used Dick’s (2008) three types of knowledge fidelity (pedagogical, mathematical, and cognitive), to
evaluate 53 VGM. His evaluation found that over half of them were considered inappropriate for
classroom use and, even for those that were considered as appropriate, teachers were still required to
make informed pedagogical decisions regarding their use to ensure their learning potential was
maximized.

To combat the paucity of quality VM available to support learning, other mathematics educators have
developed theoretical approaches that should underpin the creation of VM. For example, Ladel and
Kortenkamp (2013) created Artifact Centric Activity Theory (ACAT) as a model for the development
of quality VM and Hirsh-Pasek et al. (2015) use “four pillars of learning” as their conceptual frame
in the development of robust VM. Mathematics educators have also been active in building VGM,
for example *Cinderella* (Richter-Gebert & Kortenkamp, 2012), *Cubeling* (Etzold), or *Angles* (Etzold).
Given the mathematical education background of these designers, teachers can be more confident in
using these manipulatives to support student learning.

While these critiques are useful for teachers to use to support their decision-making process, and the
frameworks are useful for designers of VM, in this conceptual paper we want to discuss two questions
that we think remain unanswered in relation to the use of VGM.

* RQ 1. What issues arise in considering the use of VGM in educational contexts?

* RQ 2. What changes to how geometry is taught are needed to effectively use VGM?

THE PROBLEM

In addressing Question 1, we commence first with an ontological perspective and then discuss
concerns from a technological perspective. We finish this section by considering how these
perspectives impact the pedagogical decisions that teachers must make regarding the use of VGM.

Ontological concerns

Our ontological concerns are two-fold. Firstly, is it possible to accurately represent 3D mathematical
objects in 2D representations (in this case via digital 2D representations)? Any 3D object can only be
represented in 2D by projecting it onto a lower-dimensional space and, as a consequence, the
projection will not have all the properties of the original object. Thus, we cannot be sure of the
physical object without further manipulation of the 2D representation, for example, by change of the
projection. As this change will involve time as another dimension, our conclusions must always be
supported by argumentation and trust. Secondly, putting aside the ontological problem of 3D objects
being represented in 2D, how can students using VGM be confident that the designers of the software
are *truthfully* representing *reality* in digital form? It could be the case that designers are, in fact,
operating as *bad faith* actors who are either deliberately representing objects incorrectly, or are doing
so inadvertently as they do not have the necessary technical skills to represent them accurately. We
see this bad faith phenomenon evident in digital environments, where cheaply made apps dominate
the market (Larkin, 2016). This is also, in some sense, the dilemma regarding the use of Artificial
Intelligence (AI), whereby it is increasingly difficult to place trust in digital information, as we can
no longer easily identify the veracity of representations of reality presented to us. We take up this
point briefly again in the conclusion of this paper.
Let us illustrate this bad faith phenomenon: Imagine a VGM for 2D shapes that can be moved and rotated on screen, similar to a (real) manipulative of polygons made out of paper, wood, or plastic. The real manipulative supports important invariants of Euclidean Geometry: moving or rotating a shape will not change its inherent properties such as area or circumference. The VGM, however, could change the shapes slightly when they are moved or rotated, and we could experience a proof that the Pythagorean Theorem is true for any triangle, not only right-angled ones. There is no sense in creating and using such a VGM; however, the sheer possibility to create such a VGM raises the question why students should trust manipulatives, given that such manipulatives may represent mathematics incorrectly and thus generate misconceptions?

## Technological concerns

At first, it should be noted that ontological and technological concerns can be considered as intertwined, for example, the technological problem of presenting a circle in a digital (discrete) manner also relates to ontological concern about the nature of circles. In terms of this paper, we distinguish the two by considering concerns that are based on bad faith and concerns that are based on technological impossibility. Therefore, even if the developers of VGM are aware of the ontological difficulties, and try to truthfully represent geometric content in the best possible way, there are technological problems that cannot be overcome, but at best only be hidden.

We provide two specific examples: (1) The representation on screen is always based on a digital representation of either real world objects or mathematical objects. Usually, this representation consists of a computer screen with high, but finite, resolution. Imagine representing a circle around a point P with radius r. Some pixels on the screen need to be painted. Apart from the fact that these points will have a (small) area, they are not infinitely small, so their distance to the center will not exactly be r, but vary within a (small) ε-range. Consequently, they do not represent a circle in the mathematical sense. What looks like a circle on screen is actually a polygonal shape. (2) The side and the diagonal of a square are not commensurable – so we cannot use rational numbers to express them both: therefore, a triangle created by cutting a square at one diagonal cannot be represented accurately using a digital screen.

Both examples show the difficulties in accurate representation of geometric content on a screen. The introduction of very high-resolution displays hides those fundamental problems from the eye, but they still exist. We are not suggesting that VGM shouldn't be used because of these difficulties, as this problem exists in the real world as well. One could easily ask, on the (sub-)atomic level, "Is it possible to create a true square from wood?" We are just indicating the need for teachers and students to be aware of these issues.

## Pedagogical concerns

Presuming that the ontological and technological concerns are resolved, or at least minimized, a third concern remains in relation to how VGM should be used for best effect in the learning process. A common approach to the use of manipulatives is based on Bruner's (1966) suggestion that students learn through three different experiential means:

* *Enactive* (direct sensory) experience where students take an active part in their learning through the manipulation of their learning environment;
Let us illustrate this bad faith phenomenon: Imagine a VGM for 2D shapes that can be moved and rotated on screen, similar to a (real) manipulative of polygons made out of paper, wood, or plastic. The real manipulative supports important invariants of Euclidean Geometry: moving or rotating a shape will not change its inherent properties such as area or circumference. The VGM, however, could change the shapes slightly when they are moved or rotated, and we could experience a proof that the Pythagorean Theorem is true for any triangle, not only right-angled ones. There is no sense in creating and using such a VGM; however, the sheer possibility to create such a VGM raises the question why students should trust manipulatives, given that such manipulatives may represent mathematics incorrectly and thus generate misconceptions?

## Technological concerns

At first, it should be noted that ontological and technological concerns can be considered as intertwined, for example, the technological problem of presenting a circle in a digital (discrete) manner also relates to ontological concern about the nature of circles. In terms of this paper, we distinguish the two by considering concerns that are based on bad faith and concerns that are based on technological impossibility. Therefore, even if the developers of VGM are aware of the ontological difficulties, and try to truthfully represent geometric content in the best possible way, there are technological problems that cannot be overcome, but at best only be hidden.

We provide two specific examples: (1) The representation on screen is always based on a digital representation of either real world objects or mathematical objects. Usually, this representation consists of a computer screen with high, but finite, resolution. Imagine representing a circle around a point P with radius r. Some pixels on the screen need to be painted. Apart from the fact that these points will have a (small) area, they are not infinitely small, so their distance to the center will not exactly be r, but vary within a (small) ε-range. Consequently, they do not represent a circle in the mathematical sense. What looks like a circle on screen is actually a polygonal shape. (2) The side and the diagonal of a square are not commensurable – so we cannot use rational numbers to express them both: therefore, a triangle created by cutting a square at one diagonal cannot be represented accurately using a digital screen.

Both examples show the difficulties in accurate representation of geometric content on a screen. The introduction of very high-resolution displays hides those fundamental problems from the eye, but they still exist. We are not suggesting that VGM shouldn't be used because of these difficulties, as this problem exists in the real world as well. One could easily ask, on the (sub-)atomic level, "Is it possible to create a true square from wood?" We are just indicating the need for teachers and students to be aware of these issues.

## Pedagogical concerns

Presuming that the ontological and technological concerns are resolved, or at least minimized, a third concern remains in relation to how VGM should be used for best effect in the learning process. A common approach to the use of manipulatives is based on Bruner's (1966) suggestion that students learn through three different experiential means:

* *Enactive* (direct sensory) experience where students take an active part in their learning through the manipulation of their learning environment;
* *Iconic* representation of experience where enacted experiences are represented via diagrams, film clips, charts etc.; and

* *Symbolic* representation including written language symbols such as words and mathematical symbols.

What needs to be kept in mind, however, is that these are not staged levels; rather, they should be considered as equivalent in importance. The critical pedagogical element is that there should be a special focus on students transitioning between the three different ways of learning. For example, students should be able to describe symbolically their enactive experiences or find an iconic representation of a symbolic one. What is vital, in a pedagogical sense, is that teachers thoughtfully consider the use of VGM in student learning as, for example, it is problematic if teachers rely exclusively on the use of largely symbolic geometry software packages such as *Cinderella* or *GeoGebra*.

# BUILDING TRUST IN GEOMETRICAL SOLUTIONS

In this section we address Question 2 and suggest a pedagogical approach, based on the notion of trust, that we think supports students in learning geometry (and indeed mathematics more broadly). We then provide two practical examples of this approach in high school.

We commence this discussion about trust, as we did with Question 1, from an ontological position, and ask “What is trust?”. Although there are many different definitions of trust (see Viljanen, 2005), we use the definition by Amaral et al. (2019) who suggest that “trust is a central component of social life and is frequently referred to as the ‘glue of society’, vital in economics, social cooperation, organizations, groups, etc. (p. 1)”. These authors indicate that the term has been used to refer to different types of relationships, including “between software systems operating in a network” (p. 1) and that trust is “generally the basis for decision making closely related to achieving a goal” (p. 2). In this paper, when we are discussing trust, we mean the trust between a user (student) and a software system (VGM) and we suggest that this notion of trust, and how it can be established, provides a possible mechanism for resolving the ontological and technological concerns we outlined earlier. The establishment of trust in a virtual environment is based on what we will call *trust certificates*.

## Trust certificates as a mathematical and pedagogical approach

In computing contexts, the issue of trust between users and software is largely resolved using digital certificates. According to Wikipedia, a “digital certificate certifies the ownership of a public key by the named subject of the certificate, and this allows others to rely upon signatures or on assertions made about the private key that corresponds to the certified public key.” In this way, a trust relationship can be established between the subject and the party relying upon the certificate. We suggest that this approach to trust, whereby users build trust in the software via the use of trust certificates, can also be utilized in a modified way, by students as they learn about geometry.

It is important to distinguish here between simply checking an answer and using trust certificates. Students are often taught by teachers to “check their answers” – and will often complete a process to do so. We agree that this is an appropriate pedagogical approach, if implemented correctly. However, often the student’s checking is based on the inverse operation of the initial operation they performed, which might be easier or just more familiar to the students. They will check if 15 minus 7 is 8 by adding 7 to 8 to get 15. Of course, this works for correct answers; however, if children are merely
* *Iconic* representation of experience where enacted experiences are represented via diagrams, film clips, charts etc.; and

* *Symbolic* representation including written language symbols such as words and mathematical symbols.

What needs to be kept in mind, however, is that these are not staged levels; rather, they should be considered as equivalent in importance. The critical pedagogical element is that there should be a special focus on students transitioning between the three different ways of learning. For example, students should be able to describe symbolically their enactive experiences or find an iconic representation of a symbolic one. What is vital, in a pedagogical sense, is that teachers thoughtfully consider the use of VGM in student learning as, for example, it is problematic if teachers rely exclusively on the use of largely symbolic geometry software packages such as *Cinderella* or *GeoGebra*.

# BUILDING TRUST IN GEOMETRICAL SOLUTIONS

In this section we address Question 2 and suggest a pedagogical approach, based on the notion of trust, that we think supports students in learning geometry (and indeed mathematics more broadly). We then provide two practical examples of this approach in high school.

We commence this discussion about trust, as we did with Question 1, from an ontological position, and ask “What is trust?”. Although there are many different definitions of trust (see Viljanen, 2005), we use the definition by Amaral et al. (2019) who suggest that “trust is a central component of social life and is frequently referred to as the ‘glue of society’, vital in economics, social cooperation, organizations, groups, etc. (p. 1)”. These authors indicate that the term has been used to refer to different types of relationships, including “between software systems operating in a network” (p. 1) and that trust is “generally the basis for decision making closely related to achieving a goal” (p. 2). In this paper, when we are discussing trust, we mean the trust between a user (student) and a software system (VGM) and we suggest that this notion of trust, and how it can be established, provides a possible mechanism for resolving the ontological and technological concerns we outlined earlier. The establishment of trust in a virtual environment is based on what we will call *trust certificates*.

## Trust certificates as a mathematical and pedagogical approach

In computing contexts, the issue of trust between users and software is largely resolved using digital certificates. According to Wikipedia, a “digital certificate certifies the ownership of a public key by the named subject of the certificate, and this allows others to rely upon signatures or on assertions made about the private key that corresponds to the certified public key.” In this way, a trust relationship can be established between the subject and the party relying upon the certificate. We suggest that this approach to trust, whereby users build trust in the software via the use of trust certificates, can also be utilized in a modified way, by students as they learn about geometry.

It is important to distinguish here between simply checking an answer and using trust certificates. Students are often taught by teachers to “check their answers” – and will often complete a process to do so. We agree that this is an appropriate pedagogical approach, if implemented correctly. However, often the student’s checking is based on the inverse operation of the initial operation they performed, which might be easier or just more familiar to the students. They will check if 15 minus 7 is 8 by adding 7 to 8 to get 15. Of course, this works for correct answers; however, if children are merely
following a routine process, then this checking may result in correct logic but an incorrect outcome
– if 15 minus 6 is 8 **then** adding 6 to 8 gives 15. The relationship between the statements is logically
true but the answer is incorrect. By contrast, as seen in the example below, a trust certificate approach
builds *confidence* in the correctness of a result, instead of “proving” that it is correct directly. Trust
certificates for 15 - 7 = 8 *might* include:

* I “know” (from prior learning also established following a similar trust approach) that an odd number minus an odd number results in an even number; 8 is odd (✔)

* I “know” that 7 is larger than 5, so the answer will be less than 10; 8 is < 10 (✔)

* I “know” double 7 is 14, so subtracting 7 from a larger number will result in a number larger than 7, 8 is larger than 7 (✔)

* Given the three trust certificates above, I can be confident that 15 - 7 = 8.

This approach resembles unit testing (Beck, 2002), a software development method where (small)
software units are tested automatically for correctness, increasing the confidence that a function or
method of a computer program is indeed giving the correct output. The three trust certificates above
could also be formulated by students as mathematical unit tests: (1) Any solution $x$ of $z - y$ should be
even, if both $z$ and $y$ are odd. (2) Subtracting a single digit number $y$ from any number $z$ should result
in a number in the decade one less than the one of $z$. (3) If $2y < z$, then $z - y > y$.

The notion of unit testing applies equally in geometrical contexts. As one example, consider finding
the intersection of two lines, for example given by two linear equations $y = a_1x + b_1$ and $y = a_2x + b_2$. An easy way to check whether a suggested solution ($x_0, y_0$) is the intersection of two lines, we
can plug the values in each of the equations and see whether they hold. Pedagogically, we can trigger
this kind of thought by asking for evidence that a certain proposed solution is *not* the intersection.
For example, for the lines $y = 2x + 3$ and $y = -\frac{1}{2}x + \frac{1}{2}$ we can ask students to show $t$ and can ask
students to give evidence that $(-2, -1)$ is not a solution, and that $(3, -1)$ is not a solution, which
they can do by choosing the correct equation as a (counter-) certificate. A similar approach is possible
when searching for the perpendicular bisector of a segment using two circles with the same radius.

## Certificates in both real world and virtual contexts

Although our focus in this paper is on VGM, we argue that the use of trust certificates is equally valid
in both real world and virtual contexts, and that students should be concerned about the validity of
geometric representations in textbooks as well as in VGM. However, given that the ontological and
technological concerns noted earlier in this paper relate primarily to VGM, we recommend that
students first encounter the use of trust certificates in a real-world context, and then later encounter
them in a virtual context that relates to the real world one. We feel this step is necessary for students
to build trust in the representations of the same concepts presented later in VGM. However, in later
encounters this direction can be reversed. We now provide two examples of the use of trust certificates
in contexts related to the teaching of geometry.

## How might this approach work in practice?

We first discuss an implementation of the trust approach that took place in a teaching episode, which
occurred after a professional development course, where one of the authors presented a VGM that
could be used to model the Golden Gate Bridge. Here, using a photo of the bridge, a physical
following a routine process, then this checking may result in correct logic but an incorrect outcome
– if 15 minus 6 is 8 **then** adding 6 to 8 gives 15. The relationship between the statements is logically
true but the answer is incorrect. By contrast, as seen in the example below, a trust certificate approach
builds *confidence* in the correctness of a result, instead of “proving” that it is correct directly. Trust
certificates for 15 - 7 = 8 *might* include:

* I “know” (from prior learning also established following a similar trust approach) that an odd number minus an odd number results in an even number; 8 is odd (✔)

* I “know” that 7 is larger than 5, so the answer will be less than 10; 8 is < 10 (✔)

* I “know” double 7 is 14, so subtracting 7 from a larger number will result in a number larger than 7, 8 is larger than 7 (✔)

* Given the three trust certificates above, I can be confident that 15 - 7 = 8.

This approach resembles unit testing (Beck, 2002), a software development method where (small)
software units are tested automatically for correctness, increasing the confidence that a function or
method of a computer program is indeed giving the correct output. The three trust certificates above
could also be formulated by students as mathematical unit tests: (1) Any solution $x$ of $z - y$ should be
even, if both $z$ and $y$ are odd. (2) Subtracting a single digit number $y$ from any number $z$ should result
in a number in the decade one less than the one of $z$. (3) If $2y < z$, then $z - y > y$.

The notion of unit testing applies equally in geometrical contexts. As one example, consider finding
the intersection of two lines, for example given by two linear equations $y = a_1x + b_1$ and $y = a_2x + b_2$. An easy way to check whether a suggested solution ($x_0, y_0$) is the intersection of two lines, we
can plug the values in each of the equations and see whether they hold. Pedagogically, we can trigger
this kind of thought by asking for evidence that a certain proposed solution is *not* the intersection.
For example, for the lines $y = 2x + 3$ and $y = -\frac{1}{2}x + \frac{1}{2}$ we can ask students to show $t$ and can ask
students to give evidence that $(-2, -1)$ is not a solution, and that $(3, -1)$ is not a solution, which
they can do by choosing the correct equation as a (counter-) certificate. A similar approach is possible
when searching for the perpendicular bisector of a segment using two circles with the same radius.

Certificates in both real world and virtual contexts

Although our focus in this paper is on VGM, we argue that the use of trust certificates is equally valid
in both real world and virtual contexts, and that students should be concerned about the validity of
geometric representations in textbooks as well as in VGM. However, given that the ontological and
technological concerns noted earlier in this paper relate primarily to VGM, we recommend that
students first encounter the use of trust certificates in a real-world context, and then later encounter
them in a virtual context that relates to the real world one. We feel this step is necessary for students
to build trust in the representations of the same concepts presented later in VGM. However, in later
encounters this direction can be reversed. We now provide two examples of the use of trust certificates
in contexts related to the teaching of geometry.

How might this approach work in practice?

We first discuss an implementation of the trust approach that took place in a teaching episode, which
occurred after a professional development course, where one of the authors presented a VGM that
could be used to model the Golden Gate Bridge. Here, using a photo of the bridge, a physical
simulation of forces on the bridge can be projected onto the image, enabling the students to experiment with the shape of the bridge created by the forces acting on it [See Kortenkamp and Richter-Gebert (2008) for further details about the content]. It should be noted that the task reflects all the problems we raised in our first research question: The object in question is a 3D object, given as a 2D photograph, which again will be projected on a 2D screen. Students have no opportunity to interact in any way with the real object. As they only have a single photo, they cannot even choose a different projection. Also, as we are using digital imagery, we do not have a proper representation of the real bridge, and the resulting simulation is a polygonal line, instead of a parabola. It is due to the pedagogical intervention by an experienced teacher of physics and mathematics that students were helped to overcome these issues and develop confidence in the trustworthiness of the VGM. He describes this in an email (translation from German by the authors):

> [...] I immediately started to work with my group to experiment and built a [physical] model using springs and masses, took a photo and simulated the same configuration with the software. As a "confidence-building measure" I took a picture from the side and showed the students that using the right transformation the simulated curve can be projected onto the oblique picture. The culmination was your example with the Golden Gate bridge. [...]

He also sent the images (Figure 1) with a description of the technical details.

<figure><img src="image_157.png" /><figcaption>Figure 8: The simple model was built to create confidence in the simulation</figcaption></figure>

By systematically building up confidence through trust certificates (the touchable instance of a bridge model that is used with the simulation), the teacher could successfully transform the virtual manipulative from a “black box” into a trusted device that students could rightfully accept as experimental evidence.

The second example also involves the connection between the real and virtual world but reverses the direction of how trust certificates can be used. For use in acourse for pre-service teachers (PSTs), a visual proof of the convergence of the geometric series was created by the lead author (Figure 2, left), and then realized as a VGM. Instead of relying on using the digital tool alone, he also produced laser-cut squares that the PSTs could manipulate to determine how these tiles can be arranged to a converging series (Figure 2, right). We claim that the creation of real manipulatives, based on experiences in a VG, builds confidence and note that this is an increasingly affordable option in many schools. As a low-cost alternative, paper printouts from a VGM can often be fabricated to create physical manipulatives that can assist students to trust the digital representations. This notion is also implicit in the theory of Duos of Artefacts (Soury-Lavergne, 2021).
simulation of forces on the bridge can be projected onto the image, enabling the students to experiment with the shape of the bridge created by the forces acting on it [See Kortenkamp and Richter-Gebert (2008) for further details about the content]. It should be noted that the task reflects all the problems we raised in our first research question: The object in question is a 3D object, given as a 2D photograph, which again will be projected on a 2D screen. Students have no opportunity to interact in any way with the real object. As they only have a single photo, they cannot even choose a different projection. Also, as we are using digital imagery, we do not have a proper representation of the real bridge, and the resulting simulation is a polygonal line, instead of a parabola. It is due to the pedagogical intervention by an experienced teacher of physics and mathematics that students were helped to overcome these issues and develop confidence in the trustworthiness of the VGM. He describes this in an email (translation from German by the authors):

> [...] I immediately started to work with my group to experiment and built a [physical] model using springs and masses, took a photo and simulated the same configuration with the software. As a "confidence-building measure" I took a picture from the side and showed the students that using the right transformation the simulated curve can be projected onto the oblique picture. The culmination was your example with the Golden Gate bridge. [...]

He also sent the images (Figure 1) with a description of the technical details.

<figure><img src="image_164.png" /><figcaption>Figure 8: The simple model was built to create confidence in the simulation</figcaption></figure>

By systematically building up confidence through trust certificates (the touchable instance of a bridge model that is used with the simulation), the teacher could successfully transform the virtual manipulative from a “black box” into a trusted device that students could rightfully accept as experimental evidence.

The second example also involves the connection between the real and virtual world but reverses the direction of how trust certificates can be used. For use in acourse for pre-service teachers (PSTs), a visual proof of the convergence of the geometric series was created by the lead author (Figure 2, left), and then realized as a VGM. Instead of relying on using the digital tool alone, he also produced laser-cut squares that the PSTs could manipulate to determine how these tiles can be arranged to a converging series (Figure 2, right). We claim that the creation of real manipulatives, based on experiences in a VG, builds confidence and note that this is an increasingly affordable option in many schools. As a low-cost alternative, paper printouts from a VGM can often be fabricated to create physical manipulatives that can assist students to trust the digital representations. This notion is also implicit in the theory of Duos of Artefacts (Soury-Lavergne, 2021).
<figure><img src="image_158.png" /><figcaption>Figure 9: The visual proof in the VM and the laser-cut realization that can provide confidence</figcaption></figure>

Although the thrust of this article thus far has been the use of trust certificates in regard to VGM, we finish with the proposition that it is possible to use trust certificates as a tool for validating information provided by AI. In many cases AI suggests answers to mathematical questions, which appear to be correct, but are often incorrect either in terms of the overall solution, or in terms of the mathematical logic that is used to derive the solution. One of the authors has already begun to use trust certificates with PSTs to help them acquire skills they need to cope with AI generated answers submitted by students. In a similar vein, Kortenkamp and Dohrmann (2023) outline how PSTs can use AI outputs, which answer geometrical problems, as a training tool for helping their future school students generate questions that narrow the problem to understandable solution steps that can be verified independently from the AI output. Overall, the main contribution of trust certificates is that they enable students to build trust, based on verifiable mathematical understanding of the component building blocks, which they can use locally to support argumentation, and which build towards trust in the more complex solutions generated by VGM or AI.

## CONCLUSION AND WAY FORWARD

In this paper we identified three sets of concerns regarding the use of VGM. Firstly, ontological ones whereby it is perhaps impossible to faithfully represent the 3D world in 2D and the associated difficulties that can occur with bad faith designers. We then discussed the technological difficulties of creating such representations, which largely relate to discrepancies between formal mathematics and how formal understandings can be represented in digital form. Finally, both these ontological and technological concerns impinge upon the pedagogical decisions required to use VGM. We then suggested the use of trust certificates as a way forward in responding to ontological and technological concerns which, whilst not totally alleviating them, at least provide students with a procedure for developing confidence in evaluating the accuracy of representations in VGM. This is achieved by building trust in the smaller building blocks that comprise the complex solutions, often generated by ‘black boxes’, where the workings of the technology remain hidden from students, and by providing students with opportunities to replicate digital outputs in more concrete representations (albeit that these in turn are still only models of the real situation). Given the rapid proliferation of digital representations, in the form of VGM and AI, we argue that a process for determining the veracity of information is an important mathematical skill for students to learn.

## References

* Amaral, G., Sales, T. P., Guizzardi, G., & Porello, D. (2019). Towards a reference ontology of trust. In *On the Move to Meaningful Internet Systems: OTM 2019 Conferences: Confederated International Conferences: CoopIS, ODBASE, C&TC 2019, Rhodes, Greece*. Springer International Publishing.

* Beck, K. (2002). *Test-driven development by example*. Addison Wesley.
<figure><img src="image_165.png" /><figcaption>Figure 9: The visual proof in the VM and the laser-cut realization that can provide confidence</figcaption></figure>

Although the thrust of this article thus far has been the use of trust certificates in regard to VGM, we finish with the proposition that it is possible to use trust certificates as a tool for validating information provided by AI. In many cases AI suggests answers to mathematical questions, which appear to be correct, but are often incorrect either in terms of the overall solution, or in terms of the mathematical logic that is used to derive the solution. One of the authors has already begun to use trust certificates with PSTs to help them acquire skills they need to cope with AI generated answers submitted by students. In a similar vein, Kortenkamp and Dohrmann (2023) outline how PSTs can use AI outputs, which answer geometrical problems, as a training tool for helping their future school students generate questions that narrow the problem to understandable solution steps that can be verified independently from the AI output. Overall, the main contribution of trust certificates is that they enable students to build trust, based on verifiable mathematical understanding of the component building blocks, which they can use locally to support argumentation, and which build towards trust in the more complex solutions generated by VGM or AI.

## CONCLUSION AND WAY FORWARD

In this paper we identified three sets of concerns regarding the use of VGM. Firstly, ontological ones whereby it is perhaps impossible to faithfully represent the 3D world in 2D and the associated difficulties that can occur with bad faith designers. We then discussed the technological difficulties of creating such representations, which largely relate to discrepancies between formal mathematics and how formal understandings can be represented in digital form. Finally, both these ontological and technological concerns impinge upon the pedagogical decisions required to use VGM. We then suggested the use of trust certificates as a way forward in responding to ontological and technological concerns which, whilst not totally alleviating them, at least provide students with a procedure for developing confidence in evaluating the accuracy of representations in VGM. This is achieved by building trust in the smaller building blocks that comprise the complex solutions, often generated by ‘black boxes’, where the workings of the technology remain hidden from students, and by providing students with opportunities to replicate digital outputs in more concrete representations (albeit that these in turn are still only models of the real situation). Given the rapid proliferation of digital representations, in the form of VGM and AI, we argue that a process for determining the veracity of information is an important mathematical skill for students to learn.

## References

* Amaral, G., Sales, T. P., Guizzardi, G., & Porello, D. (2019). Towards a reference ontology of trust. In *On the Move to Meaningful Internet Systems: OTM 2019 Conferences: Confederated International Conferences: CoopIS, ODBASE, C&TC 2019, Rhodes, Greece*. Springer International Publishing.

* Beck, K. (2002). *Test-driven development by example*. Addison Wesley.
Bruner, J. (1966). *Toward a theory of instruction*. Oxford University Press.

Dick, T. (2008). Fidelity in Technological Tools for Mathematics Education. In G. Blume, & M. K. Reid (Eds.), *Research on Technology and the Teaching and Learning of Mathematics: Volume 2. Cases and Perspectives* (pp. 333-339). Information Age Publishing.

Etzold, H. (2013-2020). *Winkel*. https://apps.apple.com/ch/app/winkel/id715739128

Etzold, H. (2015-2022). *Cubeling*. https://apps.apple.com/bs/app/cubeling/id1027746349

Hirsh-Pasek, K., Zosh, J., Michnick Golinkoff, R., Gray, J., Robb, M., & Kaufman, J. (2015). Putting Education in “Educational” Apps: Lessons From the Science of Learning. *Psychological Science in the Public Interest, 16*(1), 3-34. https://doi.org/10.1177/1529100615569721

Ladel, S. (2009). Multiple externe Repräsentationen (MERs) und deren Verknüpfung durch Computereinsatz. *Zur Bedeutung für das Mathematiklernen im Anfangsunterricht*. Verlag Dr. Kovac.

Ladel, S., & Kortenkamp, U. (2013). An activity-theoretic approach to multi-touch tools in early maths learning. *The International Journal for Technology in Mathematics Education, 20*(1).

Larkin, K. (2016). Geometry and iPads in Primary Schools: Does their usefulness extend beyond tracing an oblong?. In: Moyer-Packenham, P. (Ed.), *International perspectives on teaching and learning mathematics with virtual manipulatives. Mathematics education in the digital era*, vol 7. Springer. https://doi.org/10.1007/978-3-319-32718-1_11

Moyer-Packenham, P. S., & Bolyard, J. J. (2016). Revisiting the definition of a virtual manipulative. In P. S. Moyer-Packenham (Ed.), *International perspectives on teaching and learning mathematics with virtual manipulatives. Mathematics education in the digital era*, vol 7. Springer. https://doi.org/10.1007/978-3-319-32718-1_11

Namukasa, I. K., Gadanidis, G., Sarina, V., Scucuglia, S., & Aryee, K. (2016). Selection of apps for teaching difficult mathematics topic: An instrument to evaluate touch-screen tablet and smartphone mathematics apps. In P. S. Moyer-Packenham (Ed.), *International perspectives on teaching and learning mathematics with virtual manipulatives. Mathematics education in the digital era*, vol 7. https://doi.org/10.1007/978-3-319-32718-1_12

Kortenkamp, U., & Dohrmann, C. (2023). Pre-service teacher training with AI: Using ChatGPT discussions to practice teacher-student discourse. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), *Proceedings of PME 46 (Vol. 3, pp. 187–194)*.

Kortenkamp, U., & Richter-Gebert, J. (2008). *Cinderella.2 – Geometrie und Physik im Dialog. Computeralgebra-Rundbrief: Sonderheft zum Jahr der Mathematik 2008*, 12–14. https://fachgruppe-computeralgebra.de/data/JdM-2008/Sonderheft.pdf

Richter-Gebert, J., & Kortenkamp, U. H. (2012). *The Cinderella.2 Manual – Working with the Interactive Geometry Software*. Springer. https://doi.org/10.1007/978-3-540-34926-6

Soury-Lavergne, S. (2021). Duos of digital and tangible artefacts in didactical situations. *Digital Experiences in Mathematics Education, 7* (p.1-21).

Viljanen, L. (2005). Towards an ontology of trust. In: Katsikas, S., López, J., Pernul, G. (Eds.), *Trust, Privacy, and Security in Digital Business. TrustBus 2005. Lecture Notes in Computer Science*, vol 3592. Springer. https://doi.org/10.1007/11537878_18
Bruner, J. (1966). *Toward a theory of instruction*. Oxford University Press.

Dick, T. (2008). Fidelity in Technological Tools for Mathematics Education. In G. Blume, & M. K. Reid (Eds.), *Research on Technology and the Teaching and Learning of Mathematics: Volume 2. Cases and Perspectives* (pp. 333-339). Information Age Publishing.

Etzold, H. (2013-2020). *Winkel*. https://apps.apple.com/ch/app/winkel/id715739128

Etzold, H. (2015-2022). *Cubeling*. https://apps.apple.com/bs/app/cubeling/id1027746349

Hirsh-Pasek, K., Zosh, J., Michnick Golinkoff, R., Gray, J., Robb, M., & Kaufman, J. (2015). Putting Education in “Educational” Apps: Lessons From the Science of Learning. *Psychological Science in the Public Interest, 16*(1), 3-34. https://doi.org/10.1177/1529100615569721

Ladel, S. (2009). Multiple externe Repräsentationen (MERs) und deren Verknüpfung durch Computereinsatz. *Zur Bedeutung für das Mathematiklernen im Anfangsunterricht*. Verlag Dr. Kovac.

Ladel, S., & Kortenkamp, U. (2013). An activity-theoretic approach to multi-touch tools in early maths learning. *The International Journal for Technology in Mathematics Education, 20*(1).

Larkin, K. (2016). Geometry and iPads in Primary Schools: Does their usefulness extend beyond tracing an oblong?. In: Moyer-Packenham, P. (Ed.), *International perspectives on teaching and learning mathematics with virtual manipulatives. Mathematics education in the digital era*, vol 7. Springer. https://doi.org/10.1007/978-3-319-32718-1_11

Moyer-Packenham, P. S., & Bolyard, J. J. (2016). Revisiting the definition of a virtual manipulative. In P. S. Moyer-Packenham (Ed.), *International perspectives on teaching and learning mathematics with virtual manipulatives. Mathematics education in the digital era*, vol 7. Springer. https://doi.org/10.1007/978-3-319-32718-1_11

Namukasa, I. K., Gadanidis, G., Sarina, V., Scucuglia, S., & Aryee, K. (2016). Selection of apps for teaching difficult mathematics topic: An instrument to evaluate touch-screen tablet and smartphone mathematics apps. In P. S. Moyer-Packenham (Ed.), *International perspectives on teaching and learning mathematics with virtual manipulatives. Mathematics education in the digital era*, vol 7. https://doi.org/10.1007/978-3-319-32718-1_12

Kortenkamp, U., & Dohrmann, C. (2023). Pre-service teacher training with AI: Using ChatGPT discussions to practice teacher-student discourse. In M. Ayalon, B. Koichu, R. Leikin, L. Rubel, & M. Tabach (Eds.), *Proceedings of PME 46 (Vol. 3, pp. 187–194)*.

Kortenkamp, U., & Richter-Gebert, J. (2008). *Cinderella.2 – Geometrie und Physik im Dialog. Computeralgebra-Rundbrief: Sonderheft zum Jahr der Mathematik 2008*, 12–14. https://fachgruppe-computeralgebra.de/data/JdM-2008/Sonderheft.pdf

Richter-Gebert, J., & Kortenkamp, U. H. (2012). *The Cinderella.2 Manual – Working with the Interactive Geometry Software*. Springer. https://doi.org/10.1007/978-3-540-34926-6

Soury-Lavergne, S. (2021). Duos of digital and tangible artefacts in didactical situations. *Digital Experiences in Mathematics Education, 7* (p.1-21).

Viljanen, L. (2005). Towards an ontology of trust. In: Katsikas, S., López, J., Pernul, G. (Eds.), *Trust, Privacy, and Security in Digital Business. TrustBus 2005. Lecture Notes in Computer Science*, vol 3592. Springer. https://doi.org/10.1007/11537878_18
# SUPPORTING GEOMETRIC REASONING FOR UNDERSERVED STUDENTS IN INDIA THROUGH CONNECTED LEARNING INITIATIVE

Ruchi S. Kumar¹ & Suchismita Srinivas²

¹ Tata Institute of Social Sciences (India), ² University of Cambridge (UK)

This study explores the impact of module-based interventions on geometric reasoning in underserved Indian students, as part of the large-scale action research project -Connected Learning Initiative (CLIx). The interventions aligned with active learning pedagogy, emphasizing exploration, discussion, and digital resource utilization. Teacher capacity was enhanced through workshops and online courses. The focus was on developing understanding of quadrilaterals. A quasi-experimental approach adopted with the sub-sample of the large-scale implementation revealed significant geometric reasoning improvements in intervention school students compared to non-intervention peers. Female students in the intervention group exhibited notable gains. Schools with higher module fidelity achieved greater success. Observations showed that intervention schools promoted discussions, while non-intervention schools leaned towards lecture-based approaches. Students in intervention schools engaged more actively. Teachers not only facilitated discussions but adapted to contingencies, fostering reflection. Besides cognitive gains, students displayed increased confidence, motivation, collaborative skills, and autonomy. The study underscores the significance of teacher capacity and resource use in under-resourced contexts. This research informs in-service teacher education and resource development for similar settings. The findings bear implications for equitable mathematics education, particularly in resource-constrained contexts.

## INTRODUCTION

Students learning, teachers' beliefs, knowledge, skills and practice and their use of resources to support students' learning are intricately connected. However, these connections are often studied in high-resource contexts and there is a dearth of studies that focus on under-resourced contexts. Adler (2000) posits that resources can be considered as an “extension of the teacher in school mathematics practice” (p. 205) which includes hybridized content and pedagogy and mediated use of the resources depending on their relative transparency to support meaning-making by the students. Brodie (2020) highlights the role of knowledge, material/logistical, affective and human resources that support teacher collaboration for professional learning in a community. The process of instrumental genesis (Béguin & Rabardel, 2000) involving instrumentalization and instrumentation is thus facilitated by both how a teacher visualizes and adapts a resource for the classroom and by the function of participation of the teacher in a community that supports this visualization. However, there is a challenge of studying the use of resources at a large scale and characterizing how the teachers' use and adaptation of the resource and school mathematics practice constitute each other.

Resources used for teaching geometry and for geometry teacher education, seem to have a disproportionate focus on the use of software, of which many are proprietary and thus inaccessible in under-resourced contexts (except GeoGebra which is an Open Educational Resource (OER)). Though there are several quality hands-on resources available for teaching geometry, there is scant literature
# SUPPORTING GEOMETRIC REASONING FOR UNDERSERVED STUDENTS IN INDIA THROUGH CONNECTED LEARNING INITIATIVE

Ruchi S. Kumar¹ & Suchismita Srinivas²

¹ Tata Institute of Social Sciences (India), ² University of Cambridge (UK)

This study explores the impact of module-based interventions on geometric reasoning in underserved Indian students, as part of the large-scale action research project -Connected Learning Initiative (CLIx). The interventions aligned with active learning pedagogy, emphasizing exploration, discussion, and digital resource utilization. Teacher capacity was enhanced through workshops and online courses. The focus was on developing understanding of quadrilaterals. A quasi-experimental approach adopted with the sub-sample of the large-scale implementation revealed significant geometric reasoning improvements in intervention school students compared to non-intervention peers. Female students in the intervention group exhibited notable gains. Schools with higher module fidelity achieved greater success. Observations showed that intervention schools promoted discussions, while non-intervention schools leaned towards lecture-based approaches. Students in intervention schools engaged more actively. Teachers not only facilitated discussions but adapted to contingencies, fostering reflection. Besides cognitive gains, students displayed increased confidence, motivation, collaborative skills, and autonomy. The study underscores the significance of teacher capacity and resource use in under-resourced contexts. This research informs in-service teacher education and resource development for similar settings. The findings bear implications for equitable mathematics education, particularly in resource-constrained contexts.

## INTRODUCTION

Students learning, teachers' beliefs, knowledge, skills and practice and their use of resources to support students' learning are intricately connected. However, these connections are often studied in high-resource contexts and there is a dearth of studies that focus on under-resourced contexts. Adler (2000) posits that resources can be considered as an “extension of the teacher in school mathematics practice” (p. 205) which includes hybridized content and pedagogy and mediated use of the resources depending on their relative transparency to support meaning-making by the students. Brodie (2020) highlights the role of knowledge, material/logistical, affective and human resources that support teacher collaboration for professional learning in a community. The process of instrumental genesis (Béguin & Rabardel, 2000) involving instrumentalization and instrumentation is thus facilitated by both how a teacher visualizes and adapts a resource for the classroom and by the function of participation of the teacher in a community that supports this visualization. However, there is a challenge of studying the use of resources at a large scale and characterizing how the teachers' use and adaptation of the resource and school mathematics practice constitute each other.

Resources used for teaching geometry and for geometry teacher education, seem to have a disproportionate focus on the use of software, of which many are proprietary and thus inaccessible in under-resourced contexts (except GeoGebra which is an Open Educational Resource (OER)). Though there are several quality hands-on resources available for teaching geometry, there is scant literature
in mathematics education research about the use of such resources for underserved populations or under-resourced contexts. Classroom discourse though focused in research, needs to be considered as a resource in supporting students’ learning and resource mediation. However, instead of focusing on only hands-on and classroom discourse for geometry learning in under-resourced classroom, it is imperative to equip the underserved population with the digital skills and the use of digital resources to address equity in mathematics education. Therefore, under-resourced classroom teachers need to learn the skill of managing the use of hands-on, digital and facilitation of classroom discourse in a coordinated manner. It highlights the significance of the selection of tools that meet the learners where they are while providing opportunities to build their skills. Additionally, the tools should support the active exploration and meaning-making of the mathematical content rather than using the digital medium to ‘sugarcoat’ the mathematics drill and practice or worse use the medium to ‘transfer’ the content to students replacing the teacher.

In this paper, we discuss some of the findings from a large-scale action research project implemented in India called as Connected Learning Initiative (CLIx). This project involved the design and implementation of blended modules (having both hands-on and digital content) to empower them to develop 21st century skills in Mathematics, Science and English and have been released as Open Educational Resources (https://demo-clix.tiss.edu/softwares/). The intervention involved developing an ecosystem to support the implementation of the modules through engaging teachers through in-person workshops and online courses. Between the period of 2016 and 2023, CLIx has been implemented in 1418 schools, 105000 students and 3200 math teachers across India approximately.

In this paper, we share findings using a subsample from the large-scale implementation from a rural region in India using a quasi-experimental approach for 10 intervention and 10 control (no CLIx intervention, teaching as usual) schools. Implications for in-service teacher education about how resources can be used for addressing and developing teachers’ knowledge and skills through the development of a professional learning community are discussed.

## THEORY OF CHANGE

The theoretical assumptions behind CLIx math interventions were to provide resources (Digital and hands on) that embed and illustrate active learning pedagogy by encouraging students to explore and develop meaning of mathematics. The resources for students were compiled in the form of an open educational resource – A geometric reasoning module for ninth-grade students and an online course for teachers. Through this course, teachers were involved in providing opportunities to engage in hands-on activities, collaborative discussion with peers, opportunities to learn through mistakes and exploring digital resources. The capacity of teachers to use such resources in the classroom and to use active learning pedagogy was enhanced through topic-specific workshops, engaging in online courses and interaction with peers and experts through mobile-based chat groups supporting the development of the community. The teachers were supported in the school through follow-ups by the team to provide technical assistance for the use of hardware and software and allocating time approved by the administration for exploring these resources. The workshops, online courses and interactions on chat groups focused on developing understanding of theoretical ideas for geometry learning like Van Hiele theory, discussion on types of mistakes and misconceptions of the students, activities that can aid in addressing misconceptions, sharing artefacts from the classrooms in form of students’ responses and constructions to delve deeper into students’ understanding.
in mathematics education research about the use of such resources for underserved populations or under-resourced contexts. Classroom discourse though focused in research, needs to be considered as a resource in supporting students’ learning and resource mediation. However, instead of focusing on only hands-on and classroom discourse for geometry learning in under-resourced classroom, it is imperative to equip the underserved population with the digital skills and the use of digital resources to address equity in mathematics education. Therefore, under-resourced classroom teachers need to learn the skill of managing the use of hands-on, digital and facilitation of classroom discourse in a coordinated manner. It highlights the significance of the selection of tools that meet the learners where they are while providing opportunities to build their skills. Additionally, the tools should support the active exploration and meaning-making of the mathematical content rather than using the digital medium to ‘sugarcoat’ the mathematics drill and practice or worse use the medium to ‘transfer’ the content to students replacing the teacher.

In this paper, we discuss some of the findings from a large-scale action research project implemented in India called as Connected Learning Initiative (CLIx). This project involved the design and implementation of blended modules (having both hands-on and digital content) to empower them to develop 21st century skills in Mathematics, Science and English and have been released as Open Educational Resources (https://demo-cli.x.tiss.edu/softwares/). The intervention involved developing an ecosystem to support the implementation of the modules through engaging teachers through in-person workshops and online courses. Between the period of 2016 and 2023, CLIx has been implemented in 1418 schools, 105000 students and 3200 math teachers across India approximately.

In this paper, we share findings using a subsample from the large-scale implementation from a rural region in India using a quasi-experimental approach for 10 intervention and 10 control (no CLIx intervention, teaching as usual) schools. Implications for in-service teacher education about how resources can be used for addressing and developing teachers’ knowledge and skills through the development of a professional learning community are discussed.

## THEORY OF CHANGE

The theoretical assumptions behind CLIx math interventions were to provide resources (Digital and hands on) that embed and illustrate active learning pedagogy by encouraging students to explore and develop meaning of mathematics. The resources for students were compiled in the form of an open educational resource – A geometric reasoning module for ninth-grade students and an online course for teachers. Through this course, teachers were involved in providing opportunities to engage in hands-on activities, collaborative discussion with peers, opportunities to learn through mistakes and exploring digital resources. The capacity of teachers to use such resources in the classroom and to use active learning pedagogy was enhanced through topic-specific workshops, engaging in online courses and interaction with peers and experts through mobile-based chat groups supporting the development of the community. The teachers were supported in the school through follow-ups by the team to provide technical assistance for the use of hardware and software and allocating time approved by the administration for exploring these resources. The workshops, online courses and interactions on chat groups focused on developing understanding of theoretical ideas for geometry learning like Van Hiele theory, discussion on types of mistakes and misconceptions of the students, activities that can aid in addressing misconceptions, sharing artefacts from the classrooms in form of students’ responses and constructions to delve deeper into students’ understanding.
GEOMETRIC REASONING MODULE

The geometric reasoning module was designed to focus on developing a foundational understanding of Quadrilaterals and their properties while the textbook chapter on Quadrilaterals focused on proofs of theorems and application of the theorems. The objective of the module was to focus on foundational concepts of geometry, providing opportunities for students to engage in problem-solving and reasoning, peer discussion and creating a learning environment that is a safe space to make mistakes. The geometric module unit is composed of 5 units designed based on van Hiele theory (Fuys, Geddes & Tischler, 1988). The focus of the unit and the types of resources created in each unit are given in Table 1 below.

<table><caption>Table 1: Units, activities and resources in the CLIx Geometric Reasoning Module</caption><thead><tr><td>Unit</td><td>Type of activities</td><td>Resources</td></tr></thead><tbody><tr><td>Concept of Shape</td><td>Creating shapes, what changes when rotation, translation, deformation</td><td>Origami, Turtle logo, Matchstick Mecano, student handbook activities</td></tr><tr><td>Analysing and Describing shapes</td><td>Identifying and Describing shapes using properties and qualifiers</td><td>PoliceQuad Game Mission 1 and 2, student handbook activities</td></tr><tr><td>Classifying and defining shapes</td><td>Classification of shapes based on similarities and differences</td><td>PoliceQuad Game Mission 3, student handbook activities</td></tr><tr><td>Property-based reasoning</td><td>Hierarchical relation between shapes, if-then reasoning and need for definitions and alternative definitions</td><td>PoliceQuad Game Mission 4, Case of multiple definitions, student handbook activities</td></tr><tr><td>Need for proof</td><td>Making conjectures, Are verifications proofs? Inductive and deductive reasoning</td><td>Geogebra activity, student handbook activities</td></tr></tbody></table>

The hands-on activities were implemented in the classroom while the digital activities were implemented in the school computer lab. The field staff helped in getting the school computer lab ready and going. Due to the large class size (60-80), the students were divided into batches for accessing the digital activities collaboratively in the computer lab with ten computers. The hands-on activities included paper folding, matchsticks Mecano and a student workbook including several tasks for reflection and discussion. The digital activities involved the use of open software like Turtle logo (Sugarlabs), GeoGebra and a digital game 'PoliceQuad' with 4 missions and 12 levels. The game PoliceQuad was designed as an OER taking the context of playing the role of police and releasing the innocent based on clues provided. The successive missions required the students to identify the 'suspect' shape based on clues describing the properties of shapes (Mission1). Mission 2 required creating the clues as questions using the properties of shapes at display and releasing the innocent
GEOMETRIC REASONING MODULE

The geometric reasoning module was designed to focus on developing a foundational understanding of Quadrilaterals and their properties while the textbook chapter on Quadrilaterals focused on proofs of theorems and application of the theorems. The objective of the module was to focus on foundational concepts of geometry, providing opportunities for students to engage in problem-solving and reasoning, peer discussion and creating a learning environment that is a safe space to make mistakes. The geometric module unit is composed of 5 units designed based on van Hiele theory (Fuys, Geddes & Tischler, 1988). The focus of the unit and the types of resources created in each unit are given in Table 1 below.

<table><caption>Table 1: Units, activities and resources in the CLIx Geometric Reasoning Module</caption><thead><tr><td>Unit</td><td>Type of activities</td><td>Resources</td></tr></thead><tbody><tr><td>Concept of Shape</td><td>Creating shapes, what changes when rotation, translation, deformation</td><td>Origami, Turtle logo, Matchstick Mecano, student handbook activities</td></tr><tr><td>Analysing and Describing shapes</td><td>Identifying and Describing shapes using properties and qualifiers</td><td>PoliceQuad Game Mission 1 and 2, student handbook activities</td></tr><tr><td>Classifying and defining shapes</td><td>Classification of shapes based on similarities and differences</td><td>PoliceQuad Game Mission 3, student handbook activities</td></tr><tr><td>Property-based reasoning</td><td>Hierarchical relation between shapes, if-then reasoning and need for definitions and alternative definitions</td><td>PoliceQuad Game Mission 4, Case of multiple definitions, student handbook activities</td></tr><tr><td>Need for proof</td><td>Making conjectures, Are verifications proofs? Inductive and deductive reasoning</td><td>Geogebra activity, student handbook activities</td></tr></tbody></table>

The hands-on activities were implemented in the classroom while the digital activities were implemented in the school computer lab. The field staff helped in getting the school computer lab ready and going. Due to the large class size (60-80), the students were divided into batches for accessing the digital activities collaboratively in the computer lab with ten computers. The hands-on activities included paper folding, matchsticks Mecano and a student workbook including several tasks for reflection and discussion. The digital activities involved the use of open software like Turtle logo (Sugarlabs), GeoGebra and a digital game 'PoliceQuad' with 4 missions and 12 levels. The game PoliceQuad was designed as an OER taking the context of playing the role of police and releasing the innocent based on clues provided. The successive missions required the students to identify the 'suspect' shape based on clues describing the properties of shapes (Mission1). Mission 2 required creating the clues as questions using the properties of shapes at display and releasing the innocent
based on a 'yes/no' response from the computer. The third mission focused on identifying similarities and differences across two groups of shapes to create a shape belonging to one of the groups approved by the computer. The fourth mission involved guessing the shape on the grid based on the special quadrilateral selected and discovering edges and vertices by clicking on various points of the enemy grid (similar to the Battleships game). The students were expected to engage in the activities in small groups and pairs and give responses after the discussion. The module is accessible in three languages (English, Hindi and Telugu) by toggling on the student platform. The quasi-experimental study reported in this paper involved the implementation of some activities from the first 3 units by intervention school teachers due to constraints of time.

## REFLECTIVE MATHEMATICS TEACHING COURSE FOR TEACHERS

The three-day in-person workshop and the 'Reflective Mathematics Teaching- Geometric Reasoning' online course (run for 3 months) for the teachers focused on the development of the teachers' beliefs, knowledge and practice for focusing on developing an understanding of geometric concepts through active exploration, collaborative math talk, learning through mistakes and classroom discussion facilitated by the teacher. The teachers engaged with the module first as learner and then thought through the perspective of the teacher for planning to teach the module and reflecting on the classroom implementation. The course provided opportunities for understanding theoretical ideas like van Hiele's Theory, types of misconceptions that students have, activities that can help students engage with geometric reasoning, pedagogy embedded in activities and resources in the module, reflection on school mathematics practice and reflection on implementation of the activities of the module. In addition to this, the teachers participating in the quasi-experimental study were given two-day refresher workshop on the module and planning of implementation in the classrooms.

## METHODOLOGY

A mixed method approach involving a quasi-experimental design was adopted for the study in the year 2018 for three months. The study was conducted in government high schools in semi-urban and rural areas of the Dhamtari district in Chhattisgarh. Ten intervention and ten control schools were selected from the larger pool of schools. Intervention schools were selected based on the condition of the computer lab. Comparability of the intervention and control schools was established using the pass percentage of the tenth-grade official certificate exam using independent samples t-test and no significant difference was found among the two groups. A pre and post-test was constructed for the students adapted from Shaughnessy and Burger (1985) with 8 multiple choice questions (MCQ) and 6 constructed response items. In this paper, we present findings from the MCQs only. Classroom and lab observations were documented using tallies for teachers and student actions and free write documenting the interactions along with the time devoted to lecture mode, individual work, group work and others. For analysis, two observations each from schools that performed best on MCQs for both intervention and control schools have been summarized. Descriptive and inferential analysis was done using the 7 MCQs data from pre and post-test. The frequency of the behaviors and actions from the observation tools was collated. 507 students from 10 intervention schools and 565 students from 9 non-intervention schools participated in the study.

In the control schools, teachers taught the chapter on quadrilaterals from the textbook which focused on the properties of the quadrilaterals, special quadrilaterals, theorems based on parallelogram and
based on a 'yes/no' response from the computer. The third mission focused on identifying similarities and differences across two groups of shapes to create a shape belonging to one of the groups approved by the computer. The fourth mission involved guessing the shape on the grid based on the special quadrilateral selected and discovering edges and vertices by clicking on various points of the enemy grid (similar to the Battleships game). The students were expected to engage in the activities in small groups and pairs and give responses after the discussion. The module is accessible in three languages (English, Hindi and Telugu) by toggling on the student platform. The quasi-experimental study reported in this paper involved the implementation of some activities from the first 3 units by intervention school teachers due to constraints of time.

## REFLECTIVE MATHEMATICS TEACHING COURSE FOR TEACHERS

The three-day in-person workshop and the 'Reflective Mathematics Teaching- Geometric Reasoning' online course (run for 3 months) for the teachers focused on the development of the teachers' beliefs, knowledge and practice for focusing on developing an understanding of geometric concepts through active exploration, collaborative math talk, learning through mistakes and classroom discussion facilitated by the teacher. The teachers engaged with the module first as learner and then thought through the perspective of the teacher for planning to teach the module and reflecting on the classroom implementation. The course provided opportunities for understanding theoretical ideas like van Hiele's Theory, types of misconceptions that students have, activities that can help students engage with geometric reasoning, pedagogy embedded in activities and resources in the module, reflection on school mathematics practice and reflection on implementation of the activities of the module. In addition to this, the teachers participating in the quasi-experimental study were given two-day refresher workshop on the module and planning of implementation in the classrooms.

## METHODOLOGY

A mixed method approach involving a quasi-experimental design was adopted for the study in the year 2018 for three months. The study was conducted in government high schools in semi-urban and rural areas of the Dhamtari district in Chhattisgarh. Ten intervention and ten control schools were selected from the larger pool of schools. Intervention schools were selected based on the condition of the computer lab. Comparability of the intervention and control schools was established using the pass percentage of the tenth-grade official certificate exam using independent samples t-test and no significant difference was found among the two groups. A pre and post-test was constructed for the students adapted from Shaughnessy and Burger (1985) with 8 multiple choice questions (MCQ) and 6 constructed response items. In this paper, we present findings from the MCQs only. Classroom and lab observations were documented using tallies for teachers and student actions and free write documenting the interactions along with the time devoted to lecture mode, individual work, group work and others. For analysis, two observations each from schools that performed best on MCQs for both intervention and control schools have been summarized. Descriptive and inferential analysis was done using the 7 MCQs data from pre and post-test. The frequency of the behaviors and actions from the observation tools was collated. 507 students from 10 intervention schools and 565 students from 9 non-intervention schools participated in the study.

In the control schools, teachers taught the chapter on quadrilaterals from the textbook which focused on the properties of the quadrilaterals, special quadrilaterals, theorems based on parallelogram and
mid-point theorem. The teachers mostly explained the content of the textbook and solved the questions given in the textbook on the board. Students were then asked to solve rest of the questions from the textbook exercise for practice. In contrast the teachers in the intervention schools focused on more foundational concepts of understanding angles, parallel lines, analysis of shapes of special quadrilaterals using activities from the modules.

## FINDINGS

The analysis of students' responses to seven geometry-related multiple-choice questions (Q1-7) revealed notable differences between the intervention and non-intervention groups. Initially, the intervention group had a lower average pre-test score (29.77%, SD = 20.03) compared to the non-intervention group (33.92%, SD = 20.31). However, after the intervention, the intervention group's post-test scores significantly improved to 43.19% (SD = 22.69), while the non-intervention group showed a more modest gain, reaching 36.73% (SD = 20.49). The pre- to post-test gain for the intervention group was substantial, reflecting a significant increase (p < .0001) in comparison to the non-intervention group. Further analysis demonstrated a significant discrepancy in the gains between the intervention and non-intervention groups (p < .001), underscoring the efficacy of the module's implementation in enhancing geometric reasoning skills among students. It is also pertinent to mention in this report that a gender-wise analysis of student performance revealed a significantly higher gain in scores for female students in the intervention group than for those in the non-intervention group (p < 0.01). The schools showing a higher extent of implementation of the geometric reasoning module and with higher fidelity to pedagogical principles underlined in the module, showed higher gains in students' learning as compared to schools with low levels of implementation or low fidelity. This outcome suggests that the module had a positive impact on students' geometric reasoning abilities, as evidenced by the substantial improvement in test scores among the intervention group compared to their non-intervention counterparts.

While 60% of the students in the intervention schools were not able to recognize the shape as square when orientation was rotated, 88% of the students were able to recognize it in the post-test. There was a 17% increase in students being able to identify a common property given a group of shapes and a 7% increase in the number of students being able to identify the class relationship that a square can also be considered as a rectangle.

Observation data indicated that Intervention schools allocated approximately 60% of class time to teacher-facilitated discussions, promoting meaningful 'math talk' between students or between the teacher and students. In contrast, non-intervention schools primarily adhered to the 'Teacher Lecture (TL)' mode, with nearly 70% of class time dedicated to passive student listening and occasional collective responses. Additionally, students in intervention schools engaged in more individual work, predominantly focused on workbook tasks designed to enhance their ability to articulate ideas and foster reasoning. This marked a substantial contrast from the practices observed in non-intervention schools.

Student behavior data from 6 best-performing intervention and control schools (on post-test) depicted in Table 2 shows a higher level of engagement and interaction of students in the classroom discussion in the intervention schools as compared to control. This is corroborated by the data about teachers' behavior in intervention schools depicted in Table 3 as the teachers in intervention schools asked for
mid-point theorem. The teachers mostly explained the content of the textbook and solved the questions given in the textbook on the board. Students were then asked to solve rest of the questions from the textbook exercise for practice. In contrast the teachers in the intervention schools focused on more foundational concepts of understanding angles, parallel lines, analysis of shapes of special quadrilaterals using activities from the modules.

## FINDINGS

The analysis of students' responses to seven geometry-related multiple-choice questions (Q1-7) revealed notable differences between the intervention and non-intervention groups. Initially, the intervention group had a lower average pre-test score (29.77%, SD = 20.03) compared to the non-intervention group (33.92%, SD = 20.31). However, after the intervention, the intervention group's post-test scores significantly improved to 43.19% (SD = 22.69), while the non-intervention group showed a more modest gain, reaching 36.73% (SD = 20.49). The pre- to post-test gain for the intervention group was substantial, reflecting a significant increase (p < .0001) in comparison to the non-intervention group. Further analysis demonstrated a significant discrepancy in the gains between the intervention and non-intervention groups (p < .001), underscoring the efficacy of the module's implementation in enhancing geometric reasoning skills among students. It is also pertinent to mention in this report that a gender-wise analysis of student performance revealed a significantly higher gain in scores for female students in the intervention group than for those in the non-intervention group (p < 0.01). The schools showing a higher extent of implementation of the geometric reasoning module and with higher fidelity to pedagogical principles underlined in the module, showed higher gains in students' learning as compared to schools with low levels of implementation or low fidelity. This outcome suggests that the module had a positive impact on students' geometric reasoning abilities, as evidenced by the substantial improvement in test scores among the intervention group compared to their non-intervention counterparts.

While 60% of the students in the intervention schools were not able to recognize the shape as square when orientation was rotated, 88% of the students were able to recognize it in the post-test. There was a 17% increase in students being able to identify a common property given a group of shapes and a 7% increase in the number of students being able to identify the class relationship that a square can also be considered as a rectangle.

Observation data indicated that Intervention schools allocated approximately 60% of class time to teacher-facilitated discussions, promoting meaningful 'math talk' between students or between the teacher and students. In contrast, non-intervention schools primarily adhered to the 'Teacher Lecture (TL)' mode, with nearly 70% of class time dedicated to passive student listening and occasional collective responses. Additionally, students in intervention schools engaged in more individual work, predominantly focused on workbook tasks designed to enhance their ability to articulate ideas and foster reasoning. This marked a substantial contrast from the practices observed in non-intervention schools.

Student behavior data from 6 best-performing intervention and control schools (on post-test) depicted in Table 2 shows a higher level of engagement and interaction of students in the classroom discussion in the intervention schools as compared to control. This is corroborated by the data about teachers' behavior in intervention schools depicted in Table 3 as the teachers in intervention schools asked for
reasons and explanations and built on students' incorrect/partially correct responses. However, they also tend to give the answers or explanations themselves rather than waiting for students to respond to the question.

<table><caption>Table 2: Comparison of student behavior in 6 of the best-performing intervention and control schools</caption><thead><tr><th rowspan="2">Student behavior</th><th rowspan="2"></th><th rowspan="2">Boys/Girls</th><th colspan="2">Average number of instances observed</th></tr><tr><th>Intervention</th><th>Control school</th></tr></thead><tbody><tr><td rowspan="2">Students gave reasons in support for his/her response</td><td>Boys</td><td>38</td><td>5</td></tr><tr><td>Girls</td><td>39</td><td>3</td></tr><tr><td rowspan="2">Students who gave short answers/ response</td><td>Boys</td><td>57</td><td>38</td></tr><tr><td>Girls</td><td>54</td><td>41</td></tr><tr><td rowspan="2">Students who gave extended responses/ answers (with Reasoning)</td><td>Boys</td><td>22</td><td>3</td></tr><tr><td>Girls</td><td>25</td><td>2</td></tr></tbody></table>

<table><caption>Table 3: Comparison of Teacher behavior in 6 of the top performing intervention and control schools</caption><thead><tr><th rowspan="2">Teacher behavior</th><th rowspan="2"></th><th rowspan="2"></th><th colspan="2">Average number of instances observed</th></tr><tr><th>Intervention</th><th>Control school</th></tr></thead><tbody><tr><td rowspan="2">Teacher asking for reasons/explanations for students' response</td><td>For correct response</td><td>60</td><td>1</td></tr><tr><td>For incorrect response</td><td>16</td><td>0</td></tr><tr><td colspan="2">Teacher him/herself provided the answer or explanation of the question</td><td>33</td><td>28</td></tr><tr><td colspan="2">Teacher builds on incorrect/ partially correct response</td><td>17</td><td>2</td></tr></tbody></table>

Tables 2 and 3 depict that there is a marked difference in the classroom environment and the discourse as the intervention schools seem to focus more on giving opportunities for students to articulate and reason. These opportunities though embedded in the tasks and activities of the geometric reasoning module, were taken as school mathematics practices by intervention school teachers.

Besides cognitive gains, there were certain non-cognitive and affective changes were also visible amongst students like increased confidence and motivation to learn especially among students who were repeating ninth grades and some girls who were earlier not motivated. Opportunities to discuss with peers and articulation in the classroom also helped in building collaborative skills and autonomy amongst students. The exposure to digital activities helped in developing skills and confidence to engage with computers and some of the students were seen to take charge of managing the maintenance of the computer lab and supporting other students to engage with digital activities by
reasons and explanations and built on students' incorrect/partially correct responses. However, they also tend to give the answers or explanations themselves rather than waiting for students to respond to the question.

<table><caption>Table 2: Comparison of student behavior in 6 of the best-performing intervention and control schools</caption><thead><tr><th rowspan="2">Student behavior</th><th rowspan="2"></th><th rowspan="2">Boys/Girls</th><th colspan="2">Average number of instances observed</th></tr><tr><th>Intervention</th><th>Control school</th></tr></thead><tbody><tr><td rowspan="2">Students gave reasons in support for his/her response</td><td>Boys</td><td>38</td><td>5</td></tr><tr><td>Girls</td><td>39</td><td>3</td></tr><tr><td rowspan="2">Students who gave short answers/ response</td><td>Boys</td><td>57</td><td>38</td></tr><tr><td>Girls</td><td>54</td><td>41</td></tr><tr><td rowspan="2">Students who gave extended responses/ answers (with Reasoning)</td><td>Boys</td><td>22</td><td>3</td></tr><tr><td>Girls</td><td>25</td><td>2</td></tr></tbody></table>

<table><caption>Table 3: Comparison of Teacher behavior in 6 of the top performing intervention and control schools</caption><thead><tr><th rowspan="2">Teacher behavior</th><th rowspan="2"></th><th rowspan="2"></th><th colspan="2">Average number of instances observed</th></tr><tr><th>Intervention</th><th>Control school</th></tr></thead><tbody><tr><td rowspan="2">Teacher asking for reasons/explanations for students' response</td><td>For correct response</td><td>60</td><td>1</td></tr><tr><td>For incorrect response</td><td>16</td><td>0</td></tr><tr><td colspan="2">Teacher him/herself provided the answer or explanation of the question</td><td>33</td><td>28</td></tr><tr><td colspan="2">Teacher builds on incorrect/ partially correct response</td><td>17</td><td>2</td></tr></tbody></table>

Tables 2 and 3 depict that there is a marked difference in the classroom environment and the discourse as the intervention schools seem to focus more on giving opportunities for students to articulate and reason. These opportunities though embedded in the tasks and activities of the geometric reasoning module, were taken as school mathematics practices by intervention school teachers.

Besides cognitive gains, there were certain non-cognitive and affective changes were also visible amongst students like increased confidence and motivation to learn especially among students who were repeating ninth grades and some girls who were earlier not motivated. Opportunities to discuss with peers and articulation in the classroom also helped in building collaborative skills and autonomy amongst students. The exposure to digital activities helped in developing skills and confidence to engage with computers and some of the students were seen to take charge of managing the maintenance of the computer lab and supporting other students to engage with digital activities by
helping them to start the computer, log in details and showing the features which can be used in different software.

## DISCUSSION

The study demonstrated a significant improvement in students' geometric reasoning abilities in intervention schools compared to non-intervention schools. This improvement was evident in various cognitive gains, including the development of shape concepts, properties of plane shapes, and hierarchical class relationships, as reflected in higher scores on post-test multiple-choice questions in schools where the intervention was reasonably implemented and maintained. Beyond conceptual understanding, the study highlighted that both the module activities and teachers' actions supported the behaviors necessary for engaging in mathematical reasoning processes. The practices adapted by the teachers in the intervention school were the result of the instrumentation process through which the teachers developed the utilization schemes for activities and tasks in geometric reasoning module. The geometric reasoning module for students and the reflective mathematics teaching online course and workshops for the teachers provided impetus for the instrumentalization that influenced the adaptation of tasks taken up in the classroom using hands-on and digital activities. Both digital and hands-on activities enabled students to articulate ideas that might remain concealed in conventional classrooms focused on repetitive procedures and definitions.

The members of the community belonging to the chat groups as well as in-person interactions with teacher educators and peer teachers provided opportunity for instrumentation by bringing together ideas from module and the teacher course and supporting teachers in identifying misconceptions and difficulties in understanding key geometric concepts. Teachers not only facilitated discussions for tasks within the module but also created tasks, examples, arguments and counterexamples depending on the contingent situations that arose as a result of students' responses. Some of the responses were unexpected for students and teachers struggled to address that appropriately. For example, one of the teachers found it difficult to address the students' question of why there are no right angles in Kite (See Thakur et al., 2020). The increase in opportunities for students to articulate and give reasons also increased the contingent situations in the classrooms, pushing the teachers to make in-the-moment decisions using subject matter knowledge and activating the pedagogical content knowledge. Some of these instances provided opportunities for teachers to re-learn and reflect more deeply on their knowledge. These included teachers coming up with counterexamples to challenge students' naïve or identifications of properties based on the prototypical figure or themselves becoming aware of their own predisposition to use prototypical figures for identifying the necessary property for defining a figure (See Kumar et al, 2019). The contingent situations acted as the catalyst for in-the-moment instrumentation. Thus, the use of artefacts from the modules and teacher course and classroom teaching practice were supported by the dialectic between instrumentalization and instrumentation process.

The instrumental genesis of the teachers in the control school is relatively rigid in comparison to the intervention schools as textbook serve as only artefact for instrumental genesis. The instrumentalization of the questions in the textbook is followed rigidly along with the instrumentation of the practices that expect student to listen and follow the procedure. The impoverished environment with lack of access to other resources and artefacts for supporting teaching, lack of interactions with peers and experts to support the instrumental genesis process has an impact on student learning
helping them to start the computer, log in details and showing the features which can be used in different software.

## DISCUSSION

The study demonstrated a significant improvement in students' geometric reasoning abilities in intervention schools compared to non-intervention schools. This improvement was evident in various cognitive gains, including the development of shape concepts, properties of plane shapes, and hierarchical class relationships, as reflected in higher scores on post-test multiple-choice questions in schools where the intervention was reasonably implemented and maintained. Beyond conceptual understanding, the study highlighted that both the module activities and teachers' actions supported the behaviors necessary for engaging in mathematical reasoning processes. The practices adapted by the teachers in the intervention school were the result of the instrumentation process through which the teachers developed the utilization schemes for activities and tasks in geometric reasoning module. The geometric reasoning module for students and the reflective mathematics teaching online course and workshops for the teachers provided impetus for the instrumentalization that influenced the adaptation of tasks taken up in the classroom using hands-on and digital activities. Both digital and hands-on activities enabled students to articulate ideas that might remain concealed in conventional classrooms focused on repetitive procedures and definitions.

The members of the community belonging to the chat groups as well as in-person interactions with teacher educators and peer teachers provided opportunity for instrumentation by bringing together ideas from module and the teacher course and supporting teachers in identifying misconceptions and difficulties in understanding key geometric concepts. Teachers not only facilitated discussions for tasks within the module but also created tasks, examples, arguments and counterexamples depending on the contingent situations that arose as a result of students' responses. Some of the responses were unexpected for students and teachers struggled to address that appropriately. For example, one of the teachers found it difficult to address the students' question of why there are no right angles in Kite (See Thakur et al., 2020). The increase in opportunities for students to articulate and give reasons also increased the contingent situations in the classrooms, pushing the teachers to make in-the-moment decisions using subject matter knowledge and activating the pedagogical content knowledge. Some of these instances provided opportunities for teachers to re-learn and reflect more deeply on their knowledge. These included teachers coming up with counterexamples to challenge students' naïve or identifications of properties based on the prototypical figure or themselves becoming aware of their own predisposition to use prototypical figures for identifying the necessary property for defining a figure (See Kumar et al, 2019). The contingent situations acted as the catalyst for in-the-moment instrumentation. Thus, the use of artefacts from the modules and teacher course and classroom teaching practice were supported by the dialectic between instrumentalization and instrumentation process.

The instrumental genesis of the teachers in the control school is relatively rigid in comparison to the intervention schools as textbook serve as only artefact for instrumental genesis. The instrumentalization of the questions in the textbook is followed rigidly along with the instrumentation of the practices that expect student to listen and follow the procedure. The impoverished environment with lack of access to other resources and artefacts for supporting teaching, lack of interactions with peers and experts to support the instrumental genesis process has an impact on student learning
outcomes as the practices do not factor in contingencies arising from students’ difficulties. On the
other hand, the presence of alternative tasks and mediums in intervention classroom supported the
instrumentalization of adaption of tasks for the classroom to elicit and support student reasoning and
responding to contingencies by the dynamic instrumentation process.

The findings have implications for the design of in-service teacher education initiatives and the design of resources for under-resourced contexts which can be implemented at scale. It is essential to note that rather than resources, the development of knowledge and skills of the teachers through adaptation of tasks and resources (instrumentalization) and using discursive practices that supported reasoning (instrumentation) contributed to significant difference in students’ understanding in intervention schools. The number of factors that support the implementation of the resources in the under-resourced classroom needs to be studied further to understand how they support or constrain students’ and teachers’ learning.

Acknowledgements

Connected Learning Initiative has been supported by Tata Trusts. The design and development of the
modules have been done in collaboration with Massachusetts Institute of Technology (USA) and
Homi Bhabha Institute of Science Education among others. We are grateful to the study participants,
especially students and teachers, and other CLIx team members for their role in the implementation
of the study.

References

Adler, J. (2000). Conceptualising resources as a theme for teacher education. *Journal of Mathematics Teacher Education, 3*(3), 205-224.

Béguin, P., & Rabardel, P. (2000). Designing for instrument-mediated activity. *Scandinavian Journal of Information Systems, 12*(1), 1.

Brodie, K. (2020). Resources for and from collaboration: A conceptual framework. In *ICMI Study 25 conference proceedings: Teachers of Mathematics Working and Learning in Collaborative Groups* (pp. 37-48). Lisbon, Portugal.

Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. *Journal for Research in Mathematics Education*. Monograph, 3, i-196.

Kumar R.S., Srinivas, S., Bose, A., Rahman, J., Thakur, S., & Bapat, A. (2019) Examining Knowledge demands for teaching class inclusion of quadrilaterals. In M. Graven, H. Venkat, A. Essien, & P. Vale (Eds.). *Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 504-510). PME.

Shaughnessy, J. M., & Burger, W. F. (1985). Spadework prior to deduction in geometry. *The Mathematics Teacher, 78*(6), 419-428.

Thakur.S, Bose. A, & Kumar, R. S. (2020). Missed opportunities: Instances from geometry lessons. In Mashood K.K., & Tathagata Sengupta (Eds.). *Proceedings of Eighth Episteme Conference 2020* (pp. 308-318). HBCSE-Mumbai.
outcomes as the practices do not factor in contingencies arising from students’ difficulties. On the
other hand, the presence of alternative tasks and mediums in intervention classroom supported the
instrumentalization of adaption of tasks for the classroom to elicit and support student reasoning and
responding to contingencies by the dynamic instrumentation process.

The findings have implications for the design of in-service teacher education initiatives and the design of resources for under-resourced contexts which can be implemented at scale. It is essential to note that rather than resources, the development of knowledge and skills of the teachers through adaptation of tasks and resources (instrumentalization) and using discursive practices that supported reasoning (instrumentation) contributed to significant difference in students’ understanding in intervention schools. The number of factors that support the implementation of the resources in the under-resourced classroom needs to be studied further to understand how they support or constrain students’ and teachers’ learning.

Acknowledgements

Connected Learning Initiative has been supported by Tata Trusts. The design and development of the
modules have been done in collaboration with Massachusetts Institute of Technology (USA) and
Homi Bhabha Institute of Science Education among others. We are grateful to the study participants,
especially students and teachers, and other CLIx team members for their role in the implementation
of the study.

References

Adler, J. (2000). Conceptualising resources as a theme for teacher education. *Journal of Mathematics Teacher Education, 3*(3), 205-224.

Béguin, P., & Rabardel, P. (2000). Designing for instrument-mediated activity. *Scandinavian Journal of Information Systems, 12*(1), 1.

Brodie, K. (2020). Resources for and from collaboration: A conceptual framework. In *ICMI Study 25 conference proceedings: Teachers of Mathematics Working and Learning in Collaborative Groups* (pp. 37-48). Lisbon, Portugal.

Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. *Journal for Research in Mathematics Education*. Monograph, 3, i-196.

Kumar R.S., Srinivas, S., Bose, A., Rahman, J., Thakur, S., & Bapat, A. (2019) Examining Knowledge demands for teaching class inclusion of quadrilaterals. In M. Graven, H. Venkat, A. Essien, & P. Vale (Eds.). *Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 504-510). PME.

Shaughnessy, J. M., & Burger, W. F. (1985). Spadework prior to deduction in geometry. *The Mathematics Teacher, 78*(6), 419-428.

Thakur.S, Bose. A, & Kumar, R. S. (2020). Missed opportunities: Instances from geometry lessons. In Mashood K.K., & Tathagata Sengupta (Eds.). *Proceedings of Eighth Episteme Conference 2020* (pp. 308-318). HBCSE-Mumbai.
# MATERIAL AND DIGITAL TOOLS FOR GEOMETRY IN MATHEMATICS LABORATORY

Michela Maschietto

Dept of Education and Human Sciences, University of Modena e Reggio Emilia

In this paper, we present two examples of activities carried out with material and digital tools following the methodology of mathematics laboratory. The aim is to show how activities with the two kinds of tools can enrich students' experience and allow them to construct mathematical meanings. These two examples contain the same artifact used with different didactical functionalities at different school grades.

## INTRODUCTION

As reported in the Discussion Document of this ICMI study, the use of material manipulatives and digital tools is widespread in teaching and learning geometry. Our perspective is the use of both material manipulatives and digital tools within the same educational context, thinking about their intertwined use and basing on their respective potential (Maschietto, 2018). Two examples are discussed in this paper. The first example concerns the content of measurement of 2D figures (area and perimeter) at primary and low secondary school; it deals with Tangram puzzles and other artifacts, in their material and digital versions (including GeoGebra); the second example focuses on conic sections at high secondary school through the use of particular artifacts for geometry, the mathematical machines (Bartolini Bussi & Maschietto, 2011). All the activities are carried out within the methodology of the mathematics laboratory (Maschietto & Trouche, 2010).

With these two examples, this paper aims to contribute to the discussion on resources (topic C) and to provide elements for answering the following questions (Discussion document, p.12):

* How can bridges be created between manipulation and visualization and mathematical thinking in geometry?

* How can students be helped to abstract from perceptive and concrete properties of physical objects?

* What is the place of language, signs, gestures in activities based on manipulatives?

## USE OF ARTIFACTS IN MATHEMATICS LABORATORY

The history of mathematics laboratory in the European culture is long and interesting (Maschietto, 2015). In Italy, at the beginning of the 21st century, the UMI-CIIM (Italian Mathematical Union, Italian Commission on Mathematical Instruction) prepared a Curriculum for Mathematics for primary and secondary schools (Anichini et al., 2004), which gave a special emphasis on mathematics laboratory in its methodological part. This emphasis is maintained in the Italian National Guidelines for Primary and Secondary Instruction, approved respectively in 2012 and 2010. Therefore, in the mathematics laboratory, relevant components are the use of tools and the interactions among people working together for the construction of mathematical meanings. These elements are central in the Theory of Semiotic Mediation (TSM, Bartolini Bussi & Mariotti, 2008), within which the
# MATERIAL AND DIGITAL TOOLS FOR GEOMETRY IN MATHEMATICS LABORATORY

Michela Maschietto

Dept of Education and Human Sciences, University of Modena e Reggio Emilia

In this paper, we present two examples of activities carried out with material and digital tools following the methodology of mathematics laboratory. The aim is to show how activities with the two kinds of tools can enrich students' experience and allow them to construct mathematical meanings. These two examples contain the same artifact used with different didactical functionalities at different school grades.

## INTRODUCTION

As reported in the Discussion Document of this ICMI study, the use of material manipulatives and digital tools is widespread in teaching and learning geometry. Our perspective is the use of both material manipulatives and digital tools within the same educational context, thinking about their intertwined use and basing on their respective potential (Maschietto, 2018). Two examples are discussed in this paper. The first example concerns the content of measurement of 2D figures (area and perimeter) at primary and low secondary school; it deals with Tangram puzzles and other artifacts, in their material and digital versions (including GeoGebra); the second example focuses on conic sections at high secondary school through the use of particular artifacts for geometry, the mathematical machines (Bartolini Bussi & Maschietto, 2011). All the activities are carried out within the methodology of the mathematics laboratory (Maschietto & Trouche, 2010).

With these two examples, this paper aims to contribute to the discussion on resources (topic C) and to provide elements for answering the following questions (Discussion document, p.12):

* How can bridges be created between manipulation and visualization and mathematical thinking in geometry?

* How can students be helped to abstract from perceptive and concrete properties of physical objects?

* What is the place of language, signs, gestures in activities based on manipulatives?

## USE OF ARTIFACTS IN MATHEMATICS LABORATORY

The history of mathematics laboratory in the European culture is long and interesting (Maschietto, 2015). In Italy, at the beginning of the 21st century, the UMI-CIIM (Italian Mathematical Union, Italian Commission on Mathematical Instruction) prepared a Curriculum for Mathematics for primary and secondary schools (Anichini et al., 2004), which gave a special emphasis on mathematics laboratory in its methodological part. This emphasis is maintained in the Italian National Guidelines for Primary and Secondary Instruction, approved respectively in 2012 and 2010. Therefore, in the mathematics laboratory, relevant components are the use of tools and the interactions among people working together for the construction of mathematical meanings. These elements are central in the Theory of Semiotic Mediation (TSM, Bartolini Bussi & Mariotti, 2008), within which the
mathematics laboratory activities reported in this paper are designed and analyzed. The main
components of this theoretical framework are:

* The teacher uses an artifact to mediate mathematical meanings.

* An artifact is analyzed in terms of semiotic potential, defined as the double link between the artifact and students’ personal senses and the artifact and mathematical knowledge; the conception of tasks for the students is based on this analysis.

* The didactical cycle of activities (working with artifacts, individual work, and collective discussion) always begins with the exploration of the artifact, according to the questions *How is the machine made?* and *What does the machine make?* (they support students’ instrumental genesis; Vérillon & Rabardel, 1995); then, it focuses on the emergence of mathematical meanings embedded in the artifact by questions such as *Why does it make it?* and *What could happen if...?* (Bartolini et al., 2011).

The artifacts present in laboratory activities, as already written, can be both material and digital.

## REFERENCES FOR LEARNING GEOMETRY

The Discussion Document highlights several components that characterize the teaching and learning of geometry. In this section, we refer to some of those components to support the analysis of the examples that will be presented in the following sections.

The first reference concerns the visualization of geometric figures. Duval (2005) distinguishes between iconic visualization and non iconic visualization. The first is the most spontaneous approach to a drawing: a subject recognizes an object because its shape is similar to an already known object. In this sense, the shape resembles the typical shape of the real object represented and the contour of the object is mainly considered. Duval claims that this is the usual way of visualizing and pupils tend to deal with drawings through an iconic visualization. Under this visualization, the shapes appear stable: it is difficult to transform them into other similar shapes (for example, a square seen as a rhombus) or different ones (for example, non-congruent figures with the same area). According to a non iconic visualization, a drawing is one of the representations of a geometrical object; the subject passes from the visual similarity of an object to a familiar shape to its geometric properties. This kind of visualization develops against the spontaneous way of seeing the shape by iconic visualization; passing from iconic visualization to non iconic visualization is neither easy nor natural, but this transition is necessary to the learning of mathematical proof in geometry. Duval explains that using drawings to solve geometry problems involves three distinct operations: mereological deconstruction, instrumental deconstruction, and dimensional deconstruction. Mereological deconstruction consists of seeing the drawing as a union of superpositions or juxtapositions of figural units of the same dimension. Under an instrumental deconstruction, a drawing is considered the result of a construction process associated with a geometrical object, with the use of tools. Dimensional deconstruction is linked to a mathematical way of seeing drawings: a geometrical figure is considered as a set of figural units related to each other by geometric properties, and the drawing is analyzed accordingly. Non- iconic visualization is a necessary condition for dimensional deconstruction to be operational.

In addition, we consider three further aspects. First, we suppose that an iconic visualization supports a global viewpoint on a drawing, while a non iconic visualization supports a local viewpoint. In this sense, relationships between global/local viewpoints should be considered in teaching and learning
mathematics laboratory activities reported in this paper are designed and analyzed. The main
components of this theoretical framework are:

* The teacher uses an artifact to mediate mathematical meanings.

* An artifact is analyzed in terms of semiotic potential, defined as the double link between the artifact and students' personal senses and the artifact and mathematical knowledge; the conception of tasks for the students is based on this analysis.

* The didactical cycle of activities (working with artifacts, individual work, and collective discussion) always begins with the exploration of the artifact, according to the questions *How is the machine made?* and *What does the machine make?* (they support students' instrumental genesis; Vérillon & Rabardel, 1995); then, it focuses on the emergence of mathematical meanings embedded in the artifact by questions such as *Why does it make it?* and *What could happen if...?* (Bartolini et al., 2011).

The artifacts present in laboratory activities, as already written, can be both material and digital.

## REFERENCES FOR LEARNING GEOMETRY

The Discussion Document highlights several components that characterize the teaching and learning of geometry. In this section, we refer to some of those components to support the analysis of the examples that will be presented in the following sections.

The first reference concerns the visualization of geometric figures. Duval (2005) distinguishes between iconic visualization and non iconic visualization. The first is the most spontaneous approach to a drawing: a subject recognizes an object because its shape is similar to an already known object. In this sense, the shape resembles the typical shape of the real object represented and the contour of the object is mainly considered. Duval claims that this is the usual way of visualizing and pupils tend to deal with drawings through an iconic visualization. Under this visualization, the shapes appear stable: it is difficult to transform them into other similar shapes (for example, a square seen as a rhombus) or different ones (for example, non-congruent figures with the same area). According to a non iconic visualization, a drawing is one of the representations of a geometrical object; the subject passes from the visual similarity of an object to a familiar shape to its geometric properties. This kind of visualization develops against the spontaneous way of seeing the shape by iconic visualization; passing from iconic visualization to non iconic visualization is neither easy nor natural, but this transition is necessary to the learning of mathematical proof in geometry. Duval explains that using drawings to solve geometry problems involves three distinct operations: mereological deconstruction, instrumental deconstruction, and dimensional deconstruction. Mereological deconstruction consists of seeing the drawing as a union of superpositions or juxtapositions of figural units of the same dimension. Under an instrumental deconstruction, a drawing is considered the result of a construction process associated with a geometrical object, with the use of tools. Dimensional deconstruction is linked to a mathematical way of seeing drawings: a geometrical figure is considered as a set of figural units related to each other by geometric properties, and the drawing is analyzed accordingly. Non- iconic visualization is a necessary condition for dimensional deconstruction to be operational.

In addition, we consider three further aspects. First, we suppose that an iconic visualization supports a global viewpoint on a drawing, while a nonICONIC visualization supports a local viewpoint. In this sense, relationships between global/local viewpoints should be considered in teaching and learning
geometry. Then, another aspect linked to visualization is the relationship between static and dynamic aspects, well highlighted by research in dynamic geometry environment. The variation in geometric drawings allows the identification of invariant properties (the “invariants”) and relations between the properties of the represented figures; for example, they are useful in the formulation of theorem statements (Baccaglini-Frank et al., 2009). Finally, the way of seeing the figures and the language to describe them are closely linked. In constructing a geometric vocabulary, the definitions and relationships between the shapes recognized by the students come into play. For example, the type of visualization and the consequent interpretation of the drawings can support exclusive or inclusive classifications of geometric figures (for example, they concern triangles and quadrilaterals, studied in primary and lower secondary schools). Several research studies have been interested in the role of definitions and the implications of these definitions in the conceptualization of geometric figures (2D and 3D).

## THE FIRST EXAMPLE: PERIMETER AND AREA AT GRADES 4-5 AND 7

The first example is taken from a collaborative project on the mathematics laboratory (https://sites.google.com/view/diffusioneculturamodena2020/home-page) between teachers and researchers. Two teaching experiments were designed and carried out on the topic of area and perimeter, one in primary school (grade 4-5, 4 classes) and one in low secondary school (grade 7, 10 classes). The choice of the topic is based on the results of research (Fandiño Pinilla & D'Amore, 2006) which highlight how these two concepts are kept separate in teaching practice (at least, in Italy) and how students (but also teachers) have difficulty relating them. In particular, the tasks for students often deal with the conservation of areas and perimeter or contextual increase/decrease of measurements without discussing the variation of the area of isoperimetric figures or the variation of the perimeter of equivalent figures. The manipulatives (in their material and digital versions) used in these experiments were the Tangram puzzle, the geoboard (grade 4-5 only), and the mathematical machine for tracing the ellipse by the gardener's way (grade 7 only).

The didactical intervention for grade 7 is composed of three phases. In the first phase, the Tangram puzzle was initially explored in its material version (Figure 1, left), with the objectives of identifying the geometrical figures that constitute it (isosceles right triangles, square, rhombus) and the property (the invariance of the area) common to all the combinations of the seven pieces of the puzzle. The square configuration was represented on paper and pencil and/or by paper folding, favoring the transition from 3D to 2D representations and implementing instrumental and dimensional deconstructions. To support these two processes, the students were asked to write the procedure for drawing the square in small groups (Figure 1) and to exchange the texts with other groups for reviewing and obtaining a correct final version. A Geogebra book with incorrect Tangram squares (Figure 1, right) was assigned to consolidate the construction procedure and the properties of the figures. The digital tool is used to propose tasks that cannot be performed with the material Tangram (GeoGebra figures offer strategy feedback by dragging some vertices) and to collect and assess qualitatively students' answers (they had to upload their answers). In this way, it is possible to assign an individual work, as set by the TSM. After this task, the students played the puzzle game in small groups; there the reproduction of the Tangram combinations is based on mereological deconstruction by juxtaposition accompanied by the gestures of moving pieces for composing new combinations.
geometry. Then, another aspect linked to visualization is the relationship between static and dynamic aspects, well highlighted by research in dynamic geometry environment. The variation in geometric drawings allows the identification of invariant properties (the “invariants”) and relations between the properties of the represented figures; for example, they are useful in the formulation of theorem statements (Baccaglini-Frank et al., 2009). Finally, the way of seeing the figures and the language to describe them are closely linked. In constructing a geometric vocabulary, the definitions and relationships between the shapes recognized by the students come into play. For example, the type of visualization and the consequent interpretation of the drawings can support exclusive or inclusive classifications of geometric figures (for example, they concern triangles and quadrilaterals, studied in primary and lower secondary schools). Several research studies have been interested in the role of definitions and the implications of these definitions in the conceptualization of geometric figures (2D and 3D).

## THE FIRST EXAMPLE: PERIMETER AND AREA AT GRADES 4-5 AND 7

The first example is taken from a collaborative project on the mathematics laboratory (https://sites.google.com/view/diffusioneculturamodena2020/home-page) between teachers and researchers. Two teaching experiments were designed and carried out on the topic of area and perimeter, one in primary school (grade 4-5, 4 classes) and one in low secondary school (grade 7, 10 classes). The choice of the topic is based on the results of research (Fandiño Pinilla & D'Amore, 2006) which highlight how these two concepts are kept separate in teaching practice (at least, in Italy) and how students (but also teachers) have difficulty relating them. In particular, the tasks for students often deal with the conservation of areas and perimeter or contextual increase/decrease of measurements without discussing the variation of the area of isoperimetric figures or the variation of the perimeter of equivalent figures. The manipulatives (in their material and digital versions) used in these experiments were the Tangram puzzle, the geoboard (grade 4-5 only), and the mathematical machine for tracing the ellipse by the gardener's way (grade 7 only).

The didactical intervention for grade 7 is composed of three phases. In the first phase, the Tangram puzzle was initially explored in its material version (Figure 1, left), with the objectives of identifying the geometrical figures that constitute it (isosceles right triangles, square, rhombus) and the property (the invariance of the area) common to all the combinations of the seven pieces of the puzzle. The square configuration was represented on paper and pencil and/or by paper folding, favoring the transition from 3D to 2D representations and implementing instrumental and dimensional deconstructions. To support these two processes, the students were asked to write the procedure for drawing the square in small groups (Figure 1) and to exchange the texts with other groups for reviewing and obtaining a correct final version. A Geogebra book with incorrect Tangram squares (Figure 1, right) was assigned to consolidate the construction procedure and the properties of the figures. The digital tool is used to propose tasks that cannot be performed with the material Tangram (GeoGebra figures offer strategy feedback by dragging some vertices) and to collect and assess qualitatively students' answers (they had to upload their answers). In this way, it is possible to assign an individual work, as set by the TSM. After this task, the students played the puzzle game in small groups; there the reproduction of the Tangram combinations is based on mereological deconstruction by juxtaposition accompanied by the gestures of moving pieces for composing new combinations.
<figure><img src="image_159.png" /><figcaption>Figure 1: group work; construction of the tangram in paper and pencil; GeoGebra book</figcaption></figure>

The activities with the Tangram puzzle allow students to pay attention to the invariance of the area for each combination of the 7 pieces, and to the measurement of this area and the area of the components using the smallest triangle taken as the unit. Furthermore, this game represents a good context to ask questions about the perimeter of the combinations and to look for combinations with minimum/maximum perimeter, already in primary school (Fig. 2). By the comparison of different combinations, the non-preservation of the perimeter emerges (Fig. 2, center and right).

<figure><img src="image_160.png" /><figcaption>Figure 2: Tangram at IWB; notebook from grade 4-5; comparison of perimeter</figcaption></figure>

The gestures performed by the Tangram puzzle related to the juxtaposition of pieces and the manipulation of equivalent combinations by moving pieces were recalled in studying the areas of quadrilaterals and obtaining their corresponding formulas (Figures 3, left and center). In this case, a digital tool is used to support collective discussion and mereological deconstruction. This work allows students to understand those formulas (Figure 3, right), which are often studied by heart.

<figure><img src="image_161.png" /><figcaption>Figure 3. Puzzle of quadrilaterals and students’ opinions on these activities</figcaption></figure>

In the second phase, the construction of the mathematical machine tracing the ellipse (“ellipsograph”) following the gardener’s way was proposed to grade 7 classes for asking questions on non-equivalent isoperimetric figures. It was built by students with recycled material (cardboard, pins, and string; Figure 4, left and center). The exploration followed the structure of the questions referring to the TSM and it was guided by a worksheet (Figure 4, center). The ellipse was also traced in the GeoGebra environment (Figure 4, right); the students were invited to explore it by varying the parameters (length
<figure><img src="image_166.png" /><figcaption>Figure 1: group work; construction of the tangram in paper and pencil; GeoGebra book</figcaption></figure>

The activities with the Tangram puzzle allow students to pay attention to the invariance of the area for each combination of the 7 pieces, and to the measurement of this area and the area of the components using the smallest triangle taken as the unit. Furthermore, this game represents a good context to ask questions about the perimeter of the combinations and to look for combinations with minimum/maximum perimeter, already in primary school (Fig. 2). By the comparison of different combinations, the non-preservation of the perimeter emerges (Fig. 2, center and right).

<figure><img src="image_167.png" /><figcaption>Figure 2: Tangram at IWB; notebook from grade 4-5; comparison of perimeter</figcaption></figure>

The gestures performed by the Tangram puzzle related to the juxtaposition of pieces and the manipulation of equivalent combinations by moving pieces were recalled in studying the areas of quadrilaterals and obtaining their corresponding formulas (Figures 3, left and center). In this case, a digital tool is used to support collective discussion and mereological deconstruction. This work allows students to understand those formulas (Figure 3, right), which are often studied by heart.

<figure><img src="image_168.png" /><figcaption>Figure 3. Puzzle of quadrilaterals and students’ opinions on these activities</figcaption></figure>

In the second phase, the construction of the mathematical machine tracing the ellipse (“ellipsograph”) following the gardener’s way was proposed to grade 7 classes for asking questions on non-equivalent isoperimetric figures. It was built by students with recycled material (cardboard, pins, and string; Figure 4, left and center). The exploration followed the structure of the questions referring to the TSM and it was guided by a worksheet (Figure 4, center). The ellipse was also traced in the GeoGebra environment (Figure 4, right); the students were invited to explore it by varying the parameters (length
of the string and focal distance) as homework. The construction in GeoGebra allows students to go
beyond the physical constraints of the mathematical machine: by varying the parameters, the students
can see different drawings of the ellipse which cannot all be drawn with the ellipsograph. This task
aims to support iconic visualization for those students who meet the curve for the first time.

<figure><img src="image_162.png" /><figcaption>Figure 4. Ellipsograph, worksheets, and GeoGebra figure</figcaption></figure>

The teaching experiment ends with the proposal of a challenge (third phase): “Build a machine that does the opposite of the ellipsograph: equivalent non-isoperimetric triangles”. This assignment requires identifying a change in perimeter and ensuring the invariance of the area; that is another variation to study, and it is a real problem to solve for the students (referring to the question *What could happen if...*). Different machines were proposed (Figure 5, left and center): some of them worked, and others did not. During the classwork, the teacher discussed the solutions, also within the GeoGebra environment, and proposed other problems on equivalent figures. In this case, too, new tasks are proposed by GeoGebra books (Figure 4, right), aiming to support a nonICONIC visualization.

<figure><img src="image_163.png" /><figcaption>Figure 5. The new machine and new problems within GeoGebra books</figcaption></figure>

THE SECOND EXAMPLE: CONIC SECTIONS AT GRADE 11

The study of conic sections is a classic mathematical content in Italian high secondary schools for grade 11 students. The National Guidelines for High Secondary School contain a precise reference to the approach to be considered, in which the conic sections must be studied from both a synthetic and analytical geometric point of view. The mathematical machines (they belong to the Museum System and Botanical Garden of the University of Modena e Reggio Emilia, www.mmlab.unimore.it) are suitable for proposing a synthetic approach (i.e., Maschietto & Bartolini Bussi, 2011). Two types of conicographs were considered in the didactical cycles (Dondi & Maschietto, 2022): conicographs for ellipse, parabola, and hyperbola functioning by taut string, and crossed-parallelogram conicographs for ellipse and hyperbola. They all embed the metric definitions (i.e., with foci and directrix).

The activities are structured in three phases: in the initial phase, all those elements considered
necessary for the exploration of the mathematical machines are presented to students; the central
phase is dedicated to the exploration of the conicographs; in the final phase, the theory of conic
sections is presented through historical references. The mathematical machines are proposed with
two different didactical functionalities (Maschietto, 2015): the taut string conicographs are used to
of the string and focal distance) as homework. The construction in GeoGebra allows students to go
beyond the physical constraints of the mathematical machine: by varying the parameters, the students
can see different drawings of the ellipse which cannot all be drawn with the ellipsograph. This task
aims to support iconic visualization for those students who meet the curve for the first time.

<figure><img src="image_169.png" /><figcaption>Figure 4. Ellipsograph, worksheets, and GeoGebra figure</figcaption></figure>

The teaching experiment ends with the proposal of a challenge (third phase): “Build a machine that does the opposite of the ellipsograph: equivalent non-isoperimetric triangles”. This assignment requires identifying a change in perimeter and ensuring the invariance of the area; that is another variation to study, and it is a real problem to solve for the students (referring to the question *What could happen if...*). Different machines were proposed (Figure 5, left and center): some of them worked, and others did not. During the classwork, the teacher discussed the solutions, also within the GeoGebra environment, and proposed other problems on equivalent figures. In this case, too, new tasks are proposed by GeoGebra books (Figure 4, right), aiming to support a nonICONIC visualization.

<figure><img src="image_170.png" /><figcaption>Figure 5. The new machine and new problems within GeoGebra books</figcaption></figure>

THE SECOND EXAMPLE: CONIC SECTIONS AT GRADE 11

The study of conic sections is a classic mathematical content in Italian high secondary schools for grade 11 students. The National Guidelines for High Secondary School contain a precise reference to the approach to be considered, in which the conic sections must be studied from both a synthetic and analytical geometric point of view. The mathematical machines (they belong to the Museum System and Botanical Garden of the University of Modena e Reggio Emilia, www.mmlab.unimore.it) are suitable for proposing a synthetic approach (i.e., Maschietto & Bartolini Bussi, 2011). Two types of conicographs were considered in the didactical cycles (Dondi & Maschietto, 2022): conicographs for ellipse, parabola, and hyperbola functioning by taut string, and crossed-parallelogram conicographs for ellipse and hyperbola. They all embed the metric definitions (i.e., with foci and directrix).

The activities are structured in three phases: in the initial phase, all those elements considered
necessary for the exploration of the mathematical machines are presented to students; the central
phase is dedicated to the exploration of the conicographs; in the final phase, the theory of conic
sections is presented through historical references. The mathematical machines are proposed with
two different didactical functionalities (Maschietto, 2015): the taut string conicographs are used to
introduce the new mathematical curves, their definitions, and properties; while the crossed-
parallelogram conicographs are used to propose a unitary vision of conic sections and to approach
the meanings of tangent and normal straight lines to a curve at a point of the curve itself. In particular,
for taut string conicographs, the definition of the geometric locus generated by the machine is
required, while for crossed-parallelogram machines the justification of the type of curve drawn is
solicited (question *Why does it make it?*). Students' work is guided by worksheets, constructed
consistently with the framework of the TSM.

We report an excerpt from the exploration of the gardener’s ellipsograph by a group of grade 11 students (called A1, A2, A3, and A4). They generally recognize the drawn curve according to an iconic and global visualization related to the curve seen at low secondary school (above all in astronomy), but the analysis of the trace and the identification of parameters, variables, and invariants for the definition of the geometric locus is not trivial for them, because it requires a nonICONIC visualization and a local viewpoint on the curve. In this excerpt, first, variables and fixed lengths are identified (#3, #4), then a variable triangle (#8) as well as its isoperimetry property (#9, #14) even if it is not explicitly expressed in these terms and does not concern the vertex P (until #15). Nevertheless, the definition of the geometric locus (local viewpoint) needs a formulation involving the point P. We note the use of the terms “the same” (#9, #12, #14) and “equal” (#15) instead of “constant”.

2 A2: So, what does it vary [], [*he forms a triangle with the thread, Figure 6 left*]?

3 A4: the distance between the two pins [...]

4 A4: this does not vary [he points to the focal distance] and this varies [he points to the other sides of the triangle while A2 moves the thread] [...]

5 A3: then it varies the distance [he tightens the string and moves it around the pins]

6 A*: the distance of P from F1 and F2 varies [the students write on their worksheet]

The students then read the notes and wondered if there were other variables. [...]

7 A2: Beyond the point [P], the length of the two sides also varies. [...]

8 A2: Let's look at it as a triangle.

9 A3: This [side of the triangle] plus this [other side of the triangle] is always the same.

10 A2: Let's see more triangles. The two angles vary, and the lengths of the sides vary... up to approximately this limit... [Figure 6, center].

11 A4: They always vary.

A2 moves the string.

13 A3: That is, if we measure this [he points to the distance between the foci and P]

15 A2: The sum of these two [sides of the triangle between the focus and point P] is equal.

Question 3 of the worksheet asked to draw three different configurations of the machine. The objective was to stimulate students to provide a graphical representation of the parameters and the invariant (i.e., by measuring the length of the string and the sides of triangles). However, all answers contain iconic and global representations (Figure 6, right), even if sometimes the invariant is written.
introduce the new mathematical curves, their definitions, and properties; while the crossed-
parallelogram conicographs are used to propose a unitary vision of conic sections and to approach
the meanings of tangent and normal straight lines to a curve at a point of the curve itself. In particular,
for taut string conicographs, the definition of the geometric locus generated by the machine is
required, while for crossed-parallelogram machines the justification of the type of curve drawn is
solicited (question *Why does it make it?*). Students' work is guided by worksheets, constructed
consistently with the framework of the TSM.

We report an excerpt from the exploration of the gardener’s ellipsograph by a group of grade 11 students (called A1, A2, A3, and A4). They generally recognize the drawn curve according to an iconic and global visualization related to the curve seen at low secondary school (above all in astronomy), but the analysis of the trace and the identification of parameters, variables, and invariants for the definition of the geometric locus is not trivial for them, because it requires a nonICONIC visualization and a local viewpoint on the curve. In this excerpt, first, variables and fixed lengths are identified (#3, #4), then a variable triangle (#8) as well as its isoperimetry property (#9, #14) even if it is not explicitly expressed in these terms and does not concern the vertex P (until #15). Nevertheless, the definition of the geometric locus (local viewpoint) needs a formulation involving the point P. We note the use of the terms “the same” (#9, #12, #14) and “equal” (#15) instead of “constant”.

2 A2: So, what does it vary [], [*he forms a triangle with the thread, Figure 6 left*]?

3 A4: the distance between the two pins [...]

4 A4: this does not vary [he points to the focal distance] and this varies [he points to the other sides of the triangle while A2 moves the thread] [...]

5 A3: then it varies the distance [he tightens the string and moves it around the pins]

6 A*: the distance of P from F1 and F2 varies [the students write on their worksheet]

The students then read the notes and wondered if there were other variables. [...]

7 A2: Beyond the point [P], the length of the two sides also varies. [...]

8 A2: Let's look at it as a triangle.

9 A3: This [side of the triangle] plus this [other side of the triangle] is always the same.

10 A2: Let's see more triangles. The two angles vary, and the lengths of the sides vary... up to approximately this limit... [Figure 6, center].

11 A4: They always vary.

A2 moves the string.

13 A3: That is, if we measure this [he points to the distance between the foci and P]

15 A2: The sum of these two [sides of the triangle between the focus and point P] is equal.

Question 3 of the worksheet asked to draw three different configurations of the machine. The objective was to stimulate students to provide a graphical representation of the parameters and the invariant (i.e., by measuring the length of the string and the sides of triangles). However, all answers contain iconic and global representations (Figure 6, right), even if sometimes the invariant is written.
<figure><img src="image_164.png" /><figcaption>Figure 6. Triangle configuration and representation of these configurations (Question 3)</figcaption></figure>

Contrary to a classical lesson in which the definitions of conic sections are stated (and the students often learn them by heart), working with the mathematical machines allows them to discover and build the definitions and to identify properties, by exploring and exploiting the machines. In the identification of parameters, variables, and invariants, the analysis shows some difficulties in referring to the meanings of terms they have already encountered in another context before the study of conic sections. The use of different machines allows students' conceptualization to be tested: for example, each conicograph offers different material constraints and different representations for the parameters of the curves. The construction of the ellipsograph in the GeoGebra environment (as in Figure 4, right), asked after the work with the material machine, highlights the need to work more on parameters, variables, and invariants: for instance, in that construction, the possibility of choosing foci as free points hides the choice of that parameter of the curve. Compared with the first example discussed in this paper, GeoGebra is used with different didactical functionalities: there the students were supported to observe the shape of the ellipse depends on the variation of its parameters, while here the students were asked to make explicit their conceptualization of variables and parameters by constructing the curve.

## CONCLUDING REMARKS

This paper presents two examples of didactical activities to contribute to the discussion about the use of material and digital tools in learning and teaching geometry. Through them, we aim to show how a bridge between the manipulation of artifacts and mathematical thinking in geometry can be created, proposing tasks by which the students describe procedures and drawings, represent the mathematical objects in different environments, and look for regularities and invariants. Concerning the use of only material artifacts, the use of digital tools allows students to perform tasks and problems that can not be proposed through the material ones; for teachers, it can be possible to structure sub-tasks and implement evaluation and strategy feedback, supporting in this way students learning by collective and individual tasks. On the other hand, the use of material artifacts encourages collaborative group work and students' exploration processes within a rich semiotic activity. From our perspective, each kind of tool offers complementary representations and approaches to mathematical objects. In the two examples, the activities carried out by the students through the digital tool often corresponded to individual activities, completing the didactical cycles of the TSM.

The artifacts are chosen following their preliminary analysis (semiotic potential in the TSM), which suggests which characteristics can be exploited with students at a certain grade and how to construct appropriate tasks. For instance, the use of the gardeners' ellipsograph at different grades is based on this kind of analysis. Furthermore, several tasks can involve the same artifact, because the emergence and the construction of mathematical meanings requires time (also for instrumental genesis), and it
<figure><img src="image_171.png" /><figcaption>Figure 6. Triangle configuration and representation of these configurations (Question 3)</figcaption></figure>

Contrary to a classical lesson in which the definitions of conic sections are stated (and the students often learn them by heart), working with the mathematical machines allows them to discover and build the definitions and to identify properties, by exploring and exploiting the machines. In the identification of parameters, variables, and invariants, the analysis shows some difficulties in referring to the meanings of terms they have already encountered in another context before the study of conic sections. The use of different machines allows students' conceptualization to be tested: for example, each conicograph offers different material constraints and different representations for the parameters of the curves. The construction of the ellipsograph in the GeoGebra environment (as in Figure 4, right), asked after the work with the material machine, highlights the need to work more on parameters, variables, and invariants: for instance, in that construction, the possibility of choosing foci as free points hides the choice of that parameter of the curve. Compared with the first example discussed in this paper, GeoGebra is used with different didactical functionalities: there the students were supported to observe the shape of the ellipse depends on the variation of its parameters, while here the students were asked to make explicit their conceptualization of variables and parameters by constructing the curve.

## CONCLUDING REMARKS

This paper presents two examples of didactical activities to contribute to the discussion about the use of material and digital tools in learning and teaching geometry. Through them, we aim to show how a bridge between the manipulation of artifacts and mathematical thinking in geometry can be created, proposing tasks by which the students describe procedures and drawings, represent the mathematical objects in different environments, and look for regularities and invariants. Concerning the use of only material artifacts, the use of digital tools allows students to perform tasks and problems that can not be proposed through the material ones; for teachers, it can be possible to structure sub-tasks and implement evaluation and strategy feedback, supporting in this way students learning by collective and individual tasks. On the other hand, the use of material artifacts encourages collaborative group work and students' exploration processes within a rich semiotic activity. From our perspective, each kind of tool offers complementary representations and approaches to mathematical objects. In the two examples, the activities carried out by the students through the digital tool often corresponded to individual activities, completing the didactical cycles of the TSM.

The artifacts are chosen following their preliminary analysis (semiotic potential in the TSM), which suggests which characteristics can be exploited with students at a certain grade and how to construct appropriate tasks. For instance, the use of the gardeners' ellipsograph at different grades is based on this kind of analysis. Furthermore, several tasks can involve the same artifact, because the emergence and the construction of mathematical meanings requires time (also for instrumental genesis), and it
is not always possible to activate and end these processes in only one or two lessons. Another common
element between the two examples is that the gestures made during a task can be performed in other
tasks or with other artifacts. For instance, the gesture of moving pieces in the Tangram puzzle for
composing new configurations is used then for quadrilaterals, and in general for transforming figures
into equivalent ones that are not congruent. For conic section tracers, the gesture of usage of
tightening a string is common to all three conicographs; in addition, it corresponds to creating a
segment (cognitive root for the straight line) and gives a strong experience of a straight line. The
analyses show that students very often have an iconic visualization, which can be questioned by tasks
involving mereological deconstruction and the comparison between the variation of geometrical
quantities and the invariant properties. Thus, the transition to a nonICONIC visualization is supported.

In our classes, the manipulation of material artifacts, intertwined with digital tools, supported
students’ engagement in geometry activities. And this is not a secondary aspect for student’ learning.

References

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is not always possible to activate and end these processes in only one or two lessons. Another common
element between the two examples is that the gestures made during a task can be performed in other
tasks or with other artifacts. For instance, the gesture of moving pieces in the Tangram puzzle for
composing new configurations is used then for quadrilaterals, and in general for transforming figures
into equivalent ones that are not congruent. For conic section tracers, the gesture of usage of
tightening a string is common to all three conicographs; in addition, it corresponds to creating a
segment (cognitive root for the straight line) and gives a strong experience of a straight line. The
analyses show that students very often have an iconic visualization, which can be questioned by tasks
involving mereological deconstruction and the comparison between the variation of geometrical
quantities and the invariant properties. Thus, the transition to a nonICONIC visualization is supported.

In our classes, the manipulation of material artifacts, intertwined with digital tools, supported
students’ engagement in geometry activities. And this is not a secondary aspect for student’ learning.

References

Anichini, G., Arzarello, F., Ciarrapico, L., Robutti, O., & Statale, L. S. (2004). *Matematica 2003. La matematica per il cittadino (Ciclo secondario)*. Matteoni Stampatore.

Baccaglini-Frank A., Mariotti M.A., & Antonini S. (2009) Different perceptions of invariants and generality of proof in dynamic geometry. In M. Tzekaki & H. Sakonidis (Eds.), *Proceedings of the 33th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 89–96). PME.

Bartolini Bussi, M.G., & Mariotti M. A. (2008). Semiotic mediation in the mathematics classroom: artefacts and signs after a Vygotskian perspective. In L. English et al. (Eds.), *Handbook of international research in mathematics education* (2nd edition, pp. 746–783). Routledge.

Bartolini Bussi, M.G., Garuti, R., Martignone, F., & Maschietto, M. (2011). Tasks for teachers in the MMLAB-ER Project. In B. Ubuz (Ed.), *Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 1, pp. 127–130). PME.

Dondi, V., & Maschietto, M. (2022). Un percorso laboratoriale sulle sezioni coniche con le macchine matematiche. *L'Insegnamento della Matematica e delle Scienze Integrate, 45B*(4), 307–336.

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.

Fandiño Pinilla, M. & D'Amore, B. (2006). *Area e perimetro. Aspetti concettuali e didattici*. Erickson.

Maschietto, M. (2015). Teachers, students and resources in mathematics laboratory. In S.J. Cho (Ed.), *Selected Regular Lectures from the 12th Inter. Congress on Math. Education* (pp. 527–546). Springer.

Maschietto, M. (2018). Chapter 11. Classical and digital technologies for the Pythagorean theorem. In L. Ball, P. Drijvers, S. Ladel, H.-S. Siller, M. Tabach & C. Vale (Eds.), *Uses of technology in primary and secondary mathematics education- ICME-13 Monographs* (pp. 203–225). Springer.

Maschietto, M. & Bartolini Bussi, M.G. (2011). Mathematical machines: From history to the mathematics classroom. In P. Sullivan & O. Zavlasky (Eds.), *Constructing knowledge for teaching secondary mathematics: Tasks to enhance prospective and practicing teacher learning* (Vol. 6, pp. 227–245). Springer.

Maschietto, M., & Trouche, L. (2010). Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories. *ZDM, 42*(1), 33–47.

Vérillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. *European Journal of Psychology of Education, 10*(1), 77–101.
# DEVELOPMENT OF RESOURCES TO STUDY GEOMETRY REUSING MATERIALS

Natalia F. Sgreccia¹,², Lucía I. Schaefer¹, Sabrina B. Grossi¹, Bianca Di Biaggio¹

¹ National University of Rosario, ² National Council for Scientific and Technical Research
Argentina

The Professorship in Mathematics of the National University of Rosario (Argentina) supports both educational research projects linked to teacher training, as well as university extension projects to articulate the work with schools in disadvantaged or vulnerable contexts through the use of physical didactic resources, in particular, for the teaching and learning of geometry. Access to these resources is not always direct, and limitations of various kinds are usually encountered. In this instance, some attempts to address such obstacles are documented, highlighting the intrinsic relationship between university extension and educational research for the improvement of educational proposals in geometry when teaching resources made by teachers and students themselves are used. As a peculiarity, it should be noted that for its elaboration it is sought to put into action the ecological 3Rs as a transversal contribution according to the Sustainable Development Goals.

## INTRODUCTION

The Professorship in Mathematics (PM) of the National University of Rosario (province of Santa Fe, Argentina) is one of the 30 Professorships of this type in the country. It has existed for 35 years at the University and is going through its third curriculum since 2018. In it, the Professional Teaching Practice (PPD) is conceived as an articulating project through training, where the link with the environment is fundamental for the delimitation of the professional profile with which the institution contributes to society. It is sustained from research projects that take as their object the training in, with, from and for PPD. For example, the current one is called “The processes of knowledge construction about the PPD in the PM” and is framed in a Research Program with other careers.

Within this framework and as a plausible form of university social commitment, for more than a decade university extension and volunteer activities have been promoted, sustained and strengthened, which, in this case, include articulated work with schools in a variety of contexts, especially disadvantaged or with some type of vulnerability. In particular, these activities are part of the mathematical work with educational resources, initially digital (within the framework of the Connect Equality Program, University Volunteering 2011), with the incorporation of analog manuals (from a University Extension Project 2013) and in turn with attention to environmental care in its design (from the Integrating Program 2018). All this gave rise to the ReMatEd team (acronym for Resources + Mathematics + Education) composed of students, graduates and teachers of the PM.

On this occasion we share the resources to teach and learn geometry that have been designed, as well as some aspects related to their elaboration that contribute to the task of the Mathematics teacher who wishes to produce them. We have been implementing these resources in schools in the city of Rosario and surroundings. About the educational levels, we are responsible for both secondary and primary,
# DEVELOPMENT OF RESOURCES TO STUDY GEOMETRY REUSING MATERIALS

Natalia F. Sgreccia¹,², Lucía I. Schaefer¹, Sabrina B. Grossi¹, Bianca Di Biaggio¹

¹ National University of Rosario, ² National Council for Scientific and Technical Research
Argentina

The Professorship in Mathematics of the National University of Rosario (Argentina) supports both educational research projects linked to teacher training, as well as university extension projects to articulate the work with schools in disadvantaged or vulnerable contexts through the use of physical didactic resources, in particular, for the teaching and learning of geometry. Access to these resources is not always direct, and limitations of various kinds are usually encountered. In this instance, some attempts to address such obstacles are documented, highlighting the intrinsic relationship between university extension and educational research for the improvement of educational proposals in geometry when teaching resources made by teachers and students themselves are used. As a peculiarity, it should be noted that for its elaboration it is sought to put into action the ecological 3Rs as a transversal contribution according to the Sustainable Development Goals.

## INTRODUCTION

The Professorship in Mathematics (PM) of the National University of Rosario (province of Santa Fe, Argentina) is one of the 30 Professorships of this type in the country. It has existed for 35 years at the University and is going through its third curriculum since 2018. In it, the Professional Teaching Practice (PPD) is conceived as an articulating project through training, where the link with the environment is fundamental for the delimitation of the professional profile with which the institution contributes to society. It is sustained from research projects that take as their object the training in, with, from and for PPD. For example, the current one is called “The processes of knowledge construction about the PPD in the PM” and is framed in a Research Program with other careers.

Within this framework and as a plausible form of university social commitment, for more than a decade university extension and volunteer activities have been promoted, sustained and strengthened, which, in this case, include articulated work with schools in a variety of contexts, especially disadvantaged or with some type of vulnerability. In particular, these activities are part of the mathematical work with educational resources, initially digital (within the framework of the Connect Equality Program, University Volunteering 2011), with the incorporation of analog manuals (from a University Extension Project 2013) and in turn with attention to environmental care in its design (from the Integrating Program 2018). All this gave rise to the ReMatEd team (acronym for Resources + Mathematics + Education) composed of students, graduates and teachers of the PM.

On this occasion we share the resources to teach and learn geometry that have been designed, as well as some aspects related to their elaboration that contribute to the task of the Mathematics teacher who wishes to produce them. We have been implementing these resources in schools in the city of Rosario and surroundings. About the educational levels, we are responsible for both secondary and primary,
and teacher training for these levels, being the secondary level (students 12-18 years old) that we will
center. Briefly, it is intended to illustrate possible routes with accessible resources to study geometry,
which can be designed by multiple actors (students, teachers, parents, researchers), with the elements
that are available according to the possibilities. This with the purpose of taking advantage of inputs
that people discard in their usual consumption, to activate the 3Rs of ecology (reduce, reuse, recycle)
that encourage to cushion the human impact on the environment through a more efficient use.

BACKGROUND

There are numerous definitions of “teaching resources”. Alsina et al. (1988) consider it as any object, device or means of communication that can help describe, understand and consolidate concepts in different phases of learning. Without differentiate resource and didactic material, we consider the didactic resource as a material that although it may have specific intentions from its design, often transcends the original intention of use and admits varied applications (Alegre et al., 2018).

On the other hand, although geometry is recognized as one of the most important components of the
school curriculum of mathematics (Atiyah, 2001; Jones et al., 2006), in secondary school it is poorly
developed and often even forgotten how to treat in teacher training (Gutiérrez, 2010). For Corica &
Marin (2014), its exclusion or superficial study obstruct different ways of reasoning, inferring and
producing properties and relationships. We consider then that the use of didactic resources in the
geometry area can be an engine of promotion for its approach in the school.

It is also necessary to address the particularities of educating in vulnerable contexts. Promoting participatory methodologies linked to the game allows overcoming master classes, capturing the attention of students in non-intimidating environments and more fun ones, as reported by Hierro & Seller (2020). From our perspective, we believe that the use of concrete manipulative resources can be especially motivating in these contexts, but we should not forget to consider the cost of them.

For more than a decade in a sustained manner, ReMatEd participated in science dissemination events in public spaces such as squares or public university premises, reaching a diversity of public and socioeconomic sectors. In addition, game kits (many of those mentioned in this paper) were developed and shared with public primary and secondary schools in the region.

CURRICULUM FRAMEWORK

As presented above, in this work a cut is made in terms of educational level. In reference to this, in
Argentina, the Secondary School corresponds to the last stage of compulsory education. In particular,
in the province of Santa Fe, it consists of five years (students are 12 to 18 years old). Likewise, of
these five years, the first two ones make up the Basic Cycle, and the last three, the Oriented Cycle.

At the national level, there is a curricular document called Priority Learning Cores (NAP), which reflects the knowledge considered key and relevant for adolescents. In turn, based on the NAP, the province of Santa Fe proposes the Curricular Guidelines at the jurisdictional level. In both cases, only work with resources of a technological nature, referring to dynamic geometry software, is mentioned, and no reference is made to resources of another type, such as manipulatives.

As for the Geometry area, the NAP promotes teaching situations that allow the use of properties of
figures and geometric bodies to solve problems, as well as the production and analysis of geometric
constructions, and the production and validation of conjectures on geometric relationships. In the
and teacher training for these levels, being the secondary level (students 12-18 years old) that we will
center. Briefly, it is intended to illustrate possible routes with accessible resources to study geometry,
which can be designed by multiple actors (students, teachers, parents, researchers), with the elements
that are available according to the possibilities. This with the purpose of taking advantage of inputs
that people discard in their usual consumption, to activate the 3Rs of ecology (reduce, reuse, recycle)
that encourage to cushion the human impact on the environment through a more efficient use.

BACKGROUND

There are numerous definitions of “teaching resources”. Alsina et al. (1988) consider it as any object, device or means of communication that can help describe, understand and consolidate concepts in different phases of learning. Without differentiate resource and didactic material, we consider the didactic resource as a material that although it may have specific intentions from its design, often transcends the original intention of use and admits varied applications (Alegre et al., 2018).

On the other hand, although geometry is recognized as one of the most important components of the
school curriculum of mathematics (Atiyah, 2001; Jones et al., 2006), in secondary school it is poorly
developed and often even forgotten how to treat in teacher training (Gutiérrez, 2010). For Corica &
Marin (2014), its exclusion or superficial study obstruct different ways of reasoning, inferring and
producing properties and relationships. We consider then that the use of didactic resources in the
geometry area can be an engine of promotion for its approach in the school.

It is also necessary to address the particularities of educating in vulnerable contexts. Promoting participatory methodologies linked to the game allows overcoming master classes, capturing the attention of students in non-intimidating environments and more fun ones, as reported by Hierro & Seller (2020). From our perspective, we believe that the use of concrete manipulative resources can be especially motivating in these contexts, but we should not forget to consider the cost of them.

For more than a decade in a sustained manner, ReMatEd participated in science dissemination events in public spaces such as squares or public university premises, reaching a diversity of public and socioeconomic sectors. In addition, game kits (many of those mentioned in this paper) were developed and shared with public primary and secondary schools in the region.

CURRICULUM FRAMEWORK

As presented above, in this work a cut is made in terms of educational level. In reference to this, in
Argentina, the Secondary School corresponds to the last stage of compulsory education. In particular,
in the province of Santa Fe, it consists of five years (students are 12 to 18 years old). Likewise, of
these five years, the first two ones make up the Basic Cycle, and the last three, the Oriented Cycle.

At the national level, there is a curricular document called Priority Learning Cores (NAP), which reflects the knowledge considered key and relevant for adolescents. In turn, based on the NAP, the province of Santa Fe proposes the Curricular Guidelines at the jurisdictional level. In both cases, only work with resources of a technological nature, referring to dynamic geometry software, is mentioned, and no reference is made to resources of another type, such as manipulatives.

As for the Geometry area, the NAP promotes teaching situations that allow the use of properties of
figures and geometric bodies to solve problems, as well as the production and analysis of geometric
constructions, and the production and validation of conjectures on geometric relationships. In the
Curricular Orientations, the importance of validating the constructions through processes that pass the empirical test and favor deductive argumentation is added. They also indicate that a dynamic geometry software would enhance formulation of conjecture, argumentation and modeling, and make explicit the integration with the axis of Algebra and Functions, where certain geometric objects are treated algebraically.

As for the priority established contents, a large part of Secondary Education is devoted to those related to synthetic geometry, the last year being the only one totally dedicated to analytical geometry. The Basic Cycle deals with the study of triangles (properties of their angles, notable points, congruence and similarity, Pythagorean theorem), angles between lines, quadrilaterals, polyhedral bodies and isometric transformations, promoting constructions with ruler and compass, demonstrations and relationships between concepts (such as, between area, perimeter and volume). In the Oriented Cycle, the study of polygons in general (similarity) and triangles, specifically, with trigonometry, is deepened. In the last year, in particular, analytical geometry is approached from the work with Cartesian plane and equations of the line, circumference and parabola.

## RESOURCES MADE WITH YOUR OWN HANDS

Below we share four manipulative resources and five adapted board games, that we have produced in greater quantity and with which we have worked frequently in secondary level institutions. These types of manipulative resources offer students the possibility of linking mathematical concepts not only through the auditory sense, but also through the visual and tactile sense. In this way, abstract interpretations are combined with concrete interpretations of the geometric definitions and relationships being worked on. Some examples of activities can be found in Sgreccia et al. (2019).

The peculiarity of our contribution lies, on the one hand, in the reused materials that we employ to produce the resources, with an ecological touch. On the other hand, we prioritize that they can be carried out by teachers and/or students, in turn activating geometric knowledge in this elaboration process. This “learning by doing” is similar to movements such as New Scholl and Maker Space, particularized here in geometry and with affordable materials.

## Geoplane

The geoplane has a great versatility that, as Villarroel & Sgreccia (2012) highlight, allows it to address varied contents linked to the plane and, in some cases, to space. It consists of a plate with fixed points (nails) in which segments, angles and geometric figures can be formed using elastic bands, which are possible to stretch achieving the desired length.

It is easy to manufacture depending on the type of material used according to the purposes for which it is intended and the resources available. It can be made of wood with nails or screws (fixed geoplane), or failing that with some firm, but perforable material, such as cork sheets or EVA foam sheet pricking with push pins (mobile geoplane).

Different working templates can be used on each geoplane depending on the location of the nails: quadrangular, isometric or circular. The mobile geoplane, although it is more unstable or less durable than the fixed one, allows to adapt the template according to the work carried out in the classroom by changing the printed paper that is placed on the cork sheet. This is one of the advantages of creating the own geoplanes, with the desired templates.
Curricular Orientations, the importance of validating the constructions through processes that pass the empirical test and favor deductive argumentation is added. They also indicate that a dynamic geometry software would enhance formulation of conjecture, argumentation and modeling, and make explicit the integration with the axis of Algebra and Functions, where certain geometric objects are treated algebraically.

As for the priority established contents, a large part of Secondary Education is devoted to those related to synthetic geometry, the last year being the only one totally dedicated to analytical geometry. The Basic Cycle deals with the study of triangles (properties of their angles, notable points, congruence and similarity, Pythagorean theorem), angles between lines, quadrilaterals, polyhedral bodies and isometric transformations, promoting constructions with ruler and compass, demonstrations and relationships between concepts (such as, between area, perimeter and volume). In the Oriented Cycle, the study of polygons in general (similarity) and triangles, specifically, with trigonometry, is deepened. In the last year, in particular, analytical geometry is approached from the work with Cartesian plane and equations of the line, circumference and parabola.

## RESOURCES MADE WITH YOUR OWN HANDS

Below we share four manipulative resources and five adapted board games, that we have produced in greater quantity and with which we have worked frequently in secondary level institutions. These types of manipulative resources offer students the possibility of linking mathematical concepts not only through the auditory sense, but also through the visual and tactile sense. In this way, abstract interpretations are combined with concrete interpretations of the geometric definitions and relationships being worked on. Some examples of activities can be found in Sgreccia et al. (2019).

The peculiarity of our contribution lies, on the one hand, in the reused materials that we employ to produce the resources, with an ecological touch. On the other hand, we prioritize that they can be carried out by teachers and/or students, in turn activating geometric knowledge in this elaboration process. This “learning by doing” is similar to movements such as New Scholl and Maker Space, particularized here in geometry and with affordable materials.

## Geoplane

The geoplane has a great versatility that, as Villarroel & Sgreccia (2012) highlight, allows it to address varied contents linked to the plane and, in some cases, to space. It consists of a plate with fixed points (nails) in which segments, angles and geometric figures can be formed using elastic bands, which are possible to stretch achieving the desired length.

It is easy to manufacture depending on the type of material used according to the purposes for which it is intended and the resources available. It can be made of wood with nails or screws (fixed geoplane), or failing that with some firm, but perforable material, such as cork sheets or EVA foam sheet pricking with push pins (mobile geoplane).

Different working templates can be used on each geoplane depending on the location of the nails: quadrangular, isometric or circular. The mobile geoplane, although it is more unstable or less durable than the fixed one, allows to adapt the template according to the work carried out in the classroom by changing the printed paper that is placed on the cork sheet. This is one of the advantages of creating the own geoplanes, with the desired templates.
We show as an example in Figure 1, the process of making geoplanes with reused wood and screws, carried out by students who are part of the ReMatEd project.

<figure><img src="image_165.png" /><figcaption>Figure 1: Elaboration of circular and quadrangular geoplanes</figcaption></figure>

Although in vulnerable contexts there is not always an adequate device (it should be noted that for more than a decade the Connect Equality Program has been providing equipment to institutions, teachers and students), another option to use this resource is through virtual geoplanes. These, in terms of digital simulation, offer the same possibilities as manipulative geoplanes by simply accessing from mobile devices, tablets or computers; for example, Geoboard by The Math Learning Center.

Among the contents of secondary school that we have addressed with the geoplane are: construction of flat figures, similarity of triangles, analysis of formulas for the calculation of surface measurements, area, perimeter and their relationship, angles, notion of parallelism, obliquity and/or perpendicularity, Pythagorean Theorem, Sine and Cosine Theorem, trigonometric ratios.

At the same time, on certain occasions we were able to detect some limitations of the resource. For example, grids do not show explicit links with Cartesian planes, a difficulty that could be overcome by combining with some other resource of a technological nature such as GeoGebra (as we have done, for example, in a sequence to introduce the sine trigonometric ratio). Likewise, we can argue that proposals that require the use of geoplanes strongly promote the motor development and creativity of students linked to specific contents of geometry. For example, the construction of a segment, closed polygonal or open polygonal in direct relation to the placement of elastic bands on the geoplane.

## Polyforms

Polyforms are recessed pieces with flaps that allow the creation of polyhedron bodies. These constructions strongly use the notion of a polyhedron's face together with that of flat development. For its manufacture, cardboard or firm plastics are enough, which can be used from product packaging such as tea or cereal boxes, disused X-ray plates, among others. Due to this characteristic, it is easy to prepare and very low cost, allowing it to be considered an object of promotion of the activity of recycling and conservation of the environment, in the key of sustainable development (Corbetta, 2022). We show in Figure 2 the process of developing this resource.

<figure><img src="image_166.png" /><figcaption>Figure 2: Elaboration of polyforms with radiographic plates and cartons in disused containers</figcaption></figure>

The pieces we have designed at the moment are regular figures, from three to five sides, although they can be made of more types (not regular, more sides). With a kit of twenty regular triangles, six regular squares and twelve regular pentagons, it is possible to construct (assembling and disassembling) the five Platonic solids. Although these regular bodies constitute an entity in themselves, the work with polyforms also allows the construction of special bodies (for example, a stellated dodecahedron) according to the free production of each student. Action that makes it
We show as an example in Figure 1, the process of making geoplanes with reused wood and screws, carried out by students who are part of the ReMatEd project.

<figure><img src="image_172.png" /><figcaption>Figure 1: Elaboration of circular and quadrangular geoplanes</figcaption></figure>

Although in vulnerable contexts there is not always an adequate device (it should be noted that for more than a decade the Connect Equality Program has been providing equipment to institutions, teachers and students), another option to use this resource is through virtual geoplanes. These, in terms of digital simulation, offer the same possibilities as manipulative geoplanes by simply accessing from mobile devices, tablets or computers; for example, Geoboard by The Math Learning Center.

Among the contents of secondary school that we have addressed with the geoplane are: construction of flat figures, similarity of triangles, analysis of formulas for the calculation of surface measurements, area, perimeter and their relationship, angles, notion of parallelism, obliquity and/or perpendicularity, Pythagorean Theorem, Sine and Cosine Theorem, trigonometric ratios.

At the same time, on certain occasions we were able to detect some limitations of the resource. For example, grids do not show explicit links with Cartesian planes, a difficulty that could be overcome by combining with some other resource of a technological nature such as GeoGebra (as we have done, for example, in a sequence to introduce the sine trigonometric ratio). Likewise, we can argue that proposals that require the use of geoplanes strongly promote the motor development and creativity of students linked to specific contents of geometry. For example, the construction of a segment, closed polygonal or open polygonal in direct relation to the placement of elastic bands on the geoplane.

## Polyforms

Polyforms are recessed pieces with flaps that allow the creation of polyhedron bodies. These constructions strongly use the notion of a polyhedron's face together with that of flat development. For its manufacture, cardboard or firm plastics are enough, which can be used from product packaging such as tea or cereal boxes, disused X-ray plates, among others. Due to this characteristic, it is easy to prepare and very low cost, allowing it to be considered an object of promotion of the activity of recycling and conservation of the environment, in the key of sustainable development (Corbetta, 2022). We show in Figure 2 the process of developing this resource.

<figure><img src="image_173.png" /><figcaption>Figure 2: Elaboration of polyforms with radiographic plates and cartons in disused containers</figcaption></figure>

The pieces we have designed at the moment are regular figures, from three to five sides, although they can be made of more types (not regular, more sides). With a kit of twenty regular triangles, six regular squares and twelve regular pentagons, it is possible to construct (assembling and disassembling) the five Platonic solids. Although these regular bodies constitute an entity in themselves, the work with polyforms also allows the construction of special bodies (for example, a stellated dodecahedron) according to the free production of each student. Action that makes it
possible to make geometric work an artistic and creative activity, strongly promoting special reasoning and the determination of conjectures about spatial relationships of objects. Likewise, if the development of the resource is carried out by the students, it allows them to work on the construction of geometric figures with a ruler and compass and all their properties.

As a possible didactic limitation of this resource can be considered the association of the notion of figure (two-dimensional) with that of the piece (three-dimensional) of the polyforms, because each piece takes the concept of figure to be considered face of the constructed body. Working in parallel with two-dimensional representations on paper can help overcome this type of obstacle. Polyforms, as a manipulative resource, give rise to a special concrete conception of the three-dimensional, promoting construction skills, which is difficult only with pencil and paper.

Among the contents that we have addressed with polyforms in different courses of secondary school are: construction of regular polyhedra bodies, recognition of faces, vertices and edges of polyhedron bodies, Euler's equation, plane developments, notion of area and volume of polyhedron bodies.

## Circular sectors

The kit of circular sectors that we have assembled is formed by six discs of the same diameter: five of them divided with the same amplitude -1) two circular sectors of 180°; 2) three of 120°; 3) four of 90°; 4) six of 60°; 5) eight of 45°- and one whole (without cuts). This resource has a total of 24 pieces.

For its elaboration, the materials have to allow an easy cut, because having to round the discs becomes very expensive the task. To avoid this cumbersome procedure and avoid messiness or imprecision, we make them with disused compact discs and on these we simply make straight cuts to create the circular sectors. Then they are covered with EVA foam sheets, also preferably reused, of different colors. As mentioned for other resources, its processing cost is almost zero and enables awareness spaces about the ecological 3Rs. We share in Figure 3 the process of developing these circular sectors.

<figure><img src="image_167.png" /><figcaption>Figure 3: Construction process of circular sectors with disused CD and EVA foam sheets</figcaption></figure>

As for the mathematical contents promoted by the work with this resource, one can clearly allude to the study of angles, but also to the notion of circular sectors, central angles, circumference and circle, part-whole relationship, equivalent fractions, trigonometric ratios, among others. Working with this type of materials allows one to interpret and identify geometric concepts through the action of assembling-disassembling, relating and comparing amplitudes, a much less intuitive activity that is carried out only with pencil and paper. On the other hand, constructing the circular sectors themselves allows choosing the number of pieces or measurement of the angles according to the objectives of the teaching activity, as well as, if created by the students, encouraging the measuring of angles and the construction of figures with ruler and compass.

## Tangram

Tangram is a very popular educational resource as a puzzle game for all ages. Its objective is to create certain geometric shapes or specific silhouettes with small figures (regular and irregular). There is an immense variety of types of tangram according to the figures that make up each one. Among them:
possible to make geometric work an artistic and creative activity, strongly promoting special reasoning and the determination of conjectures about spatial relationships of objects. Likewise, if the development of the resource is carried out by the students, it allows them to work on the construction of geometric figures with a ruler and compass and all their properties.

As a possible didactic limitation of this resource can be considered the association of the notion of figure (two-dimensional) with that of the piece (three-dimensional) of the polyforms, because each piece takes the concept of figure to be considered face of the constructed body. Working in parallel with two-dimensional representations on paper can help overcome this type of obstacle. Polyforms, as a manipulative resource, give rise to a special concrete conception of the three-dimensional, promoting construction skills, which is difficult only with pencil and paper.

Among the contents that we have addressed with polyforms in different courses of secondary school are: construction of regular polyhedra bodies, recognition of faces, vertices and edges of polyhedron bodies, Euler's equation, plane developments, notion of area and volume of polyhedron bodies.

## Circular sectors

The kit of circular sectors that we have assembled is formed by six discs of the same diameter: five of them divided with the same amplitude -1) two circular sectors of 180°; 2) three of 120°; 3) four of 90°; 4) six of 60°; 5) eight of 45°- and one whole (without cuts). This resource has a total of 24 pieces.

For its elaboration, the materials have to allow an easy cut, because having to round the discs becomes very expensive the task. To avoid this cumbersome procedure and avoid messiness or imprecision, we make them with disused compact discs and on these we simply make straight cuts to create the circular sectors. Then they are covered with EVA foam sheets, also preferably reused, of different colors. As mentioned for other resources, its processing cost is almost zero and enables awareness spaces about the ecological 3Rs. We share in Figure 3 the process of developing these circular sectors.

<figure><img src="image_174.png" /><figcaption>Figure 3: Construction process of circular sectors with disused CD and EVA foam sheets</figcaption></figure>

As for the mathematical contents promoted by the work with this resource, one can clearly allude to the study of angles, but also to the notion of circular sectors, central angles, circumference and circle, part-whole relationship, equivalent fractions, trigonometric ratios, among others. Working with this type of materials allows one to interpret and identify geometric concepts through the action of assembling-disassembling, relating and comparing amplitudes, a much less intuitive activity that is carried out only with pencil and paper. On the other hand, constructing the circular sectors themselves allows choosing the number of pieces or measurement of the angles according to the objectives of the teaching activity, as well as, if created by the students, encouraging the measuring of angles and the construction of figures with ruler and compass.

## Tangram

Tangram is a very popular educational resource as a puzzle game for all ages. Its objective is to create certain geometric shapes or specific silhouettes with small figures (regular and irregular). There is an immense variety of types of tangram according to the figures that make up each one. Among them:
Chinese (the most popular), Ovoid, Cardiotangram, Brügner, Fletcher, Lloyd, Stomachion, Triangular, Pentagonal, Hexagonal, Pythagorean, Spatial, as reported in Villarroel & Sgreccia (2012) and exemplified with activities in teacher training in Dominguez (2021).

For its elaboration plates of a material that is easy to cut but resistant to deformations are required. We usually use EVA foam sheets to make the different pieces of each tangram, as they are accessible from diverse discarded inputs in good condition or, failing that, in bookstores at affordable costs.

The richness of this manipulative in terms of learning geometry is well recognized. It allows to develop visual, drawing and construction skills when reproducing models and given data, communication when interpreting geometric information, and reasoning when exploring concepts and regularities. It promotes geometric learning not only during the same game of construction of the desired figure but also can take advantage of the same process of elaboration of the material by the students. For this construction, various geometric reasoning is put into play combining concepts such as: angle measurement, bisector, mediatrix, perpendicularity, midpoint, rational numbers.

As with the geoplane, the tangram also has digital versions that promote geometric skills; for example, Tangram Online for Kids in Cokitos. There are numerous options, varying in complexity, so they can be used for different levels of schooling and intentionalities.

## Adapted board games

Board games are those that, as the name implies, can be played on the table, although any space where a group of people can meet can also be used where the participants are arranged surrounding the game. They are also versatile in terms of their elaboration, that is, anyone from some recycled materials can recreate and customize them according to their preferences. This represents a benefit in contexts of vulnerability and has allowed us to create copies for various schools without making a significant expense. Even, on more than one occasion, the same institutional actors were the ones who elaborated them from collecting disused materials for reuse. The elaboration of the resources themselves determines a moment of reflection on the concepts addressed, allowing different mathematical notions to be put into criticism, relationship and contrast.

Board games adapted to mathematical content are a tool that can be used from the initial level to the secondary level and even higher. These games allow us to address a wide range of mathematical concepts since they are easily adapted and customized according to the specific learning objectives. Some of the geometric contents that can be addressed are area and perimeter of plane figures, geometric bodies, angles, classification of triangles, similarity of triangles, among others. Particularly, the board games that we have adapted are Domino, Bingo, Ludo, Memotest and Game of the Goose (Figure 4a, b, c, d & e); for all ones were designed the same criteria as the original game.

<figure><img src="image_168.png" /><figcaption>Figure 4: Adapted board games: a) domino; b) bingo; c) ludo; d) memotest; e) goose game</figcaption></figure>

Adapted board games have the advantage of being versatile; that is, they can be adapted to any content, in particular geometric ones. They offer an interactive and relaxed way to learn mathematical concepts and allow students to develop mathematical skills in a playful way. They are likely to encourage the participation of students favoring their progression and the consolidation of content.
Chinese (the most popular), Ovoid, Cardiotangram, Brügner, Fletcher, Lloyd, Stomachion, Triangular, Pentagonal, Hexagonal, Pythagorean, Spatial, as reported in Villarroel & Sgreccia (2012) and exemplified with activities in teacher training in Domínguez (2021).

For its elaboration plates of a material that is easy to cut but resistant to deformations are required. We usually use EVA foam sheets to make the different pieces of each tangram, as they are accessible from diverse discarded inputs in good condition or, failing that, in bookstores at affordable costs.

The richness of this manipulative in terms of learning geometry is well recognized. It allows to develop visual, drawing and construction skills when reproducing models and given data, communication when interpreting geometric information, and reasoning when exploring concepts and regularities. It promotes geometric learning not only during the same game of construction of the desired figure but also can take advantage of the same process of elaboration of the material by the students. For this construction, various geometric reasoning is put into play combining concepts such as: angle measurement, bisector, mediatrix, perpendicularity, midpoint, rational numbers.

As with the geoplane, the tangram also has digital versions that promote geometric skills; for example, Tangram Online for Kids in Cokitos. There are numerous options, varying in complexity, so they can be used for different levels of schooling and intentionalities.

## Adapted board games

Board games are those that, as the name implies, can be played on the table, although any space where a group of people can meet can also be used where the participants are arranged surrounding the game. They are also versatile in terms of their elaboration, that is, anyone from some recycled materials can recreate and customize them according to their preferences. This represents a benefit in contexts of vulnerability and has allowed us to create copies for various schools without making a significant expense. Even, on more than one occasion, the same institutional actors were the ones who elaborated them from collecting disused materials for reuse. The elaboration of the resources themselves determines a moment of reflection on the concepts addressed, allowing different mathematical notions to be put into criticism, relationship and contrast.

Board games adapted to mathematical content are a tool that can be used from the initial level to the secondary level and even higher. These games allow us to address a wide range of mathematical concepts since they are easily adapted and customized according to the specific learning objectives. Some of the geometric contents that can be addressed are area and perimeter of plane figures, geometric bodies, angles, classification of triangles, similarity of triangles, among others. Particularly, the board games that we have adapted are Domino, Bingo, Ludo, Memotest and Game of the Goose (Figure 4a, b, c, d & e); for all ones were designed the same criteria as the original game.

<figure><img src="image_175.png" /><figcaption>Figure 4: Adapted board games: a) domino; b) bingo; c) ludo; d) memotest; e) goose game</figcaption></figure>

Adapted board games have the advantage of being versatile; that is, they can be adapted to any content, in particular geometric ones. They offer an interactive and relaxed way to learn mathematical concepts and allow students to develop mathematical skills in a playful way. They are likely to encourage the participation of students favoring their progression and the consolidation of content.
## CONCLUSIONS

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

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

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

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

From ReMatEd we have gone through different contexts: from closets full of abandoned resources and without knowing their existence by the members of the institution, to schools with the intention