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_44.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>(b)</figcaption></figure>

<caption>Figure 1: (a) Visual diagram of the GP process (b) An example of prediction open problem</caption>

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
named Giulia, who at the time of the interview had just graduated with a master's degree in mathematics.

This paper presents excerpts from Giulia's interview as an *instrumental case study* (Stake, 2003), since it provides a telling example of GP processes in a quite short time span. The selected excerpts allow us to navigate the multifaceted aspects of the GP model in depth. More specifically, excerpts report on Giulia's resolution of the prediction task shown in Figure 1b. Interpreting the resolution process using only figural manipulations could be quite challenging. Indeed, the solvers need to consider what has to remain invariant (the right angle at C, the length of AB, the hypotenuse is at AB) while other elements can be modified (the position of C, the legs length); this involves many theoretical aspects. For example, four positions of C can be figurally reached by imagining symmetries, but the right-angled and isosceles triangle can only be considered as a possible configuration by means of a conceptual elaboration.

During the resolution of this task, several GP processes can be accomplished, leading the solver to describe possible alternative configurations. We have called *configuration* an instance of a geometrical figure expressed by a drawing and/or a gesture (see $C_i$ in Figure 2A). Excerpts are analyzed using the GP model (Figure 1a), in order to describe in greater depth how figural manipulation and theoretical elements come into play. The unit of analyses is the interview while the solver is solving each task. Each transcript is divided into segments, delimited by two configurations given or elaborated by the solver. We focus on the segments in which one or more products of GP are communicated by the solver. A *product of GP* is labeled using a progressive number ($GP_i$) when the solver refers to elements (a geometrical object or part of it) which are not present in the drawing or to a new arrangement of the configuration without drawing anything. Gesture and speech are considered jointly as a window onto the solver's processes. A wider description of data analyses along with the analytical scheme is available in (Miragliotta, 2022).

## ANALYSIS

Two selected excerpts from Giulia's interview are presented here and analyzed using the GP model.

### First excepts: Giulia communicates $GP_1$ and $GP_2$

The excerpt starts right after the first question (see Figure 1b and watch the video here).

Giulia: [She points at C] That I can move C so that it makes...[using her finger, she traces an arc from C to A and then a semicircle, starting from A, passing through C and ending at B, see Figure 2B-a]...a half circle [she repeats the same gesture as in Figure 2B-a].

Giulia: Where does this semicircle [she moves her finger upon AB from A to B] has its center...[she points at a position on AB between A and B, see Figure 2B-b]...at the midpoint of AB [she points at A and B one after the other].

Giulia: There, the hypotenuse AB always has a fixed length, because I am only moving C.

Giulia: Since C is...the vertex of a triangle that lies on a semicircle circumscribed to the triangle, it is always right, so the triangle always stays... right. In any place I move C [using her finger, she outlines the beginning of a circular path].

The first product of prediction (see $GP_1$ in Figure 2A) is reached quite soon in the first line. At the beginning Giulia looks at the drawing silently, while she points at C. The initial stage of interpretation of the givens is so fast and silent that we cannot gather information on how the configuration is
named Giulia, who at the time of the interview had just graduated with a master's degree in mathematics.

This paper presents excerpts from Giulia's interview as an *instrumental case study* (Stake, 2003), since it provides a telling example of GP processes in a quite short time span. The selected excerpts allow us to navigate the multifaceted aspects of the GP model in depth. More specifically, excerpts report on Giulia's resolution of the prediction task shown in Figure 1b. Interpreting the resolution process using only figural manipulations could be quite challenging. Indeed, the solvers need to consider what has to remain invariant (the right angle at C, the length of AB, the hypotenuse is at AB) while other elements can be modified (the position of C, the legs length); this involves many theoretical aspects. For example, four positions of C can be figurally reached by imagining symmetries, but the right-angled and isosceles triangle can only be considered as a possible configuration by means of a conceptual elaboration.

During the resolution of this task, several GP processes can be accomplished, leading the solver to describe possible alternative configurations. We have called *configuration* an instance of a geometrical figure expressed by a drawing and/or a gesture (see $C_i$ in Figure 2A). Excerpts are analyzed using the GP model (Figure 1a), in order to describe in greater depth how figural manipulation and theoretical elements come into play. The unit of analyses is the interview while the solver is solving each task. Each transcript is divided into segments, delimited by two configurations given or elaborated by the solver. We focus on the segments in which one or more products of GP are communicated by the solver. A *product of GP* is labeled using a progressive number ($GP_i$) when the solver refers to elements (a geometrical object or part of it) which are not present in the drawing or to a new arrangement of the configuration without drawing anything. Gesture and speech are considered jointly as a window onto the solver's processes. A wider description of data analyses along with the analytical scheme is available in (Miragliotta, 2022).

## ANALYSIS

Two selected excerpts from Giulia's interview are presented here and analyzed using the GP model.

### First excepts: Giulia communicates $GP_1$ and $GP_2$

The excerpt starts right after the first question (see Figure 1b and watch the video here).

* Giulia: [She points at C] That I can move C so that it makes...[using her finger, she traces an arc from C to A and then a semicircle, starting from A, passing through C and ending at B, see Figure 2B-a]...a half circle [she repeats the same gesture as in Figure 2B-a].

* Giulia: Where does this semicircle [she moves her finger upon AB from A to B] has its center...[she points at a position on AB between A and B, see Figure 2B-b]...at the midpoint of AB [she points at A and B one after the other].

* Giulia: There, the hypotenuse AB always has a fixed length, because I am only moving C.

* Giulia: Since C is...the vertex of a triangle that lies on a semicircle circumscribed to the triangle, it is always right, so the triangle always stays... right. In any place I move C [using her finger, she outlines the beginning of a circular path].

The first product of prediction (see $GP_1$ in Figure 2A) is reached quite soon in the first line. At the beginning Giulia looks at the drawing silently, while she points at C. The initial stage of interpretation of the givens is so fast and silent that we cannot gather information on how the configuration is
interpreted; however, looking at the whole excerpt we can say that an interpretation took place and the solver refers to the figural components of the figural concepts in focus using specific theoretical elements (i.e., hypothenuse, right triangle, right angle). The most observable part of the process is the continuous manipulation of figural elements (point C). Giulia engages in a dialogue with the given drawing: after she mimics the semicircle for the first time, she looks at the drawing silently; then she concludes that C “makes a half circle” and simultaneously she repeats the circular gesture. We can infer that the theoretical control intervenes for checking the coherence of the figural manipulation outcome (i.e., the circular path) and then recalling the most suitable theoretical counterpart (i.e., the semicircle). The dynamic and silent gestures coupled with the following utterance suggest that the outcome of the process is not a mere recollection from long-term memory of a theoretical element in a crystallized form (as she will do later), but an actual construction of a new figural concept, i.e. a prediction. She repeats the same gesture twice (Figure 2B-a), but with slight differences: the first gesture is performed silently and slowly, it is fragmented and quite tentative; the second gesture is fast, confident, and coupled with speech. These differences might correspond to different roles. The first gesture seems to be performed in order to look at a possible manipulation; so far, the outcome is quite fuzzy, and the gesture provides figural support to solver’s processes of prediction. The second gesture seems to be made only to emphasize the solver’s verbal production and to confirm her now less vague prediction. Finally, the gesture serves to crystallize the product of Giulia’s prediction: the dyad composed of the drawing and the circular gesture is the initial configuration ($C_2$ in Figure 2A) for the next GP processes. Indeed, she keeps talking about the semicircle as if it is actually part of the drawing (see the deixis “this” in the second utterance of the excerpt).

<figure><img src="image_40.png" /><figcaption>Figure 2: (A) Overview of Giulia’s GP processes and their connections, each $GP_i$ is a reformulation of the solver’s productions; (B) instances of Giulia’s gestural production</figcaption></figure>

The second process shows an intense interplay between figural and theoretical elements. A new figural element is discretely introduced (a point on *AB*) and it is seen as part of different figural concepts, as demonstrated by the introduction of two theoretical elements (the circle center, the midpoint of *AB*). The theoretical control allows the solver to manage this complexity. The utterance shows the intertwinement of theoretical and figural components of the figural concepts in focus, which leads the solver to refine the first prediction by communicating additional geometrical properties of the semicircle: its center and radius. $GP_1$ is refined in $GP_2$ (see Figure 2A) through the top-down imposition of additional properties. This GP process is condensed into a single static
interpreted; however, looking at the whole excerpt we can say that an interpretation took place and the solver refers to the figural components of the figural concepts in focus using specific theoretical elements (i.e., hypothenuse, right triangle, right angle). The most observable part of the process is the continuous manipulation of figural elements (point C). Giulia engages in a dialogue with the given drawing: after she mimics the semicircle for the first time, she looks at the drawing silently; then she concludes that C “makes a half circle” and simultaneously she repeats the circular gesture. We can infer that the theoretical control intervenes for checking the coherence of the figural manipulation outcome (i.e., the circular path) and then recalling the most suitable theoretical counterpart (i.e., the semicircle). The dynamic and silent gestures coupled with the following utterance suggest that the outcome of the process is not a mere recollection from long-term memory of a theoretical element in a crystallized form (as she will do later), but an actual construction of a new figural concept, i.e. a prediction. She repeats the same gesture twice (Figure 2B-a), but with slight differences: the first gesture is performed silently and slowly, it is fragmented and quite tentative; the second gesture is fast, confident, and coupled with speech. These differences might correspond to different roles. The first gesture seems to be performed in order to look at a possible manipulation; so far, the outcome is quite fuzzy, and the gesture provides figural support to solver’s processes of prediction. The second gesture seems to be made only to emphasize the solver’s verbal production and to confirm her now less vague prediction. Finally, the gesture serves to crystallize the product of Giulia’s prediction: the dyad composed of the drawing and the circular gesture is the initial configuration ($C_2$ in Figure 2A) for the next GP processes. Indeed, she keeps talking about the semicircle as if it is actually part of the drawing (see the deixis “this” in the second utterance of the excerpt).

<figure><img src="image_46.png" /><figcaption>Figure 2: (A) Overview of Giulia’s GP processes and their connections, each $GP_i$ is a reformulation of the solver’s productions; (B) instances of Giulia’s gestural production</figcaption></figure>

The second process shows an intense interplay between figural and theoretical elements. A new figural element is discretely introduced (a point on *AB*) and it is seen as part of different figural concepts, as demonstrated by the introduction of two theoretical elements (the circle center, the midpoint of *AB*). The theoretical control allows the solver to manage this complexity. The utterance shows the intertwinement of theoretical and figural components of the figural concepts in focus, which leads the solver to refine the first prediction by communicating additional geometrical properties of the semicircle: its center and radius. $GP_1$ is refined in $GP_2$ (see Figure 2A) through the top-down imposition of additional properties. This GP process is condensed into a single static
utterance (“this semicircle has its center at the midpoint of AB”), but the claims that follow suggest a more dynamic dimension that is only verbally recalled (“I am only moving C”) and not gesturally performed. Indeed, Giulia says that when C is moving “there” on the semicircle she has imagined, the hypothenuse AB “has a fixed length”, stressing that the given constraint on AB is maintained.

The tone of the voice and the long pauses suggest that Giulia is quite hesitant, until she recalls a theorem that connects the properties of right angles, circles, and triangles. It is not formally stated, rather Giulia simply recalls the theoretical elements that support her prediction. The tone of her voice suggests that she is recalling a piece of previous knowledge, suggesting the intervention of theoretical control. We notice that Giulia uses a statically stated theorem to justify a dynamically reached prediction. As already stressed, although the positions of C were dynamically explored, when the semicircle emerges as a product of prediction it becomes part of the configuration, to the extent that the point now “lies on” the semicircle that was created by that point instead. In the same way, Giulia seems to start from a generic triangle that is inscribed on a semicircle: the right-angled triangle, which was a given, becomes a result that is drawn from the rightness of the angle at C. Therefore, the product of GP becomes a hypothesis, while the right angle at C is the thesis.

## Excerpt 2: Giulia undertakes two additional prediction processes that lead to $GP_3$ and $GP_4$

A new process of prediction starts when the interviewer asks for positions for point C (video here).

**Giulia:** So that it maintains this configuration? If I draw...[using her finger, she points at C and traces a line downwards to a specific point, see Figure 2B-c]...point transformed from C [she points at A and B] with respect to AB, so I send the perpendicular from C to AB [she points at C and traces a line towards AB], that is I draw [she uses two fingers, see Figure 2B-d] a segment that is always perpendicular to AB of the same length [she uses two fingers between C and AB and then as in Figure 2B-d] of the one that I drew before.

**Giulia:** So, C can...can also stay on the opposite side [she traces a small arc from A to the imagined symmetric point, Figure 2B-e]. And so, in the end C can be on the...[she traces a circle, see Figure 2B-f]...can be, yes, lies on a circle with center at the midpoint of AB.

$GP_3$ is reached quite quickly and it incorporates all the theoretical and figural elements that were part of $GP_1$ and $GP_2$. Giulia starts reconsidering all the given constraints (see the first interrogative statement), she focuses on the figural element point C, and mimics the motion of C towards and below AB (Figure 2B-c). The gesture follows continuously a straight trajectory (as if the solver were holding a fictional pencil), but it is quite fast and confident as she were anticipating the final destination, so in this sense the manipulation is also discrete. The final pointing gesture communicates $GP_3$, which is silently contemplated for a while. There is a silent top-down and bottom-up interaction with the drawing, and finally the verbal production informs us that the new point is “transformed with respect to AB”, that is according to a line symmetry. We can infer that the theoretical control supports the solver in evaluating the coherence of this manipulation, before continuing the exploration. The theoretical control also supports the introduction of new theoretical elements which add detail to the prediction. Indeed, Giulia claims that the new point is obtained by a geometric transformation of C and then she starts explaining how the symmetric point can be geometrically constructed. In doing this, she introduces a number of new theoretical elements (perpendicular line, length transport, opposite position), coupled with new figural elements, which blend together in a dense interplay. Here again some figural elements are interpreted as part of different figural concepts: now the hypothenuse and diameter AB is conceived as an axis for the line symmetry.
utterance (“this semicircle has its center at the midpoint of AB”), but the claims that follow suggest a more dynamic dimension that is only verbally recalled (“I am only moving C”) and not gesturally performed. Indeed, Giulia says that when C is moving “there” on the semicircle she has imagined, the hypothenuse AB “has a fixed length”, stressing that the given constraint on AB is maintained.

The tone of the voice and the long pauses suggest that Giulia is quite hesitant, until she recalls a theorem that connects the properties of right angles, circles, and triangles. It is not formally stated, rather Giulia simply recalls the theoretical elements that support her prediction. The tone of her voice suggests that she is recalling a piece of previous knowledge, suggesting the intervention of theoretical control. We notice that Giulia uses a statically stated theorem to justify a dynamically reached prediction. As already stressed, although the positions of C were dynamically explored, when the semicircle emerges as a product of prediction it becomes part of the configuration, to the extent that the point now “lies on” the semicircle that was created by that point instead. In the same way, Giulia seems to start from a generic triangle that is inscribed on a semicircle: the right-angled triangle, which was a given, becomes a result that is drawn from the rightness of the angle at C. Therefore, the product of GP becomes a hypothesis, while the right angle at C is the thesis.

## Excerpt 2: Giulia undertakes two additional prediction processes that lead to $GP_3$ and $GP_4$

A new process of prediction starts when the interviewer asks for positions for point C (video here).

**Giulia:** So that it maintains this configuration? If I draw...[using her finger, she points at C and traces a line downwards to a specific point, see Figure 2B-c]...point transformed from C [she points at A and B] with respect to AB, so I send the perpendicular from C to AB [she points at C and traces a line towards AB], that is I draw [she uses two fingers, see Figure 2B-d] a segment that is always perpendicular to AB of the same length [she uses two fingers between C and AB and then as in Figure 2B-d] of the one that I drew before.

**Giulia:** So, C can...can also stay on the opposite side [she traces a small arc from A to the imagined symmetric point, Figure 2B-e]. And so, in the end C can be on the...[she traces a circle, see Figure 2B-f]...can be, yes, lies on a circle with center at the midpoint of AB.

$GP_3$ is reached quite quickly and it incorporates all the theoretical and figural elements that were part of $GP_1$ and $GP_2$. Giulia starts reconsidering all the given constraints (see the first interrogative statement), she focuses on the figural element point C, and mimics the motion of C towards and below AB (Figure 2B-c). The gesture follows continuously a straight trajectory (as if the solver were holding a fictional pencil), but it is quite fast and confident as she were anticipating the final destination, so in this sense the manipulation is also discrete. The final pointing gesture communicates $GP_3$, which is silently contemplated for a while. There is a silent top-down and bottom-up interaction with the drawing, and finally the verbal production informs us that the new point is “transformed with respect to AB”, that is according to a line symmetry. We can infer that the theoretical control supports the solver in evaluating the coherence of this manipulation, before continuing the exploration. The theoretical control also supports the introduction of new theoretical elements which add detail to the prediction. Indeed, Giulia claims that the new point is obtained by a geometric transformation of C and then she starts explaining how the symmetric point can be geometrically constructed. In doing this, she introduces a number of new theoretical elements (perpendicular line, length transport, opposite position), coupled with new figural elements, which blend together in a dense interplay. Here again some figural elements are interpreted as part of different figural concepts: now the hypothenuse and diameter AB is conceived as an axis for the line symmetry.
The process moves in a constant interplay between the left and right side of the model, driven by
theoretical control. The output of this process, which blends the recollection of theoretical elements
and the manipulation of figural elements, culminates in GP₃.

The products of all these prediction processes flow into a final one (see Figure 2A). Right after the
symmetric point is described, Giulia summarizes her findings (“So, C can [...] and so, in the end”).
She starts from the last GP, stressing the new position of C, and outlines a small arc with her finger
(see Figure 2B-f). This gesture blends the initial locus (GP₁, GP₂) and the symmetric position of C
(GP₃) in a new locus. GP₄ constitutes a refined and extended version of the previously communicated
predictions (see Figure 2A). Finally, we observe that the last claim on C is purified of any dynamic
dimension and crystallized into a static statement. Indeed, at the very beginning of the interview, the
solver has stated that she “can move C so that it makes” a certain locus, while now, at the end of the
exploration, C “lies on” a certain fixed locus.

DISCUSSION AND CONCLUSION

We set out to describe *GP* as a bridging process between spatial reasoning and geometric
understanding in order to gain a deep insight into their interplay in the case of Euclidean geometry.

Following the GP model while an expert solver is solving an open problem, we can see how the
process contains a constant interplay between the theoretical and the figural elements as they are
embedded into the two sides of the model. In particular, we can identify two catalyzers of the GP
process: the prominent role played by the theoretical aspects and the dynamic approach. Indeed, from
the beginning Giulia’s figural manipulations seem to be prompted by the aim of looking at possible
positions of point C; this dynamic view guides her GP processes. At the same time, she needs fixed
products of the manipulation to be reinvested into a new process, giving rise to chains of GP processes
(Figure 2A). We can confirm that GP processes involve many components of spatial reasoning, but
only a strong geometric understanding can support and suggest spatial transformations that are
effective and controlled. This is particularly evident when Giulia recalls theoretical elements that are
part of a theorem she knows and when she describes the isometric transformation of point C. On the
other hand, figural manipulations could guide the introduction of new theoretical elements,
suggesting that spatial transformations might affect geometric understanding. Indeed, before
introducing the theoretical elements of the theorem in a crystallized form, Giulia seems to need the
figural manipulation of point C. Moreover, very often the figural component, that is mainly
communicated gesturally, precedes the verbal description of the theoretical counterpart of the figural
concepts at stake.

More generally, expert solvers demonstrate a productive combination of theoretical and figural
aspects, which come together into play in an immediate and condensed processes of GP. An essential
part of the process consists in giving (multiple) geometrical interpretations to spatial elements.
Indeed, looking diachronically at the elements in focus during the four GP processes, gestural and
verbal productions reveal that the figural elements (e.g., AB) change their theoretical status (e.g.,
hypothenuse or side of a right triangle, diameter of a circle, axis of a line symmetry). The graphical
element is always the same (e.g., the straight line between A and B), but each time it is addressed as
a different geometrical object. Giving different geometric meanings to the same object could involve
the spatial transformations (e.g., decomposing, rearranging), but it deals also with the geometrical
The process moves in a constant interplay between the left and right side of the model, driven by
theoretical control. The output of this process, which blends the recollection of theoretical elements
and the manipulation of figural elements, culminates in GP₃.

The products of all these prediction processes flow into a final one (see Figure 2A). Right after the
symmetric point is described, Giulia summarizes her findings (“So, C can [...] and so, in the end”).
She starts from the last GP, stressing the new position of C, and outlines a small arc with her finger
(see Figure 2B-f). This gesture blends the initial locus (GP₁, GP₂) and the symmetric position of C
(GP₃) in a new locus. GP₄ constitutes a refined and extended version of the previously communicated
predictions (see Figure 2A). Finally, we observe that the last claim on C is purified of any dynamic
dimension and crystallized into a static statement. Indeed, at the very beginning of the interview, the
solver has stated that she “can move C so that it makes” a certain locus, while now, at the end of the
exploration, C “lies on” a certain fixed locus.

DISCUSSION AND CONCLUSION

We set out to describe *GP* as a bridging process between spatial reasoning and geometric
understanding in order to gain a deep insight into their interplay in the case of Euclidean geometry.

Following the GP model while an expert solver is solving an open problem, we can see how the
process contains a constant interplay between the theoretical and the figural elements as they are
embedded into the two sides of the model. In particular, we can identify two catalyzers of the GP
process: the prominent role played by the theoretical aspects and the dynamic approach. Indeed, from
the beginning Giulia's figural manipulations seem to be prompted by the aim of looking at possible
positions of point C; this dynamic view guides her GP processes. At the same time, she needs fixed
products of the manipulation to be reinvested into a new process, giving rise to chains of GP processes
(Figure 2A). We can confirm that GP processes involve many components of spatial reasoning, but
only a strong geometric understanding can support and suggest spatial transformations that are
effective and controlled. This is particularly evident when Giulia recalls theoretical elements that are
part of a theorem she knows and when she describes the isometric transformation of point C. On the
other hand, figural manipulations could guide the introduction of new theoretical elements,
suggesting that spatial transformations might affect geometric understanding. Indeed, before
introducing the theoretical elements of the theorem in a crystallized form, Giulia seems to need the
figural manipulation of point C. Moreover, very often the figural component, that is mainly
communicated gesturally, precedes the verbal description of the theoretical counterpart of the figural
concepts at stake.

More generally, expert solvers demonstrate a productive combination of theoretical and figural
aspects, which come together into play in an immediate and condensed processes of GP. An essential
part of the process consists in giving (multiple) geometrical interpretations to spatial elements.
Indeed, looking diachronically at the elements in focus during the four GP processes, gestural and
verbal productions reveal that the figural elements (e.g., AB) change their theoretical status (e.g.,
hypothenuse or side of a right triangle, diameter of a circle, axis of a line symmetry). The graphical
element is always the same (e.g., the straight line between A and B), but each time it is addressed as
a different geometrical object. Giving different geometric meanings to the same object could involve
the spatial transformations (e.g., decomposing, rearranging), but it deals also with the geometrical
interpretation of the configuration and, in particular, with the (meta)cognitive act of seeing the same figural element as part of different figural concepts. In the case of GP processes, theoretical control collects many metacognitive acts: monitoring and checking the coherence of a figural manipulations, managing the temporality (from dynamism to crystallization and back), supporting the selection of theoretical elements to be recalled and recognizing invariants, (re)interpreting given figural elements, considering or rejecting the outcome of a manipulations as suitable for a new dynamic process.

Zooming out from the particular case in focus and considering the GP model as a prototypical example of bridging processes, we can say that the interplay between spatial transformations and geometric understanding is driven by specific metacognitive acts that deserve our explicit attention in order to be further explored and identified; the GP model allows us to highlight that theoretical control is one of them. Therefore, the study shows the explanatory potential of analyzing processes which lie at the intersection between different cognitive activities, but further research is needed in order to identify other bridging processes within and beyond Euclidean geometry.

From the educational point of view, this study provides further evidence of the importance of supporting the blending of theoretical and figural components of figural concepts. This must become an educational goal. Since open problems seem to spontaneously trigger processes that elicit a dialectic between the two components, we can advance the hypothesis that their use in geometrical activity could help educators in pursuing this goal. Investigations on open problem is not new, however the new perspectives on spatial reasoning can stimulate a new fresh insight into their use both for researcher and educators.

## References

* Davis, B., & Spatial Reasoning Study Group, (2015). *Spatial reasoning in the early years: principles, assertions, and speculations* (1st ed.). Routledge.

* Fischbein, E. (1993). The theory of figural concepts. *Educational Studies in Mathematics*, 24(2), 139–162.

* Gal, H., & Linchevski, L. (2010). To see or not to see: analyzing difficulties in geometry from the perspective of visual perception. *Educational Studies in Mathematics*, 74(2), 163–183.

* Mariotti, M. A. (1995). Images and concepts in geometrical reasoning. In R. Sutherland & J. Mason (Eds.), *Exploiting mental imagery with computers in mathematics education* (pp. 97–116). Springer.

* Mariotti, M.A., & Baccaglini-Frank, A. (2018). Developing geometrical exploration skills through dynamic geometry. In S. Carreira, N. Amado, & K. Jones (Eds.), *Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect* (pp. 153–176). Springer.

* Mariotti, M. A., Bartolini Bussi, M. G., Boero, P., Ferri, F. E., & Garuti, R. (1997). Approaching geometry theorems in contexts: from history and epistemology to cognition. In E. Pehkonen (Ed.), *Proceedings of the 21st PME Conference* (Vol. 1, pp. 180–195). PME.

* Miragliotta, E. (2022). Geometric prediction: A framework to gain insight into solvers’ geometrical reasoning. *The Journal of Mathematical Behavior*, 65, 100927.

* Presmeg, N.C. (1986). Visualisation in high school mathematics. *For the Learning of Mathematics*, 6(3), 42–46.

* Stake, R. (2003). Case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), *Strategies of qualitative inquiry* (2nd ed.) (pp. 134–164). Sage.
interpretation of the configuration and, in particular, with the (meta)cognitive act of seeing the same figural element as part of different figural concepts. In the case of GP processes, theoretical control collects many metacognitive acts: monitoring and checking the coherence of a figural manipulations, managing the temporality (from dynamism to crystallization and back), supporting the selection of theoretical elements to be recalled and recognizing invariants, (re)interpreting given figural elements, considering or rejecting the outcome of a manipulations as suitable for a new dynamic process.

Zooming out from the particular case in focus and considering the GP model as a prototypical example of bridging processes, we can say that the interplay between spatial transformations and geometric understanding is driven by specific metacognitive acts that deserve our explicit attention in order to be further explored and identified; the GP model allows us to highlight that theoretical control is one of them. Therefore, the study shows the explanatory potential of analyzing processes which lie at the intersection between different cognitive activities, but further research is needed in order to identify other bridging processes within and beyond Euclidean geometry.

From the educational point of view, this study provides further evidence of the importance of supporting the blending of theoretical and figural components of figural concepts. This must become an educational goal. Since open problems seem to spontaneously trigger processes that elicit a dialectic between the two components, we can advance the hypothesis that their use in geometrical activity could help educators in pursuing this goal. Investigations on open problem is not new, however the new perspectives on spatial reasoning can stimulate a new fresh insight into their use both for researcher and educators.

## References

* Davis, B., & Spatial Reasoning Study Group, (2015). *Spatial reasoning in the early years: principles, assertions, and speculations* (1st ed.). Routledge.

* Fischbein, E. (1993). The theory of figural concepts. *Educational Studies in Mathematics*, 24(2), 139–162.

* Gal, H., & Linchevski, L. (2010). To see or not to see: analyzing difficulties in geometry from the perspective of visual perception. *Educational Studies in Mathematics*, 74(2), 163–183.

* Mariotti, M. A. (1995). Images and concepts in geometrical reasoning. In R. Sutherland & J. Mason (Eds.), *Exploiting mental imagery with computers in mathematics education* (pp. 97–116). Springer.

* Mariotti, M.A., & Baccaglini-Frank, A. (2018). Developing geometrical exploration skills through dynamic geometry. In S. Carreira, N. Amado, & K. Jones (Eds.), *Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect* (pp. 153–176). Springer.

* Mariotti, M. A., Bartolini Bussi, M. G., Boero, P., Ferri, F. E., & Garuti, R. (1997). Approaching geometry theorems in contexts: from history and epistemology to cognition. In E. Pehkonen (Ed.), *Proceedings of the 21st PME Conference* (Vol. 1, pp. 180–195). PME.

* Miragliotta, E. (2022). Geometric prediction: A framework to gain insight into solvers’ geometrical reasoning. *The Journal of Mathematical Behavior*, 65, 100927.

* Presmeg, N.C. (1986). Visualisation in high school mathematics. *For the Learning of Mathematics*, 6(3), 42–46.

* Stake, R. (2003). Case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), *Strategies of qualitative inquiry* (2nd ed.) (pp. 134–164). Sage.
# BUILDING MATHEMATICAL MATURITY THROUGH ALGEBRAIC TOPOLOGY CHALLENGING PATHS

Annamaria Miranda

University of Salerno

A central educational goal of higher-level mathematics undergraduate courses is to foster the development of competences that enable students to use mathematical abstraction and formalism and, at the same time, to develop mathematical creativity. Does it mean mathematical maturity? Research in university mathematics education suggests that to grasp the characteristics of maturity, it might be beneficial to introduce students to theoretical objects by engaging them in the autonomous construction of examples, conjecturing, and proving, especially in geometry. This approach is generally fruitful, but students sometimes have difficulty providing counterexamples to false conjectures, formulating conjectures, or proving statements. Starting from the point that there is no way to acquire mathematical maturity except by doing math, we introduce a path that stimulates students to construct knowledge by themselves in the realm of topology. We face the issue of promoting students' maturity through designing paths that stimulate geometric reasoning. We present the first findings on some students' development and self-perceptions of their maturity.

## INTRODUCTION

Mathematical maturity is one of those expectations in higher education that is often mentioned but not yet clearly defined (Steen, 1983; Lew, 2019). There is no empirically based description of mathematical maturity currently. Lew explores existing descriptions of mathematical maturity as well as descriptions of the related concepts of mathematical intuition and mathematical beliefs, investigating how mathematicians describe mathematical maturity. Interviewed mathematicians report using this term in various ways: ways of thinking about mathematics, mathematical intuition, and comfort with and competence in mathematics. Even more obscure is how exactly to reach mathematical maturity, both for educators and students. Research in mathematics education agrees that encouraging students to work on their own to generate examples of mathematical concepts, to conjecture, to prove or refute a statement, or to prove a theorem is an effective way to reach maturity. We firmly believe that two components, both stimulating conceptual understanding, can decisively contribute to its achievement at every school level: the first one concerns the topic, and a powerful candidate is geometry, which with its visual push can be an excellent springboard, and the second one deals with the teaching practices to be implemented, designing suitable paths that start from the construction of examples and pass through conjecture, proving or disproving, and giving alternative proofs of known theorems. In geometry, properties of figures derive from definitions within an axiomatic system: a figure is “controlled by its definition” (Fischbein, 1993). The construction of a visual representation of a key definition or concept in the statement of a theorem can lead to the recognition of the key idea of the proof (Gallagher & Infante, 2022). A primary goal of advanced mathematics courses at the university level is to have students become proficient at writing proofs. However, research has reported that this goal is rarely met. The difficulty seems to be in passing from
# BUILDING MATHEMATICAL MATURITY THROUGH ALGEBRAIC TOPOLOGY CHALLENGING PATHS

Annamaria Miranda

University of Salerno

A central educational goal of higher-level mathematics undergraduate courses is to foster the development of competences that enable students to use mathematical abstraction and formalism and, at the same time, to develop mathematical creativity. Does it mean mathematical maturity? Research in university mathematics education suggests that to grasp the characteristics of maturity, it might be beneficial to introduce students to theoretical objects by engaging them in the autonomous construction of examples, conjecturing, and proving, especially in geometry. This approach is generally fruitful, but students sometimes have difficulty providing counterexamples to false conjectures, formulating conjectures, or proving statements. Starting from the point that there is no way to acquire mathematical maturity except by doing math, we introduce a path that stimulates students to construct knowledge by themselves in the realm of topology. We face the issue of promoting students' maturity through designing paths that stimulate geometric reasoning. We present the first findings on some students' development and self-perceptions of their maturity.

## INTRODUCTION

Mathematical maturity is one of those expectations in higher education that is often mentioned but not yet clearly defined (Steen, 1983; Lew, 2019). There is no empirically based description of mathematical maturity currently. Lew explores existing descriptions of mathematical maturity as well as descriptions of the related concepts of mathematical intuition and mathematical beliefs, investigating how mathematicians describe mathematical maturity. Interviewed mathematicians report using this term in various ways: ways of thinking about mathematics, mathematical intuition, and comfort with and competence in mathematics. Even more obscure is how exactly to reach mathematical maturity, both for educators and students. Research in mathematics education agrees that encouraging students to work on their own to generate examples of mathematical concepts, to conjecture, to prove or refute a statement, or to prove a theorem is an effective way to reach maturity. We firmly believe that two components, both stimulating conceptual understanding, can decisively contribute to its achievement at every school level: the first one concerns the topic, and a powerful candidate is geometry, which with its visual push can be an excellent springboard, and the second one deals with the teaching practices to be implemented, designing suitable paths that start from the construction of examples and pass through conjecture, proving or disproving, and giving alternative proofs of known theorems. In geometry, properties of figures derive from definitions within an axiomatic system: a figure is “controlled by its definition” (Fischbein, 1993). The construction of a visual representation of a key definition or concept in the statement of a theorem can lead to the recognition of the key idea of the proof (Gallagher & Infante, 2022). A primary goal of advanced mathematics courses at the university level is to have students become proficient at writing proofs. However, research has reported that this goal is rarely met. The difficulty seems to be in passing from
the stage in which the operational aspects of a mathematical concept and, more generally, the
procedural knowledge of mathematics is gained to the stage at which the structural aspects and
conceptual knowledge of mathematics are acquired (Sfard, 1991). Tall and Vinner (1981) relate the
understanding of a concept to the distinct notions of concept image and concept definition. Moore
integrates the construct with the notion of concept usage, "which refers to the ways one operates with
the concepts in doing proofs" (Moore, 1994). An efficient way to develop conceptual understanding
is to approach it through examples. The exploration of example spaces is essential to learning
mathematics, and this power manifests particularly in geometry. Watson and Mason (2005) propose
that "examples can be perceived or experienced as members of structured spaces" and introduce the
term "example space." Example space exploration and extension activities play an important role in
students' mathematical thinking as they make formal mathematics more familiar. Dahlberg and
Housman (1997) argue that creating examples can be a powerful "learning event" in which students
make real progress in understanding and that it might be beneficial to introduce students to new
concepts by starting with a concept definition and going on to enrich the concept image. Conceptual
understanding is necessary for advanced mathematical thinking (Tall, 1992) and creativity, two
essential characteristics of maturity. We could say that:

mathematical maturity refers to cognitive, communicative, metacognitive, and affective problem-solving competences dealing with generating examples and counterexamples, conjecturing, proving, and mathematical creativity in constructing original examples, modifying some already known or generating new ones, producing original proofs of already known theorems using different techniques or new theorems, and, in the meantime, to a sense of self-efficacy and mastery, even in collective discussions.

Once given an attempt at the definition of maturity, it comes naturally to face the following questions:
how can we encourage students' mathematical maturity? What specific paths or trajectories during
the process of learning could help them to be able to independently learn and apply geometry? In
what ways can students be engaged in formulating conjectures and doing geometric proofs? Our
attempt at an answer lies in the design of specific paths that engage students in self-constructing
knowledge. We address the issue of promoting students' maturity by designing suitable paths in the
realm of geometry, which in our setting is topology. Topology is a tool encouraging undergraduate
students' maturity, as counterexample constructions (Steen & Seebach, 1970), intuition, abstraction,
and generalization are highly valued. Its dual figural-conceptual power arises from the impulse to
abstract essential features from complex situations and then to let our curiosity roam while striving
to truly understand what is essential about fundamental ideas (Fischbein, 1993; Tall, 1992; Weber &
Alcock, 2004; Selden & Selden, 2013). The designed path aims to engage students in example
construction, conjecturing, and proving in topology, helping them to make deeper sense of known
theorems and techniques and to discover new ones. In the paper, we give a taste of students' answers
to highlight the figural-conceptual power of topology and analyze how students show and perceive
their own maturity after the implemented path.

RESEARCH FRAMEWORK AND RESEARCH QUESTIONS

The theoretical model to which we refer to design the path is that of *concept usage* (Moore, 1994), in the form proposed by Dahlberg and Housman (1997), integrated with the construct of *example space* (Watson & Mason, 2005) and with the notion of *figural concept* (Fishbein, 1993), essential in the realm of geometry. Dahlberg and Housman assign students some exploration "pages," starting from a concept definition, to develop their own related concept images: to define the concept (*definition*
the stage in which the operational aspects of a mathematical concept and, more generally, the
procedural knowledge of mathematics is gained to the stage at which the structural aspects and
conceptual knowledge of mathematics are acquired (Sfard, 1991). Tall and Vinner (1981) relate the
understanding of a concept to the distinct notions of concept image and concept definition. Moore
integrates the construct with the notion of concept usage, "which refers to the ways one operates with
the concepts in doing proofs" (Moore, 1994). An efficient way to develop conceptual understanding
is to approach it through examples. The exploration of example spaces is essential to learning
mathematics, and this power manifests particularly in geometry. Watson and Mason (2005) propose
that "examples can be perceived or experienced as members of structured spaces" and introduce the
term "example space." Example space exploration and extension activities play an important role in
students' mathematical thinking as they make formal mathematics more familiar. Dahlberg and
Housman (1997) argue that creating examples can be a powerful "learning event" in which students
make real progress in understanding and that it might be beneficial to introduce students to new
concepts by starting with a concept definition and going on to enrich the concept image. Conceptual
understanding is necessary for advanced mathematical thinking (Tall, 1992) and creativity, two
essential characteristics of maturity. We could say that:

mathematical maturity refers to cognitive, communicative, metacognitive, and affective problem-solving competences dealing with generating examples and counterexamples, conjecturing, proving, and mathematical creativity in constructing original examples, modifying some already known or generating new ones, producing original proofs of already known theorems using different techniques or new theorems, and, in the meantime, to a sense of self-efficacy and mastery, even in collective discussions.

Once given an attempt at the definition of maturity, it comes naturally to face the following questions:
how can we encourage students' mathematical maturity? What specific paths or trajectories during
the process of learning could help them to be able to independently learn and apply geometry? In
what ways can students be engaged in formulating conjectures and doing geometric proofs? Our
attempt at an answer lies in the design of specific paths that engage students in self-constructing
knowledge. We address the issue of promoting students' maturity by designing suitable paths in the
realm of geometry, which in our setting is topology. Topology is a tool encouraging undergraduate
students' maturity, as counterexample constructions (Steen & Seebach, 1970), intuition, abstraction,
and generalization are highly valued. Its dual figural-conceptual power arises from the impulse to
abstract essential features from complex situations and then to let our curiosity roam while striving
to truly understand what is essential about fundamental ideas (Fischbein, 1993; Tall, 1992; Weber &
Alcock, 2004; Selden & Selden, 2013). The designed path aims to engage students in example
construction, conjecturing, and proving in topology, helping them to make deeper sense of known
theorems and techniques and to discover new ones. In the paper, we give a taste of students' answers
to highlight the figural-conceptual power of topology and analyze how students show and perceive
their own maturity after the implemented path.

RESEARCH FRAMEWORK AND RESEARCH QUESTIONS

The theoretical model to which we refer to design the path is that of *concept usage* (Moore, 1994), in the form proposed by Dahlberg and Housman (1997), integrated with the construct of *example space* (Watson & Mason, 2005) and with the notion of *figural concept* (Fishbein, 1993), essential in the realm of geometry. Dahlberg and Housman assign students some exploration "pages," starting from a concept definition, to develop their own related concept images: to define the concept (*definition*
page); to generate two examples of objects, one satisfying that definition and the other not (generation page); to verify whether some given objects do or do not satisfy the definition (verification page); and to explore some statements linking that concept to others (conjecture page). We make Dahlberg and Housman's model more powerful by adding a page named the bridge page (Miranda, 2023b), requiring the example space extension through bridging examples suitable to overcome students' difficulty in providing counterexamples and in formulating conjectures. The lens with which we look at students' maturity and how students perceive it refers to the descriptions reported by Lew (2019). In this proposal, considering how mathematicians describe mathematical maturity (Lew, 2019), we give a taste of students' answers in a particular step of the path, the last proving/disproving conjecturing phase, highlighting the power of figural and conceptual aspects in developing an aware concept usage and investigating students' perceived maturity. We address this research question: *What is the impact of such an approach on students' outcomes and perceptions of advanced mathematics and their personal maturity in knowing or constructing it?*

## METHODOLOGY

The research is part of a project concerning the transition to advanced mathematical thinking (Miranda, 2023a), supported by paths starting with the example of space extensions and going towards generalization and abstraction (Miranda, 2023b). The implementation, held in the academic years 2021–22 and 2022–23, involved 34 students attending an introductory algebraic topology course within a Bachelor of Mathematics, in which we gave definitions of algebraic objects that are topological invariants, such as the fundamental group. The path was designed to encourage the transition from informal to formal, from intuition to logic deduction, from empirical to theoretical thinking, and to stimulate those thinking processes that are activated when facing a conjecture. Our design choices were inspired by the learning model of *concept image/concept definition/concept usage* (Moore, 1994; Tall & Vinner, 1981), the *example space* construct (Watson & Mason, 2005), and the theory of figural concepts (Fischbein, 1993). The path was packaged by integrating the pages (definition, generation, verification, and conjecture) of the Dahlberg and Housman task model (1997) with another page, the *bridge page*. The path is composed of three phases. In the first phase, the example phase, it starts with a given definition and requires generating examples and non-examples satisfying the definition (generation page) and verifying if a given object belongs or not to the example space (verification page). The bridge page connects the example phase with the conjecturing phase, requiring comparing the definition with some constraints that must or must not be satisfied and the conjecturing phase. The path concludes with proving or disproving a given statement (conjecture page). In the bridge page, assuming that one of the constraints is the beginning definition, we associate with each constraint the value 1 if it must be satisfied and the value 0 if it must not:

> *Give an example, if possible, of a mathematical object satisfying/not satisfying Constraint 1, Constraint 2, Constraint 3.*

This requirement generates a matrix containing eight rows, corresponding to eight subtasks, and three columns. Students explore the subtasks to solve them. If, in a particular case, they fail to find an example, they must understand why they cannot do so. It could be difficult to find an example or impossible to generate one. In the latter case, it is necessary to prove this impossibility. The students worked individually on each task in the first phase and collectively in the second phase to discuss, compare, or improve their solutions. Students constructed personal examples and proofs. They were
page); to generate two examples of objects, one satisfying that definition and the other not (generation page); to verify whether some given objects do or do not satisfy the definition (verification page); and to explore some statements linking that concept to others (conjecture page). We make Dahlberg and Housman's model more powerful by adding a page named the bridge page (Miranda, 2023b), requiring the example space extension through bridging examples suitable to overcome students' difficulty in providing counterexamples and in formulating conjectures. The lens with which we look at students' maturity and how students perceive it refers to the descriptions reported by Lew (2019). In this proposal, considering how mathematicians describe mathematical maturity (Lew, 2019), we give a taste of students' answers in a particular step of the path, the last proving/disproving conjecturing phase, highlighting the power of figural and conceptual aspects in developing an aware concept usage and investigating students' perceived maturity. We address this research question: *What is the impact of such an approach on students' outcomes and perceptions of advanced mathematics and their personal maturity in knowing or constructing it?*

## METHODOLOGY

The research is part of a project concerning the transition to advanced mathematical thinking (Miranda, 2023a), supported by paths starting with the example of space extensions and going towards generalization and abstraction (Miranda, 2023b). The implementation, held in the academic years 2021–22 and 2022–23, involved 34 students attending an introductory algebraic topology course within a Bachelor of Mathematics, in which we gave definitions of algebraic objects that are topological invariants, such as the fundamental group. The path was designed to encourage the transition from informal to formal, from intuition to logic deduction, from empirical to theoretical thinking, and to stimulate those thinking processes that are activated when facing a conjecture. Our design choices were inspired by the learning model of *concept image/concept definition/concept usage* (Moore, 1994; Tall & Vinner, 1981), the *example space* construct (Watson & Mason, 2005), and the theory of figural concepts (Fischbein, 1993). The path was packaged by integrating the pages (definition, generation, verification, and conjecture) of the Dahlberg and Housman task model (1997) with another page, the *bridge page*. The path is composed of three phases. In the first phase, the example phase, it starts with a given definition and requires generating examples and non-examples satisfying the definition (generation page) and verifying if a given object belongs or not to the example space (verification page). The bridge page connects the example phase with the conjecturing phase, requiring comparing the definition with some constraints that must or must not be satisfied and the conjecturing phase. The path concludes with proving or disproving a given statement (conjecture page). In the bridge page, assuming that one of the constraints is the beginning definition, we associate with each constraint the value 1 if it must be satisfied and the value 0 if it must not:

> *Give an example, if possible, of a mathematical object satisfying/not satisfying Constraint 1, Constraint 2, Constraint 3.*

This requirement generates a matrix containing eight rows, corresponding to eight subtasks, and three columns. Students explore the subtasks to solve them. If, in a particular case, they fail to find an example, they must understand why they cannot do so. It could be difficult to find an example or impossible to generate one. In the latter case, it is necessary to prove this impossibility. The students worked individually on each task in the first phase and collectively in the second phase to discuss, compare, or improve their solutions. Students constructed personal examples and proofs. They were
then asked to explain and comment on their solving process to the class, explaining the reasons why a particular example does or does not satisfy the required property, whether an example exists, and why no example exists. Personal examples were collectively analyzed initially to compare all the outputs and identify common aspects and differences, and then a second time to identify interesting features, such as accessibility, and to choose a representative of a larger class. The first phase concludes with the collective choice of the less complex example, if it exists, for every row. If they found a difficulty, all the components acted as prompts to modify an example or recall a useful result, with the aim of constructing the solution together. The second phase deals with conjecturing by looking at the matrix columns. The collective discourse is a very strong pedagogical tool because it encourages questions and discussions (Sfard, 1996) to explore the links between the properties expressed by the constraints, to discover whether these properties are independent, to open a conjecture-oriented exploration, and to foster generalizations. As an example: Why is this an example? Why is there no example? Does a mathematical object that satisfies constraint *i* satisfy constraint *j*? And vice versa? Some examples then become counterexamples, showing that certain implications are false. Others generate conjectures. Finding that a set of constraints seems mutually incompatible is an excellent way to generate a conjecture. Looking at the columns and comparing them, some conjectures are produced. The last phase deals with proving or disproving given conjectures. The model represents a teaching-learning opportunity to explore students' understanding and usage of definitions. It aims to encourage fluency, flexibility, and originality through different solution paths and making use of different representations (Duval, 2006), therefore shaping maturity.

## About a specific path

The definition running through the selected path (Table 1) is simply connectedness (*c2*), a concept in which two complementary operational-structural views and figural aspects coexist and can be combined to ensure its reification (Sfard, 1991; Fishbein, 1993).

<table><caption>Table 1: A specific path from the example phase to the conjecturing phase</caption><thead><tr><th></th><th>Definition page</th><th>Bridge page</th></tr></thead><tbody><tr><th scope="row">Definition</th><td>You will have a few minutes to study the following definition. A subspace of a topological space is said to be simply connected if it path connected and has a trivial fundamental group.</td><td>Consider conditions c1, c2, c3, associated with the value of 1 if you require that it be satisfied by the example you are looking for and the value 0 if it shouldn't be. Construct examples, if they exist, according to the requests of the following dichotomous matrix. Choose, comparing collectively, a generic example. What can you conjecture by comparing the columns?</td></tr><tr><th scope="row">Generation page</th><td>Answer the following questions.<ol><li>Give an example of a space that is simply connected and explain why it is.</li><li>Give an example of a space that is not simply connected and explain why it is not.</li><li>In your own words and/or images, simply explain what a simply connected space is.</li></ol></td><td><table><thead><tr><th>c1 : X contraibile</th><th>c2 : X semplicemente connesso</th><th>c3 : X stellato</th><th>Example, if it exists</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>1</td><td>1</td><td>0</td><td></td></tr><tr><td>0</td><td>1</td><td>1</td><td></td></tr><tr><td>1</td><td>0</td><td>1</td><td></td></tr><tr><td>1</td><td>0</td><td>0</td><td></td></tr><tr><td>0</td><td>1</td><td>0</td><td></td></tr><tr><td>0</td><td>0</td><td>1</td><td></td></tr><tr><td>0</td><td>0</td><td>0</td><td></td></tr></tbody></table></td></tr><tr><th scope="row">Verification page</th><td>Determine which of the following spaces is a space simply connected. Explain why.<ol><li>R</li><li>{(x, cos x): x ∈ R}</li><li>S<sup>1</sup></li><li>S<sup>2</sup></li></ol></td><td><table><thead><tr><th>c1 : X contraibile</th><th>c2 : X semplicemente connesso</th><th>c3 : X stellato</th><th>Example, if it exists</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>1</td><td>1</td><td>0</td><td></td></tr><tr><td>0</td><td>1</td><td>1</td><td></td></tr><tr><td>1</td><td>0</td><td>1</td><td></td></tr><tr><td>1</td><td>0</td><td>0</td><td></td></tr><tr><td>0</td><td>1</td><td>0</td><td></td></tr><tr><td>0</td><td>0</td><td>1</td><td></td></tr><tr><td>0</td><td>0</td><td>0</td><td></td></tr></tbody></table></td></tr></tbody></table>

Explored the first three pages, the bridge task requires comparing the property *c2: simply connected* with *c1: contractible* and *c3: star-shaped*: give an example, if possible, of a topological space satisfying/not satisfying constraints *c1, c2, c3*. In some cases, it is possible to generate examples, but this is not possible for subtasks such as (0,1,1) because “a star-shaped space is contractible.” Different
then asked to explain and comment on their solving process to the class, explaining the reasons why a particular example does or does not satisfy the required property, whether an example exists, and why no example exists. Personal examples were collectively analyzed initially to compare all the outputs and identify common aspects and differences, and then a second time to identify interesting features, such as accessibility, and to choose a representative of a larger class. The first phase concludes with the collective choice of the less complex example, if it exists, for every row. If they found a difficulty, all the components acted as prompts to modify an example or recall a useful result, with the aim of constructing the solution together. The second phase deals with conjecturing by looking at the matrix columns. The collective discourse is a very strong pedagogical tool because it encourages questions and discussions (Sfard, 1996) to explore the links between the properties expressed by the constraints, to discover whether these properties are independent, to open a conjecture-oriented exploration, and to foster generalizations. As an example: Why is this an example? Why is there no example? Does a mathematical object that satisfies constraint *i* satisfy constraint *j*? And vice versa? Some examples then become counterexamples, showing that certain implications are false. Others generate conjectures. Finding that a set of constraints seems mutually incompatible is an excellent way to generate a conjecture. Looking at the columns and comparing them, some conjectures are produced. The last phase deals with proving or disproving given conjectures. The model represents a teaching-learning opportunity to explore students' understanding and usage of definitions. It aims to encourage fluency, flexibility, and originality through different solution paths and making use of different representations (Duval, 2006), therefore shaping maturity.

## About a specific path

The definition running through the selected path (Table 1) is simply connectedness (*c2*), a concept in which two complementary operational-structural views and figural aspects coexist and can be combined to ensure its reification (Sfard, 1991; Fishbein, 1993).

<table><caption>Table 1: A specific path from the example phase to the conjecturing phase</caption><thead><tr><th>Definition page</th><th>Bridge page</th></tr></thead><tbody><tr><td><u>Instructions.</u> You will have a few minutes to study the following definition.<br><b>Definition</b> A subspace of a topological space is said <i>to be simply connected if it path connected and has a trivial fundamental group.</i></td><td><u>Instructions.</u> Consider conditions <i>c1, c2, c3</i>, associated with the value of 1 if you require that it be satisfied by the example you are looking for and the value 0 if it shouldn't be. Construct examples, if they exist, according to the requests of the following dichotomous matrix. Choose, comparing collectively, a generic example. What can you conjecture by comparing the columns?</td></tr><tr><td><b>Generation page</b><br><u>Instructions.</u> Answer the following questions.<br>1. Give an example of a space that is simply connected and explain why it is.<br>2. Give an example of a space that is not simply connected and explain why it is not.<br>3. In your own words and/or images, simply explain what a simply connected space is.</td><td><table><thead><tr><th colspan="2">1</th><th colspan="2">2</th><th colspan="2">3</th></tr></thead><tbody><tr><td>1</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>2</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>3</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>4</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>5</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>6</td><td></td><td>1</td><td></td><td>1</td><td></td></tr></tbody></table></td></tr><tr><td><b>Verification page</b><br><u>Instructions.</u> Determine which of the following spaces is a space simply connected. Explain why.<br>1. R<br>2. {<i>x</i>, cos <i>x</i>}: <i>x</i> &isin; R<br>3. S<sup>1</sup></td><td><table><thead><tr><th colspan="2">1</th><th colspan="2">2</th><th colspan="2">3</th></tr></thead><tbody><tr><td>1</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>2</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>3</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>4</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>5</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>6</td><td></td><td>1</td><td></td><td>1</td><td></td></tr></tbody></table></td></tr></tbody></table>

Explored the first three pages, the bridge task requires comparing the property *c2: simply connected* with *c1: contractible* and *c3: star-shaped*: give an example, if possible, of a topological space satisfying/not satisfying constraints *c1, c2, c3*. In some cases, it is possible to generate examples, but this is not possible for subtasks such as (0,1,1) because “a star-shaped space is contractible.” Different
kinds of tasks entail different processes and, consequently, different behaviors. Looking at the
columns, students are required to generate conjectures about the relationships between the constraints.
Exploring the links, they could ask themselves: "Is a contractible space simply connected?" The
bridge page opens the door to the conjecture that "a contractible space is simply connected". In the
last conjecture page, students are involved in a prove-disprove activity dealing with the concepts of
contractibility and simple connectedness (Table 2), intuitive concepts sometimes wrongly identified.

**Data collection and data analyses**

All data related to the training activity were digitally stored via the Moodle platform: written notes, recorded discussion, and a feedback questionnaire. To shed light on the impact that the practice had on the students' development and self-perception of mathematical maturity, we focused our analyses on some selected answers to conjecture page and questions 1 and 2 of the feedback questionnaire.

**Conjecture page (Prove/disproving) Instructions.** Say which of the following statements are true. Justify the answer.

*If a topological space is contractible then it is simply connected.*

*If a topological space is simply connected, then it is contractible.*

*The quotient of a simply connected space is simply connected.*

*Identity id: $S^1 \to S^1$ can be continuously extended on disk D.²*

We searched for sentences providing evidence about the characteristics of conceptual understanding (concept usage) and the meaningful processes towards maturity students experienced and perceived.

**Question 1.** The activities included the construction of examples, counterexamples, and proofs. Have they been useful for you to interpret a definition, to acquire a concept, or to acquire autonomy in proving? Did you happen to produce a proof by yourself? What has changed compared to previous experiences?

**Question 2** Beyond the exam, what do you have left of these experiences? Would you recommend a friend to have similar experiences? Why?

**FIRST FINDINGS**

Due to space restrictions, we confine our analysis to some selected answers to the mathematical problem in the last conjecture page (see Table 2) and to two questions of the feedback questionnaire. Students' first attention is focused on the topological object and properties that must be visualized or empirically produced to sketch the first attempt at a proof; they, through the graphical representations, explored and found a preliminary proof. This was the prelude to its formal proof. We search where concepts have been used in the sense of Moore (1993), with what improvement towards conceptual understanding, and which processes have determined maturity.

In Fig 1. St5 starts from the intuitive idea that a contractible space is a space collapsing into a point and then recalls the notion of deformation and gives a logical proof. She uses her prior knowledge (definitions) to create a new one, a logic proof (concept usage). 2. The graphical register through which the simple connectedness of the sphere can be intuitively deduced is convincing and preludes to its proving. She uses graphical and then verbal register (concept usage) to prove that the implication is false. In 3. StT3, mimics the passage to the quotient while explaining and says: 'It is as if you consider a string and join it to the extremes' (empirically reasoning), and makes previsions (geometric predictions). Then she justifies formally and generalizes (concept usage). Some difficulties can be seen in other protocols: StT10 seems to confuse the properties of a quotient of topologic space with those of the quotient group of its fundamental group and deduce that the quotient of a simply connected space has a trivial fundamental group by quotienting the group. Finally, to prove that *the*
kinds of tasks entail different processes and, consequently, different behaviors. Looking at the
columns, students are required to generate conjectures about the relationships between the constraints.
Exploring the links, they could ask themselves: "Is a contractible space simply connected?" The
bridge page opens the door to the conjecture that "a contractible space is simply connected". In the
last conjecture page, students are involved in a prove-disprove activity dealing with the concepts of
contractibility and simple connectedness (Table 2), intuitive concepts sometimes wrongly identified.

**Data collection and data analyses**

All data related to the training activity were digitally stored via the Moodle platform: written notes, recorded discussion, and a feedback questionnaire. To shed light on the impact that the practice had on the students' development and self-perception of mathematical maturity, we focused our analyses on some selected answers to conjecture page and questions 1 and 2 of the feedback questionnaire.

**Conjecture page (Prove/disproving) Instructions.** Say which of the following statements are true. Justify the answer.

*If a topological space is contractible then it is simply connected.*

*If a topological space is simply connected, then it is contractible.*

*The quotient of a simply connected space is simply connected.*

*Identity id: $S^1 \to S^1$ can be continuously extended on disk D.²*

We searched for sentences providing evidence about the characteristics of conceptual understanding (concept usage) and the meaningful processes towards maturity students experienced and perceived.

**Question 1.** The activities included the construction of examples, counterexamples, and proofs. Have they been useful for you to interpret a definition, to acquire a concept, or to acquire autonomy in proving? Did you happen to produce a proof by yourself? What has changed compared to previous experiences?

**Question 2** Beyond the exam, what do you have left of these experiences? Would you recommend a friend to have similar experiences? Why?

**FIRST FINDINGS**

Due to space restrictions, we confine our analysis to some selected answers to the mathematical problem in the last conjecture page (see Table 2) and to two questions of the feedback questionnaire. Students' first attention is focused on the topological object and properties that must be visualized or empirically produced to sketch the first attempt at a proof; they, through the graphical representations, explored and found a preliminary proof. This was the prelude to its formal proof. We search where concepts have been used in the sense of Moore (1993), with what improvement towards conceptual understanding, and which processes have determined maturity.

In Fig 1. St5 starts from the intuitive idea that a contractible space is a space collapsing into a point and then recalls the notion of deformation and gives a logical proof. She uses her prior knowledge (definitions) to create a new one, a logic proof (concept usage). 2. The graphical register through which the simple connectedness of the sphere can be intuitively deduced is convincing and preludes to its proving. She uses graphical and then verbal register (concept usage) to prove that the implication is false. In 3. StT3, mimics the passage to the quotient while explaining and says: 'It is as if you consider a string and join it to the extremes' (empirically reasoning), and makes previsions (geometric predictions). Then she justifies formally and generalizes (concept usage). Some difficulties can be seen in other protocols: StT10 seems to confuse the properties of a quotient of topologic space with those of the quotient group of its fundamental group and deduce that the quotient of a simply connected space has a trivial fundamental group by quotienting the group. Finally, to prove that *the*
identity id: $S^1 \to S^1$ is not continuously extendable to the disk, StT13 translates the negation of the statement in terms of retraction to explicit semantics (Weber & Alcock, 2004) and produces a contradiction. This reveals a crucial choice, giving impetus to a syntactically valid proof (concept usage) different from other students' attempts. At the beginning of the proof, which is a delicate step (Moore, 1994), some students need to review the definitions. We observe that students who reinterpreted the meaning reached proof success (StT13), while StT6 invoked the *Brouwer Fixed Point Theorem* without arguing correctly. Some findings delineating signs of maturity emerged from the last step of the path: students' use of different semiotic representations (Duval, 2006), among which the figural one (Fishbein, 1993) is strategic to apply geometric reasoning to prove the conjecture; the transition from an intuitive to a more structural approach to understanding (Sfard, 1991).

<table><caption>Table 2: Some excerpts selected from the students' answers to the proving/disproving task</caption><tr><td>1.</td><td>St.5.: True. If X is a contractible topological space, then it has as a deformation retract a point, so it is homotopy equivalent to a point (since the retraction is a homotopic equivalence); therefore, X has the same fundamental group as the point which is the trivial one. Furthermore is path connected, as you see in the figure.<br>St. 4: True: A topological space, which is homotopy equivalent to a point, has its fundamental group. Furthermore, the path connection is a homotopy invariant, so since the point is connected by paths, X is too. It follows that X is simply connected.</td></tr><tr><td>2.</td><td>StT3: False. The n-sphere is simply connected (it is path connected and has trivial fundamental group for n≥2), but it is not contractible.<br>St. 6: False. The surface of any polyhedron is simply connected but it is not contractible.</td></tr><tr><td>3.</td><td>StT3: False. The quotient space of a simply connected space is simply connected<br>False. Interval [0,1] is simply connected, but its quotient [0,1]/{0,1} ~ S<sup>1</sup> (the quotient map identifies the extremes) is not.<br>St. 5: False. The square is simply connected but the torus, the cylinder, the projective plane are not.<br>St.10 True. The quotient of a trivial group is trivial, therefore it is simply connected</td></tr><tr><td>4. StT6: False. ∄ a way extending the identity map id<sub>S<sup>1</sup></sub> to a map D<sup>2</sup> → S<sup>1</sup>, if it was possible to extend it, it would have no fixed points ⇒ I would violate Brouwer's fixed-point theorem</td><td>StT13: False. If id: S<sup>1</sup> → S<sup>1</sup> was continuously extendable over the disk D<sup>2</sup>, then S<sup>1</sup> would be a retract of D<sup>2</sup>, which is a simply connected space, and therefore S<sup>1</sup> should be simply connected, too, but it is a contradiction.</td></tr></table>

## Identifying some characteristics of maturity from answers to question 1 of the questionnaire

The protocols show that the activities were very useful to internalize a definition, to acquire awareness (StT7), autonomy, mastery, self-confidence, and self-efficacy, and to develop a natural propensity towards abstraction and formalization (StT1).

StT1: [...] were of fundamental importance for the focus of the notions. Without them, I probably would not have grasped the more sophisticated and interesting aspects of some concepts. [...] In many cases, *intuition* was a guide, and *formalization* was quite natural.
identity id: $S^1 \to S^1$ is not continuously extendable to the disk, StT13 translates the negation of the statement in terms of retraction to explicit semantics (Weber & Alcock, 2004) and produces a contradiction. This reveals a crucial choice, giving impetus to a syntactically valid proof (concept usage) different from other students' attempts. At the beginning of the proof, which is a delicate step (Moore, 1994), some students need to review the definitions. We observe that students who reinterpreted the meaning reached proof success (StT13), while StT6 invoked the *Brouwer Fixed Point Theorem* without arguing correctly. Some findings delineating signs of maturity emerged from the last step of the path: students' use of different semiotic representations (Duval, 2006), among which the figural one (Fishbein, 1993) is strategic to apply geometric reasoning to prove the conjecture; the transition from an intuitive to a more structural approach to understanding (Sfard, 1991).

<table><caption>Table 2: Some excerpts selected from the students' answers to the proving/disproving task</caption><tr><td>1.</td><td>St.5.: True. If X is a contractible topological space, then it has as a deformation retract a point, so it is homotopy equivalent to a point (since the retraction is a homotopic equivalence); therefore, X has the same fundamental group as the point which is the trivial one. Furthermore is path connected, as you see in the figure.<br>St. 4: True: A topological space, which is homotopy equivalent to a point, has its fundamental group. Furthermore, the path connection is a homotopy invariant, so since the point is connected by paths, X is too. It follows that X is simply connected.</td></tr><tr><td>2.</td><td>StT3: False. The n-sphere is simply connected (it is path connected and has trivial fundamental group for n≥2), but it is not contractible.<br>St. 6: False. The surface of any polyhedron is simply connected but it is not contractible.</td></tr><tr><td>3.</td><td>StT3: False. The quotient space of a simply connected space is simply connected<br>False. Interval [0,1] is simply connected, but its quotient [0,1]/{0,1} ~ S<sup>1</sup> (the quotient map identifies the extremes) is not.<br>St. 5: False. The square is simply connected but the torus, the cylinder, the projective plane are not.<br>St.10 True. The quotient of a trivial group is trivial, therefore it is simply connected</td></tr><tr><td>4. StT6: False. ∄ a way extending the identity map id<sub>S<sup>1</sup></sub> to a map D<sup>2</sup> → S<sup>1</sup>, if it was possible to extend it, <b>it would have no fixed points ⇒ I would violate Brouwer's fixed-point theorem</b></td><td>StT13: False. If id: S<sup>1</sup> → S<sup>1</sup> was continuously extendable over the disk D<sup>2</sup>, then S<sup>1</sup> would be a retract of D<sup>2</sup>, which is a simply connected space, and therefore S<sup>1</sup> should be simply connected, too, but it is a contradiction.</td></tr></table>

## Identifying some characteristics of maturity from answers to question 1 of the questionnaire

The protocols show that the activities were very useful to internalize a definition, to acquire awareness (StT7), autonomy, mastery, self-confidence, and self-efficacy, and to develop a natural propensity towards abstraction and formalization (StT1).

StT1: [...] were of fundamental importance for the focus of the notions. Without them, I probably would not have grasped the more sophisticated and interesting aspects of some concepts. [...] In many cases, *intuition* was a guide, and *formalization* was quite natural.
StT7: Absolutely, there is no better way to *internalize a definition* in mathematics than by producing examples and counterexamples. [...] The activities were very important to acquire greater awareness of the topics covered in the course. [...] Thanks to them, I was able to have more *mastery of many concepts* and *proof techniques*.

The vision of the proof changes. The term “methodical” in the following excerpt would seem to recall an idea of a reproductive, procedural approach to proof, with respect to which the student applies according to a procedure, while the term “reasoned” is linked to a productive idea with respect to which the student autonomously reason by himself and therefore has an active thinking approach:

StT1: [...] Compared to previous experiences, I had a less *methodical* and more *reasoned* approach to proof. [...]

Another fundamental ingredient to deepen maturity is inspired by the discourse, communication, negotiation, and participation in a mathematical community that are dominant (Sfard, 1996):

StT7: The experiences that will surely remain more impressive are the *independent reflections* and the *moments of collective discussion*. The former because they fully immerse the student in the subject, while the latter allow for *many different ideas* and *points of view*, even just in *understanding where two ideas of different proofs* for the same result come from. The *discourse* is fundamental, I believe, a comparison I found in many moments of the path, and these are formative experiences for the our career.

## Identifying some characteristics of maturity from answers to question 2 of the questionnaire

From the first analysis, it seems that the students perceived the experience as very demanding but received many benefits. They declare that they have lived a rewarding experience, with serenity and as a protagonist (StT1), leaving the *students' comfort zone*:

StT1: This experience has been challenging for me, but really *rewarding*. I am satisfied with the contents studied and the method used. I lived the journey with serenity, never feeling the weight of the study load or the pressure of deadlines. [...] I would recommend similar experiences to a friend because they allow you to live a course fully, keeping in mind the objective of passing the exam, but making you the real *protagonist of the path* made for the mere pleasure of studying and *doing mathematics*.

StT3: Personally, the more we got to the heart of the course, the more stimulating it was to be engaged in the activities [...].

StT2: Surely being *out of the comfort zone* of the student and having put his hand to a topic not mediated by a professor was beneficial. [...] to exhibit their work was very interesting, and certainly I would recommend someone with similar experiences.

StT3: I would recommend it to colleagues; indeed, *I would also* include it in the other courses.

Students recognize that it was a formative experience although difficult:

StT5: Pages paths are a great idea; the difficulty of some tasks should be moderated, but I think that stops are very formative for a math student and quite pleasant.

## CONCLUSIONS

Topology is a topic in which visual reasoning is a prelude to maturity because it allows the key idea to be identified to go towards formal reasoning and proving. Generating examples reveals itself as a good way to achieve an understanding of definitions; exploring the links between constraints is an excellent way to generate conjectures; and, finally, transporting personal example choices to a
StT7: Absolutely, there is no better way to *internalize a definition* in mathematics than by producing examples and counterexamples. [...] The activities were very important to acquire greater awareness of the topics covered in the course. [...] Thanks to them, I was able to have more *mastery of many concepts* and *proof techniques*.

The vision of the proof changes. The term “methodical” in the following excerpt would seem to recall an idea of a reproductive, procedural approach to proof, with respect to which the student applies according to a procedure, while the term “reasoned” is linked to a productive idea with respect to which the student autonomously reason by himself and therefore has an active thinking approach:

StT1: [...] Compared to previous experiences, I had a less *methodical* and more *reasoned* approach to proof. [...]

Another fundamental ingredient to deepen maturity is inspired by the discourse, communication, negotiation, and participation in a mathematical community that are dominant (Sfard, 1996):

StT7: The experiences that will surely remain more impressive are the *independent reflections* and the *moments of collective discussion*. The former because they fully immerse the student in the subject, while the latter allow for *many different ideas* and *points of view*, even just in *understanding where two ideas of different proofs* for the same result come from. The *discourse* is fundamental, I believe, a comparison I found in many moments of the path, and these are formative experiences for the our career.

## Identifying some characteristics of maturity from answers to question 2 of the questionnaire

From the first analysis, it seems that the students perceived the experience as very demanding but received many benefits. They declare that they have lived a rewarding experience, with serenity and as a protagonist (StT1), leaving the *students' comfort zone*:

StT1: This experience has been challenging for me, but really *rewarding*. I am satisfied with the contents studied and the method used. I lived the journey with serenity, never feeling the weight of the study load or the pressure of deadlines. [...] I would recommend similar experiences to a friend because they allow you to live a course fully, keeping in mind the objective of passing the exam, but making you the real *protagonist of the path* made for the mere pleasure of studying and *doing mathematics*.

StT3: Personally, the more we got to the heart of the course, the more stimulating it was to be engaged in the activities [...].

StT2: Surely being *out of the comfort zone* of the student and having put his hand to a topic not mediated by a professor was beneficial. [...] to exhibit their work was very interesting, and certainly I would recommend someone with similar experiences.

StT3: I would recommend it to colleagues; indeed, *I would also* include it in the other courses.

Students recognize that it was a formative experience although difficult:

StT5: Pages paths are a great idea; the difficulty of some tasks should be moderated, but I think that stops are very formative for a math student and quite pleasant.

## CONCLUSIONS

Topology is a topic in which visual reasoning is a prelude to maturity because it allows the key idea to be identified to go towards formal reasoning and proving. Generating examples reveals itself as a good way to achieve an understanding of definitions; exploring the links between constraints is an excellent way to generate conjectures; and, finally, transporting personal example choices to a
collective discussion forces students to become aware of a more general class of examples and aids
proof productions. Our results confirm that example-space activities are crucial in developing
students' theoretical thinking and supporting their undergraduate-graduate transition to maturity. We
find that the collective manipulation of the personal examples contributes to a deeper understanding
and shapes the premises for abstracting and generalizing. The value of geometry for enhancing
competences in conjecturing and proving came out. However, we believe that further comparative
studies are needed at every level to improve the design of activities and the power of geometry to
develop a structural approach to knowledge towards maturity. The question of how to define
mathematical maturity and encourage students to build it remains open.

References

Dahlberg, R. P., & Housman, D. L. (1997). Facilitating learning events through example generation. *Educational Studies in Mathematics, **33***, 283–299.

Duval, R. (2006). Transformations of semiotic representations and thought praxis in mathematics. *Mathematics and its Didactics, 4*, 585–619.

Fischbein, E. (1993). The theory of figural concepts. *Educational Studies in Mathematics, **24*** (2),

Gallagher, K., Infante, N.E. (2022). A case study of undergraduates' proving behaviors and uses of visual representations in identification of key ideas in topology. *Int. J. Res. Undergrad. Math. Ed.* 8, 176–210.

Lew, K. (2019). How do mathematicians describe mathematical maturity? *Cognition and Instruction, **37*** (2), 121-142.

Miranda A. (2023a). The bridging role of the conjecture in the transition to advanced mathematical thinking. *Proceedings 46th Conference of the International Group for the PME*, (Vol. 1, p. 292). PME.

Miranda A. (2023b). Extending example spaces in topology to aid undergraduate students' transition to generalization and abstraction. In P. Drijvers, C. Csapodi, H. Palmér, K. Gosztonyi, & E. Kónya (Eds.), *Proceedings CERME 13* (pp. 2461–2462). Alfréd Rényi Institute of Mathematics and ERME.

Moore, R. C. (1994). Making the transition to formal proof. *Educational Studies in Mathematics, **27***, 249–266.

Selden, A. & Selden, J. (2013). Proof and Problem Solving at University Level. *The Mathematics Enthusiast, 10, 1*, 303-333.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. *Educational Studies in Mathematics, **22***, 1–36.

Sfard, A. (1996). On acquisition metaphor and participation metaphor for mathematics learning. In C. Alsina, J. M. Alvarez, B. Hodgson, C. Laborde, & A. Pèrez (Eds.), *ICME. Selected Lectures* (pp. 397–411). S.A.E.M. 'THALES'.

Steen L. A. (1983). Developing Mathematical Maturity. *The Future of College Mathematics: Proceedings of Conference on the First Two Years of College Mathematics* (pp. 99–110). Springer.

Steen, L. A. & Seebach J.A. (1970). *Counterexamples in Topology*, Springer-Verlag.

Tall, D. (1992). The psychology of advanced mathematical thinking. In D. Tall (Ed.), *Advanced Mathematical Thinking* (pp. 3-21). Kluwer.

Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics, with particular reference to limits and continuity. *Educational Studies in Mathematics, 12*, 151–169.

Watson, A. & Mason, J. (2005). *Mathematics as a constructive activity: Learners generating examples*. Lawrence Erlbaum Associates.

Weber, K. & Alcock, L.J. (2004). Semantic and syntactic proof productions. *Educational Studies in Mathematics, 56*, 209-23.
collective discussion forces students to become aware of a more general class of examples and aids
proof productions. Our results confirm that example-space activities are crucial in developing
students' theoretical thinking and supporting their undergraduate-graduate transition to maturity. We
find that the collective manipulation of the personal examples contributes to a deeper understanding
and shapes the premises for abstracting and generalizing. The value of geometry for enhancing
competences in conjecturing and proving came out. However, we believe that further comparative
studies are needed at every level to improve the design of activities and the power of geometry to
develop a structural approach to knowledge towards maturity. The question of how to define
mathematical maturity and encourage students to build it remains open.

References

Dahlberg, R. P., & Housman, D. L. (1997). Facilitating learning events through example generation. *Educational Studies in Mathematics, 33*, 283–299.

Duval, R. (2006). Transformations of semiotic representations and thought praxis in mathematics. *Mathematics and its Didactics, 4*, 585–619.

Fischbein, E. (1993). The theory of figural concepts. *Educational Studies in Mathematics, 24*(2),

Gallagher, K., Infante, N.E. (2022). A case study of undergraduates' proving behaviors and uses of visual representations in identification of key ideas in topology. *Int. J. Res. Undergrad. Math. Ed.* 8, 176–210.

Lew, K. (2019). How do mathematicians describe mathematical maturity? *Cognition and Instruction, 37*(2), 121-142.

Miranda A. (2023a). The bridging role of the conjecture in the transition to advanced mathematical thinking. *Proceedings 46th Conference of the International Group for the PME*, (Vol. 1, p. 292). PME.

Miranda A. (2023b). Extending example spaces in topology to aid undergraduate students' transition to generalization and abstraction. In P. Drijvers, C. Csapodi, H. Palmér, K. Gosztonyi, & E. Kónya (Eds.), *Proceedings CERME 13* (pp. 2461–2462). Alfréd Rényi Institute of Mathematics and ERME.

Moore, R. C. (1994). Making the transition to formal proof. *Educational Studies in Mathematics, 27*, 249–266.

Selden, A. & Selden, J. (2013). Proof and Problem Solving at University Level. *The Mathematics Enthusiast, 10*, 1–303-333.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. *Educational Studies in Mathematics, 22*, 1–36.

Sfard, A. (1996). On acquisition metaphor and participation metaphor for mathematics learning. In C. Alsina, J. M. Alvarez, B. Hodgson, C. Laborde, & A. Pèrez (Eds.), *ICME. Selected Lectures* (pp. 397–411). S.A.E.M. 'THALES'.

Steen L. A. (1983). Developing Mathematical Maturity. *The Future of College Mathematics: Proceedings of Conference on the First Two Years of College Mathematics* (pp. 99–110). Springer.

Steen, L. A. & Seebach J.A. (1970). *Counterexamples in Topology*, Springer-Verlag.

Tall, D. (1992). The psychology of advanced mathematical thinking. In D. Tall (Ed.), *Advanced Mathematical Thinking* (pp. 3-21). Kluwer.

Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics, with particular reference to limits and continuity. *Educational Studies in Mathematics, 12*, 151–169.

Watson, A. & Mason, J. (2005). *Mathematics as a constructive activity: Learners generating examples*. Lawrence Erlbaum Associates.

Weber, K. & Alcock, L.J. (2004). Semantic and syntactic proof productions. *Educational Studies in Mathematics, 56*, 209-23.
# ON THE ROLE OF SHIFTS OF ATTENTION AND FIGURAL APPREHENSION IN THE EVOLUTION OF GEOMETRIC PERCEPTION

Alik Palatnik

The Hebrew University of Jerusalem, The Seymour Fox School of Education, Israel

Geometry learning pivots on perceiving and reasoning upon geometrical objects and relationships. This paper develops theoretical and didactical ideas on ways to evolve students' perceptive and analytical discernment ability for learning spatial geometry. I locally integrate Duval's framework on geometric apprehension with Mason's theory on shifts of attention for analyzing 10th-grade student R's experience with spatial geometry problem-solving facilitated by a 3D pen. This analysis suggests that active engagement with a 3D medium and tools, such as the construction and manipulation of models, play a vital role in the evolution of students' geometric perception and reasoning. Shifts in attention lead to the need for different forms of figural apprehension. This study underscores the need for enactive, dynamic learning experiences in geometry education.

## ENCULTURATING PERCEPTION AS A POSSIBLE KEY TO GEOMETRY LEARNING

Perception is critical in learning, especially in geometry. Johnston-Wilder and Mason (2005) articulate that the crux of geometric learning is rooted in the ability to “see,” i.e., to identify and comprehend geometrical objects and their relationships. However, what enables to educate this ability of perceptive and analytical discernment, this learning to “see,” remains less understood.

Bryson (1988) posits that learning involves enculturation of the senses. He argues that one's perception is not directly interacting with the environment but is mediated by a system of socially constructed symbols and discourses. This notion resonates with Gibson's (2014) ecological approach to perception, in particular the concept of affordances as potentialities for action, and with scholars of cognitive and learning sciences addressing the construct of culturally conditioned vision—‘professional vision’ (Goodwin, 2018), ‘educated perception’ (Goldstone et al., 2010). Goldstone et al. (2010) claimed, “A credible and worthy hope for education is to teach students to take the natural affordances of our long-tuned perceptual systems, which are at their core spatial and dynamic, and retask them for new purposes” (p.280). This contribution aims to develop theoretical and didactical ideas on possible ways to retask students’ perception for spatial geometry learning.

The geometric characteristics of the environment, such as right angles and parallel lines, are not inherently obvious. Instead, they become apparent through interacting with and manipulating cultural artifacts (Bartolini Bussi & Mariotti, 2008). Consequently, adopting these cultural mediators alters one's perception (Goodwin, 2018). Johnston-Wilder and Mason (2005) recount the case of a young girl struggling to recognize a specifically oriented wooden shape as a triangle (Figure 1a). This narrative exemplifies one of the ways in which geometric perception may be educated: “I’m not sure, but it is a triangle for her’, pointing to her friend on the other side of the desk.” (p. 3). By acknowledging a different perspective, the student comes closer to grasping the invariant essence of geometric forms, such as triangles. Our preliminary research (Palatnik & Abrahamson, 2022)
# ON THE ROLE OF SHIFTS OF ATTENTION AND FIGURAL APPREHENSION IN THE EVOLUTION OF GEOMETRIC PERCEPTION

Alik Palatnik

The Hebrew University of Jerusalem, The Seymour Fox School of Education, Israel

Geometry learning pivots on perceiving and reasoning upon geometrical objects and relationships. This paper develops theoretical and didactical ideas on ways to evolve students' perceptive and analytical discernment ability for learning spatial geometry. I locally integrate Duval's framework on geometric apprehension with Mason's theory on shifts of attention for analyzing 10th-grade student R's experience with spatial geometry problem-solving facilitated by a 3D pen. This analysis suggests that active engagement with a 3D medium and tools, such as the construction and manipulation of models, play a vital role in the evolution of students' geometric perception and reasoning. Shifts in attention lead to the need for different forms of figural apprehension. This study underscores the need for enactive, dynamic learning experiences in geometry education.

## ENCULTURATING PERCEPTION AS A POSSIBLE KEY TO GEOMETRY LEARNING

Perception is critical in learning, especially in geometry. Johnston-Wilder and Mason (2005) articulate that the crux of geometric learning is rooted in the ability to “see,” i.e., to identify and comprehend geometrical objects and their relationships. However, what enables to educate this ability of perceptive and analytical discernment, this learning to “see,” remains less understood.

Bryson (1988) posits that learning involves enculturation of the senses. He argues that one's perception is not directly interacting with the environment but is mediated by a system of socially constructed symbols and discourses. This notion resonates with Gibson's (2014) ecological approach to perception, in particular the concept of affordances as potentialities for action, and with scholars of cognitive and learning sciences addressing the construct of culturally conditioned vision—‘professional vision’ (Goodwin, 2018), ‘educated perception’ (Goldstone et al., 2010). Goldstone et al. (2010) claimed, “A credible and worthy hope for education is to teach students to take the natural affordances of our long-tuned perceptual systems, which are at their core spatial and dynamic, and retask them for new purposes” (p.280). This contribution aims to develop theoretical and didactical ideas on possible ways to retask students’ perception for spatial geometry learning.

The geometric characteristics of the environment, such as right angles and parallel lines, are not inherently obvious. Instead, they become apparent through interacting with and manipulating cultural artifacts (Bartolini Bussi & Mariotti, 2008). Consequently, adopting these cultural mediators alters one's perception (Goodwin, 2018). Johnston-Wilder and Mason (2005) recount the case of a young girl struggling to recognize a specifically oriented wooden shape as a triangle (Figure 1a). This narrative exemplifies one of the ways in which geometric perception may be educated: “I’m not sure, but it is a triangle for her’, pointing to her friend on the other side of the desk.” (p. 3). By acknowledging a different perspective, the student comes closer to grasping the invariant essence of geometric forms, such as triangles. Our preliminary research (Palatnik & Abrahamson, 2022)
exemplified how students solved a spatial geometry problem (Figure 1b) by shifting their perception through moving, sketching, manipulating the model, and being attentive to others' actions and discourse (Figure 1c), thereby enculturating each other perception and reconciling their spatial and analytic reasoning (Fujita et al., 2020).

<figure><img src="image_41.png" /><figcaption>Figure 1: Learning to perceive a(n equilateral) triangle</figcaption></figure>

## Role of figural apprehension in learning to perceive

Raymond Duval's seminal work provides a useful lens on the mechanisms of seeing and interacting with figures mathematically (Duval, 1995, 2005, 2017). Duval (1995) distinguishes four distinct types of apprehending geometric figures: perceptual, sequential, discursive, and operative. According to Duval, *perceptual apprehension* is the most basic level and pertains to recognizing geometric figures and their constituent parts at first glance. *Sequential apprehension* focuses on the procedural aspects of how a geometric shape is formed and the order in which the figure's elements are assembled, relates to the figure's construction, and depends on technical constraints that change with construction tools. *Discursive apprehension* involves recognizing and processing attributes of geometric figures mediated through language, definitions, and mathematical properties. It requires the learner to make discursive statements that define and determine the figure's properties. According to Duval, *operative apprehension* is the most advanced level of apprehension. It requires learners to actively engage with a figure by modifying it mentally or physically, often to gain insights into problem solving or proving. Duval asserts that learners must distinguish and coordinate between these different forms of apprehension to understand geometric figures comprehensively. However, various forms of apprehension may collide. For example, emphasizing discursive apprehension may impede the insights obtained through operative apprehension. These classifications provide a useful framework to study the cognitive processes involved in learning geometry and to design environments facilitating students' coordination of different forms of apprehension.

Developing Duval's (1995, 2017) ideas, Mithalal and Balacheff (2019) explored conditions in which construction tasks in 3D DGE (dynamic geometry environment) stimulate students' transition from working with drawings to perceiving geometric properties of figures. Recently, Palatnik (in press) adopted a figural apprehension framework to characterize the affordances of the different novel medium—3D pen (a hand-held device enabling spatial sketching with hot, fast-hardening plastic) for spatial geometry learning. Preliminary findings suggest that 3D sketching as a form of embodied and enactive math activity (Ng & Ferrara, 2020; Palatnik et al., 2023) may evoke coordination between various types of apprehension, thus facilitating learning and teaching spatial geometry. The importance of operative apprehension aligns with other key ideas from Duval's (2017) framework.
exemplified how students solved a spatial geometry problem (Figure 1b) by shifting their perception through moving, sketching, manipulating the model, and being attentive to others’ actions and discourse (Figure 1c), thereby enculturating each other perception and reconciling their spatial and analytic reasoning (Fujita et al., 2020).

<figure><img src="image_47.png" /><figcaption>Figure 1: Learning to perceive a(n equilateral) triangle</figcaption></figure>

## Role of figural apprehension in learning to perceive

Raymond Duval’s seminal work provides a useful lens on the mechanisms of seeing and interacting with figures mathematically (Duval, 1995, 2005, 2017). Duval (1995) distinguishes four distinct types of apprehending geometric figures: perceptual, sequential, discursive, and operative. According to Duval, *perceptual apprehension* is the most basic level and pertains to recognizing geometric figures and their constituent parts at first glance. *Sequential apprehension* focuses on the procedural aspects of how a geometric shape is formed and the order in which the figure’s elements are assembled, relates to the figure’s construction, and depends on technical constraints that change with construction tools. *Discursive apprehension* involves recognizing and processing attributes of geometric figures mediated through language, definitions, and mathematical properties. It requires the learner to make discursive statements that define and determine the figure’s properties. According to Duval, *operative apprehension* is the most advanced level of apprehension. It requires learners to actively engage with a figure by modifying it mentally or physically, often to gain insights into problem solving or proving. Duval asserts that learners must distinguish and coordinate between these different forms of apprehension to understand geometric figures comprehensively. However, various forms of apprehension may collide. For example, emphasizing discursive apprehension may impede the insights obtained through operative apprehension. These classifications provide a useful framework to study the cognitive processes involved in learning geometry and to design environments facilitating students’ coordination of different forms of apprehension.

Developing Duval’s (1995, 2017) ideas, Mithalal and Balacheff (2019) explored conditions in which construction tasks in 3D DGE (dynamic geometry environment) stimulate students’ transition from working with drawings to perceiving geometric properties of figures. Recently, Palatnik (in press) adopted a figural apprehension framework to characterize the affordances of the different novel medium—3D pen (a hand-held device enabling spatial sketching with hot, fast-hardening plastic) for spatial geometry learning. Preliminary findings suggest that 3D sketching as a form of embodied and enactive math activity (Ng & Ferrara, 2020; Palatnik et al., 2023) may evoke coordination between various types of apprehension, thus facilitating learning and teaching spatial geometry. The importance of operative apprehension aligns with other key ideas from Duval’s (2017) framework.
Specifically, Duval proposes two figural operations that facilitate students' progression from "discriminated recognition of forms to the identification of the objects to see" (Duval, 2005, p. 13, translated by Herbst et al., 2017, p.86): the mereological division of shape (transformation of the geometric figure into figural units of the same dimension) and the dimensional deconstruction of shape (decomposition into elements of dimension lower than in given figure). However, Duval remarks that introducing a new element into a figure does not necessarily mean the student becomes aware of a new property. This insight mirrors John Mason's (2008) ideas about attention and awareness: "someone may be attending to something in a particular way but unaware explicitly of the what or the how" (p. 12). The What and the How pertain to the focus and structures of attention.

## Role of shifts of attention in learning to perceive

Mason (2010) describes learning as a transformation of attention that involves 'shifts in the form as well as the focus of attention' (p. 24). Mason (2008) distinguishes five different structures of attention: (1) holding wholes, (2) discerning details, (3) recognizing relationships, (4) perceiving properties, and (5) reasoning based on perceived properties. The structure of *holding wholes* is related to a macroscopic view, where learners consider the overall structure or context. *Discerning details* involves zooming in on specific elements, which might be akin to focusing on individual vertices or edges in a geometric model. *Recognizing relationships* emphasizes the interconnections between the elements, such as equal sides in a geometric shape. *Perceiving properties* pertains to an active search of the elements possessing the inherent characteristics. *Reasoning based on perceived properties* is the advanced structure of attention, where learners make logical conclusions or predictions based on selected properties. Mason's framework is frequently used to understand problem-solving processes.

Palatnik (2022) applied shifts of attention as an analytical framework for investigating spatial geometry learning in the context of collaborative geometric activity in which students constructed tangible models of a geometric object on different scales. This study documented shifts between foci of attention and the shifts between all five attention structures. These shifts in attention were linked to students' visual and tactile senses, proprioception, and physical interactions with the models. For example, by positioning the model at its vertex, students could visually segment the icosahedron into three distinct sets, leading to a breakthrough in establishing the number of edges.

Next, I will illustrate the idea of learning geometry as the evolution of geometric perception by analyzing an episode involving a 10th-grade student who tackled a spatial geometry question while utilizing a 3D pen. This episode is part of a larger data set from an extensive research project focusing on the design of embodied learning for spatial geometry (e.g., Palatnik & Abrahamson, 2022). In this analysis, two theoretical frameworks will be integrated: Mason's (2008, 2010) theory on shifts of attention and Duval's (1995) framework concerning the cognitive apprehension of geometric figures. By employing a local integration of these disparate viewpoints (Bikner-Ahsbahs & Prediger, 2014), I aim to provide a theoretical foundation for learning geometry as the evolution of students' geometric perception in the context of spatial geometry problem-solving activities facilitated by 3D sketching.

## THE CASE STUDY

The episode illustrating the evolution of student geometric perception in spatial geometry problem solving presents the student R, facing the Triangle task (Figure 2). While R did not formally study spatial geometry, he had knowledge of planar geometry for the presented challenges. When facing
Specifically, Duval proposes two figural operations that facilitate students' progression from "discriminated recognition of forms to the identification of the objects to see" (Duval, 2005, p. 13, translated by Herbst et al., 2017, p.86): the mereological division of shape (transformation of the geometric figure into figural units of the same dimension) and the dimensional deconstruction of shape (decomposition into elements of dimension lower than in given figure). However, Duval remarks that introducing a new element into a figure does not necessarily mean the student becomes aware of a new property. This insight mirrors John Mason's (2008) ideas about attention and awareness: "someone may be attending to something in a particular way but unaware explicitly of the what or the how" (p. 12). The What and the How pertain to the focus and structures of attention.

## Role of shifts of attention in learning to perceive

Mason (2010) describes learning as a transformation of attention that involves 'shifts in the form as well as the focus of attention' (p. 24). Mason (2008) distinguishes five different structures of attention: (1) holding wholes, (2) discerning details, (3) recognizing relationships, (4) perceiving properties, and (5) reasoning based on perceived properties. The structure of *holding wholes* is related to a macroscopic view, where learners consider the overall structure or context. *Discerning details* involves zooming in on specific elements, which might be akin to focusing on individual vertices or edges in a geometric model. *Recognizing relationships* emphasizes the interconnections between the elements, such as equal sides in a geometric shape. *Perceiving properties* pertains to an active search of the elements possessing the inherent characteristics. *Reasoning based on perceived properties* is the advanced structure of attention, where learners make logical conclusions or predictions based on selected properties. Mason's framework is frequently used to understand problem-solving processes.

Palatnik (2022) applied shifts of attention as an analytical framework for investigating spatial geometry learning in the context of collaborative geometric activity in which students constructed tangible models of a geometric object on different scales. This study documented shifts between foci of attention and the shifts between all five attention structures. These shifts in attention were linked to students' visual and tactile senses, proprioception, and physical interactions with the models. For example, by positioning the model at its vertex, students could visually segment the icosahedron into three distinct sets, leading to a breakthrough in establishing the number of edges.

Next, I will illustrate the idea of learning geometry as the evolution of geometric perception by analyzing an episode involving a 10th-grade student who tackled a spatial geometry question while utilizing a 3D pen. This episode is part of a larger data set from an extensive research project focusing on the design of embodied learning for spatial geometry (e.g., Palatnik & Abrahamson, 2022). In this analysis, two theoretical frameworks will be integrated: Mason's (2008, 2010) theory on shifts of attention and Duval's (1995) framework concerning the cognitive apprehension of geometric figures. By employing a local integration of these disparate viewpoints (Bikner-Ahsbahs & Prediger, 2014), I aim to provide a theoretical foundation for learning geometry as the evolution of students' geometric perception in the context of spatial geometry problem-solving activities facilitated by 3D sketching.

## THE CASE STUDY

The episode illustrating the evolution of student geometric perception in spatial geometry problem solving presents the student R, facing the Triangle task (Figure 2). While R did not formally study spatial geometry, he had knowledge of planar geometry for the presented challenges. When facing
the previous tasks, he already used 3D sketching to facilitate problem solving (see Palatnik, in press). He also knew that due to the structure of the task (succession of four questions), he may change his initial answer to the true/false questions. The whole episode took little less than 5 minutes. The first T/F question in the task was: CB’D’ is a right-angled triangle. The right angle is ____?

<figure><img src="image_42.png" /></figure>

ABCDA'B'C'D' is a cube. Answer the true/ false questions and explain your reasoning.

1. CB'D' is a right-angled triangle. The right angle is ____?

2. B'D' is the shortest side of the triangle CB'D'.

3. Triangle CB'D' has an obtuse angle.

4. In triangle CB'D', all angles are equal.

<caption>Figure 2: Triangle task (adopted from Widder et al., 2019)</caption>

## Interaction with a 3D model conflicts with the perception of a 2D diagram

R started by observing the illustration on the computer screen while tilting a head in different directions. He pointed with a 3D pen at a model's vertices C, B', and D'. Then, he drew an imaginary triangle by outlining segments CB', B'D' (Figure 3a), and D'C on a model.

R: I don't even need to (3D) sketch this to see that this is a right angle. Cause it seems... [draws imaginary angle CB'D' on a model]. Wait a minute... [tilts a head]

I: You should (sketch), I think.

R: Well, here, actually, I should. But also on the diagram, it is a right angle. [starts sketching CB'.] But it doesn't say anything. [sketches] But... I assume it is true [the angle is right]

<figure><img src="image_43.png" /><figcaption>Figure 3: Actions advancing learning to perceive: gesturing, sketching, rotating the model (red arrows are added to show the movement)</figcaption></figure>

**Analysis.** In terms of shifts of attention, R's attention is split between two main foci: a diagram and a model. On a diagram, he discerns a detail—angle B' and identifies it as a right angle. R's actions—tilting his head, pointing with a 3D pen, and drawing the imaginary angle and triangle—help him establish a correspondence between the two focal objects. Through these actions, he discerns details on the model, too: vertices of a triangle, its sides, and eventually an angle CB'D' as a candidate for a right angle. In terms of figural apprehension, perceptual apprehension is dominant (mostly vision within a flat and static medium). In addition, R starts to apprehend the problem sequentially (the imaginary construction of a triangle on a cube's model is sequential).

## Sketching and rotating the model shapes the perception of a geometric problem

R: [sketches CB' and then B'D' (Figure 3b)] Ok. Now it becomes a little bit different. [continues to sketch D'C]. Nice! It is good that I sketched. It is a right-angled triangle, but the right angle is not this, but instead, in fact... Wait a minute... [fixes the sketch]
the previous tasks, he already used 3D sketching to facilitate problem solving (see Palatnik, in press). He also knew that due to the structure of the task (succession of four questions), he may change his initial answer to the true/false questions. The whole episode took little less than 5 minutes. The first T/F question in the task was: CB’D’ is a right-angled triangle. The right angle is ____?

<figure><img src="image_48.png" /></figure>

ABCDA'B'C'D' is a cube. Answer the true/ false questions and explain your reasoning.

1. CB'D' is a right-angled triangle. The right angle is ____?

2. B'D' is the shortest side of the triangle CB'D'.

3. Triangle CB'D' has an obtuse angle.

4. In triangle CB'D', all angles are equal.

<caption>Figure 2: Triangle task (adopted from Widder et al., 2019)</caption>

## Interaction with a 3D model conflicts with the perception of a 2D diagram

R started by observing the illustration on the computer screen while tilting a head in different directions. He pointed with a 3D pen at a model's vertices C, B', and D'. Then, he drew an imaginary triangle by outlining segments CB', B'D' (Figure 3a), and D'C on a model.

R: I don't even need to (3D) sketch this to see that this is a right angle. Cause it seems... [draws imaginary angle CB'D' on a model]. Wait a minute... [tilts a head]

I: You should (sketch), I think.

R: Well, here, actually, I should. But also on the diagram, it is a right angle. [starts sketching CB)]. But it doesn't say anything. [sketches] But... I assume it is true [the angle is right]

<figure><img src="image_49.png" /><figcaption>Figure 3: Actions advancing learning to perceive: gesturing, sketching, rotating the model (red arrows are added to show the movement)</figcaption></figure>

**Analysis.** In terms of shifts of attention, R's attention is split between two main foci: a diagram and a model. On a diagram, he discerns a detail—angle B' and identifies it as a right angle. R's actions—tilting his head, pointing with a 3D pen, and drawing the imaginary angle and triangle—help him establish a correspondence between the two focal objects. Through these actions, he discerns details on the model, too: vertices of a triangle, its sides, and eventually an angle CB'D' as a candidate for a right angle. In terms of figural apprehension, perceptual apprehension is dominant (mostly vision within a flat and static medium). In addition, R starts to apprehend the problem sequentially (the imaginary construction of a triangle on a cube's model is sequential).

## Sketching and rotating the model shapes the perception of a geometric problem

R: [sketches CB' and then B'D' (Figure 3b)] Ok. Now it becomes a little bit different. [continues to sketch D'C]. Nice! It is good that I sketched. It is a right-angled triangle, but the right angle is not this, but instead, in fact... Wait a minute... [fixes the sketch]
...

R: [inclines a head]. Now, according to how it looks now, well... [points with a pen] D'CB' is a right angle. Well, I don't have to explain why.

I: Yes, it is better to explain.

R: Well, what? Acute [points at the angle D'], acute [points at the angle B'], right [points at C]. Nu, [points with a pen] C B' D'...

I: You can rotate the trian(gle), the square, the cube if you want (to show me).

R: Ok, so I will rotate it [starts to rotate the model, looks at the model while rotating it back and forth]. Now it looks different. Now it looks as if it is not like this. [tilts a model] Wait a minute. [rotates a model again, changing axes of rotation]. Ah. No! It is not a right-angled [triangle]. It is, in fact, isosceles. [gestures two sides with two strokes of a pen].

R: [rotates a model, Figure 3c]. It is confusing. [looks at the model] No! It is not a right-angled (triangle). False.

**Analysis.** R is focused mainly on a model. His exclamation while sketching: "Now it becomes a little bit different," indicates that his perception takes shape through the action of sketching. This process is gradual (R's first candidate for a right angle now looks acute to him). Further action of rotation provides additional perspectives and reshapes perception further: "Now it looks as if it is not like this". R discerns new details. His attention is now shifted from angles to the sides (his stroking gesture indicates this), probably by the action of sketching. R establishes a relationship of equality between two sides and perceives the triangle as isosceles. In terms of figural apprehension, sketching provides both perceptual and sequential apprehension. The 3D pen tool affords immediate materialization of an imagined geometric element (the way R sketches emulates the sequence of his previous imaginary drawings). The rotations accompanying sketching and further exploration of the model add operational apprehension. R states, "it is confusing", and we can attribute this confusion to incoherence between various forms of figural apprehension.

**Consolidated forms of figural apprehension support reasoning**

R: [Checks the question on the screen] Where we were at? B'D' is the shortest side (T/F?) in the triangle C, B', D' [points on model's vertices with a pen]. No. These are diagonals. These are diagonals of the cube (faces). So, they all are equal. In fact, it looks now as an equilateral triangle. Yes. Well, it is [B'D'] not the shortest side because they are all equal.

R: [reads the next question] (Does) Triangle CB'D' have an obtuse angle?

R: No. There is no obtuse angle. After all, it is an equilateral triangle, so 60, 60, 60. All the angles are equal, so 60, 60, 60. That's it.

**Analysis.** R is focused solely on the model. He continues with discerning details—the triangle's sides. But structures of attention evolve further, first to recognizing relationships "These are diagonals of the cube", then to perceiving properties "...they all are equal". R's reasoning is based on perceived properties—"[B'D'] is not the shortest side". From a perspective of figural apprehension types, the utterance "In fact, it looks now as an equilateral triangle" indicates that the perceptual, operational, and discursive apprehensions are "now" coherent.

Following a protocol, each task was ended with a question about a confidence level (out of five) in the correctness of the answer.

R: Between 3 and 4. Cause, I don't have backing for this, except for vision.
...

R: [inclines a head]. Now, according to how it looks now, well... [points with a pen] D'CB' is a right angle. Well, I don't have to explain why.

I: Yes, it is better to explain.

R: Well, what? Acute [points at the angle D'], acute [points at the angle B'], right [points at C]. Nu, [points with a pen] C B' D'...

I: You can rotate the trian(gle), the square, the cube if you want (to show me).

R: Ok, so I will rotate it [starts to rotate the model, looks at the model while rotating it back and forth]. Now it looks different. Now it looks as if it is not like this. [tilts a model] Wait a minute. [rotates a model again, changing axes of rotation]. Ah. No! It is not a right-angled [triangle]. It is, in fact, isosceles. [gestures two sides with two strokes of a pen].

R: [rotates a model, Figure 3c]. It is confusing. [looks at the model] No! It is not a right-angled (triangle). False.

**Analysis.** R is focused mainly on a model. His exclamation while sketching: "Now it becomes a little bit different," indicates that his perception takes shape through the action of sketching. This process is gradual (R's first candidate for a right angle now looks acute to him). Further action of rotation provides additional perspectives and reshapes perception further: "Now it looks as if it is not like this". R discerns new details. His attention is now shifted from angles to the sides (his stroking gesture indicates this), probably by the action of sketching. R establishes a relationship of equality between two sides and perceives the triangle as isosceles. In terms of figural apprehension, sketching provides both perceptual and sequential apprehension. The 3D pen tool affords immediate materialization of an imagined geometric element (the way R sketches emulates the sequence of his previous imaginary drawings). The rotations accompanying sketching and further exploration of the model add operational apprehension. R states, "it is confusing", and we can attribute this confusion to incoherence between various forms of figural apprehension.

**Consolidated forms of figural apprehension support reasoning**

R: [Checks the question on the screen] Where we were at? B'D' is the shortest side (T/F?) in the triangle C, B', D' [points on model's vertices with a pen]. No. These are diagonals. These are diagonals of the cube (faces). So, they all are equal. In fact, it looks now as an equilateral triangle. Yes. Well, it is [B'D'] not the shortest side because they are all equal.

R: [reads the next question] (Does) Triangle CB'D' have an obtuse angle?

R: No. There is no obtuse angle. After all, it is an equilateral triangle, so 60, 60, 60. All the angles are equal, so 60, 60, 60. That's it.

**Analysis.** R is focused solely on the model. He continues with discerning details—the triangle's sides. But structures of attention evolve further, first to recognizing relationships "These are diagonals of the cube", then to perceiving properties "...they all are equal". R's reasoning is based on perceived properties—"[B'D'] is not the shortest side". From a perspective of figural apprehension types, the utterance "In fact, it looks now as an equilateral triangle" indicates that the perceptual, operational, and discursive apprehensions are "now" coherent.

Following a protocol, each task was ended with a question about a confidence level (out of five) in the correctness of the answer.

R: Between 3 and 4. Cause, I don't have backing for this, except for vision.
I: Ok.

R: Well, I can bring a ruler, put out (the segments), and measure [gestures with a finger an upper diagonal]. But what (for)? Again, [takes a model in hand and rotates, Figure 3d] it is a cube, after all. And these are its diagonals. Thus, these (triangle) sides are equal.

I: Ok. So?

R: Thus, it is an equilateral triangle

I: Are you sure about this? 100%?

R: I think that, yes, I'm sure of it.

**Analysis.** This segment shows how R consolidates different forms of figural apprehension of triangle CB'D'. The perceptual form of apprehension is still very dominant: "I don't have backing for this, except for vision". By his relatively low initial self-confidence assessment, R acknowledges a deficiency of a visual argument. However, he is aware of the possibility of establishing the equality of segments by measuring, and he rotates the model for alternative perspectives (both actions are forms of operative apprehension). His concluding deductive chain is correct and adds a discursive apprehension. This argument structure follows R's previous structures of attention shifts: from holding the wholes (a cube with a triangle) to discerning details—cube's (faces) diagonals as sides of a triangle to recognizing relationship "equal" to confident reasoning, "it is an equilateral triangle".

## DISCUSSION

R answered the questions correctly, and his argumentation was valid. In studying spatial reasoning skills about 2D representations of 3D geometrical shapes, Fujita et al. (2020) reported just a 52.4% success rate for Japanese 9th graders (N=225) on a similar task. It is easy to dismiss R's success with the task as trivial. After all, he had additional tools—a 3D model and a sketching pen. However, as Shvarts et al. (2021) note, simply providing culturally developed tools is insufficient for students to discover their affordances (Gibson, 2014), nor does the existence of motor-perceptual systems guarantee the realization of their body potentialities. Drawing from Duval's figural apprehension and Mason's shifts of attention frameworks offers insights into his success and, in general, "how students chose particular spatial reasoning skills and domain-specific knowledge" (Fujita et al., 2020, p. 254).

Contesting the initial and misleading perceptual apprehension of a problem was challenging for the student in agreement with the findings of Widder et al. (2019). While the tools afford active ways to interact with a diagram, the student was initially reluctant (see also Palatnik & Abrahamson, 2022). Thus, the interventions of the interviewer: "You should (sketch)..." and "You can rotate..." served as action-perception catalysts for shifts in foci and forms of the student's attention and triggered other types of apprehension. These 'calls for action' with an intention to change perception appeared thought-provoking. The need to coordinate a 2D diagram with a 3D model by mapping a vertex to vertex and side to side through 3D sketching led to alternate attention foci and the occurrence of unusual dual sequential apprehension. Moreover, the continuous change of a 3D display during sketching and rotating contrasted to a static 2D diagram let R notice the inconsistency between 2D and 3D representations, creating a productive doubt in perceptual apprehension. When backing based on one of the apprehension types is discredited, a need for other types of apprehension arises.

The nature of the final attention structure—reasoning based on perceived properties (Mason, 2008) may appear as a purely cognitive act contrasting perceptually and action-oriented holding, discerning,
I: Ok.

R: Well, I can bring a ruler, put out (the segments), and measure [gestures with a finger an upper diagonal]. But what (for)? Again, [takes a model in hand and rotates, Figure 3d] it is a cube, after all. And these are its diagonals. Thus, these (triangle) sides are equal.

I: Ok. So?

R: Thus, it is an equilateral triangle

I: Are you sure about this? 100%?

R: I think that, yes, I'm sure of it.

**Analysis.** This segment shows how R consolidates different forms of figural apprehension of triangle CB'D'. The perceptual form of apprehension is still very dominant: "I don't have backing for this, except for vision". By his relatively low initial self-confidence assessment, R acknowledges a deficiency of a visual argument. However, he is aware of the possibility of establishing the equality of segments by measuring, and he rotates the model for alternative perspectives (both actions are forms of operative apprehension). His concluding deductive chain is correct and adds a discursive apprehension. This argument structure follows R's previous structures of attention shifts: from holding the wholes (a cube with a triangle) to discerning details—cube's (faces) diagonals as sides of a triangle to recognizing relationship "equal" to confident reasoning, "it is an equilateral triangle".

## DISCUSSION

R answered the questions correctly, and his argumentation was valid. In studying spatial reasoning skills about 2D representations of 3D geometrical shapes, Fujita et al. (2020) reported just a 52.4% success rate for Japanese 9th graders (N=225) on a similar task. It is easy to dismiss R's success with the task as trivial. After all, he had additional tools—a 3D model and a sketching pen. However, as Shvarts et al. (2021) note, simply providing culturally developed tools is insufficient for students to discover their affordances (Gibson, 2014), nor does the existence of motor-perceptual systems guarantee the realization of their body potentialities. Drawing from Duval's figural apprehension and Mason's shifts of attention frameworks offers insights into his success and, in general, "how students chose particular spatial reasoning skills and domain-specific knowledge" (Fujita et al., 2020, p. 254).

Contesting the initial and misleading perceptual apprehension of a problem was challenging for the student in agreement with the findings of Widder et al. (2019). While the tools afford active ways to interact with a diagram, the student was initially reluctant (see also Palatnik & Abrahamson, 2022). Thus, the interventions of the interviewer: "You should (sketch)..." and "You can rotate..." served as action-perception catalysts for shifts in foci and forms of the student's attention and triggered other types of apprehension. These 'calls for action' with an intention to change perception appeared thought-provoking. The need to coordinate a 2D diagram with a 3D model by mapping a vertex to vertex and side to side through 3D sketching led to alternate attention foci and the occurrence of unusual dual sequential apprehension. Moreover, the continuous change of a 3D display during sketching and rotating contrasted to a static 2D diagram let R notice the inconsistency between 2D and 3D representations, creating a productive doubt in perceptual apprehension. When backing based on one of the apprehension types is discredited, a need for other types of apprehension arises.

The nature of the final attention structure—*reasoning based on perceived properties* (Mason, 2008) may appear as a purely cognitive act contrasting perceptually and action-oriented holding, discerning,
recognizing, and perceiving. However, an enactivist stance in which cognition is perceived as an active, ongoing engagement with the world: "All doing is knowing, and all knowing is doing" (Maturana & Varela, 1987, p. 26) resolves a dichotomy of perceptual and cognitive aspects of attention structures. Within this framework, "perception consists of perceptually guided action," and "cognitive structures emerge from the recurrent sensorimotor patterns that enable action to be perceptually guided." (Varela, 1999, p.12). Indeed, R's actions led to shifts in perceptual structures of attention, which informed his further actions. A shift to discerning details (vertices, sides of the triangle) was associated with a figural operation of dimensional deconstruction. 3D sketching and rotating actions were related to sequential and operative apprehension, shifting the student's attention to the triangle sides and enabling a discursive apprehension of them as diagonals of the cube's faces. When all forms of figural apprehension appear coherent, they support geometric reasoning.

The exchange concluding the episode demonstrates that an enactive approach goes beyond the ability to recognize the model's geometric properties empirically. The student briefly considered measuring the segments but chose an alternative path. He starts the argument with a rotation of a model, helping him relieve his previous actions and perceptions and situate the application of relevant geometric knowledge. However, this material scaffolding fades out as R constructs an almost complete chain of deductions, increasing his confidence level about the correctness of his answer.

The presented episode contrasts with R's (and most of our students) usual form of learning geometry, which involves merely observing and rarely interacting with any form of diagram. It was a *learning* activity since it allowed R to use his senses to guide his actions, change his perspective on a problem, and recall adequate geometric content to improve his perception of it. To solve a problem, R overcame initially deceptive perceptions and combined different kinds of apprehension. By designing such experiences and letting students and teachers reflect on them, noticing and guiding shifts of attention, we may facilitate learning geometry by evolving our geometric perception.

**Acknowledgments**

This research was supported by the Israeli Science Foundation (ISF), Grant 1323/23.

**References**

Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. D. English et al., (Eds.), *Handbook of international research in mathematics education, 2nd revised edition* (pp. 720–749). Lawrence Erlbaum.

Bikner-Ahsbahs, A., & Prediger, S. (Eds.), (2014). *Networking of theories as a research practice in mathematics education*. Springer.

Bryson, N. (1988). The gaze in the expanded field. In H. Foster (Ed.), *Vision and visuality* (pp. 87–108). The New Press.

Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processes. In R. Sutherland & J. Mason (Eds.), *Exploiting mental imagery with computers in mathematics education* (pp.142–157). Springer.

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

Duval, R. (2017). *Understanding the mathematical way of thinking—the registers of semiotic representations* (edited by Tania M. M. Campos). Springer.
recognizing, and perceiving. However, an enactivist stance in which cognition is perceived as an active, ongoing engagement with the world: "All doing is knowing, and all knowing is doing" (Maturana & Varela, 1987, p. 26) resolves a dichotomy of perceptual and cognitive aspects of attention structures. Within this framework, "perception consists of perceptually guided action," and "cognitive structures emerge from the recurrent sensorimotor patterns that enable action to be perceptually guided." (Varela, 1999, p.12). Indeed, R's actions led to shifts in perceptual structures of attention, which informed his further actions. A shift to discerning details (vertices, sides of the triangle) was associated with a figural operation of dimensional deconstruction. 3D sketching and rotating actions were related to sequential and operative apprehension, shifting the student's attention to the triangle sides and enabling a discursive apprehension of them as diagonals of the cube's faces. When all forms of figural apprehension appear coherent, they support geometric reasoning.

The exchange concluding the episode demonstrates that an enactive approach goes beyond the ability to recognize the model's geometric properties empirically. The student briefly considered measuring the segments but chose an alternative path. He starts the argument with a rotation of a model, helping him relieve his previous actions and perceptions and situate the application of relevant geometric knowledge. However, this material scaffolding fades out as R constructs an almost complete chain of deductions, increasing his confidence level about the correctness of his answer.

The presented episode contrasts with R's (and most of our students) usual form of learning geometry, which involves merely observing and rarely interacting with any form of diagram. It was a *learning* activity since it allowed R to use his senses to guide his actions, change his perspective on a problem, and recall adequate geometric content to improve his perception of it. To solve a problem, R overcame initially deceptive perceptions and combined different kinds of apprehension. By designing such experiences and letting students and teachers reflect on them, noticing and guiding shifts of attention, we may facilitate learning geometry by evolving our geometric perception.

## Acknowledgments

This research was supported by the Israeli Science Foundation (ISF), Grant 1323/23.

## References

* Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. D. English et al., (Eds.), *Handbook of international research in mathematics education, 2nd revised edition* (pp. 720–749). Lawrence Erlbaum.

* Bikner-Ahsbahs, A., & Prediger, S. (Eds.), (2014). *Networking of theories as a research practice in mathematics education*. Springer.

* Bryson, N. (1988). The gaze in the expanded field. In H. Foster (Ed.), *Vision and visuality* (pp. 87–108). The New Press.

* Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processes. In R. Sutherland & J. Mason (Eds.), *Exploiting mental imagery with computers in mathematics education* (pp.142–157). Springer.

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

* Duval, R. (2017). *Understanding the mathematical way of thinking—the registers of semiotic representations* (edited by Tania M. M. Campos). Springer.
Fujita, T., Kondo, Y., Kumakura, H., Kunimune, S., & Jones, K. (2020). Spatial reasoning skills about 2D representations of 3D geometrical shapes in grades 4 to 9. *Mathematics Education Research Journal, 32*, 235–255.

Gibson, J. J. (1986/ 2014). *The ecological approach to visual perception: Classic edition*. Taylor & Francis.

Goldstone, R. L., Landy, D. H., & Son, J. Y. (2010). The education of perception. *Topics in Cognitive Science*, 2(2), 265–284.

Goodwin, C. (2018). *Co-operative action*. Cambridge University Press.

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

Johnston-Wilder, S., & Mason, J. (2005). *Developing thinking in geometry*. Sage.

Mason, J. (2008). Being mathematical with and in front of learners: Attention, awareness, and attitude as sources of differences between teacher educators, teachers & learners. In T. Wood & B. Jaworski (Eds.), *International handbook of mathematics teacher education* (Vol. 4, pp. 31–56). Sense Publishers.

Mason, J. (2010). Attention and intention in learning about teaching through teaching. In R. Leikin & R. Zazkis (Eds.), *Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice* (pp. 23–47). Springer.

Maturana, H. R., & Varela, F. J. (1987). *The tree of knowledge: The biological roots of human understanding*. New Science Library/Shambhala Publications.

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

Ng, O., & Ferrara, F. (2020). Towards a materialist vision of ‘learning as Making’: The case of 3D Printing Pens in school mathematics. *International Journal of Science and Mathematics Education, 18*, 925–944.

Palatnik, A. & Abrahamson, D. (2022). Escape from Plato’s cave: An enactivist argument for learning 3D geometry by constructing tangible models. In G. Bolondi, F. Ferretti, & C. Spagnolo (Eds.), *Proceedings of the CERME12*, (pp. 742–749). ERME.

Palatnik, A. (2022). Students’ exploration of tangible geometric models: Focus on shifts of attention. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.). *Proceedings of the 45th PME*, (Vol. 3, pp. 275–282). PME.

Palatnik, A. (in press). Interaction of spatial and theoretical aspects in spatial geometry problem solving through construction: the case of 3D sketching. In *Proceedings of the Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13)*. ERME.

Palatnik, A., Abrahamson, D., Baccaglini-Frank, A., Ng, O., Shvarts, A., & Swidan, O. (2023). Theory and practice of designing embodied mathematics learning. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel., & M. Tabach (Eds.). (2023). *Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education*, (Vol. 1, pp. 159–189). PME.

Shvarts, A., Alberto, R., Bakker, A., Doorman, M., & Drijvers, P. (2021). Embodied instrumentation in learning mathematics as the genesis of a body-artifact functional system. *Educational Studies in Mathematics*, 107(3), 447–469.

Varela, F. J. (1999). *Ethical know-how: Action, wisdom, and cognition*. Stanford University Press.

Widder, M., Berman, A., & Koichu, B. (2019). An a priori measure of visual difficulty of 2-D sketches depicting 3-D Objects. *Journal for Research in Mathematics Education, 50*(5), 489–528.
Fujita, T., Kondo, Y., Kumakura, H., Kunimune, S., & Jones, K. (2020). Spatial reasoning skills about 2D representations of 3D geometrical shapes in grades 4 to 9. *Mathematics Education Research Journal, 32*, 235–255.

Gibson, J. J. (1986/ 2014). *The ecological approach to visual perception: Classic edition*. Taylor & Francis.

Goldstone, R. L., Landy, D. H., & Son, J. Y. (2010). The education of perception. *Topics in Cognitive Science*, 2(2), 265–284.

Goodwin, C. (2018). *Co-operative action*. Cambridge University Press.

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

Johnston-Wilder, S., & Mason, J. (2005). *Developing thinking in geometry*. Sage.

Mason, J. (2008). Being mathematical with and in front of learners: Attention, awareness, and attitude as sources of differences between teacher educators, teachers & learners. In T. Wood & B. Jaworski (Eds.), *International handbook of mathematics teacher education* (Vol. 4, pp. 31–56). Sense Publishers.

Mason, J. (2010). Attention and intention in learning about teaching through teaching. In R. Leikin & R. Zazkis (Eds.), *Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice* (pp. 23–47). Springer.

Maturana, H. R., & Varela, F. J. (1987). *The tree of knowledge: The biological roots of human understanding*. New Science Library/Shambhala Publications.

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

Ng, O., & Ferrara, F. (2020). Towards a materialist vision of ‘learning as Making’: The case of 3D Printing Pens in school mathematics. *International Journal of Science and Mathematics Education, 18*, 925–944.

Palatnik, A. & Abrahamson, D. (2022). Escape from Plato’s cave: An enactivist argument for learning 3D geometry by constructing tangible models. In G. Bolondi, F. Ferretti, & C. Spagnolo (Eds.), *Proceedings of the CERME12*, (pp. 742–749). ERME.

Palatnik, A. (2022). Students’ exploration of tangible geometric models: Focus on shifts of attention. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.). *Proceedings of the 45th PME*, (Vol. 3, pp. 275–282). PME.

Palatnik, A. (in press). Interaction of spatial and theoretical aspects in spatial geometry problem solving through construction: the case of 3D sketching. In *Proceedings of the Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13)*. ERME.

Palatnik, A., Abrahamson, D., Baccaglini-Frank, A., Ng, O., Shvarts, A., & Swidan, O. (2023). Theory and practice of designing embodied mathematics learning. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel., & M. Tabach (Eds.). (2023). *Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education*, (Vol. 1, pp. 159–189). PME.

Shvarts, A., Alberto, R., Bakker, A., Doorman, M., & Drijvers, P. (2021). Embodied instrumentation in learning mathematics as the genesis of a body-artifact functional system. *Educational Studies in Mathematics*, 107(3), 447–469.

Varela, F. J. (1999). *Ethical know-how: Action, wisdom, and cognition*. Stanford University Press.

Widder, M., Berman, A., & Koichu, B. (2019). An a priori measure of visual difficulty of 2-D sketches depicting 3-D Objects. *Journal for Research in Mathematics Education, 50*(5), 489–528.
# SPATIAL REASONING INTERVENTIONS AND TRANSFER TO GEOMETRY: WHAT WE KNOW ABOUT MECHANISM

Ilyse Resnick, Jonathan Adams

University of Canberra

There is growing evidence that spatial reasoning learning causally supports improvements in geometry performance. However, the mechanism that explains this transfer is not yet well understood. The current study provides a systematic literature review on all existing experimental studies that examine the impact of spatial learning on mathematics. We have found that, despite an increasing interest in mechanism and uptake of experimental studies, only a few studies have explicitly assessed mechanistic pathways. The two most promising pathways are the use of specific spatial skills within specific geometric tasks and visuospatial working memory. Although changes in strategy is understudied, this may be another pathway. We identify process models, mixed-methods (including qualitative research), and mediation/moderation as important aspects of future experimental work.

## SPATIAL REASONING SKILLS SUPPORT GEOMETRY LEARNING

There is growing evidence that improving spatial reasoning skills causally transfers to improvements in mathematics achievement, including within the field of geometry (Adams et al., 2022). For example, a recent meta-analysis of 28 spatial intervention studies found that spatial reasoning learning consistently improves mathematics performance with average effect sizes between .20 and .42, depending on transfer distance (Hawes et al., 2022). Despite the strong evidence supporting spatial transfer to mathematics, the mechanisms behind transfer are not yet clear (Young et al., 2018) – that is, *why* does spatial learning support mathematics achievement?

Understanding why spatial reasoning supports geometry learning has theoretical and practical implications for a range of research disciplines (e.g., mathematics education, cognition, development, neuropsychology), which are interested in characterizing the relation between the constructs. For example, there are currently qualitatively and functionally different theoretical models (described below). Characterizing an accurate theoretical model of mechanism would, in turn, inform the development of more targeted and effective mathematics learning interventions. That is, the value in understanding spatial mechanism for research on the teaching, learning, and understanding of geometry, is the optimization of instruction strategies (e.g., what spatial techniques are the most beneficial); effective integration of technology (e.g., what spatial barriers may students face); and the development of individualized learning plans, valid assessments, and curriculum.

The current study provides a systematic review of existing experimental research, to characterize what we know about why spatial reasoning learning transfers to support geometry and mathematics achievement. Correlational research was not included because, by design, it is unable to establish causal mechanisms (Rohrer, 2018). The current study then suggests innovative methodologies for capturing mechanism in future research, which includes mechanisms not yet considered.
# SPATIAL REASONING INTERVENTIONS AND TRANSFER TO GEOMETRY: WHAT WE KNOW ABOUT MECHANISM

Ilyse Resnick, Jonathan Adams

University of Canberra

There is growing evidence that spatial reasoning learning causally supports improvements in geometry performance. However, the mechanism that explains this transfer is not yet well understood. The current study provides a systematic literature review on all existing experimental studies that examine the impact of spatial learning on mathematics. We have found that, despite an increasing interest in mechanism and uptake of experimental studies, only a few studies have explicitly assessed mechanistic pathways. The two most promising pathways are the use of specific spatial skills within specific geometric tasks and visuospatial working memory. Although changes in strategy is understudied, this may be another pathway. We identify process models, mixed-methods (including qualitative research), and mediation/moderation as important aspects of future experimental work.

## SPATIAL REASONING SKILLS SUPPORT GEOMETRY LEARNING

There is growing evidence that improving spatial reasoning skills causally transfers to improvements in mathematics achievement, including within the field of geometry (Adams et al., 2022). For example, a recent meta-analysis of 28 spatial intervention studies found that spatial reasoning learning consistently improves mathematics performance with average effect sizes between .20 and .42, depending on transfer distance (Hawes et al., 2022). Despite the strong evidence supporting spatial transfer to mathematics, the mechanisms behind transfer are not yet clear (Young et al., 2018) – that is, *why* does spatial learning support mathematics achievement?

Understanding why spatial reasoning supports geometry learning has theoretical and practical implications for a range of research disciplines (e.g., mathematics education, cognition, development, neuropsychology), which are interested in characterizing the relation between the constructs. For example, there are currently qualitatively and functionally different theoretical models (described below). Characterizing an accurate theoretical model of mechanism would, in turn, inform the development of more targeted and effective mathematics learning interventions. That is, the value in understanding spatial mechanism for research on the teaching, learning, and understanding of geometry, is the optimization of instruction strategies (e.g., what spatial techniques are the most beneficial); effective integration of technology (e.g., what spatial barriers may students face); and the development of individualized learning plans, valid assessments, and curriculum.

The current study provides a systematic review of existing experimental research, to characterize what we know about why spatial reasoning learning transfers to support geometry and mathematics achievement. Correlational research was not included because, by design, it is unable to establish causal mechanisms (Rohrer, 2018). The current study then suggests innovative methodologies for capturing mechanism in future research, which includes mechanisms not yet considered.
Spatial reasoning, spatial learning, and geometry

Spatial reasoning is an umbrella term for a wide range of different, but interrelated, skills that involve thinking about two-dimensional and three-dimensional relations between and within objects. For example, being able to imagine an object rotating (mental rotation) or a perspective different from your own (perspective-taking) are two separate spatial reasoning skills (Hegarty & Waller, 2004). Spatial visualization is also a kind of spatial reasoning skill that refers to the multi-step mental manipulation and transformation of spatial information.

There is extensive empirical evidence spanning the past three decades that show spatial reasoning skills, and in particular spatial visualization and mental rotation, support geometry performance (e.g., Clements & Battista, 1992). This relation is consistent, predictive, and strengthens over time (Resnick et al., 2019). More recently, there has been experimental evidence that shows improving spatial reasoning skills causally improves geometry understanding. Some spatial learning interventions have included targeted spatial learning opportunities, such as origami (Arıcı & Aslan-Tutak, 2015; Boakes, 2009), mental rotation or spatial scaling training (Gilligan et al., 2020); whereas others have included a wider range of spatial learning opportunities, such as mental rotation, spatial orientation, spatial visualization, and the integration of these spatial skills (Adams et al., 2022; Lowrie et al., 2017, 2019).

Current theories of mechanism

We are not aware of any broad theoretical models that characterize potential pathways between spatial reasoning and geometry, and so we will focus here on general mathematics achievement. Existing theoretical models are based primarily on correlational research (Battista et al., 2018). Hawes and Ansari (2020) have identified four possible mechanisms based on this work: (a) individuals map number onto space to better understand them (spatial representation of numbers); (b) completing both spatial and numerical tasks involve the same brain areas (shared neural processing); (c) individuals use spatial visualization skills as a “mental blackboard” to organize and manipulate mathematics problems (spatial modelling); and (d) spatial reasoning is a proxy for other cognitively demanding skills (working memory). Notably, these models are not mutually exclusive and may reflect varying levels of analysis (e.g., activation in the intra-parietal sulcus during spatial and geometric tasks may be functionally equivalent to using spatial visualization skills to complete geometric tasks).

Interestingly, none of the Hawes and Ansari's (2020) models positioned mathematics as inherently spatial (e.g., numerical-spatial mappings serve as a tool for understanding; cognitive architecture originally specialized for interacting with the physical world was co-opted to understand mathematics). This may be due to their focus on numerical reasoning rather than geometry. That geometry is inherently spatial is a common description in geometry literature (e.g., Battista et al., 2018; Boakes, 2009). Although such an explanation highlights the affordances embedded within task demands, it falls short of providing a causal mechanism. That is, it does not answer questions of how, when, or which kind of spatial skills are employed to complete different geometry tasks.

To address the complexity of multiple, overlapping mechanisms, Mix (2019) characterizes broader attributes of mechanism. They argue that the relation between spatial reasoning and mathematics is general, and not limited to specific spatial and mathematics sub-skills. However, elsewhere they acknowledge that this relation is fragile and difficult to replicate. In contrast, most research to date assumes that specific spatial skills underlie specific mathematics tasks (e.g., Gilligan et al., 2018).
Spatial reasoning, spatial learning, and geometry

Spatial reasoning is an umbrella term for a wide range of different, but interrelated, skills that involve thinking about two-dimensional and three-dimensional relations between and within objects. For example, being able to imagine an object rotating (mental rotation) or a perspective different from your own (perspective-taking) are two separate spatial reasoning skills (Hegarty & Waller, 2004). Spatial visualization is also a kind of spatial reasoning skill that refers to the multi-step mental manipulation and transformation of spatial information.

There is extensive empirical evidence spanning the past three decades that show spatial reasoning skills, and in particular spatial visualization and mental rotation, support geometry performance (e.g., Clements & Battista, 1992). This relation is consistent, predictive, and strengthens over time (Resnick et al., 2019). More recently, there has been experimental evidence that shows improving spatial reasoning skills causally improves geometry understanding. Some spatial learning interventions have included targeted spatial learning opportunities, such as origami (Arıcı & Aslan-Tutak, 2015; Boakes, 2009), mental rotation or spatial scaling training (Gilligan et al., 2020); whereas others have included a wider range of spatial learning opportunities, such as mental rotation, spatial orientation, spatial visualization, and the integration of these spatial skills (Adams et al., 2022; Lowrie et al., 2017, 2019).

Current theories of mechanism

We are not aware of any broad theoretical models that characterize potential pathways between spatial reasoning and geometry, and so we will focus here on general mathematics achievement. Existing theoretical models are based primarily on correlational research (Battista et al., 2018). Hawes and Ansari (2020) have identified four possible mechanisms based on this work: (a) individuals map number onto space to better understand them (spatial representation of numbers); (b) completing both spatial and numerical tasks involve the same brain areas (shared neural processing); (c) individuals use spatial visualization skills as a “mental blackboard” to organize and manipulate mathematics problems (spatial modelling); and (d) spatial reasoning is a proxy for other cognitively demanding skills (working memory). Notably, these models are not mutually exclusive and may reflect varying levels of analysis (e.g., activation in the intra-parietal sulcus during spatial and geometric tasks may be functionally equivalent to using spatial visualization skills to complete geometric tasks).

Interestingly, none of the Hawes and Ansari's (2020) models positioned mathematics as inherently spatial (e.g., numerical-spatial mappings serve as a tool for understanding; cognitive architecture originally specialized for interacting with the physical world was co-opted to understand mathematics). This may be due to their focus on numerical reasoning rather than geometry. That geometry is inherently spatial is a common description in geometry literature (e.g., Battista et al., 2018; Boakes, 2009). Although such an explanation highlights the affordances embedded within task demands, it falls short of providing a causal mechanism. That is, it does not answer questions of how, when, or which kind of spatial skills are employed to complete different geometry tasks.

To address the complexity of multiple, overlapping mechanisms, Mix (2019) characterizes broader attributes of mechanism. They argue that the relation between spatial reasoning and mathematics is general, and not limited to specific spatial and mathematics sub-skills. However, elsewhere they acknowledge that this relation is fragile and difficult to replicate. In contrast, most research to date assumes that specific spatial skills underlie specific mathematics tasks (e.g., Gilligan et al., 2018).
Mix (2019) suggested that spatial skills can perform several functions in real-time problem solving, such as decoding, mapping referents, and the use of mental models. Although there is some automatic processing between spatial reasoning and mathematics, strategic recruitment of spatial processes is more predictive of achievement. Finally, they find that the relation between spatial reasoning and mathematics is consistent but may change qualitatively as both familiarity and task demands change.

## Methods for characterizing mechanism

Most research characterizing the relation between spatial reasoning and geometry (and mathematics) has been correlational. These approaches can involve completing of a range of different spatial and mathematics tasks to examine the relative correlations and emergence of factors (Young et al., 2018). For example, mental rotation has been found to predict performance in geometry and not algebra, whereas visuospatial working memory (the ability to keep spatial information active in the mind) had the opposite relation (Kyttälä & Lehto, 2008). Although it is speculated that geometry involves reasoning about transformations whereas algebra can involve mental operations (Newcombe et al., 2019), correlational analyses are unable to establish causal mechanisms (Rohrer, 2018), and factor analytic approaches have practical, statistical, and interpretative limitations (Young et al., 2018).

Experimental research is required to empirically assess hypothesized mechanisms. Unfortunately, such research is limited, especially in geometry. There are currently only nine existing experimental studies connecting spatial learning with geometry. The primary aim of these studies was to develop effective spatial learning materials to support geometry learning. Subsequently, what these studies can tell us about mechanism (*why* the spatial learning materials help) may be limited in scope. Nevertheless, these studies, and the broader context of experimental spatial learning studies, are an important starting point to examine the mechanism connecting spatial reasoning within geometry.

## METHODS

A systematic literature review was conducted to characterize how experimental studies, examining the causal effects of spatial reasoning learning on mathematics achievement, have considered mechanism in their justification, experimental design, and explanation of findings.

### Data collection and inclusion criteria

A recent meta-analysis by Hawes et al. (2022) identified 28 experimental studies that examine the causal effect of spatial training studies on mathematics achievement. To be included in the Hawes et al. meta-analysis, studies were required to (a) involve humans, (b) be published in English, (c) report on behavioral outcomes, (d) use a pretest-training-posttest design, (e) have at least one mathematics outcome measure, (f) include sufficient spatial training, and (g) compare a spatial training group with a control group. Although Hawes et al. included unpublished findings and doctoral theses in their analysis, we have included only those publications that have undertaken peer-review (n = 23). An additional literature search was conducted using the same criteria, to identify studies published after the Hawes et al. study (n = 7). Taken together, a total of 30 studies were included in this review.

### Data coding

A thematic analysis (Guest et al., 2012) was used to identify and characterize mechanistic explanations for how spatial learning may support mathematics learning. The operational definition of mechanism used in this study was phrases that considered how or why spatial reasoning may
Mix (2019) suggested that spatial skills can perform several functions in real-time problem solving, such as decoding, mapping referents, and the use of mental models. Although there is some automatic processing between spatial reasoning and mathematics, strategic recruitment of spatial processes is more predictive of achievement. Finally, they find that the relation between spatial reasoning and mathematics is consistent but may change qualitatively as both familiarity and task demands change.

## Methods for characterizing mechanism

Most research characterizing the relation between spatial reasoning and geometry (and mathematics) has been correlational. These approaches can involve completing of a range of different spatial and mathematics tasks to examine the relative correlations and emergence of factors (Young et al., 2018). For example, mental rotation has been found to predict performance in geometry and not algebra, whereas visuospatial working memory (the ability to keep spatial information active in the mind) had the opposite relation (Kyttälä & Lehto, 2008). Although it is speculated that geometry involves reasoning about transformations whereas algebra can involve mental operations (Newcombe et al., 2019), correlational analyses are unable to establish causal mechanisms (Rohrer, 2018), and factor analytic approaches have practical, statistical, and interpretative limitations (Young et al., 2018).

Experimental research is required to empirically assess hypothesized mechanisms. Unfortunately, such research is limited, especially in geometry. There are currently only nine existing experimental studies connecting spatial learning with geometry. The primary aim of these studies was to develop effective spatial learning materials to support geometry learning. Subsequently, what these studies can tell us about mechanism (*why* the spatial learning materials help) may be limited in scope. Nevertheless, these studies, and the broader context of experimental spatial learning studies, are an important starting point to examine the mechanism connecting spatial reasoning within geometry.

## METHODS

A systematic literature review was conducted to characterize how experimental studies, examining the causal effects of spatial reasoning learning on mathematics achievement, have considered mechanism in their justification, experimental design, and explanation of findings.

### Data collection and inclusion criteria

A recent meta-analysis by Hawes et al. (2022) identified 28 experimental studies that examine the causal effect of spatial training studies on mathematics achievement. To be included in the Hawes et al. meta-analysis, studies were required to (a) involve humans, (b) be published in English, (c) report on behavioral outcomes, (d) use a pretest-training-posttest design, (e) have at least one mathematics outcome measure, (f) include sufficient spatial training, and (g) compare a spatial training group with a control group. Although Hawes et al. included unpublished findings and doctoral theses in their analysis, we have included only those publications that have undertaken peer-review (n = 23). An additional literature search was conducted using the same criteria, to identify studies published after the Hawes et al. study (n = 7). Taken together, a total of 30 studies were included in this review.

### Data coding

A thematic analysis (Guest et al., 2012) was used to identify and characterize mechanistic explanations for how spatial learning may support mathematics learning. The operational definition of mechanism used in this study was phrases that considered how or why spatial reasoning may
support mathematics learning. For example, the phrase, "*children might use spatial visualization... to picture and rotate the shapes presented to count the number of sides (faces) on the shape.*," (Gilligan et al., 2020, p. 15) would be counted as a mechanistic explanation because it explains *how* a spatial reasoning skill may be used to solve mathematics problems. In contrast, the phrase, "*Spatial ability and visualization have been shown to be related with academic success in mathematics and geometry,*" (Arıcı & Aslan-Tutak, 2015, p. 181) would not be counted as a mechanistic explanation because it does not provide an explanation of *why* this relation is observed.

An initial reviewer coded each instance of mechanistic explanation for emergent themes within the literature review, methods, and discussion sections (results sections were not coded because they do not include explanation or connections to the literature). New codes were created as required. Once the full set of codes were identified, the initial reviewer re-coded the full sample to ensure a subsequent code wasn't more appropriate. In a final step, two new reviewers independently coded the full sample of mechanistic explanations using the developed coding system. Although new codes could be created during this step, none were required. One code was renamed to capture that theme more accurately and another code was split into two for greater specificity. Discrepancies were resolved through discussion, resulting in 100% interrater reliability.

## RESULTS

Of the 30 studies examined, 76% justified their study (Introduction section), and 66% explained the results from their study (Discussion section), by speculating a causal mechanism connecting spatial reasoning and mathematics achievement. Explored in depth below, is that these mechanistic models (a) varied, (b) were largely absent from the experimental design, and (c) changed over time. A summary of the findings where methodological design considers mechanism is also included.

### Varied mechanisms that have been considered (description of emergent themes)

The most prevalent mechanistic pathways involved using improved spatial skills to complete mathematics tasks (60%). This could include using proportional reasoning skills to complete quantity processing tasks (Gilligan et al., 2020), spatial visualization to create mental models (Lowrie et al., 2019), form perception to interpret shapes and graphs (Burte et al., 2017), or mental number line to compare magnitude (Mix et al., 2021). All these specific connections reflect the idea that there are specific spatial skills that are recruited to complete specific mathematics tasks.

Visuospatial working memory (vsWM) was also identified frequently as required to complete mathematics tasks (37%). In this view, keeping in mind a spatial organization of sequences of moves and transformations when completing multi-step problems would require vsWM. Some related discussion referenced working memory and executive function skills more broadly (7%). We separated out this coding to distinguish between the two explanations.

Although 30% of studies acknowledge the possibility of priming spatial strategies, this discussion was usually framed around not being able to rule this explanation out, while favoring an improved spatial skill or vsWM model. Only two studies (7%) described strategy change as a main outcome. Another common explanation was that mathematics is inherently spatial (27%), though notably this does not actually address mechanism. Shared cognitive resources (i.e., the same brain area is responsible for completing spatial and mathematics tasks) and consideration of moderators (e.g., how mechanisms may vary based on age or task demands) were also commonly mentioned (23%).
support mathematics learning. For example, the phrase, "*children might use spatial visualization... to picture and rotate the shapes presented to count the number of sides (faces) on the shape.*," (Gilligan et al., 2020, p. 15) would be counted as a mechanistic explanation because it explains *how* a spatial reasoning skill may be used to solve mathematics problems. In contrast, the phrase, "*Spatial ability and visualization have been shown to be related with academic success in mathematics and geometry,*" (Arıcı & Aslan-Tutak, 2015, p. 181) would not be counted as a mechanistic explanation because it does not provide an explanation of *why* this relation is observed.

An initial reviewer coded each instance of mechanistic explanation for emergent themes within the literature review, methods, and discussion sections (results sections were not coded because they do not include explanation or connections to the literature). New codes were created as required. Once the full set of codes were identified, the initial reviewer re-coded the full sample to ensure a subsequent code wasn't more appropriate. In a final step, two new reviewers independently coded the full sample of mechanistic explanations using the developed coding system. Although new codes could be created during this step, none were required. One code was renamed to capture that theme more accurately and another code was split into two for greater specificity. Discrepancies were resolved through discussion, resulting in 100% interrater reliability.

## RESULTS

Of the 30 studies examined, 76% justified their study (Introduction section), and 66% explained the results from their study (Discussion section), by speculating a causal mechanism connecting spatial reasoning and mathematics achievement. Explored in depth below, is that these mechanistic models (a) varied, (b) were largely absent from the experimental design, and (c) changed over time. A summary of the findings where methodological design considers mechanism is also included.

### Varied mechanisms that have been considered (description of emergent themes)

The most prevalent mechanistic pathways involved using improved spatial skills to complete mathematics tasks (60%). This could include using proportional reasoning skills to complete quantity processing tasks (Gilligan et al., 2020), spatial visualization to create mental models (Lowrie et al., 2019), form perception to interpret shapes and graphs (Burte et al., 2017), or mental number line to compare magnitude (Mix et al., 2021). All these specific connections reflect the idea that there are specific spatial skills that are recruited to complete specific mathematics tasks.

Visuospatial working memory (vsWM) was also identified frequently as required to complete mathematics tasks (37%). In this view, keeping in mind a spatial organization of sequences of moves and transformations when completing multi-step problems would require vsWM. Some related discussion referenced working memory and executive function skills more broadly (7%). We separated out this coding to distinguish between the two explanations.

Although 30% of studies acknowledge the possibility of priming spatial strategies, this discussion was usually framed around not being able to rule this explanation out, while favoring an improved spatial skill or vsWM model. Only two studies (7%) described strategy change as a main outcome. Another common explanation was that mathematics is inherently spatial (27%), though notably this does not actually address mechanism. Shared cognitive resources (i.e., the same brain area is responsible for completing spatial and mathematics tasks) and consideration of moderators (e.g., how mechanisms may vary based on age or task demands) were also commonly mentioned (23%).
Experimental evidence for mechanism

There were seven studies that were able to address mechanism through their experimental design. Five of these studies had separate conditions to compare the effects of training different spatial skills, finding broadly that specific spatial skills support specific mathematics tasks. Only two of these studies had geometry outcomes. Gilligan et al. (2020) found that, although mental rotation supported missing term problems and spatial scaling supported number line, improvements in both skills supported geometry performance. Gilligan et al. (2020) suggest that form perception might be useful for distinguishing between different symbols and identifying information presented in charts. Adams et al. (2022) found that spatial learning (which included a range of different visualization-based skills) transfers to geometry outcomes and not algebra outcomes. Adams et al. (2022) suggest that spatial visualization might be useful in geometry because it involves reasoning about transformations.

Visuospatial working memory (vsWM) may also provide support. Both mental rotation and vsWM training equally transferred to improved mathematics performance, with no transfer on one another (Zhang et al., 2021). Notably, Mix et al. (2020) found that age moderates the relation between spatial learning and mathematics, with spatial skills being more important early on and vsWM more important later. These contributions appear to be unique to vsWM and not improvements in general cognitive processing, with one study including cognitive skills as co-variates (Cornu et al., 2019) and another not finding transfer to general skills (Cheung et al., 2019).

There were four studies that considered mechanism in their research aims, but did not have a sufficient experimental design to adequately address it. Three of these studies, instead, addressed best teaching practices, finding benefits of embedded (Lowrie & Logan, 2023) and embodied (Gilligan et al., 2023) approaches that include strong pedagogy (Mulligan et al., 2020). There were also three studies that found socio-economic status (SES) mediated the impact of training. Although this does not provide a mechanism, it highlights the importance of including contextual factors as well as potential mediators / moderators when characterizing mechanistic pathways.

How consideration of mechanism has changed over the last three decades

Across the 1990's and early 2000's, there was no experimental research examining the impact of
spatial learning on mathematics, despite a large literature showing strong correlations between the
two constructs. Experimental studies begin to emerge starting in 2009. However, discussion of
mechanism is limited; from 2009 to 2016, seven studies do not consider mechanism at all (with four
justifying their study by characterizing mathematics as inherently spatial). Only two studies identify
the potential role of vsWM and one study identifies spatial visualization. From 2017 to present, there
has been a marked increased interest in mechanism, both in frequency and range of explanation.
However, explanations have focused on skills and vsWM, with only one study (Lowrie & Logan,
2023) centering the possibility of changes in strategy.

DISCUSSION

As research into the relation between spatial reasoning and mathematics has progressed, researchers have shown increasing interest in the mechanisms supporting this connection. Earlier studies focused on establishing correlations, and later verifying that spatial learning causally supports improvements in mathematics. Although studies have provided speculative explanations by drawing on correlational work, only more recently have a few studies sought to assess this directly within their methods.
Experimental evidence for mechanism

There were seven studies that were able to address mechanism through their experimental design. Five of these studies had separate conditions to compare the effects of training different spatial skills, finding broadly that specific spatial skills support specific mathematics tasks. Only two of these studies had geometry outcomes. Gilligan et al. (2020) found that, although mental rotation supported missing term problems and spatial scaling supported number line, improvements in both skills supported geometry performance. Gilligan et al. (2020) suggest that form perception might be useful for distinguishing between different symbols and identifying information presented in charts. Adams et al. (2022) found that spatial learning (which included a range of different visualization-based skills) transfers to geometry outcomes and not algebra outcomes. Adams et al. (2022) suggest that spatial visualization might be useful in geometry because it involves reasoning about transformations.

Visuospatial working memory (vsWM) may also provide support. Both mental rotation and vsWM training equally transferred to improved mathematics performance, with no transfer on one another (Zhang et al., 2021). Notably, Mix et al. (2020) found that age moderates the relation between spatial learning and mathematics, with spatial skills being more important early on and vsWM more important later. These contributions appear to be unique to vsWM and not improvements in general cognitive processing, with one study including cognitive skills as co-variates (Cornu et al., 2019) and another not finding transfer to general skills (Cheung et al., 2019).

There were four studies that considered mechanism in their research aims, but did not have a sufficient experimental design to adequately address it. Three of these studies, instead, addressed best teaching practices, finding benefits of embedded (Lowrie & Logan, 2023) and embodied (Gilligan et al., 2023) approaches that include strong pedagogy (Mulligan et al., 2020). There were also three studies that found socio-economic status (SES) mediated the impact of training. Although this does not provide a mechanism, it highlights the importance of including contextual factors as well as potential mediators / moderators when characterizing mechanistic pathways.

How consideration of mechanism has changed over the last three decades

Across the 1990's and early 2000's, there was no experimental research examining the impact of
spatial learning on mathematics, despite a large literature showing strong correlations between the
two constructs. Experimental studies begin to emerge starting in 2009. However, discussion of
mechanism is limited; from 2009 to 2016, seven studies do not consider mechanism at all (with four
justifying their study by characterizing mathematics as inherently spatial). Only two studies identify
the potential role of vsWM and one study identifies spatial visualization. From 2017 to present, there
has been a marked increased interest in mechanism, both in frequency and range of explanation.
However, explanations have focused on skills and vsWM, with only one study (Lowrie & Logan,
2023) centering the possibility of changes in strategy.

DISCUSSION

As research into the relation between spatial reasoning and mathematics has progressed, researchers have shown increasing interest in the mechanisms supporting this connection. Earlier studies focused on establishing correlations, and later verifying that spatial learning causally supports improvements in mathematics. Although studies have provided speculative explanations by drawing on correlational work, only more recently have a few studies sought to assess this directly within their methods.
The most common explanations for transfer are built on the idea that specific spatial skills are utilised within specific mathematics tasks. Although this contrasts Mix (2019), who argues for a general connection, Mix’s (2019) work is based on correlational research. The experimental work described here suggests there may be specific connections, including within geometry (Adams et al., 2022; Gilligan et al., 2020). This experimental work also highlights a separate role for vsWM. Hawes and Ansari (2020) consider vsWM in the broader context of improvements in general cognition; however, this is not supported by the experimental findings, which instead finds a unique role for vsWM separate from improved spatial skills (e.g., Zhang et al., 2021). Spatial skills and vsWM may work together by providing a larger “mental blackboard” to keep more spatial information in mind as well as the skills required to mentally manipulate that spatial information. Given the small number of experimental studies, and the inclusion of broad geometry measures, more experimental research is needed to characterize the kinds of spatial skills that support specific geometry tasks.

## A new model for mechanism – the role of strategic choice

Independent of any gains due to improved execution of spatial skills and vsWM (as described above), it is possible that spatial learning might encourage students to apply different strategic approaches when solving geometry problems. Changes in strategy are largely absent from correlational and experimental literatures (Lowrie et al., 2021; 2023), and, when included, described as a priming effect that cannot be ruled out (e.g., Gilligan et al., 2020). However, students know multiple ways of solving problems and will balance efficiency and accuracy when selecting an appropriate strategy (Siegler, 2006). These strategic decisions are influenced by the nature of the problem, their familiarity with different strategies, and their beliefs about potential success. This is aligned with the finding that spatial skills versus vsWM are relatively more important when solving mathematics problems as familiarity and task demands change (Mix et al., 2020), and that strategic processing of spatial-mathematics relations is more predictive of mathematics performance than automatic processing (Mix, 2019). Importantly, completing geometry problems can involve a multi-step problem-solving process, and students may use different cognitive and meta-cognitive strategies at any given step.

## Future directions – how to capture mechanism

We suggest that it is possible to characterize mechanism with experimental research explicitly designed to assess potential mechanisms. Such work would be benefited from process models (Young et al., 2018) to identify the specific tasks involved in geometry (previous work often uses broad measures). Mixed-methods approaches would also be helpful, with qualitative methods well suited to characterize when and how strategies may change. Also missing from the literature is analysis of changes in quality, accuracy, and relevancy of (external and internal) representations, which would inform how spatial learning supports mathematics performance. For example, future research may explore if spatial learning leads to representations becoming more detailed, abstract, or focused on relevant spatial information; or perhaps the quality of representation remains the same, but the ability to manipulate it improves. Another key approach to determining mechanism is the inclusion of key mediators and moderators to parse out how basic cognitive skills may manifest in different mathematics contexts (Resnick & Stieff, 2024).