of the class), photographic (the image of the cube to obtain) and especially scenographic (anticipation of how to fold the cube). Students are in the micro-space; they mainly deal with the 2D-to-3D passage, but to identify the corresponding semicircles, they must also resort to the study of segments (1D). In addition, activity lies within the dynamic/intrinsic quadrant since students analyze the relationships that each of the segments has with the others forming the whole, the cube. Usually, of the eleven developments in the cube, students see the one shown in Figure 3a, but as they complete this activity, they will explore the eleven nets and understand how each one forms the cube. This more dynamic and spatial way of approaching this mathematical concept seems to be conducive to a richer understanding of nets.
## CONCLUSION

Given the important role that dynamic thinking plays in SR, which in turn correlates strongly with mathematical success, we are interested in not only spatializing the curriculum, but also dynamizing it. With a specific focus on geometry, we have provided two examples in this paper of ways to render well-known primary school tasks more dynamic. One involved inviting students to work with dynamic objects, with the help of digital tools, while the other involved drawing on pedagogical strategies to increase students' dynamic thinking. In both examples, we highlighted the presence of archeological, photographic and scenographic forms of SR. We also highlighted opportunities for working across dimensions. Both activities include direct feedback—in the first case from the dynamic environment itself and in the second case from self-checking at the front of the classroom—which we see as a significant aspect of task design, as per Laborde (2001). In our presentation, we will provide other examples of dynamising geometric tasks for primary school learners, including ones not confined to the micro-space, and share results of our classroom experimentations.

## Acknowledgements

This work was supported by a Social Sciences and Humanities Research Council of Canada (SSHRC) Insight grant.

## References

* Anderson, T. & Shattuck, J. (2012). Design-Based Research: A Decade of Progress in Education Research? *Sage Journals, 41*(1), 16-25. https://doi.org/10.3102/0013189X1142881

* Barsalou, L. (2020). Challenges and opportunities for grounding cognition. *Journal of Cognition, 3*(1), 1-24.

* Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester, Jr. (Ed.), *Second handbook of research on mathematics teaching and learning* (pp. 843-908). National Council of Teachers of Mathematics.

* Battista, M. T. (2008). Development of the Shape Makers Geometry Microworld. In G. W. Blume & M.K. Heid (Ed.), *Research on technology and the teaching and learning of mathematics: Vol. 2, Cases and perspectives* (pp. 131-156). Information Age.

* Berthelot, R. & Salin, M.-H. (1999-2000). L'enseignement de l'espace à l'école primaire. *Grand N, 65*, 37-59.

* Braconne-Michoux, A. & Marchand, P. (2021). La géométrie dans l'espace: une piste d'interventions auprès des élèves en difficulté? In P. Marchand, A. Adihou, J. Koudogbo, D. Gauthier et C. Bisson (eds.), *La recherche en didactique des mathématiques et les élèves en difficulté: Quels enjeux et quelles perspectives?* (pp. 200-221). Les éditions JFD.

* Brousseau, G. (2001). Les propriétés didactiques de la géométrie élémentaire. L'étude de l'espace et de la géométrie. *Actes du Séminaire de didactique des Mathématiques* (pp. 67-83). https://hal.archives-ouvertes.fr/hal-00515110/document.

* Brousseau, G., (1983), Étude de questions d'enseignement. Un exemple: la géométrie, *Séminaire de Didactique des mathématiques et de l'informatique, 45*, LSD IMAG et Université Joseph Fourier.

* Bruce, C., Moss, J., et Ross, J. (2012). *Survey of JK to Grade 2 teachers in Ontario, Canada: Report to the Literacy and Numeracy Secretariat of the Ministry of Education*. Ministère de l'Éducation de l'Ontario.

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

* Davis, B, and The Spatial Reasoning Study Group (2015). *Spatial reasoning in the early years: Principles, assertions, and speculations*. Routledge.
## CONCLUSION

Given the important role that dynamic thinking plays in SR, which in turn correlates strongly with mathematical success, we are interested in not only spatializing the curriculum, but also dynamizing it. With a specific focus on geometry, we have provided two examples in this paper of ways to render well-known primary school tasks more dynamic. One involved inviting students to work with dynamic objects, with the help of digital tools, while the other involved drawing on pedagogical strategies to increase students' dynamic thinking. In both examples, we highlighted the presence of archeological, photographic and scenographic forms of SR. We also highlighted opportunities for working across dimensions. Both activities include direct feedback—in the first case from the dynamic environment itself and in the second case from self-checking at the front of the classroom—which we see as a significant aspect of task design, as per Laborde (2001). In our presentation, we will provide other examples of dynamising geometric tasks for primary school learners, including ones not confined to the micro-space, and share results of our classroom experimentations.

## Acknowledgements

This work was supported by a Social Sciences and Humanities Research Council of Canada (SSHRC) Insight grant.

## References

* Anderson, T. & Shattuck, J. (2012). Design-Based Research: A Decade of Progress in Education Research? *Sage Journals, 41*(1), 16-25. https://doi.org/10.3102/0013189X1142881

* Barsalou, L. (2020). Challenges and opportunities for grounding cognition. *Journal of Cognition, 3*(1), 1-24.

* Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester, Jr. (Ed.), *Second handbook of research on mathematics teaching and learning* (pp. 843-908). National Council of Teachers of Mathematics.

* Battista, M. T. (2008). Development of the Shape Makers Geometry Microworld. In G. W. Blume & M.K. Heid (Ed.), *Research on technology and the teaching and learning of mathematics: Vol. 2, Cases and perspectives* (pp. 131-156). Information Age.

* Berthelot, R. & Salin, M.-H. (1999-2000). L'enseignement de l'espace à l'école primaire. *Grand N, 65*, 37-59.

* Braconne-Michoux, A. & Marchand, P. (2021). La géométrie dans l'espace: une piste d'interventions auprès des élèves en difficulté? In P. Marchand, A. Adihou, J. Koudogbo, D. Gauthier et C. Bisson (eds.), *La recherche en didactique des mathématiques et les élèves en difficulté: Quels enjeux et quelles perspectives?* (pp. 200-221). Les éditions JFD.

* Brousseau, G. (2001). Les propriétés didactiques de la géométrie élémentaire. L'étude de l'espace et de la géométrie. *Actes du Séminaire de didactique des Mathématiques* (pp. 67-83). https://hal.archives-ouvertes.fr/hal-00515110/document.

* Brousseau, G., (1983), Étude de questions d'enseignement. Un exemple: la géométrie, *Séminaire de Didactique des mathématiques et de l'informatique, 45*, LSD IMAG et Université Joseph Fourier.

* Bruce, C., Moss, J., et Ross, J. (2012). *Survey of JK to Grade 2 teachers in Ontario, Canada: Report to the Literacy and Numeracy Secretariat of the Ministry of Education*. Ministère de l'Éducation de l'Ontario.

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

* Davis, B, and The Spatial Reasoning Study Group (2015). *Spatial reasoning in the early years: Principles, assertions, and speculations*. Routledge.
De Freitas, E. & Sinclair, N. (2014). *Mathematics and the body: Material entanglements in the classroom*. Cambridge University Press.

Gilligan-Lee, K.A., Hawes, Z.C.K. & Mix, K.S. (2022). Spatial thinking as the missing piece in mathematics curricula. *Npj Science of Learning, 7*, 10. https://doi.org/10.1038/s41539-022-00128-9.

Giofrè, D., Mammarella, I. C., Ronconi, L., & Cornoldi, C. (2013). Visuospatial working memory in intuitive geometry, and in academic achievement in geometry. *Learning and Individual Differences, 23*, 114–122.

Hawes, Z., LeFevre, J.A., Xu, C. & Bruce, C. (2015). Mental rotation with tangible tree-dimensional objects: a new measure sensitive to developmental differences in 4- to 8-year-old children. *Mind, Brain and Education, 9*(1), 10-18.

Laborde, C. (2001). Integration of technology in the design of geometry tasks with Cabri-Geometry. *International Journal of Computers for Mathematical Learning, 6*(3), 283–317.

Lowrie, T., Logan, T., & Ramful, A. (2017). Visuospatial training improves elementary students’ mathematics performance. *British Journal of Education Psychology, 87*, 170–186.

Marchand, P. (2006). Comment développer les images mentales reliées à l’apprentissage de l’espace en trois dimensions? *Annales de didactique et des sciences cognitives, 11*, 103-121.

Marchand, P. (2009). L’enseignement du sens spatial au secondaire: Analyse de deux leçons de troisième secondaire, *The Canadian Journal of Science, Mathematics and Technology Education, 9*(1), 30-49.

Marchand, P. (2020). Quelques assises pour valoriser le développement des connaissances spatiales à l'école primaire. *Recherches en Didactique des Mathématiques, 40*(2), 1-44.

Ministère de l'Éducation, du Loisir et du Sport (MELS). (2003). *Programme de formation de l'école québécoise*. Bibliothèque nationale du Québec.

Moss, J., Bruce, C. D., Caswell, B, Flynn, T., Hawes, Z. (2016). *Taking Shape. Activities to develop Geometric and Spatial Thinking*. Grades K-2. Pearson Canada.

Mulligan, J., Woolcott, G., Mitchelmore, M., Busatto, S., Lai, J. & Davis, B. (2020). Evaluating the impact of a spatial Reasoning Mathematics Program (SRMP) intervention in the primary school. *Mathematics Education Research Journal, 32*, 285-305. https://doi.org/10.1007/s13394-020-00324-z

Ng, O.-L., & Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment. *ZDM Mathematics Education, 47*(3), 421-434. DOI 10.1007/s11858-014-0660-5

Sinclair, N., Bartolini Bussi, M. G., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2016). Recent research on geometry education: an ICME-13 survey team report. *ZDM Mathematics Education, 48*(5), 691-719. DOI 10.1007/s11858-016-0796-6

Sinclair, N., & Bruce, C. (2015). New opportunities in geometry education at the primary school. *ZDM Mathematics Education 47*(3), 319-329. DOI 10.1007/s11858-015-0693-4.

Sinclair, N., Pimm, D. & Skelin, M. (2012). *Developing Essential Understanding of Geometry for Teaching Mathematics in Grades 5–8*. Essential Understanding Series. National Council of Teachers of Mathematics.

Sinclair, N., & Yurita, (2008). To be or to become: How dynamic geometry changes discourse. *Research in Mathematics Education, 10*(2), 135-150.

Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C, Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. *Psychological Bulletin, 139*(2), 352-402.

Yackel, E. & Wheatley, G. H. (1990). Promoting visual imagery in young pupils. *Arithmetic Teacher, 37*(6), 52-58.
De Freitas, E. & Sinclair, N. (2014). *Mathematics and the body: Material entanglements in the classroom*. Cambridge University Press.

Gilligan-Lee, K.A., Hawes, Z.C.K. & Mix, K.S. (2022). Spatial thinking as the missing piece in mathematics curricula. *Npj Science of Learning, 7*, 10. https://doi.org/10.1038/s41539-022-00128-9.

Giofrè, D., Mammarella, I. C., Ronconi, L., & Cornoldi, C. (2013). Visuospatial working memory in intuitive geometry, and in academic achievement in geometry. *Learning and Individual Differences, 23*, 114–122.

Hawes, Z., LeFevre, J.A., Xu, C. & Bruce, C. (2015). Mental rotation with tangible tree-dimensional objects: a new measure sensitive to developmental differences in 4- to 8-year-old children. *Mind, Brain and Education, 9*(1), 10-18.

Laborde, C. (2001). Integration of technology in the design of geometry tasks with Cabri-Geometry. *International Journal of Computers for Mathematical Learning, 6*(3), 283–317.

Lowrie, T., Logan, T., & Ramful, A. (2017). Visuospatial training improves elementary students’ mathematics performance. *British Journal of Education Psychology, 87*, 170–186.

Marchand, P. (2006). Comment développer les images mentales reliées à l’apprentissage de l’espace en trois dimensions? *Annales de didactique et des sciences cognitives, 11*, 103-121.

Marchand, P. (2009). L’enseignement du sens spatial au secondaire: Analyse de deux leçons de troisième secondaire, *The Canadian Journal of Science, Mathematics and Technology Education, 9*(1), 30-49.

Marchand, P. (2020). Quelques assises pour valoriser le développement des connaissances spatiales à l'école primaire. *Recherches en Didactique des Mathématiques, 40*(2), 1-44.

Ministère de l'Éducation, du Loisir et du Sport (MELS). (2003). *Programme de formation de l'école québécoise*. Bibliothèque nationale du Québec.

Moss, J., Bruce, C. D., Caswell, B, Flynn, T., Hawes, Z. (2016). *Taking Shape. Activities to develop Geometric and Spatial Thinking*. Grades K-2. Pearson Canada.

Mulligan, J., Woolcott, G., Mitchelmore, M., Busatto, S., Lai, J. & Davis, B. (2020). Evaluating the impact of a spatial Reasoning Mathematics Program (SRMP) intervention in the primary school. *Mathematics Education Research Journal, 32*, 285-305. https://doi.org/10.1007/s13394-020-00324-z

Ng, O.-L., & Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment. *ZDM Mathematics Education, 47*(3), 421-434. DOI 10.1007/s11858-014-0660-5

Sinclair, N., Bartolini Bussi, M. G., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2016). Recent research on geometry education: an ICME-13 survey team report. *ZDM Mathematics Education, 48*(5), 691-719. DOI 10.1007/s11858-016-0796-6

Sinclair, N., & Bruce, C. (2015). New opportunities in geometry education at the primary school. *ZDM Mathematics Education 47*(3), 319-329. DOI 10.1007/s11858-015-0693-4.

Sinclair, N., Pimm, D. & Skelin, M. (2012). *Developing Essential Understanding of Geometry for Teaching Mathematics in Grades 5–8*. Essential Understanding Series. National Council of Teachers of Mathematics.

Sinclair, N., & Yurita, (2008). To be or to become: How dynamic geometry changes discourse. *Research in Mathematics Education, 10*(2), 135-150.

Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C, Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. *Psychological Bulletin, 139*(2), 352-402.

Yackel, E. & Wheatley, G. H. (1990). Promoting visual imagery in young pupils. *Arithmetic Teacher, 37*(6), 52-58.
# GEOMETRIC PREDICTION AS A BRIDGING PROCESS BETWEEN TRANSFORMING AND UNDERSTANDING

Elisa Miragliotta

Department of Mathematics “F. Casorati”, University of Pavia

> Building on the theory of figural concepts, this paper describes geometric prediction as a process that lies at the interplay between the pure manipulation of figural elements and the pure selection and use of theoretical ones. In this paper I use a geometric prediction model to analyze the cognitive processes of an expert solver, while she is solving a geometric open problem, showing how it is instrumental in addressing the issue of zooming into the tension between spatial reasoning and geometric understanding.

## SPATIAL REASONING BETWEEN TRANSFORMING AND UNDERSTANDING

Spatial reasoning is strongly intertwined with geometrical activity, as highlighted by the amount of research that lies at the intersection of these two topics. As for research in mathematics education, almost invariably, the definitions of *spatial reasoning* have in common the activity of imagining static or dynamic objects and interacting with them through mental transformations (e.g., rotation, stretch, reflection, to name a few). However, during the resolution of a geometric task, mental or physical transformations alone may not be sufficient, and they need to be accompanied by “analytical thought processes” (Presmeg 1986, p. 45). This is particularly evident when the solver must manage spatial and theoretical dimensions, for instance in the case of *open problems*, where the solvers are free to explore the problem and draw their own conclusions. In such problems the solution process often ends with the formulation of a conjecture after a (physical or mental) exploration of the situation (Mariotti et al., 1997), but the complexity of the processes elicited by the task cannot be fully explained by the intervention of mental transformations alone (Mariotti & Baccaglini-Frank, 2018).

Also for this reason, recently researchers have taken a transdisciplinary perspective and defined *spatial reasoning* as an overarching term used to refer to several dynamic processes that benefit from the constant challenge between two major dimensions: *transforming* and *understanding* (Davis & Spatial Reasoning Study Group, 2015, p. 140). Although this is a fresh take on a traditional theme of the research in mathematics education, the intertwining of spatial reasoning and geometric understanding has not yet been fully unraveled. In this paper I tackle such an issue within the domain of 2D Euclidean geometry.

The main assumption is that specific *bridging processes* lie at the interplay between spatial transformations and geometric understanding. *Geometric prediction* (GP) is one of such bridging processes. It builds on the *Theory of figural concepts* (Fischbein, 1993) and some of its effects can be captured as solvers deal with geometric open problems. This paper focuses on unraveling aspects of the intertwining of transforming and understanding thanks to a GP model (originally introduced in Miragliotta, 2020) developed specifically to analyze the prediction processes of an expert solver.
# GEOMETRIC PREDICTION AS A BRIDGING PROCESS BETWEEN TRANSFORMING AND UNDERSTANDING

Elisa Miragliotta

Department of Mathematics “F. Casorati”, University of Pavia

> Building on the theory of figural concepts, this paper describes geometric prediction as a process that lies at the interplay between the pure manipulation of figural elements and the pure selection and use of theoretical ones. In this paper I use a geometric prediction model to analyze the cognitive processes of an expert solver, while she is solving a geometric open problem, showing how it is instrumental in addressing the issue of zooming into the tension between spatial reasoning and geometric understanding.

## SPATIAL REASONING BETWEEN TRANSFORMING AND UNDERSTANDING

Spatial reasoning is strongly intertwined with geometrical activity, as highlighted by the amount of research that lies at the intersection of these two topics. As for research in mathematics education, almost invariably, the definitions of *spatial reasoning* have in common the activity of imagining static or dynamic objects and interacting with them through mental transformations (e.g., rotation, stretch, reflection, to name a few). However, during the resolution of a geometric task, mental or physical transformations alone may not be sufficient, and they need to be accompanied by “analytical thought processes” (Presmeg 1986, p. 45). This is particularly evident when the solver must manage spatial and theoretical dimensions, for instance in the case of *open problems*, where the solvers are free to explore the problem and draw their own conclusions. In such problems the solution process often ends with the formulation of a conjecture after a (physical or mental) exploration of the situation (Mariotti et al., 1997), but the complexity of the processes elicited by the task cannot be fully explained by the intervention of mental transformations alone (Mariotti & Baccaglini-Frank, 2018).

Also for this reason, recently researchers have taken a transdisciplinary perspective and defined *spatial reasoning* as an overarching term used to refer to several dynamic processes that benefit from the constant challenge between two major dimensions: *transforming* and *understanding* (Davis & Spatial Reasoning Study Group, 2015, p. 140). Although this is a fresh take on a traditional theme of the research in mathematics education, the intertwining of spatial reasoning and geometric understanding has not yet been fully unraveled. In this paper I tackle such an issue within the domain of 2D Euclidean geometry.

The main assumption is that specific *bridging processes* lie at the interplay between spatial transformations and geometric understanding. *Geometric prediction* (GP) is one of such bridging processes. It builds on the *Theory of figural concepts* (Fischbein, 1993) and some of its effects can be captured as solvers deal with geometric open problems. This paper focuses on unraveling aspects of the intertwining of transforming and understanding thanks to a GP model (originally introduced in Miragliotta, 2020) developed specifically to analyze the prediction processes of an expert solver.
## THEORETICAL FRAMEWORK AND RESEARCH QUESTION

In the case of Euclidean geometry, the tension between transforming and understating is particularly intrigued, since it mirrors the well-known tension between spatial and theoretical aspects (Mariotti, 1995). The *Theory of figural concepts* (Fischbein, 1993) capitalizes on this tension by considering the cognitive counterpart of a geometrical object as comprising a *theoretical* and a *figural* component. Indeed, as part of a mathematical theory it is ideal, abstract, perfect, and general (conceptual or theoretical component), but it also reflects spatial properties like shape, position, and magnitude (figural component). Ideally, figural and conceptual components should be fused together in the *figural concept*, that is the actual object we are reasoning upon. During the resolution of a geometric task, the solver might alternatively operate on the figural components, through manipulations, or they might consider the theoretical nature of the objects in focus; however, it can be difficult to identify the contribution of each component when they act in harmony (Mariotti, 1995). In light of this theory, in the following we operationally consider:

* *transforming* as manipulating figural components through physical or mental transformations, that can be more or less isomorphic to geometrical transformations,

* (*geometric*) *understanding* as the act of recalling and using suitable theoretical components which belong to the mathematical reference theory at stake,

* in order to manage the complex mutual relationships between figural and theoretical components that comprise the figural concepts in focus.

In geometry, one of the processes at the interplay between figural manipulation and theoretical elaboration, triggered by *open problems*, takes place when the solver's exploration is carried out by mentally transforming a figure and imagining how it will change given certain constraints (Mariotti & Baccaglini-Frank, 2018). When this happens within the specific domain of Euclidean geometry, we have identified the process of *Geometric Prediction (GP)* as the generation of a new figural concept through the manipulation of its figural elements that maintain invariant certain theoretical elements belonging to the solver's mathematical reference theory (Miragliotta, 2022, p. 16). The process has been modeled, as shown in the visual diagram in Figure 1a, in order to highlight its main components and all their possible connections (see Miragliotta, 2022, for further details).

In a nutshell, the model is composed of two main sides, connected in the center through a sort of funnel, which metaphorically shows the dynamic interaction of the elements that flow into a GP process, producing a new object that is more than the sum of the original elements, but which retains some traces of both. On the left side of the visual diagram, there are *theoretical elements* recalled by the solver; they include all the properties that solvers attribute to the figure (or part of it), theorems and mathematical results. On the right side there are *figural elements*, on which the solver can focus, and which can be manipulated; these are elements of the solver's productions that belong to the figural domain at a specific moment as the solver looks at the figure represented.

Manipulations can be accomplished in two ways: *continuously*, that is solvers can imagine, perform, or mimic a continuous movement of one or more parts of the configuration (i.e., points, segments); *discretely*, that is solvers can locate these parts at a specific position on the plane and reconstruct the corresponding configuration. The solver's interaction with the drawing (if any) can be direct mainly by *bottom-up* (perceptually driven) or *top-down* (theory driven) processes (Gal & Linchevski, 2010).
THEORETICAL FRAMEWORK AND RESEARCH QUESTION

In the case of Euclidean geometry, the tension between transforming and understating is particularly intrigued, since it mirrors the well-known tension between spatial and theoretical aspects (Mariotti, 1995). The *Theory of figural concepts* (Fischbein, 1993) capitalizes on this tension by considering the cognitive counterpart of a geometrical object as comprising a *theoretical* and a *figural* component. Indeed, as part of a mathematical theory it is ideal, abstract, perfect, and general (conceptual or theoretical component), but it also reflects spatial properties like shape, position, and magnitude (figural component). Ideally, figural and conceptual components should be fused together in the *figural concept*, that is the actual object we are reasoning upon. During the resolution of a geometric task, the solver might alternatively operate on the figural components, through manipulations, or they might consider the theoretical nature of the objects in focus; however, it can be difficult to identify the contribution of each component when they act in harmony (Mariotti, 1995). In light of this theory, in the following we operationally consider:

* *transforming* as manipulating figural components through physical or mental transformations, that can be more or less isomorphic to geometrical transformations,

* (*geometric*) *understanding* as the act of recalling and using suitable theoretical components which belong to the mathematical reference theory at stake,

* in order to manage the complex mutual relationships between figural and theoretical components that comprise the figural concepts in focus.

In geometry, one of the processes at the interplay between figural manipulation and theoretical elaboration, triggered by *open problems*, takes place when the solver's exploration is carried out by mentally transforming a figure and imagining how it will change given certain constraints (Mariotti & Baccaglini-Frank, 2018). When this happens within the specific domain of Euclidean geometry, we have identified the process of *Geometric Prediction* (GP) as the generation of a new figural concept through the manipulation of its figural elements that maintain invariant certain theoretical elements belonging to the solver's mathematical reference theory (Miragliotta, 2022, p. 16). The process has been modeled, as shown in the visual diagram in Figure 1a, in order to highlight its main components and all their possible connections (see Miragliotta, 2022, for further details).

In a nutshell, the model is composed of two main sides, connected in the center through a sort of funnel, which metaphorically shows the dynamic interaction of the elements that flow into a GP process, producing a new object that is more than the sum of the original elements, but which retains some traces of both. On the left side of the visual diagram, there are *theoretical elements* recalled by the solver; they include all the properties that solvers attribute to the figure (or part of it), theorems and mathematical results. On the right side there are *figural elements*, on which the solver can focus, and which can be manipulated; these are elements of the solver's productions that belong to the figural domain at a specific moment as the solver looks at the figure represented.

Manipulations can be accomplished in two ways: *continuously*, that is solvers can imagine, perform, or mimic a continuous movement of one or more parts of the configuration (i.e., points, segments); *discretely*, that is solvers can locate these parts at a specific position on the plane and reconstruct the corresponding configuration. The solver's interaction with the drawing (if any) can be direct mainly by *bottom-up* (perceptually driven) or *top-down* (theory driven) processes (Gal & Linchevski, 2010).
Speech and gestures can occur at any time, shaping the process and providing the researcher windows onto the elements in focus.

<figure><img src="image_39.png" /><figcaption>Figure 1: (a) Visual diagram of the GP process (b) An example of prediction open problem</figcaption></figure>

The solver's act of mentally imposing onto the figure theoretical elements that are coherent in the mathematical reference theory at stake, i.e. *theoretical control* (Mariotti & Baccaglini-Frank, 2018), is central and guides each phase. In order to reach a coherent product of GP, a solver has to cover the whole path: starting by recalling theoretical elements referred to particular figural elements, and by continuing to manipulate the corresponding figural elements, under the strict supervision of the theoretical control. The product of GP is a new figural concept (for the given task) enriched by the figural and theoretical components that emerged during the process.

Although prediction processes may be different because their components may have different interactions – resulting in multiple paths through the model – they are all characterized by the interplay between pure manipulation of figural elements and pure recollection and use of theoretical elements.

## Research question

Given the theoretical framework outlined above, this paper addresses the following question: What insight does the model of geometric prediction provide into the intertwining of spatial reasoning and geometric understanding, in the case of dynamic processes involved in the resolution of geometric open problems?

## METHODS

The data analyzed was part of a larger data collection aimed at observing and describing prediction processes of volunteer solvers with a mathematical background, i.e. *expert solvers*. Individual task-based interviews were carried out and focused on *prediction tasks*, that are particular kind of open problems in which the solver is asked to describe possible alternative arrangements of a geometric configuration (imagined, given by a drawing and/or by a step-by-step construction) maintaining given properties. Each solver spent 60 min with the interviewer and worked through as many prediction tasks as they could. Data was collected in the form of video recordings behind the solver, in order to capture all of their productions (verbal, gestural, diagrammatic). Here we focus on a solver, fictionally