sections of path along which there are different landmarks. Finally, the survey knowledge is the
development of a personal and cognitive map.

Navigation in a real or virtual space relies also on spatial orientation and mobilization of different frames of reference, in which we distinguish between egocentric and allocentric points of view. In an egocentric frame of reference relations between objects of the environment are encoded from the person (Chen & Stanney, 2002), while in an allocentric one, these relations are encoded from a point of the space independent of the person (Pennel & al., 2001). The control of these two types of frame of reference and their coordination is crucial in real life and in several professions, e.g. in construction, for builders, plumbers or electricians (Bessot et al., 1992). According to Bell (2002) their efficient control brings a structuring of spatial information and contribute to the development of spatial knowledge.

On another level, it is important to categorize the large variety of spaces to which spatial knowledge
may refer as well as our relation to these spaces. Brousseau (1983) followed by Berthelot and Salin
(1999) define three different types of space according to the size of the space seen from the
perspective of an observer. The micro-space is characterized by the fact that the observer can see it
straight at one glance (like a worksheet or small objects). The macro-space, on the other hand, refers
to an area that the observer can only see entirely through movement and displacement. Thus, in order
to build a representation of the whole space, one needs to do recollections that require processing
partial reference points as well as stored and memorized information (neighborhood, school). The
meso-space refers to an in-between, a space too wide to be seen at one glance, but which only requires
a shift of the observer's gaze. It is typically the classroom or the school courtyard that can be fully
seen by turning the head. Developing activities in the macro-space is a key issue, in order to suitably
develop good bases for landmark, route and survey knowledge. Yet, to design activities in a macro-
space necessitates to take students outside school, which is usually difficult or even impossible,
beyond the nearby area of school in the usual conditions of teaching. Therefore, the use of technology
for the development of spatial knowledge can be a relevant option for the use of a virtual macro-space
to create activities training navigation or rotation abilities. This was our motivation in the use of
SPAGEO city, in the part related to didactics of mathematics in this project.

The next part of our contribution presents some characteristics of SPAGEO City and our questioning regarding a specific activity tested during the end of the third year of our study. One can find more examples in Frauchiger et al. (to be published), Matri et al. (2023), Coutat et al. (2022).

OUR QUESTIONING

The technological environment allows different representations of a same position in SPAGEO City.
The immersive aspect of the environment offers an immersive 3D view with an egocentric frame of
reference where it is possible to spin around (Fig. 1). In that view, all the landmarks around the
position are visible. It is possible to identify relations with visible landmarks and the position. This is
the perspective closest to a real situation of a person walking in a city. The device however allows
another point of view using a map of SPAGEO City, with an allocentric frame of reference (Fig. 2).
This view is a model of the city and like all modeling, it has its own codes and gives only partial
information. So that, it is necessary to know the different codes or representations to be able to use
the map. For example, it is crucial to be able to distinguish the road from constructions. On another
sections of path along which there are different landmarks. Finally, the survey knowledge is the
development of a personal and cognitive map.

Navigation in a real or virtual space relies also on spatial orientation and mobilization of different frames of reference, in which we distinguish between egocentric and allocentric points of view. In an egocentric frame of reference relations between objects of the environment are encoded from the person (Chen & Stanney, 2002), while in an allocentric one, these relations are encoded from a point of the space independent of the person (Pennel & al., 2001). The control of these two types of frame of reference and their coordination is crucial in real life and in several professions, e.g. in construction, for builders, plumbers or electricians (Bessot et al., 1992). According to Bell (2002) their efficient control brings a structuring of spatial information and contribute to the development of spatial knowledge.

On another level, it is important to categorize the large variety of spaces to which spatial knowledge
may refer as well as our relation to these spaces. Brousseau (1983) followed by Berthelot and Salin
(1999) define three different types of space according to the size of the space seen from the
perspective of an observer. The micro-space is characterized by the fact that the observer can see it
straight at one glance (like a worksheet or small objects). The macro-space, on the other hand, refers
to an area that the observer can only see entirely through movement and displacement. Thus, in order
to build a representation of the whole space, one needs to do recollections that require processing
partial reference points as well as stored and memorized information (neighborhood, school). The
meso-space refers to an in-between, a space too wide to be seen at one glance, but which only requires
a shift of the observer's gaze. It is typically the classroom or the school courtyard that can be fully
seen by turning the head. Developing activities in the macro-space is a key issue, in order to suitably
develop good bases for landmark, route and survey knowledge. Yet, to design activities in a macro-
space necessitates to take students outside school, which is usually difficult or even impossible,
beyond the nearby area of school in the usual conditions of teaching. Therefore, the use of technology
for the development of spatial knowledge can be a relevant option for the use of a virtual macro-space
to create activities training navigation or rotation abilities. This was our motivation in the use of
SPAGEO city, in the part related to didactics of mathematics in this project.

The next part of our contribution presents some characteristics of SPAGEO City and our questioning regarding a specific activity tested during the end of the third year of our study. One can find more examples in Frauchiger et al. (to be published), Matri et al. (2023), Coutat et al. (2022).

OUR QUESTIONING

The technological environment allows different representations of a same position in SPAGEO City. The immersive aspect of the environment offers an immersive 3D view with an egocentric frame of reference where it is possible to spin around (Fig. 1). In that view, all the landmarks around the position are visible. It is possible to identify relations with visible landmarks and the position. This is the perspective closest to a real situation of a person walking in a city. The device however allows another point of view using a map of SPAGEO City, with an allocentric frame of reference (Fig. 2). This view is a model of the city and like all modeling, it has its own codes and gives only partial information. So that, it is necessary to know the different codes or representations to be able to use the map. For example, it is crucial to be able to distinguish the road from constructions. On another
hand, some landmarks have to or could deliberately be omitted on the map, like traffic lights or
pedestrian crossings. The use of an allocentric frame of reference could influence or support the
elaboration of the cognitive map because it gives the relations between the different areas of the city,
which are not as easy to see from in the immersive egocentric view. Therefore, the use of the virtual
environment without a map, only from an egocentric frame of reference, forces students to construct
their own relations between the different locations, an important component of their cognitive map.
In our didactical sequences we decided to use only the immersive view during the first year (Grade
2). This was coherent with the curriculum in mathematics in Geneva which does not introduce maps
until Grade 3. During the second year, the map (on paper) and the immersive view (on the computer)
were accessible at the same time. Finally, in the last year, we introduced a map on the computer, but
we chose that it would be impossible to access to the map and the immersive view simultaneously.
On the computer screen, either the map or the immersive view of the city appears. Students switch
from one view to the other by pressing a specific button on the joystick. The objective was to constrain
memorization of information from a frame of reference to use it in the other. A specific activity was
proposed in the third year. They had to plan a path between 2 positions with the help of the map and
realize it in the immersive view. The study of one of this activity is guide by these questions: how do
the students plan a path between two positions and how students use the access of a map when they
have to plan a path between two positions?

THE TASK AND THE DIDACTICAL CHOICES

Among the many tasks proposed to students throughout the didactical sequences during the 3 years,
we chose to present and analyze in this paper a path planning task, proposed during the final
evaluation of the third year (grade 4). A similar task, with different path to plan, had been worked
with the students during the second year with simultaneous access to the map (on paper) and the
immersive view (on the computer). During third year, this path planning task had been worked two
times. Students are expected to make progress on this task over the years. On a practical viewpoint
(see Fig. 1 & 2), when one launches the task in the environment, the map appears. It indicates the
starting position (yellow dot) and a position (here numbered 1) that one must find. The instructions
are written on the top of the page: "Go into the city, stand in exactly the same place as the place
indicated on the map". Two situations were proposed, in the first one (Figure 1 1) the target position
is in the city center (with lots of landmarks) while the second one (Figure 2) is located in the
residential area with very few landmarks and identical houses.

<figure><img src="image_143.png" /><figcaption>Figure 1: Immersive view on the computer</figcaption></figure>

<figure><img src="image_144.png" /><figcaption>Figure 2: Map of the residential area</figcaption></figure>

We made some didactical choices to constrain students in planning their path. First of all, we chose
to indicate the starting position on the map, but not the orientation. In the city center case, students
hand, some landmarks have to or could deliberately be omitted on the map, like traffic lights or
pedestrian crossings. The use of an allocentric frame of reference could influence or support the
elaboration of the cognitive map because it gives the relations between the different areas of the city,
which are not as easy to see from in the immersive egocentric view. Therefore, the use of the virtual
environment without a map, only from an egocentric frame of reference, forces students to construct
their own relations between the different locations, an important component of their cognitive map.
In our didactical sequences we decided to use only the immersive view during the first year (Grade
2). This was coherent with the curriculum in mathematics in Geneva which does not introduce maps
until Grade 3. During the second year, the map (on paper) and the immersive view (on the computer)
were accessible at the same time. Finally, in the last year, we introduced a map on the computer, but
we chose that it would be impossible to access to the map and the immersive view simultaneously.
On the computer screen, either the map or the immersive view of the city appears. Students switch
from one view to the other by pressing a specific button on the joystick. The objective was to constrain
memorization of information from a frame of reference to use it in the other. A specific activity was
proposed in the third year. They had to plan a path between 2 positions with the help of the map and
realize it in the immersive view. The study of one of this activity is guide by these questions: how do
the students plan a path between two positions and how students use the access of a map when they
have to plan a path between two positions?

THE TASK AND THE DIDACTICAL CHOICES

Among the many tasks proposed to students throughout the didactical sequences during the 3 years,
we chose to present and analyze in this paper a path planning task, proposed during the final
evaluation of the third year (grade 4). A similar task, with different path to plan, had been worked
with the students during the second year with simultaneous access to the map (on paper) and the
immersive view (on the computer). During third year, this path planning task had been worked two
times. Students are expected to make progress on this task over the years. On a practical viewpoint
(see Fig. 1 & 2), when one launches the task in the environment, the map appears. It indicates the
starting position (yellow dot) and a position (here numbered 1) that one must find. The instructions
are written on the top of the page: "Go into the city, stand in exactly the same place as the place
indicated on the map". Two situations were proposed, in the first one (Figure 1 1) the target position
is in the city center (with lots of landmarks) while the second one (Figure 2) is located in the
residential area with very few landmarks and identical houses.

<figure><img src="image_150.png" /><figcaption>Figure 1: Immersive view on the computer</figcaption></figure>

<figure><img src="image_151.png" /><figcaption>Figure 2: Map of the residential area</figcaption></figure>

We made some didactical choices to constrain students in planning their path. First of all, we chose
to indicate the starting position on the map, but not the orientation. In the city center case, students
are facing north (the top of the map), in the residential area, they are facing south. So, once in the immersive view, students must choose the correct direction to start their path, without knowing on the map which is their initial orientation. Then, when students start to move, we have chosen not to update their position on the map. Therefore, they have to memorize their position using the various landmarks which are present on the map. This is fairly easy in the city center case, but quite a bit more complicated in the residential area, where counting the interaction is about the only possible strategy, as well as making use of global landmarks, like the mountains visible in the far distance.

To reach the target position, students can switch from the immersive view to the map back and forth at any time. However, to force a minimal memorization, we have chosen to restrict to 4 switches (students were warned). If this number is reached, students could not access the map anymore and had to remain in the immersive view of the city. When students think they have reached the target point, they have to validate the position and they got a retroaction: if it is correct, they can move to the next task, if not they are brought back to the starting point (yellow dot) and must start again. After 3 failures, a path is drawn on the ground in the immersive view so that they are guided to the target position. The maximum of 4 switches to the map were counted for all the trials.

This task aims to get students to plan a path between two given positions on the map and carried it out in the immersive environment. We are therefore very close to a real situation when using a map in a real city to go to target place. The use of technology here transposes the task in a virtual environment, but it also allows us some didactical artifices: the fact that the map is not visible simultaneously to the immersive view and the number of switches is limited top 4. Those choices operate as constrains on the didactical situation, proper to create better conditions for acquiring the knowledge we aim at.

## SOME OF STUDENTS' ANSWERS

This task was proposed to 80 students, out of which 77 were finally exploitable. The software keep data from student's activity like the time spend on the map, number of switches, position every second. 58 (respectively 61) students succeeded on the first go in the city center (respectively in the residential area). So altogether, at the end of our engineering, we can see that such a task is successful. Moreover, we want to analyze deeper the results, to do this, we're looking at how students use the map.

First, Table 1 gives the number of wrong answers crossed with the number of switches to the map used over all the trials.

<table><caption>Table 1: Scores for finding a position indicated on the map</caption><thead><tr><th rowspan="2">Nb</th><th colspan="4">Number of wrong</th><th rowspan="2">Total</th><th colspan="4">Number of wrong</th><th rowspan="2">Total</th></tr><tr><th>0</th><th>1</th><th>2</th><th>3</th><th>0</th><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><td>0</td><td>12</td><td></td><td></td><td></td><td>12</td><td>8</td><td></td><td></td><td>8</td></tr><tr><td>1</td><td>12</td><td>2</td><td></td><td></td><td>14</td><td>18</td><td>1</td><td></td><td>19</td></tr><tr><td>2</td><td>15</td><td>1</td><td>1</td><td></td><td>17</td><td>22</td><td>4</td><td></td><td>26</td></tr><tr><td>3</td><td>11</td><td>2</td><td></td><td></td><td>13</td><td>11</td><td>1</td><td>1</td><td>2</td><td>15</td></tr><tr><td>4</td><td>8</td><td>5</td><td>2</td><td>6</td><td>21</td><td>2</td><td>2</td><td>1</td><td>4</td><td>9</td></tr><tr><td>Total</td><td>58</td><td>10</td><td>3</td><td>6</td><td>77</td><td>61</td><td>7</td><td>3</td><td>6</td><td>77</td></tr></tbody></table>
are facing north (the top of the map), in the residential area, they are facing south. So, once in the immersive view, students must choose the correct direction to start their path, without knowing on the map which is their initial orientation. Then, when students start to move, we have chosen not to update their position on the map. Therefore, they have to memorize their position using the various landmarks which are present on the map. This is fairly easy in the city center case, but quite a bit more complicated in the residential area, where counting the interaction is about the only possible strategy, as well as making use of global landmarks, like the mountains visible in the far distance.

To reach the target position, students can switch from the immersive view to the map back and forth at any time. However, to force a minimal memorization, we have chosen to restrict to 4 switches (students were warned). If this number is reached, students could not access the map anymore and had to remain in the immersive view of the city. When students think they have reached the target point, they have to validate the position and they got a retroaction: if it is correct, they can move to the next task, if not they are brought back to the starting point (yellow dot) and must start again. After 3 failures, a path is drawn on the ground in the immersive view so that they are guided to the target position. The maximum of 4 switches to the map were counted for all the trials.

This task aims to get students to plan a path between two given positions on the map and carried it out in the immersive environment. We are therefore very close to a real situation when using a map in a real city to go to target place. The use of technology here transposes the task in a virtual environment, but it also allows us some didactical artifices: the fact that the map is not visible simultaneously to the immersive view and the number of switches is limited top 4. Those choices operate as constrains on the didactical situation, proper to create better conditions for acquiring the knowledge we aim at.

## SOME OF STUDENTS' ANSWERS

This task was proposed to 80 students, out of which 77 were finally exploitable. The software keep data from student's activity like the time spend on the map, number of switches, position every second. 58 (respectively 61) students succeeded on the first go in the city center (respectively in the residential area). So altogether, at the end of our engineering, we can see that such a task is successful. Moreover, we want to analyze deeper the results, to do this, we're looking at how students use the map.

First, Table 1 gives the number of wrong answers crossed with the number of switches to the map used over all the trials.

<table><caption>Table 1: Scores for finding a position indicated on the map</caption><thead><tr><th rowspan="2">Nb</th><th colspan="4">Number of wrong</th><th rowspan="2">Total</th><th colspan="4">Number of wrong</th><th rowspan="2">Total</th></tr><tr><th>0</th><th>1</th><th>2</th><th>3</th><th>0</th><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><td>0</td><td>12</td><td></td><td></td><td></td><td>12</td><td>8</td><td></td><td></td><td>8</td></tr><tr><td>1</td><td>12</td><td>2</td><td></td><td></td><td>14</td><td>18</td><td>1</td><td></td><td>19</td></tr><tr><td>2</td><td>15</td><td>1</td><td>1</td><td></td><td>17</td><td>22</td><td>4</td><td></td><td>26</td></tr><tr><td>3</td><td>11</td><td>2</td><td></td><td></td><td>13</td><td>11</td><td>1</td><td>1</td><td>2</td><td>15</td></tr><tr><td>4</td><td>8</td><td>5</td><td>2</td><td>6</td><td>21</td><td>2</td><td>2</td><td>1</td><td>4</td><td>9</td></tr><tr><td>Total</td><td>58</td><td>10</td><td>3</td><td>6</td><td>77</td><td>61</td><td>7</td><td>3</td><td>6</td><td>77</td></tr></tbody></table>
For both positions 6 students commit 3 failures. However, they are not the same students. Only 1 student commits 3 failures in the 2 cases. In the city center, 12 students managed to position themselves in the right place on the first go without having to switch again to the map. Among these 12 students, 10 also manage to find the position in the residential area on the first go but with switch at least once on the map. On the opposite, among the 8 students who found the position 2 without error and without switching to the map, there were 6 who also found position 1 without error. Only 2 students manage to find the two positions without error and without switching.

Now, if we look at the number of switches between the map and the immersive view, 21 students used 4 switches for position 1 while only 9 needed 4 switches for position 2.

The two tasks, although different, are performed equally well by the students. But do they have identical strategies between the two districts? If we look at the video recordings from the action cameras that the students had on their head, we can analyze some gestural information. Only 57 students out of 77 have a complete video recording of the exercise. Of these 57 students, it appears that 30 students point out intersections at least in one of the two cases (city center and residential area). We can then assume that these students used a numerical strategy, counting intersections to code their path. On another hand, the videos show that 3 students draw the route with their finger on the map. This shows that these students plan their route on the map before they execute it in the immersive environment. Yet, it is not possible to know how they memorize at which crossroad they have to change direction and in which sense. The 24 remaining students are motionless and silent in front of the map for a few minutes then switch to the immersive view to initiate a movement. For these students no hypothesis about their strategy can be made based on their actions.

Let us now go in deeper analyses for the students with a counting strategy: It seems that memorizing information is not the same for all of them and also differ for both situations. Indeed, looking at how these students use the switch to the map, it seems that distinctions appear. Among the 30 students who count the crossroads, only 8 mobilize their first switch to the map under the same conditions in both situations: at the starting point (2 students), on the path (1 student) at a crossroads with change of direction (4 students) or at the target position (1 student). For the 22 other students, the choices are different with almost all possible combinations. For example, 5 students do not need a switch to the map before getting to the target point in the city center, but they need it in the residential area at the start point (2 students), on the path (2 students) or at a crossroads with change of direction (1 student).

To identify whether these variations are also present in the other students, we organized the students according to their use of the switch to the map. We thus measured the difference between the total number of switches mobilized in the city center and the total number of switches in the residential area. It turns out that 3 profiles appear. A first category of 35 students out of the 77 make more switches to the map when they are in the city center. Among these students, at least 15 used counting at least in one case. The second category is made of 27 students who make more switches to the map in the city center than in the residential area. Among them, at least 13 counted the intersections (from what we can infer from the videos). Finally, 15 students use the same number of switches in the two cases. At least 2 of them count the intersections. Thus the 30 students who use a counting strategy are divided into the three categories of students. We can assume that other elements complete this strategy of coding the path when it is carried out in the immersive view.
For both positions 6 students commit 3 failures. However, they are not the same students. Only 1 student commits 3 failures in the 2 cases. In the city center, 12 students managed to position themselves in the right place on the first go without having to switch again to the map. Among these 12 students, 10 also manage to find the position in the residential area on the first go but with switch at least once on the map. On the opposite, among the 8 students who found the position 2 without error and without switching to the map, there were 6 who also found position 1 without error. Only 2 students manage to find the two positions without error and without switching.

Now, if we look at the number of switches between the map and the immersive view, 21 students used 4 switches for position 1 while only 9 needed 4 switches for position 2.

The two tasks, although different, are performed equally well by the students. But do they have identical strategies between the two districts? If we look at the video recordings from the action cameras that the students had on their head, we can analyze some gestural information. Only 57 students out of 77 have a complete video recording of the exercise. Of these 57 students, it appears that 30 students point out intersections at least in one of the two cases (city center and residential area). We can then assume that these students used a numerical strategy, counting intersections to code their path. On another hand, the videos show that 3 students draw the route with their finger on the map. This shows that these students plan their route on the map before they execute it in the immersive environment. Yet, it is not possible to know how they memorize at which crossroad they have to change direction and in which sense. The 24 remaining students are motionless and silent in front of the map for a few minutes then switch to the immersive view to initiate a movement. For these students no hypothesis about their strategy can be made based on their actions.

Let us now go in deeper analyses for the students with a counting strategy: It seems that memorizing information is not the same for all of them and also differ for both situations. Indeed, looking at how these students use the switch to the map, it seems that distinctions appear. Among the 30 students who count the crossroads, only 8 mobilize their first switch to the map under the same conditions in both situations: at the starting point (2 students), on the path (1 student) at a crossroads with change of direction (4 students) or at the target position (1 student). For the 22 other students, the choices are different with almost all possible combinations. For example, 5 students do not need a switch to the map before getting to the target point in the city center, but they need it in the residential area at the start point (2 students), on the path (2 students) or at a crossroads with change of direction (1 student).

To identify whether these variations are also present in the other students, we organized the students according to their use of the switch to the map. We thus measured the difference between the total number of switches mobilized in the city center and the total number of switches in the residential area. It turns out that 3 profiles appear. A first category of 35 students out of the 77 make more switches to the map when they are in the city center. Among these students, at least 15 used counting at least in one case. The second category is made of 27 students who make more switches to the map in the city center than in the residential area. Among them, at least 13 counted the intersections (from what we can infer from the videos). Finally, 15 students use the same number of switches in the two cases. At least 2 of them count the intersections. Thus the 30 students who use a counting strategy are divided into the three categories of students. We can assume that other elements complete this strategy of coding the path when it is carried out in the immersive view.
## CONCLUSION

As we mentioned it, one of our main initial questions was to know how students plan a path between two positions and how they use the access to the map. It is difficult to answer these questions in the sense that in our experiment students were on their own and therefore did not speak. However, regarding the use of the map, it seems that for the vast majority, it is a real support to plan their route and then realize it in the immersive environment. Moreover, it seems that students are able to use the map adequately to reach the target position. It also seems that having access to the map contributes to the realization of the planned path, even if their position is not updated on the map. It can thus be assumed that students are able to memorize their position as they move in the immersive environment. However, other parameters can also come into play.

Concerning the path planning procedure, even if our data are limited, it seems that several students use the map to plan their route and code it by counting the intersections. Moreover, even if the counting strategy is valid both in the city center and in the residential area, it seems that the specificities of the area and the number of landmarks influence the realization of the planned path and the use of this counting strategy. One might wonder what influence play the landmarks in the city center and to what extent they can be distractors or aids when traveling?

For the students who mobilized more switches to the map in the city center, one could assume that this provided a first experience which influenced what they did in the second case. They are thus more prepared and anticipate their path planning better when in the residential area.

The city center has many landmarks that can be used to "control" the path. Students can thus use switches to the map at specific landmarks to make sure that they are on the correct path. Moreover, the absence of local landmarks in the residential area can have two opposing effects on students. On one side, students may feel less tempted to frequently check if they are on the right path by switching to the map. On the other side, since students do not have reference points in order to control their path, they can use more switches to the map to control their progress. Thus, they compensate the lack of landmarks to control their path by switching more often to the map.

This gives a very limited view of all what we tested over the three years in our experiment. This study on path planning especially echoes the study of another task, that we experimented with our students; indicating on the map landmarks that were seen in the immersive. These two tasks are dual in the sense that they are similar but with the inversion of the role of the reference frames. We are now finalizing the analyses of most of the various activities we experimented over the three years. Our next goal will be to select the most significant activities in order to design a sequence that could be implemented in classes and to organize an in-service teachers' training to help teachers invest in the use of SPAGEO city in the real conditions of everyday teaching,

## References

* Bell, S. (2002). Spatial cognition and scale: A child's perspective. *Journal of Environmental Psychology*, 22(1-2), 9-27. https://doi.org/10.1006/jevp.2002.0250

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

* Bessot, A., Deprez, S., Eberhard, M., & Gomas, B. (1992). Le sens des graphismes techniques dans la formation professionnelle de base pour les travailleurs du bâtiment : Une approche didactique. *Education Permanente*, 111, 87-105.
## CONCLUSION

As we mentioned it, one of our main initial questions was to know how students plan a path between two positions and how they use the access to the map. It is difficult to answer these questions in the sense that in our experiment students were on their own and therefore did not speak. However, regarding the use of the map, it seems that for the vast majority, it is a real support to plan their route and then realize it in the immersive environment. Moreover, it seems that students are able to use the map adequately to reach the target position. It also seems that having access to the map contributes to the realization of the planned path, even if their position is not updated on the map. It can thus be assumed that students are able to memorize their position as they move in the immersive environment. However, other parameters can also come into play.

Concerning the path planning procedure, even if our data are limited, it seems that several students use the map to plan their route and code it by counting the intersections. Moreover, even if the counting strategy is valid both in the city center and in the residential area, it seems that the specificities of the area and the number of landmarks influence the realization of the planned path and the use of this counting strategy. One might wonder what influence play the landmarks in the city center and to what extent they can be distractors or aids when traveling?

For the students who mobilized more switches to the map in the city center, one could assume that this provided a first experience which influenced what they did in the second case. They are thus more prepared and anticipate their path planning better when in the residential area.

The city center has many landmarks that can be used to "control" the path. Students can thus use switches to the map at specific landmarks to make sure that they are on the correct path. Moreover, the absence of local landmarks in the residential area can have two opposing effects on students. On one side, students may feel less tempted to frequently check if they are on the right path by switching to the map. On the other side, since students do not have reference points in order to control their path, they can use more switches to the map to control their progress. Thus, they compensate the lack of landmarks to control their path by switching more often to the map.

This gives a very limited view of all what we tested over the three years in our experiment. This study on path planning especially echoes the study of another task, that we experimented with our students; indicating on the map landmarks that were seen in the immersive. These two tasks are dual in the sense that they are similar but with the inversion of the role of the reference frames. We are now finalizing the analyses of most of the various activities we experimented over the three years. Our next goal will be to select the most significant activities in order to design a sequence that could be implemented in classes and to organize an in-service teachers' training to help teachers invest in the use of SPAGEO city in the real conditions of everyday teaching,

## References

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

* Bessot, A., Deprez, S., Eberhard, M., & Gomas, B. (1992). Le sens des graphismes techniques dans la formation professionnelle de base pour les travailleurs du bâtiment : Une approche didactique. *Education Permanente*, 111, 87-105.
Brousseau, G. (1983). Étude de questions d’enseignement. L’exemple de la géométrie. *Séminaire de Didactique des Mathématiques et de l’Informatique*.

Carroll, J. B. (1993). *Human cognitive abilities. A survey of factor-analytic studies* (1st ed.). Cambridge University Press.

Chen, J., & Stanney, K. (2002). A theoretical model of wayfinding in a virtual environments: Proposed strategies for navigational aiding. *Presence: Teleoperators and Virtual Environments, 8*(6), 671-685.

Coutat, S., & Berney, S. (2023). Connaissances spatiales et habiletés visuo-spatiales – croisement de deux paradigmes pour mieux comprendre les enjeux d’enseignement et d’apprentissage en géométrie. In F. Vandebrouck & M.-L. Gardes (Éds.), *Nouvelles persepctives en didactique des mathématiques : La preuve, la modélisation et les technologies numériques (volume 2)* (p. 59-68). IREM de Paris. https://hal.science/hal-04050994

Coutat, S., Dorier, J.-L., Frauchiger, T., & Matri, S. (2022). Analyse de messages écrits dans une situation de communication de trajets suivis dans une ville virtuelle. [Communication orale] *In Acfas Congress*. 9 mai 2022.

Coutat, S., Dorier, J.-L., & Matri, S. (to be published). Learning spatial orientation in primary school with use of a virtual city. In *Proceedings of 13th Congress of the European Society for Research in Mathematics Education (CERME13)*. Budapest (Hungary) 10 - 14 July 2023.

Coutat, S., Matri, S., & Vendeira, C. (2022). Puzzle de formes géométriques et navigation dans une ville virtuelle, deux activités distinctes pour des apprentissages communs au cycle 1. In *Actes du 5ème Colloque Des Didactiques Disciplinaires* (pp. 234-241).

Duroisin, N. (2015). *Quelle place pour les apprentissages spatiaux à l'école ? Etude expérimentale du développement des compétences spatiales des élèves âgés de 6 à 15 ans*. Université de Mons, Faculté de psychologie et sciences de l'éducation.

Frauchiger, T., Matri, S., Coutat, S., & Dorier, J.-L. (to be published). Typologie de messages écrits pour communiquer un trajet dans une ville virtuelle. In A. Adihou (Éd.), *Actes du colloque Espace Mathématique Francophone EMF2022 – Cotonou (Bénin)- 12-16 Décembre 2002*.

Golledge, R. G. (1999). *Wayfinding behavior: Cognitive mapping and other spatial processes*. JHU Press.

Marchand, P. (2020). Quelques assises pour valoriser le développement des connaissances spatiales à l'école primaire. *Recherches en Didactique des Mathématiques, 40*(2), 135-178.

Matri, S., Dorier, J.-L., & Diouf, F.-M. (2023). Teaching sequences on spatial orientation using a virtual city on computer in grades 2-4 in Geneva. [Workshop] *In EARLI Congress*. Thessaloniki, Greece, August 22-26 2023.

Pennel, I., Coello, Y., & Orliaguet, J.-P. (2001). Les relations perception-actions face aux nouvelles technologies : Adaptation et apprentissage en situation de télémanipulation. In Y. Coello & J. Honoré (Éds.), *Percevoir, s'orienter et agir dans l'espace: approche pluridisciplinaire des relations perception-action*. Solal.

Pick, H. L., Wagner, D., Garing, A. E., & Rieser, J. J. (1999). The Recalibration of Rotational Locomotion. *Human Perception and Performance, 25*(5), 1179-1188.

Ruginski, I. T., Creem-Regehr, S. H., Stefanucci, J. K., & Cashdan, E. (2019). GPS use negatively affects environmental learning through spatial transformation abilities. *Journal of Environmental Psychology, 64*, 12-20.

Siegel, A. W., & White, S. H. (1975). The development of spatial representations of large-scale environments. *European Journal of Cognitive Psychology, 10*, 9-55. https://doi.org/10.1016/s0065-2407(08)60007-5

Spence, I., & Feng, J. (2010). Video games and spatial cognition. *Review of General Psychology, 14*(2), 92-104.
Brousseau, G. (1983). Étude de questions d’enseignement. L’exemple de la géométrie. *Séminaire de Didactique des Mathématiques et de l’Informatique*.

Carroll, J. B. (1993). *Human cognitive abilities. A survey of factor-analytic studies* (1st ed.). Cambridge University Press.

Chen, J., & Stanney, K. (2002). A theoretical model of wayfinding in a virtual environments: Proposed strategies for navigational aiding. *Presence: Teleoperators and Virtual Environments, 8*(6), 671-685.

Coutat, S., & Berney, S. (2023). Connaissances spatiales et habiletés visuo-spatiales – croisement de deux paradigmes pour mieux comprendre les enjeux d’enseignement et d’apprentissage en géométrie. In F. Vandebrouck & M.-L. Gardes (Éds.), *Nouvelles persepctives en didactique des mathématiques : La preuve, la modélisation et les technologies numériques (volume 2)* (p. 59-68). IREM de Paris. https://hal.science/hal-04050994

Coutat, S., Dorier, J.-L., Frauchiger, T., & Matri, S. (2022). Analyse de messages écrits dans une situation de communication de trajets suivis dans une ville virtuelle. [Communication orale] *In Acfas Congress*. 9 mai 2022.

Coutat, S., Dorier, J.-L., & Matri, S. (to be published). Learning spatial orientation in primary school with use of a virtual city. In *Proceedings of 13th Congress of the European Society for Research in Mathematics Education (CERME13)*. Budapest (Hungary) 10 - 14 July 2023.

Coutat, S., Matri, S., & Vendeira, C. (2022). Puzzle de formes géométriques et navigation dans une ville virtuelle, deux activités distinctes pour des apprentissages communs au cycle 1. In *Actes du 5ème Colloque Des Didactiques Disciplinaires* (pp. 234-241).

Duroisin, N. (2015). *Quelle place pour les apprentissages spatiaux à l'école ? Etude expérimentale du développement des compétences spatiales des élèves âgés de 6 à 15 ans*. Université de Mons, Faculté de psychologie et sciences de l'éducation.

Frauchiger, T., Matri, S., Coutat, S., & Dorier, J.-L. (to be published). Typologie de messages écrits pour communiquer un trajet dans une ville virtuelle. In A. Adihou (Éd.), *Actes du colloque Espace Mathématique Francophone EMF2022 – Cotonou (Bénin)- 12-16 Décembre 2002*.

Golledge, R. G. (1999). *Wayfinding behavior: Cognitive mapping and other spatial processes*. JHU Press.

Marchand, P. (2020). Quelques assises pour valoriser le développement des connaissances spatiales à l'école primaire. *Recherches en Didactique des Mathématiques, 40*(2), 135-178.

Matri, S., Dorier, J.-L., & Diouf, F.-M. (2023). Teaching sequences on spatial orientation using a virtual city on computer in grades 2-4 in Geneva. [Workshop] *In EARLI Congress*. Thessaloniki, Greece, August 22-26 2023.

Pennel, I., Coello, Y., & Orliaguet, J.-P. (2001). Les relations perception-actions face aux nouvelles technologies : Adaptation et apprentissage en situation de télémanipulation. In Y. Coello & J. Honoré (Éds.), *Percevoir, s'orienter et agir dans l'espace: approche pluridisciplinaire des relations perception-action*. Solal.

Pick, H. L., Wagner, D., Garing, A. E., & Rieser, J. J. (1999). The Recalibration of Rotational Locomotion. *Human Perception and Performance, 25*(5), 1179-1188.

Ruginski, I. T., Creem-Regehr, S. H., Stefanucci, J. K., & Cashdan, E. (2019). GPS use negatively affects environmental learning through spatial transformation abilities. *Journal of Environmental Psychology, 64*, 12-20.

Siegel, A. W., & White, S. H. (1975). The development of spatial representations of large-scale environments. *European Journal of Cognitive Psychology, 10*, 9-55. https://doi.org/10.1016/s0065-2407(08)60007-5

Spence, I., & Feng, J. (2010). Video games and spatial cognition. *Review of General Psychology, 14*(2), 92-104.
# LEARNING COORDINATE GEOMETRY WITH SCRATCH: TASK DESIGN FROM AN EMBODIED AND APOS APPROACH

Zhihao Cui<fnref n="1" />, Oi-Lam Ng<fnref n="1" />, Chung Man Koo<fnref n="2" />

This paper presents the designs of a series programming-based mathematical tasks targeted on coordinate geometry learning. Grounded in APOS theory and embodied cognition, we elaborate on how the task activities via Scratch, a block-based visual programming environment, support students' mathematical development alongside programming concepts and skills. We highlight the embodied experience prompted by the task design, as well as the progression from concrete actions to abstract mathematical thinking, thereby fostering a rich and holistic understanding of coordinate geometry.

## INTRODUCTION

Our work builds on the second author's previously developed conception of “learning as Making” (Ng & Chan, 2019; Ng & Ferrara, 2020) to envision a computationally enhanced mathematics education. In particular, we were interested in screen-based programming as a form of digital making to support mathematical problem-solving in fundamental ways. Programming-based mathematical activities involves students' active creation of programming artefacts; it transforms mathematical problem-solving from merely using formulae and performing arithmetic calculations procedurally to endangering CT concepts and practices such as sequences, variable-naming, abstraction, algorithmic thinking, decomposing, and iterating, through which scientific inquiry, mathematical thinking, and engineering design can also be exhibited as integrated STEM learning (Ng & Cui, 2021).

In the past three years, the research team as led by the second author has made significant efforts and progress in designing problem-based mathematical activities with content appropriate to senior primary and junior secondary mathematics curricula in Hong Kong. This paper aims to instantiate such an exploration by delving into the intersection between coordinate geometry learning and computer programming. Specifically, we discuss our recent and novel design of regarding Scratch, a block-based programming environment, as a boundary object (Ng et al., 2023a), which mediates between coordinate geometry and computer science concepts. In this paper, we discuss the designs of these programming-based mathematical tasks, which aimed to support students' mathematical development alongside programming concepts and skills while using Scratch as a necessary cognitive and physical tool in the task solutions. Our aim for this paper is to facilitate discussion regarding the potential for a pedagogical approach which conceptualizes embodiment and the progression from concrete actions to abstract mathematical thinking in coordinate geometry via Scratch.

## THEORETICAL FRAMEWORK

### APOS for the integration of programming and mathematics

The APOS theory, an acronym for Action, Process, Object, and Schema, is a cognitive framework

<footnote n="1">¹ Department of Curriculum and Instruction, The Chinese University of Hong Kong</footnote>

<footnote n="2">² Hong Kong Taoist Association The Yuen Yuen Institute No. 2 Secondary School</footnote>
# LEARNING COORDINATE GEOMETRY WITH SCRATCH: TASK DESIGN FROM AN EMBODIED AND APOS APPROACH

Zhihao Cui<fnref n="1" />, Oi-Lam Ng<fnref n="1" />, Chung Man Koo<fnref n="2" />

This paper presents the designs of a series programming-based mathematical tasks targeted on coordinate geometry learning. Grounded in APOS theory and embodied cognition, we elaborate on how the task activities via Scratch, a block-based visual programming environment, support students' mathematical development alongside programming concepts and skills. We highlight the embodied experience prompted by the task design, as well as the progression from concrete actions to abstract mathematical thinking, thereby fostering a rich and holistic understanding of coordinate geometry.

## INTRODUCTION

Our work builds on the second author's previously developed conception of “learning as Making” (Ng & Chan, 2019; Ng & Ferrara, 2020) to envision a computationally enhanced mathematics education. In particular, we were interested in screen-based programming as a form of digital making to support mathematical problem-solving in fundamental ways. Programming-based mathematical activities involves students’ active creation of programming artefacts; it transforms mathematical problem-solving from merely using formulae and performing arithmetic calculations procedurally to endangering CT concepts and practices such as sequences, variable-naming, abstraction, algorithmic thinking, decomposing, and iterating, through which scientific inquiry, mathematical thinking, and engineering design can also be exhibited as integrated STEM learning (Ng & Cui, 2021).

In the past three years, the research team as led by the second author has made significant efforts and progress in designing problem-based mathematical activities with content appropriate to senior primary and junior secondary mathematics curricula in Hong Kong. This paper aims to instantiate such an exploration by delving into the intersection between coordinate geometry learning and computer programming. Specifically, we discuss our recent and novel design of regarding Scratch, a block-based programming environment, as a boundary object (Ng et al., 2023a), which mediates between coordinate geometry and computer science concepts. In this paper, we discuss the designs of these programming-based mathematical tasks, which aimed to support students’ mathematical development alongside programming concepts and skills while using Scratch as a necessary cognitive and physical tool in the task solutions. Our aim for this paper is to facilitate discussion regarding the potential for a pedagogical approach which conceptualizes embodiment and the progression from concrete actions to abstract mathematical thinking in coordinate geometry via Scratch.

## THEORETICAL FRAMEWORK

### APOS for the integration of programming and mathematics

The APOS theory, an acronym for Action, Process, Object, and Schema, is a cognitive framework

<footnote n="1">¹ Department of Curriculum and Instruction, The Chinese University of Hong Kong</footnote>

<footnote n="2">² Hong Kong Taoist Association The Yuen Yuen Institute No. 2 Secondary School</footnote>
from the constructivist tradition that elucidates the stages of thinking and learning mathematical concepts. Originating from the research of Dubinsky (1991), this approach postulates that a learner's comprehension of mathematical ideas evolves through distinct phases. It suggests that, initially, individuals perform algorithmic actions (Action) without intrinsic understanding. As they progress, these actions are internalized into coherent, repeatable patterns (Process). Further abstraction sees these processes conceived as holistic, manipulable entities (Object). Ultimately, learners integrate these objects into broader interconnected networks of understanding (Schema). Further supported by studies such as Arnon et al. (2014), this theory offers educators a robust framework for assessing and facilitating mathematical learning.

Taking consideration of a programming context, we suggest that the APOS approach is particularly suitable for the integration of programming and mathematics. We take the block-based programming environment, Scratch, as an example. Programming always starts with concrete actions. Users drag and drop blocks, each of the motion corresponding to specific commands. Users can see the output of these commands without further explaining, which is exactly the action stage. By exploring different commands and construct different codes, the users may see some common features or patterns between the commands and their outputs, which align with the transition from using individual commands (actions) to forming the flow and logic of codes (process). Meanwhile, pattern recognition is an important construct of computational thinking (Weintrop & Wilensky, 2015); it supports the development to process stage. Advanced features in Scratch, such as creating custom blocks (My Block) or cloning sprites allow users to encapsulate scripts or behaviors as singular objects. This movement from seeing codes as a process to understanding it as modular, reusable objects mirrors the Object stage in APOS. As users become proficient in programming, they begin to see the interconnectedness of various programming concepts, such as the integration of loops, and conditionals with events and variables. Moreover, they might advance their thinking to conceptualize broader projects through "remixing" (Brennan & Resnick, 2012). This holistic understanding aligns with the Schema stage of APOS.

## Embodied cognition

Embodied cognition is a transformative perspective in cognitive science which posits that the mind is not just a product of the brain but is deeply intertwined with the body's interactions with its environment. This theory challenges traditional cognitive views that treat mental processes as abstract and detached from physical experiences. Grounded in the works of scholars like Varela et al. (1991) and furthered by Clark (1999), the essence of embodied cognition is the assertion that our bodily experiences, sensorimotor activities, and environmental interactions play a crucial role in shaping cognitive processes, from perception to reasoning.

Recent studies highlight the successful incorporation of embodied learning in various classroom environments, including mathematics classrooms. For example, scholars have introduced both individualized and collective interventions that encourage bodily engagements in the learning process (e.g., Nemirovsky et al., 2020), and to explore abstract concepts physically (e.g., Ng & Ferrara, 2020). Designing learning environments and tasks that effectively create the conditions for embodied learning is receiving increasing attentions. Abrahamson (2014) introduced an action-based embodied design for mathematics learning by intertwining both intuitive kinesthetic interactions and scientific forms. This approach aligns with Dourish (2001)'s views on the third wave human-computer
from the constructivist tradition that elucidates the stages of thinking and learning mathematical concepts. Originating from the research of Dubinsky (1991), this approach postulates that a learner's comprehension of mathematical ideas evolves through distinct phases. It suggests that, initially, individuals perform algorithmic actions (Action) without intrinsic understanding. As they progress, these actions are internalized into coherent, repeatable patterns (Process). Further abstraction sees these processes conceived as holistic, manipulable entities (Object). Ultimately, learners integrate these objects into broader interconnected networks of understanding (Schema). Further supported by studies such as Arnon et al. (2014), this theory offers educators a robust framework for assessing and facilitating mathematical learning.

Taking consideration of a programming context, we suggest that the APOS approach is particularly suitable for the integration of programming and mathematics. We take the block-based programming environment, Scratch, as an example. Programming always starts with concrete actions. Users drag and drop blocks, each of the motion corresponding to specific commands. Users can see the output of these commands without further explaining, which is exactly the action stage. By exploring different commands and construct different codes, the users may see some common features or patterns between the commands and their outputs, which align with the transition from using individual commands (actions) to forming the flow and logic of codes (process). Meanwhile, pattern recognition is an important construct of computational thinking (Weintrop & Wilensky, 2015); it supports the development to process stage. Advanced features in Scratch, such as creating custom blocks (My Block) or cloning sprites allow users to encapsulate scripts or behaviors as singular objects. This movement from seeing codes as a process to understanding it as modular, reusable objects mirrors the Object stage in APOS. As users become proficient in programming, they begin to see the interconnectedness of various programming concepts, such as the integration of loops, and conditionals with events and variables. Moreover, they might advance their thinking to conceptualize broader projects through "remixing" (Brennan & Resnick, 2012). This holistic understanding aligns with the Schema stage of APOS.

## Embodied cognition

Embodied cognition is a transformative perspective in cognitive science which posits that the mind is not just a product of the brain but is deeply intertwined with the body's interactions with its environment. This theory challenges traditional cognitive views that treat mental processes as abstract and detached from physical experiences. Grounded in the works of scholars like Varela et al. (1991) and furthered by Clark (1999), the essence of embodied cognition is the assertion that our bodily experiences, sensorimotor activities, and environmental interactions play a crucial role in shaping cognitive processes, from perception to reasoning.

Recent studies highlight the successful incorporation of embodied learning in various classroom environments, including mathematics classrooms. For example, scholars have introduced both individualized and collective interventions that encourage bodily engagements in the learning process (e.g., Nemirovsky et al., 2020), and to explore abstract concepts physically (e.g., Ng & Ferrara, 2020). Designing learning environments and tasks that effectively create the conditions for embodied learning is receiving increasing attentions. Abrahamson (2014) introduced an action-based embodied design for mathematics learning by intertwining both intuitive kinesthetic interactions and scientific forms. This approach aligns with Dourish (2001)'s views on the third wave human-computer
interaction (HCI), focusing on embodied interaction with a new wave of digital tools, such as wearable devices and augmented reality. In this study, we adopt the perspectives of embodied cognition and HCI to create and provide learning conditions fundamentally anchored in real-time and space, where embodiment is not just central but forms the very essence of the learning experience.

# TASK DESIGN WITH AN EMBODIED AND APOS APPROACH

We will first briefly introduce the features of Scratch which are relevant to the study. Then, we present four sequential examples of our designed tasks of integrating Scratch into coordinate geometry learning and explain them from both embodied and APOS perspectives.

## Introduction to Scratch

In the vast realm of programming languages and environments, Scratch stands out with its unique approach to introducing computational concepts to novices. Central to its distinctiveness are two interactive features (Figure 1): the stage and the characters (often referred to as " sprites "). These features not only set Scratch apart from traditional text-based programming languages but also provide an intuitive platform that resonates with the way humans naturally process information—visually and narratively. While most programming environments focus on abstract code structures and command-line outputs, Scratch embraces a more tangible (drag-and-drop), visual, and story-driven paradigm.

Stage is a space where commands can be executed as visual repercussions. It is not just a blank canvas; instead, it is expressed as a coordinate system. Centered at (0,0), the stage spans its x-coordinates horizontally (-240 ≤ x ≤ 240) and y-coordinates vertically -180 ≤ y ≤ 180. Importantly, the visual and dynamic representations of programming (Ng et al., 2023b) combined with the coordinate system inherently makes it possible to teaching and learning coordinate geometry.

On the other hand, the sprites, or the use of characters, can make the programming outputs animate and actionable. These characters can be anything—a cat, a ball, a person, or even custom creations. Each sprite operates like an independent actor by executing the scripts that it corresponds. They can interact with one another, react to user inputs, or even change their appearance and behavior based on the code behind them. By programming these sprites, learners are essentially directing a play, making coding an act of storytelling. The affordances of sprites lie not just in their dynamic nature, but crucially, in their ability to prompt embodied learning. As learners code the behaviours of sprites, they are not merely executing abstract commands; they're engaging in a form of digital puppetry, imbuing characters with life and narrative. This act of animating sprites fosters a deeper, more visceral understanding of code, as users see and "feel" the effects of their programming.

<figure><img src="image_145.png" /><figcaption>Figure 1: Stage and character of Scratch</figcaption></figure>
interaction (HCI), focusing on embodied interaction with a new wave of digital tools, such as wearable devices and augmented reality. In this study, we adopt the perspectives of embodied cognition and HCI to create and provide learning conditions fundamentally anchored in real-time and space, where embodiment is not just central but forms the very essence of the learning experience.

# TASK DESIGN WITH AN EMBODIED AND APOS APPROACH

We will first briefly introduce the features of Scratch which are relevant to the study. Then, we present four sequential examples of our designed tasks of integrating Scratch into coordinate geometry learning and explain them from both embodied and APOS perspectives.

## Introduction to Scratch

In the vast realm of programming languages and environments, Scratch stands out with its unique approach to introducing computational concepts to novices. Central to its distinctiveness are two interactive features (Figure 1): the stage and the characters (often referred to as " sprites "). These features not only set Scratch apart from traditional text-based programming languages but also provide an intuitive platform that resonates with the way humans naturally process information—visually and narratively. While most programming environments focus on abstract code structures and command-line outputs, Scratch embraces a more tangible (drag-and-drop), visual, and story-driven paradigm.

Stage is a space where commands can be executed as visual repercussions. It is not just a blank canvas; instead, it is expressed as a coordinate system. Centered at (0,0), the stage spans its x-coordinates horizontally (-240 ≤ x ≤ 240) and y-coordinates vertically -180 ≤ y ≤ 180. Importantly, the visual and dynamic representations of programming (Ng et al., 2023b) combined with the coordinate system inherently makes it possible to teaching and learning coordinate geometry.

On the other hand, the sprites, or the use of characters, can make the programming outputs animate and actionable. These characters can be anything—a cat, a ball, a person, or even custom creations. Each sprite operates like an independent actor by executing the scripts that it corresponds. They can interact with one another, react to user inputs, or even change their appearance and behavior based on the code behind them. By programming these sprites, learners are essentially directing a play, making coding an act of storytelling. The affordances of sprites lie not just in their dynamic nature, but crucially, in their ability to prompt embodied learning. As learners code the behaviours of sprites, they are not merely executing abstract commands; they're engaging in a form of digital puppetry, imbuing characters with life and narrative. This act of animating sprites fosters a deeper, more visceral understanding of code, as users see and "feel" the effects of their programming.

<figure><img src="image_152.png" /><figcaption>Figure 1: Stage and character of Scratch</figcaption></figure>
## Task 1: From movement to coordinate system

In the following sections, we illustrate four task designs that uniquely support mathematical thinking in the area of coordinate geometry, specifically in the junior high school level. The aim of the first task is to familiarize learners with the Scratch environment, particularly the stage, and to introduce the concept of movement in relation to the coordinate system.

*Action*. First, we would guide the student to use three different ways of cause movement of a sprite (Figure 2a): move towards certain direction; change x or y; go to x, y.

*Process*. There are two sub-tasks. First, we challenge students to predict where the sprite will land if given a set of coordinates. Besides, we ask them to describe the sprite's movement in relation to the stage's grid. They will observe how the three ways to move are interchangeable. Second, students are guided to use as many ways as possible to draw a right triangle and a square (Figure 2b-c).

*Object and schema extensions*. We introduce “My block” to chunk the drawing of a right triangle and a square, so that we can easily draw these geometric shapes by recalling the function, “My Block,” and can draw them at different location of the canvas. We would encourage students to visualize the entire journey of the sprite, understanding its path as a holistic representation of a sequence of coordinates. Finally, we prompt students to abstract the concept that any feature of the geometric shapes in the stage are related to and can be represented with coordinates.

*Embodiment embedded*. Throughout the task, we would encourage students to map the movement of the “Sprite” with their physical movement such as gesture or moving in the classroom. For example, in the task of using multiple ways to draw a square, figure 2d presents three different common approaches. Students can simulate the action of move and turn (such as walking) and moving left and right but keep their heading unchanged (change x or y).

<figure><img src="image_146.png" /><figcaption>Figure 2: Different ways of movement and construct simple geometric shapes</figcaption></figure>

## Task 2: Parallel lines and slope as paths

Drawing upon the foundational understanding from Task 1, our next exploration dives deeper into the nuances of slopes and parallel lines in coordinate geometry through an embodied programming experience in the *xy*-coordinate stage in Scratch.

*Action*. We will ask students to draw a right triangle using two characters (Figure 3a); for example, the cat represents the right-angle edge and the bear for the hypotenuse. The key code for the bear is “point towards [cat]”, which can determine the direction of the bear’s movement.

*Process*. There are three sub-tasks. First, we will prompt the students to manipulate the triangles they have created. The students will need to change the starting and final positions of the sprites by using the codes “go to x, y” and “move [ ] steps”. They will observe how the direction of the bear changes according to the cat’s movement. We will also prompt them to try negative input values (Figure 3b).
**Task 1: From movement to coordinate system**

In the following sections, we illustrate four task designs that uniquely support mathematical thinking in the area of coordinate geometry, specifically in the junior high school level. The aim of the first task is to familiarize learners with the Scratch environment, particularly the stage, and to introduce the concept of movement in relation to the coordinate system.

*Action*. First, we would guide the student to use three different ways of cause movement of a sprite (Figure 2a): move towards certain direction; change x or y; go to x, y.

*Process*. There are two sub-tasks. First, we challenge students to predict where the sprite will land if given a set of coordinates. Besides, we ask them to describe the sprite's movement in relation to the stage's grid. They will observe how the three ways to move are interchangeable. Second, students are guided to use as many ways as possible to draw a right triangle and a square (Figure 2b-c).

*Object and schema extensions*. We introduce “My block” to chunk the drawing of a right triangle and a square, so that we can easily draw these geometric shapes by recalling the function, “My Block,” and can draw them at different location of the canvas. We would encourage students to visualize the entire journey of the sprite, understanding its path as a holistic representation of a sequence of coordinates. Finally, we prompt students to abstract the concept that any feature of the geometric shapes in the stage are related to and can be represented with coordinates.

*Embodiment embedded*. Throughout the task, we would encourage students to map the movement of the “Sprite” with their physical movement such as gesture or moving in the classroom. For example, in the task of using multiple ways to draw a square, figure 2d presents three different common approaches. Students can simulate the action of move and turn (such as walking) and moving left and right but keep their heading unchanged (change x or y).

<figure><img src="image_153.png" /><figcaption>Figure 2: Different ways of movement and construct simple geometric shapes</figcaption></figure>

**Task 2: Parallel lines and slope as paths**

Drawing upon the foundational understanding from Task 1, our next exploration dives deeper into the nuances of slopes and parallel lines in coordinate geometry through an embodied programming experience in the *xy*-coordinate stage in Scratch.

*Action*. We will ask students to draw a right triangle using two characters (Figure 3a); for example, the cat represents the right-angle edge and the bear for the hypotenuse. The key code for the bear is “point towards [cat]”, which can determine the direction of the bear’s movement.

*Process*. There are three sub-tasks. First, we will prompt the students to manipulate the triangles they have created. The students will need to change the starting and final positions of the sprites by using the codes “go to x, y” and “move [ ] steps”. They will observe how the direction of the bear changes according to the cat’s movement. We will also prompt them to try negative input values (Figure 3b).
<figure><img src="image_147.png" /><figcaption>Figure 3. Different movement settings and directions of hypotenuse</figcaption></figure>

Second, the students will draw another right triangle upon step 1 (Figure 4a). There are two requirements for the second triangle: (1) the path lengths must be different from the first one, and (2) the bear should move in the same direction as the first. A sample set of codes is presented in Figure 4b. When following the codes, there will be two cycles of sequential movements of “cat run and rise” and “bear climbing.” Students may try different value settings and observe the “bear direction”. Finally, the students are challenged to draw two bears’ movement at different starting locations such that the two bears will never meet if they continue the movements in the same direction (Figure 4c).

<figure><img src="image_148.png" /><figcaption>Figure 4. Different size and position of triangles with same slope</figcaption></figure>

These tasks were meant to prompt the students to reflect on what lengths of paths travelled (the “rise” and “run”) would make the triangles similar (or the lines parallel). In other words, they are intended to foster the mental construction of the concept of slope in a few ways: (a) Slope determines the direction of a line. Defining a “rise” and a “run” determines the slope of a line; (b) three points are collinear if they have the same slope; (c) line segments of different lengths at different locations can have the same slope, making them parallel to each other; (d) slope is determined by the ratio of the changes in y and x, not the absolute magnitude; and (e) positive and negative changes in y and x make positive and negative slopes. A final extension task can be to explore the relationship between slopes of horizontal and vertical lines, as well as slopes of perpendicular lines. They can do so by being tasked with drawing different paths of the bear that are horizontal, vertical, and perpendicular.

*Embodiment embedded.* We will connect the cat’s movement with real-life situation such as staircases, in which “rise” and “run” are embedded. Also, we will connect to the bear’s movement with real-life situation such as walking up/down a hill, in which “rise” and “run” are embedded. We will draw attention to the embodied experience of how they go up or down, forward or backward, as constructing the real-life meanings of slope, and what makes it positive or negative.

**Task 3: Distance formula and pythagorean theorem as a racing game**

As we continue the journey through the world of coordinate geometry using Scratch, the next waypoint focuses on two fundamental concepts: position and distance. This task aims to introduce and deepen learners' understanding of the relationship between position on the Scratch stage and the
<figure><img src="image_154.png" /><figcaption>Figure 3. Different movement settings and directions of hypotenuse</figcaption></figure>

Second, the students will draw another right triangle upon step 1 (Figure 4a). There are two requirements for the second triangle: (1) the path lengths must be different from the first one, and (2) the bear should move in the same direction as the first. A sample set of codes is presented in Figure 4b. When following the codes, there will be two cycles of sequential movements of “cat run and rise” and “bear climbing.” Students may try different value settings and observe the “bear direction”. Finally, the students are challenged to draw two bears’ movement at different starting locations such that the two bears will never meet if they continue the movements in the same direction (Figure 4c).

<figure><img src="image_155.png" /><figcaption>Figure 4. Different size and position of triangles with same slope</figcaption></figure>

These tasks were meant to prompt the students to reflect on what lengths of paths travelled (the “rise” and “run”) would make the triangles similar (or the lines parallel). In other words, they are intended to foster the mental construction of the concept of slope in a few ways: (a) Slope determines the direction of a line. Defining a “rise” and a “run” determines the slope of a line; (b) three points are collinear if they have the same slope; (c) line segments of different lengths at different locations can have the same slope, making them parallel to each other; (d) slope is determined by the ratio of the changes in y and x, not the absolute magnitude; and (e) positive and negative changes in y and x make positive and negative slopes. A final extension task can be to explore the relationship between slopes of horizontal and vertical lines, as well as slopes of perpendicular lines. They can do so by being tasked with drawing different paths of the bear that are horizontal, vertical, and perpendicular.

*Embodiment embedded.* We will connect the cat’s movement with real-life situation such as staircases, in which “rise” and “run” are embedded. Also, we will connect to the bear’s movement with real-life situation such as walking up/down a hill, in which “rise” and “run” are embedded. We will draw attention to the embodied experience of how they go up or down, forward or backward, as constructing the real-life meanings of slope, and what makes it positive or negative.

**Task 3: Distance formula and pythagorean theorem as a racing game**

As we continue the journey through the world of coordinate geometry using Scratch, the next waypoint focuses on two fundamental concepts: position and distance. This task aims to introduce and deepen learners' understanding of the relationship between position on the Scratch stage and the
distance between sprites or points. Through this, students will also begin to grasp foundational concepts in analytic geometry.

*Action and Process*. There are two sub-tasks. The first sub-task is plotting points and distance. We instruct students to plot two separate points and use the function "distance to" to observe the distance between them (Figure 5a). Then, they shall devise two ways to begin the sprites' journeys (Figure 5b). The first way is changing *x* and *y*, and draw the right-angled sides of the triangle. The other way is to use "point direction" and "move steps". We will ask students to manipulate the values and observe the distance from the starting point (for the first approach), and the final *x* and *y* positions (for point and move approach).

The second sub-task is shading the squares. We ask students to draw a shaded square with repeated move and turn (Figure 5c), during which they will see the actions as a repeated process of moving and turning, and competent in the process of a 2-dimensional shaded region as a set of paths.

<figure><img src="image_149.png" /><figcaption>Figure 5. Move and turn, distance between points</figcaption></figure>

*Object*: First, we will have the students encapsulate the steps into a single program using "My Block" so that they can use the self-constructed function to paint a square in any position, direction, and size. Having encapsulated the process of shading squares as an object, "shaded square," we can then pose a challenge to the students to shade three solid squares formed by each of the three sides of the right triangle, and to observe the progress of the journeys (Figure 6). They will be asked why the painting will finish at the same time in relation to the squares' area. Meanwhile, we will have the students relate the side lengths of the triangles to generate the distance formula, and use the relationship founded in the observed time taken for the sprites' journeys to generate the Pythagorean Theorem.

<figure><img src="image_150.png" /><figcaption>Figure 6. The “painting race”</figcaption></figure>

*Embodiment embedded*. Students will use gestures and physical movement to simulate the turning and movement in the painting solid square task. Observe the “racing” between the two characters provides the situated experiences of how the sum of the two smaller squares equals to the largest square rather than just an abstract formula.
distance between sprites or points. Through this, students will also begin to grasp foundational concepts in analytic geometry.

*Action and Process*. There are two sub-tasks. The first sub-task is plotting points and distance. We instruct students to plot two separate points and use the function "distance to" to observe the distance between them (Figure 5a). Then, they shall devise two ways to begin the sprites' journeys (Figure 5b). The first way is changing *x* and *y*, and draw the right-angled sides of the triangle. The other way is to use "point direction" and "move steps". We will ask students to manipulate the values and observe the distance from the starting point (for the first approach), and the final *x* and *y* positions (for point and move approach).

The second sub-task is shading the squares. We ask students to draw a shaded square with repeated move and turn (Figure 5c), during which they will see the actions as a repeated process of moving and turning, and competent in the process of a 2-dimensional shaded region as a set of paths.

<figure><img src="image_156.png" /><figcaption>Figure 5. Move and turn, distance between points</figcaption></figure>

*Object*: First, we will have the students encapsulate the steps into a single program using "My Block" so that they can use the self-constructed function to paint a square in any position, direction, and size. Having encapsulated the process of shading squares as an object, "shaded square," we can then pose a challenge to the students to shade three solid squares formed by each of the three sides of the right triangle, and to observe the progress of the journeys (Figure 6). They will be asked why the painting will finish at the same time in relation to the squares' area. Meanwhile, we will have the students relate the side lengths of the triangles to generate the distance formula, and use the relationship founded in the observed time taken for the sprites' journeys to generate the Pythagorean Theorem.

<figure><img src="image_157.png" /><figcaption>Figure 6. The “painting race”</figcaption></figure>

*Embodiment embedded*. Students will use gestures and physical movement to simulate the turning and movement in the painting solid square task. Observe the "racing" between the two characters provides the situated experiences of how the sum of the two smaller squares equals to the largest square rather than just an abstract formula.
## Task 4: Plotting and shading as stamping

Building upon the earlier tasks, the last exploration links the coordinate system with functions. Conditional will be used to serve as the basis of plotting points or graphing. The aim of this task is to introduce students to the representation of mathematical conditions and functions on the Scratch stage, guiding them to visually and interactively experience the relationship between functions and graphing.

**Action.** Students will revise their previous program to shade a solid square to paint the whole stage of Scratch. Then, they will use coordinate rather than turn and move to realize the function of paint the whole stage (Figure 7a). Instead of painting the whole stage, the students will be prompted to paint certain areas such as the one shown in Figure 7 (b-d). This will help the students to become familiarized with graphing with conditions. For examples, as the character is navigating throughout the stage, rather than leaving a “stamp” (or a mark) at every position of the stage, the character will only stamp when particular conditions are met (e.g., x position = 0 in Figure 7b).

**Process.** The students will explore different ways to shade/graph only half of the stage (Figure 7c) by creating conditions that achieve such outcome. They will manipulate their graphing into different shapes, positions (e.g., Figure 7c and 7d). Prompt them to observe how they are changing the value and construct of the conditions.

<figure><img src="image_151.png" /><figcaption>Figure 7. Painting the whole stage and painting certain areas</figcaption></figure>

**Object.** First, we prompt the students to reconsider their navigating program as an object, in which it serves as a function to draw any figures according to multiple conditional statements. Then, we will have the students link the graphing tasks with the concept of slope that students just experienced. They shall connect the concept of slope with the conditions used in graphing. For example, to create the graph in Figure 7d, a typical condition could be "y position < 3 * x position", where the value of 3 represents the slope of the line which defines the graph. Finally, they will construct the linear functions in the form of $y = mx + b$ and manipulate the parameters m and b to observe their effects on the graphs.

**Schema.** Prompt the students to explore different conditions that represent the relationships between x and y positions in the conditions. The aim is to prompt students' thinking of how functions can represent different geometric shapes in coordinate systems.

**Embodiment embedded.** The action of “stamping” when meeting specific conditions situates the abstract concept of function into a motion of the characters in Scratch. Students will be prompted to think about “what should I do if I am the cat (bear)”. These kind of situated simulation and embodied experience grounds the abstract idea of coordinates and functions in tangible actions, enhancing the embodied understanding.

## DISCUSSION AND CONCLUSION

The exploration of integrating Scratch into mathematics education marks a shift from traditional
**Task 4: Plotting and shading as stamping**

Building upon the earlier tasks, the last exploration links the coordinate system with functions. Conditional will be used to serve as the basis of plotting points or graphing. The aim of this task is to introduce students to the representation of mathematical conditions and functions on the Scratch stage, guiding them to visually and interactively experience the relationship between functions and graphing.

***Action.*** Students will revise their previous program to shade a solid square to paint the whole stage of Scratch. Then, they will use coordinate rather than turn and move to realize the function of paint the whole stage (Figure 7a). Instead of painting the whole stage, the students will be prompted to paint certain areas such as the one shown in Figure 7 (b-d). This will help the students to become familiarized with graphing with conditions. For examples, as the character is navigating throughout the stage, rather than leaving a “stamp” (or a mark) at every position of the stage, the character will only stamp when particular conditions are met (e.g., x position = 0 in Figure 7b).

***Process.*** The students will explore different ways to shade/graph only half of the stage (Figure 7c) by creating conditions that achieve such outcome. They will manipulate their graphing into different shapes, positions (e.g., Figure 7c and 7d). Prompt them to observe how they are changing the value and construct of the conditions.

<figure><img src="image_158.png" /><figcaption>Figure 7. Painting the whole stage and painting certain areas</figcaption></figure>

***Object.*** First, we prompt the students to reconsider their navigating program as an object, in which it serves as a function to draw any figures according to multiple conditional statements. Then, we will have the students link the graphing tasks with the concept of slope that students just experienced. They shall connect the concept of slope with the conditions used in graphing. For example, to create the graph in Figure 7d, a typical condition could be "y position < 3 * x position", where the value of 3 represents the slope of the line which defines the graph. Finally, they will construct the linear functions in the form of $y = mx + b$ and manipulate the parameters m and b to observe their effects on the graphs.

***Schema.*** Prompt the students to explore different conditions that represent the relationships between x and y positions in the conditions. The aim is to prompt students’ thinking of how functions can represent different geometric shapes in coordinate systems.

***Embodiment embedded.*** The action of “stamping” when meeting specific conditions situates the abstract concept of function into a motion of the characters in Scratch. Students will be prompted to think about “what should I do if I am the cat (bear)”. These kind of situated simulation and embodied experience grounds the abstract idea of coordinates and functions in tangible actions, enhancing the embodied understanding.

**DISCUSSION AND CONCLUSION**

The exploration of integrating Scratch into mathematics education marks a shift from traditional
mathematics instructional paradigms. We suggest that combining programming and mathematics doesn't merely offer an engaging platform for students; it also nurtures deeper cognitive links through embodied experiences and the gradual construction of abstract concepts. The melding of the APOS framework with embodied cognition in a digital environment like Scratch promotes algorithmic thinking and grounds abstract mathematical concepts in tangible, real-world actions. However, while the potential of this approach is vast, effective implementation necessitates an understanding of both teachers' and students' capabilities. Further empirical research is needed to evaluate the effectiveness of such designs.

## References

Abrahamson, D. (2014). Building educational activities for understanding: An elaboration on the embodied-design framework and its epistemic grounds. *International Journal of Child-Computer Interaction, 2*, 1-16.

Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (2014). Introduction. In Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (Eds.), *APOS theory. A framework for research and curriculum development in mathematics education* (pp. 5-15). Springer.

Brennan, K., & Resnick, M. (2012, April). New frameworks for studying and assessing the development of computational thinking. In *Proceedings of the 2012 Annual Meeting of the American Educational Research Association* (Vol. 1, p. 25). AERA.

Clark, A. (1999). An embodied cognitive science? *Trends in Cognitive Sciences, 3*(9), 345-351.

Dourish, P. (2001). *Where the action is: the foundations of embodied interaction*. MIT Press.

Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), *Advanced mathematical thinking* (pp. 95-126). Springer Netherlands.

Nemirovsky, R., Ferrara, F., Ferrari, G., & Adamuz-Povedano, N. (2020). Body motion, early algebra, and the colours of abstraction. *Educational Studies in Mathematics, 104*, 261-283.

Ng, O. L., & Chan, T. (2019). Learning as Making: Using 3D computer-aided design to enhance the learning of shape and space in STEM-integrated ways. *British Journal of Educational Technology, 50*(1), 294-308.

Ng, O. L., & Cui, Z. (2021). Examining primary students' mathematical problem-solving in a programming context: Towards computationally enhanced mathematics education. *ZDM-Mathematics Education, 53*, 847-860.

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

Ng, O. L., Leung, A., & Ye, H. (2023). Exploring computational thinking as a boundary object between mathematics and computer programming for STEM teaching and learning. *ZDM-Mathematics Education, 1*-15.

Ng, O. L., Sinclair, N., Ferrara, F., & Liang, B. (2023). Transforming arithmetic through digital resource. In B. Pepin, G. Gueudet, & J. Choppin (Eds.), *Handbook of Digital (Curriculum). Resources in Mathematics Education*. Springer.

Varela, F. J., Thompson, E., Rosch, E., & Kabat-Zinn, J. (2017). *The embodied mind*. The MIT Press.

Weintrop, D., & Wilensky, U. (2015). Using commutative assessments to compare conceptual understanding in blocks-based and text-based programs. In *Proceedings of the 11th Annual International Conference on International Computing Education Research* (pp. 101-110).
mathematics instructional paradigms. We suggest that combining programming and mathematics doesn't merely offer an engaging platform for students; it also nurtures deeper cognitive links through embodied experiences and the gradual construction of abstract concepts. The melding of the APOS framework with embodied cognition in a digital environment like Scratch promotes algorithmic thinking and grounds abstract mathematical concepts in tangible, real-world actions. However, while the potential of this approach is vast, effective implementation necessitates an understanding of both teachers' and students' capabilities. Further empirical research is needed to evaluate the effectiveness of such designs.

## References

Abrahamson, D. (2014). Building educational activities for understanding: An elaboration on the embodied-design framework and its epistemic grounds. *International Journal of Child-Computer Interaction, 2*, 1-16.

Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (2014). Introduction. In Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (Eds.), *APOS theory. A framework for research and curriculum development in mathematics education* (pp. 5-15). Springer.

Brennan, K., & Resnick, M. (2012, April). New frameworks for studying and assessing the development of computational thinking. In *Proceedings of the 2012 Annual Meeting of the American Educational Research Association* (Vol. 1, p. 25). AERA.

Clark, A. (1999). An embodied cognitive science? *Trends in Cognitive Sciences*, 3(9), 345-351.

Dourish, P. (2001). *Where the action is: the foundations of embodied interaction*. MIT Press.

Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), *Advanced mathematical thinking* (pp. 95-126). Springer Netherlands.

Nemirovsky, R., Ferrara, F., Ferrari, G., & Adamuz-Povedano, N. (2020). Body motion, early algebra, and the colours of abstraction. *Educational Studies in Mathematics*, 104, 261-283.

Ng, O. L., & Chan, T. (2019). Learning as Making: Using 3D computer-aided design to enhance the learning of shape and space in STEM-integrated ways. *British Journal of Educational Technology*, 50(1), 294-308.

Ng, O. L., & Cui, Z. (2021). Examining primary students' mathematical problem-solving in a programming context: Towards computationally enhanced mathematics education. *ZDM-Mathematics Education*, 53, 847-860.

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

Ng, O. L., Leung, A., & Ye, H. (2023). Exploring computational thinking as a boundary object between mathematics and computer programming for STEM teaching and learning. *ZDM-Mathematics Education*, 1-15.

Ng, O. L., Sinclair, N., Ferrara, F., & Liang, B. (2023). Transforming arithmetic through digital resource. In B. Pepin, G. Gueudet, & J. Choppin (Eds.), *Handbook of Digital (Curriculum). Resources in Mathematics Education*. Springer.

Varela, F. J., Thompson, E., Rosch, E., & Kabat-Zinn, J. (2017). *The embodied mind*. The MIT Press.

Weintrop, D., & Wilensky, U. (2015). Using commutative assessments to compare conceptual understanding in blocks-based and text-based programs. In *Proceedings of the 11th Annual International Conference on International Computing Education Research* (pp. 101-110).
# CONTRIBUTION OF GESTURES TO THE ACQUISITION OF GEOMETRIC PROPERTIES

Jacques Douaire¹,², Fabien Emprin¹,³,⁴

¹ERMEL Team – IFé, ²LDAR, ³CEREP, ⁴URCA (University of Reims, Champagne-Ardenne)

This paper examines how gestures made by pupils aged between 5 and 11 during the process of drawing straight lines can help them discover the characteristics of plane figures. We also explore how this knowledge evolves throughout elementary school. The aim of our research is to show that it is possible to design problems that involve the manipulation of 2D or 3D geometric objects, thereby contributing to the conceptualization of their properties.

## PRESENTATION

This text proposes how to more specifically answer the following research questions: how it is possible to ensure that students' knowledge derived from their actions toward objects contributes to their learning of figure properties in the sense of Fischbein (1993)? What problems can be used to support this conceptualization?

To address these points, we will draw on experimental results concerning the following:

* Knowledge of the properties of geometric objects, which ranges from shape comparison conducted in the “Grande Section” (GS in France: 5-6-year-old students) to figure analysis and reasoning based on answering the question "Can a triangle have two right angles?" in the “Cours Moyen” during the second year (CM2 in France: 10-11-year-old students).

* Knowledge of straight lines, which ranges from learning to draw straight lines in the “Cours Préparatoire” (CP in France: 6-7-year-old students) to answering the question "How many points are needed to draw a straight line?" in the “Cours Moyen” during the first year (CM1 in France: 9-10-year-old students).

## OBSERVATIONS ON GEOMETRY TEACHING

Geometry teaching in primary and secondary schools has a dual purpose. On the one hand, it provides students with the knowledge they need to master their relationship with space and, in particular, to model phenomena covered in various other disciplines. On the other hand, it enables students to gradually discover this field of mathematics—with its objects of knowledge and its methods of proof (Balacheff, 2020)—the status of which will change over the course of their schooling (Houdement & Kuzniak, 2003). Our interests are in line with those expressed by Sinclair and Bruce (2015 p 320) who analyses a group of articles that highlight current trends of research: “(1) the role of spatial reasoning and its connection to school mathematics in general and school geometry in particular; (2) the function of drawing in the construction of geometric meaning;” and “(5) extending primary school geometry from its typical passive emphasis on vocabulary (naming and sorting shapes by properties) to a more active meaning-making orientation to geometry (including composing/decomposing,
# CONTRIBUTION OF GESTURES TO THE ACQUISITION OF GEOMETRIC PROPERTIES

Jacques Douaire¹, ², Fabien Emprin¹, ³, ⁴

¹ERMEL Team – IFé, ²LDAR, ³CEREP, ⁴URCA (University of Reims, Champagne-Ardenne)

This paper examines how gestures made by pupils aged between 5 and 11 during the process of drawing straight lines can help them discover the characteristics of plane figures. We also explore how this knowledge evolves throughout elementary school. The aim of our research is to show that it is possible to design problems that involve the manipulation of 2D or 3D geometric objects, thereby contributing to the conceptualization of their properties.

## PRESENTATION

This text proposes how to more specifically answer the following research questions: how it is possible to ensure that students' knowledge derived from their actions toward objects contributes to their learning of figure properties in the sense of Fischbein (1993)? What problems can be used to support this conceptualization?

To address these points, we will draw on experimental results concerning the following:

* Knowledge of the properties of geometric objects, which ranges from shape comparison conducted in the “Grande Section” (GS in France: 5-6-year-old students) to figure analysis and reasoning based on answering the question "Can a triangle have two right angles?" in the “Cours Moyen” during the second year (CM2 in France: 10-11-year-old students).

* Knowledge of straight lines, which ranges from learning to draw straight lines in the “Cours Préparatoire” (CP in France: 6-7-year-old students) to answering the question "How many points are needed to draw a straight line?" in the “Cours Moyen” during the first year (CM1 in France: 9-10-year-old students).

## OBSERVATIONS ON GEOMETRY TEACHING

Geometry teaching in primary and secondary schools has a dual purpose. On the one hand, it provides students with the knowledge they need to master their relationship with space and, in particular, to model phenomena covered in various other disciplines. On the other hand, it enables students to gradually discover this field of mathematics—with its objects of knowledge and its methods of proof (Balacheff, 2020)—the status of which will change over the course of their schooling (Houdement & Kuzniak, 2003). Our interests are in line with those expressed by Sinclair and Bruce (2015 p 320) who analyses a group of articles that highlight current trends of research: “(1) the role of spatial reasoning and its connection to school mathematics in general and school geometry in particular; (2) the function of drawing in the construction of geometric meaning;” and “(5) extending primary school geometry from its typical passive emphasis on vocabulary (naming and sorting shapes by properties) to a more active meaning-making orientation to geometry (including composing/decomposing,
classifying, mapping and orienting, comparing and mentally manipulating two- and three-
dimensional figures).”

A short analysis of easily available French resources for teaching geometry to young pupils shows that two main goals are pursued: teaching the vocabulary of geometry and drawing abilities. Spatial activities also exist, but their sole purpose seems to be manipulation itself, without this being a means of solving problems. These activities also seem unrelated to the pupil's initial knowledge or mathematical reasoning. Teachers also frequently complain that activities are somewhat repetitive from one year to the next starting at the beginning of primary school and moving forward. We therefore have to identify the components of learning and the knowledge initially available to pupils.

RESEARCH BY THE ERMEL TEAM

The research conducted by the ERMEL team (a team associated with the French Institute of Education at the “École Normale Supérieure” in Lyon) on learning mathematics and problem-solving in primary school in France has two goals. Once the needs are identified, such research involves several steps: (1) An analysis of the mathematical knowledge (problems, properties...) at stake, as well as students’ knowledge and abilities; (2) An explanation of educational issues and the organizing of the study of the different notions throughout the years; (3) Development of teaching situations and tests in several classrooms. These last three components interact: the identification of students’ abilities is the outcome of experiments conducted. (4) Writing a book for teachers and trainers with an explanation of the issues of learning and teaching, a description of the learning situations identified, a reasoned choice of learning roadmap and syllabus planning accompanied by analyses and theoretical contributions (Douaire et al. 2020, 2023).

The results of these experiments, which have been carried out in ten and, in some cases, thirty classes over several years (1998 to 2019), have led us to question some of the assumptions on which certain learning activities are based and to modify both the related situations and the organization of learning. In this presentation, we do not detail the many changes needed in the description of the situations related to the successive and necessary experiments necessary for students to produce the specified procedures. Our goal is to build a proven, complete and reliable approach to teaching engineering. These publications are designed to offer a coherent vision of spatial and geometric learning in terms of both content and the role of problem-solving. All the tasks the teacher has to carry out have been precisely described.

TAKING STUDENT KNOWLEDGE INTO ACCOUNT

Since early childhood, pupils have acquired knowledge of objects in space, for example, through 3D construction games and 2D shape assemblies. The gestures they make while using these objects ("sliding", "turning", "flipping", "nesting"...) constitute actions that the child develops with the precise intention of completing a task. We will distinguish such gestures from those that are used to communicate (Petitfour, 2017), those used with instruments, or even "routine" gestures made when efficiently using familiar objects (e.g., unfolding a sheet of paper, etc.). These gestures apply to objects of the same type, for instance, flat polygonal shapes and polyhedral solids, which may have different appearances. Gestures are therefore common to several activities. Pupils use these gestures to successfully juxtapose, nest, and adjust the relative positions of two or more objects, as well as to compare or assemble them. These gestures are anticipated and followed by an assessment of their
classifying, mapping and orienting, comparing and mentally manipulating two- and three-
dimensional figures).”

A short analysis of easily available French resources for teaching geometry to young pupils shows that two main goals are pursued: teaching the vocabulary of geometry and drawing abilities. Spatial activities also exist, but their sole purpose seems to be manipulation itself, without this being a means of solving problems. These activities also seem unrelated to the pupil's initial knowledge or mathematical reasoning. Teachers also frequently complain that activities are somewhat repetitive from one year to the next starting at the beginning of primary school and moving forward. We therefore have to identify the components of learning and the knowledge initially available to pupils.

RESEARCH BY THE ERMEL TEAM

The research conducted by the ERMEL team (a team associated with the French Institute of Education at the “École Normale Supérieure” in Lyon) on learning mathematics and problem-solving in primary school in France has two goals. Once the needs are identified, such research involves several steps: (1) An analysis of the mathematical knowledge (problems, properties...) at stake, as well as students’ knowledge and abilities; (2) An explanation of educational issues and the organizing of the study of the different notions throughout the years; (3) Development of teaching situations and tests in several classrooms. These last three components interact: the identification of students’ abilities is the outcome of experiments conducted. (4) Writing a book for teachers and trainers with an explanation of the issues of learning and teaching, a description of the learning situations identified, a reasoned choice of learning roadmap and syllabus planning accompanied by analyses and theoretical contributions (Douaire et al. 2020, 2023).

The results of these experiments, which have been carried out in ten and, in some cases, thirty classes over several years (1998 to 2019), have led us to question some of the assumptions on which certain learning activities are based and to modify both the related situations and the organization of learning. In this presentation, we do not detail the many changes needed in the description of the situations related to the successive and necessary experiments necessary for students to produce the specified procedures. Our goal is to build a proven, complete and reliable approach to teaching engineering. These publications are designed to offer a coherent vision of spatial and geometric learning in terms of both content and the role of problem-solving. All the tasks the teacher has to carry out have been precisely described.

TAKING STUDENT KNOWLEDGE INTO ACCOUNT

Since early childhood, pupils have acquired knowledge of objects in space, for example, through 3D construction games and 2D shape assemblies. The gestures they make while using these objects ("sliding", "turning", "flipping", "nesting"...) constitute actions that the child develops with the precise intention of completing a task. We will distinguish such gestures from those that are used to communicate (Petitfour, 2017), those used with instruments, or even "routine" gestures made when efficiently using familiar objects (e.g., unfolding a sheet of paper, etc.). These gestures apply to objects of the same type, for instance, flat polygonal shapes and polyhedral solids, which may have different appearances. Gestures are therefore common to several activities. Pupils use these gestures to successfully juxtapose, nest, and adjust the relative positions of two or more objects, as well as to compare or assemble them. These gestures are anticipated and followed by an assessment of their
effects. These assessments of success or failure are generally not accompanied by any verbalization
other than "it is working" or "it is not working". In this sense, these gestures constitute procedures.
"The gesture - or action (sequence of gestures) - is defined not by the movements performed, but by
the intention of its author, who decides on its execution as a function of its goal or purpose. He
represents its effects and controls its execution... Every gesture has two components, one motor and
the other cognitive. The cognitive aspects of gestures include all preparatory aspects of the action.
But often only the intention to act is conscious: the subject knows and represents the result, the effects
of his act... At this stage, the representation does not contain details of the precise way to execute the
movements in each sequence". (Mazeau & Pouhet, 2014 p114). In the context of a didactic situation,
gestures can be made explicit, and their effects can be analyzed. However, this knowledge is rarely
considered in teaching methods at the start of elementary school; rather, gestures made with objects
are often wrongly perceived as "manipulations" devoid of anticipation or intentionality. In the field
of numerical learning, however, the knowledge that pupils have before tackling a new notion has
been analyzed for several decades by numerous studies in the fields of psychology and didactics.
Hence, in geometry, teachers are often at a loss to recognize pupils' potential, identify their procedures
and exploit the observations they make about their productions.

A PROBLEM TO ENCOURAGE THE DISTINCTION BETWEEN SQUARES, RECTANGLES AND RHOMBUSES

Starting in kindergarten, pupils become familiar with real or pictured objects, and they use global perception to identify such objects with the support of a metaphorical vocabulary (e.g., "mountain" for pyramid, Coutat & Vendeira, 2019). Naming confusion is common between neighboring objects. In fact, numerous studies have shown that such confusion continues to persist following not only teaching that is focused on the simple and correct naming of the geometric shapes of physical objects or their graphic representations but also tasks involving the classification of objects according to instructions such as "put together what goes together". In other words, the geometric properties enunciated on these occasions do not actually become operative; the old criteria for recognizing shapes, based on similarities with real-world objects, remain. For example, we have found that a rectangle whose length and width are close to each other is still often referred to as a square by pupils in "Cours Élémentaire, first year" (CE1, students aged 7-8) and even in "Cours Élémentaire, second year" (CE2, students aged 8-9). The knowledge of the properties of figures and solids must, however, appear to pupils as necessity and not convention. "There is a growing amount of evidence both from psychology and mathematics education showing that children come to school with a great deal of informal geometric understanding". (Sinclair & Bruce, 2015, p 322). To overcome the ambiguities of perception, one of our hypotheses is that highlighting the properties needed to distinguish between neighboring objects with differences that are difficult to discern is better than asking students to classify objects that are already perceptually different.

Square or quasi-squares’ situation

In the "square or quasisquare" situation, 6-7-year-old pupils have to produce an assembly of squares,
rhombuses and rectangles whose dimensions are very similar (4 cm for square and rhombus sides, 4
cm ± 3 mm for rectangle sides, and an angle of 90 ± 4° for rhombus corners). Experiments have
shown that these pieces are too similar to be distinguished when side by side; for the students, such
differences are interpreted as cutting-related inaccuracies. One of our hypotheses is that by
effects. These assessments of success or failure are generally not accompanied by any verbalization
other than "it is working" or "it is not working". In this sense, these gestures constitute procedures.
"The gesture - or action (sequence of gestures) - is defined not by the movements performed, but by
the intention of its author, who decides on its execution as a function of its goal or purpose. He
represents its effects and controls its execution... Every gesture has two components, one motor and
the other cognitive. The cognitive aspects of gestures include all preparatory aspects of the action.
But often only the intention to act is conscious: the subject knows and represents the result, the effects
of his act... At this stage, the representation does not contain details of the precise way to execute the
movements in each sequence". (Mazeau & Pouhet, 2014 p114). In the context of a didactic situation,
gestures can be made explicit, and their effects can be analyzed. However, this knowledge is rarely
considered in teaching methods at the start of elementary school; rather, gestures made with objects
are often wrongly perceived as "manipulations" devoid of anticipation or intentionality. In the field
of numerical learning, however, the knowledge that pupils have before tackling a new notion has
been analyzed for several decades by numerous studies in the fields of psychology and didactics.
Hence, in geometry, teachers are often at a loss to recognize pupils' potential, identify their procedures
and exploit the observations they make about their productions.

A PROBLEM TO ENCOURAGE THE DISTINCTION BETWEEN SQUARES, RECTANGLES AND RHOMBUSES

Starting in kindergarten, pupils become familiar with real or pictured objects, and they use global perception to identify such objects with the support of a metaphorical vocabulary (e.g., "mountain" for pyramid, Coutat & Vendeira, 2019). Naming confusion is common between neighboring objects. In fact, numerous studies have shown that such confusion continues to persist following not only teaching that is focused on the simple and correct naming of the geometric shapes of physical objects or their graphic representations but also tasks involving the classification of objects according to instructions such as "put together what goes together". In other words, the geometric properties enunciated on these occasions do not actually become operative; the old criteria for recognizing shapes, based on similarities with real-world objects, remain. For example, we have found that a rectangle whose length and width are close to each other is still often referred to as a square by pupils in "Cours Élémentaire, first year" (CE1, students aged 7-8) and even in "Cours Élémentaire, second year" (CE2, students aged 8-9). The knowledge of the properties of figures and solids must, however, appear to pupils as necessity and not convention. "There is a growing amount of evidence both from psychology and mathematics education showing that children come to school with a great deal of informal geometric understanding". (Sinclair & Bruce, 2015, p 322). To overcome the ambiguities of perception, one of our hypotheses is that highlighting the properties needed to distinguish between neighboring objects with differences that are difficult to discern is better than asking students to classify objects that are already perceptually different.

Square or quasi-squares’ situation

In the "square or quasisquare" situation, 6-7-year-old pupils have to produce an assembly of squares,
rhombuses and rectangles whose dimensions are very similar (4 cm for square and rhombus sides, 4
cm ± 3 mm for rectangle sides, and an angle of 90 ± 4° for rhombus corners). Experiments have
shown that these pieces are too similar to be distinguished when side by side; for the students, such
differences are interpreted as cutting-related inaccuracies. One of our hypotheses is that by
assembling these shapes, the characteristics of the square, rhombus and rectangle could be better identified (Douaire et al., 2023).

In the first problem, students are asked to produce an assembly of four pieces taken from a set of eight pieces, namely, four squares and four rhombuses, without leaving any gaps. They then make a record of the assembly by tracing its outline. In a second problem, the pupils have to try to construct an assembly of three squares and one rhombus. After several attempts, they find that it is impossible to produce an assembly that does not leave an empty space between the four shapes. Many students discover that the angles of the rhombus are not the same and that they differ from those of the square. They also find that if a square is turned, the assembly remained unchanged, whereas by turning a rhombus, a gap or overlap is formed. In a third problem, pupils are asked to align rectangles and squares to produce the longest possible assembly. Pupils express observations about either their actions, such as "When you turn the rectangle, it gets longer", or their properties, such as "There's a side where it is long, and a side where it is short". Some students discover differences between a square and a rectangle: "If you turn them [squares], they always look the same"; "A square is not like a rectangle, turning it does not lengthen the line (implied)". This makes it clear that, unlike a rectangle, the sides of a square are of equal length.

## Findings

Four results can be stated: (1) Object properties are identified and formulated by the students. This problem contributes to a shift from the global recognition of shapes perceived as similar to their identification through the analysis of distinct geometric properties. (2) The production of solutions is made possible by the ability to place, juxtapose and swap pieces, acquired in games since kindergarten. (3) It was the assembling of similar-shaped pieces that made visible differences that the pupils had not initially perceived. (4) To overcome the ambiguities of perception, highlighting the properties needed to distinguish between neighboring objects whose differences are difficult to discern is more effective than asking students to classify objects that are already perceptually different.

## A long-term evolution of knowledge

By solving these problems, the criteria for distinguishing between neighboring objects will evolve between the ages of 5 and 7 years old. First, designations may be based on a known object from the physical world ("a pyramid"); later, designations may be made by pointing out a local physical feature (e.g., "that one is pointed and the other one is not") and ultimately by geometric criteria. From the age of 7 years old and onward, the gradual technical mastery of drawing straight lines enables students to work on figures rather than shapes. Drawing these figures enables the study of relationships (perpendicularity, parallelism, symmetry, etc.) to complete a drawing so that it, e.g., represents a rectangle.

## FROM PRODUCING LINES TO ANALYZING THE GEOMETRIC PROPERTIES OF THE STRAIGHT LINE

Approaching the concept of a straight line in elementary school involves several aspects related to the questions posed in this subtheme. At this level, we cannot yet talk about the "notion of a straight line", and the idea that a straight line can be "extended to infinity" is not yet accessible. At the beginning of elementary school, straight lines are often introduced as tools for reproducing geometric
assembling these shapes, the characteristics of the square, rhombus and rectangle could be better identified (Douaire et al., 2023).

In the first problem, students are asked to produce an assembly of four pieces taken from a set of eight pieces, namely, four squares and four rhombuses, without leaving any gaps. They then make a record of the assembly by tracing its outline. In a second problem, the pupils have to try to construct an assembly of three squares and one rhombus. After several attempts, they find that it is impossible to produce an assembly that does not leave an empty space between the four shapes. Many students discover that the angles of the rhombus are not the same and that they differ from those of the square. They also find that if a square is turned, the assembly remained unchanged, whereas by turning a rhombus, a gap or overlap is formed. In a third problem, pupils are asked to align rectangles and squares to produce the longest possible assembly. Pupils express observations about either their actions, such as "When you turn the rectangle, it gets longer", or their properties, such as "There's a side where it is long, and a side where it is short". Some students discover differences between a square and a rectangle: "If you turn them [squares], they always look the same"; "A square is not like a rectangle, turning it does not lengthen the line (implied)". This makes it clear that, unlike a rectangle, the sides of a square are of equal length.

## Findings

Four results can be stated: (1) Object properties are identified and formulated by the students. This problem contributes to a shift from the global recognition of shapes perceived as similar to their identification through the analysis of distinct geometric properties. (2) The production of solutions is made possible by the ability to place, juxtapose and swap pieces, acquired in games since kindergarten. (3) It was the assembling of similar-shaped pieces that made visible differences that the pupils had not initially perceived. (4) To overcome the ambiguities of perception, highlighting the properties needed to distinguish between neighboring objects whose differences are difficult to discern is more effective than asking students to classify objects that are already perceptually different.

## A long-term evolution of knowledge

By solving these problems, the criteria for distinguishing between neighboring objects will evolve between the ages of 5 and 7 years old. First, designations may be based on a known object from the physical world ("a pyramid"); later, designations may be made by pointing out a local physical feature (e.g., "that one is pointed and the other one is not") and ultimately by geometric criteria. From the age of 7 years old and onward, the gradual technical mastery of drawing straight lines enables students to work on figures rather than shapes. Drawing these figures enables the study of relationships (perpendicularity, parallelism, symmetry, etc.) to complete a drawing so that it, e.g., represents a rectangle.

## FROM PRODUCING LINES TO ANALYZING THE GEOMETRIC PROPERTIES OF THE STRAIGHT LINE

Approaching the concept of a straight line in elementary school involves several aspects related to the questions posed in this subtheme. At this level, we cannot yet talk about the "notion of a straight line", and the idea that a straight line can be "extended to infinity" is not yet accessible. At the beginning of elementary school, straight lines are often introduced as tools for reproducing geometric
drawings; these practices do not always allow pupils' different initial conceptions to express themselves, whether they are erroneous or potentially useful. How can these conceptions be considered and taught in a way that challenges students if necessary? Should the concept of a straight line be approached on the basis of one of the spatial experiences that pupils may have encountered, e.g., a taut wire, the edge of a solid shape, a ray of light, a fold in a sheet of paper, the rectilinear trajectory of a moving object, all of which would be modeled by a line? Or is it better to define a straight line from two distinct points?

After conducting several experiments with 7-year-old students based on one or more of these choices, we found that the transition from solving spatial problems in which a straight line was the solution for aligning objects to modeling them on a sheet of paper presented several difficulties. On the one hand, this modeling was handled by the selected device and not by the student. In other words, it was the teacher who had to state that the e.g., taut wire, , was represented by a straight line. On the other hand, the students drew broken lines without realizing why their solutions were wrong. In computer simulations using 3D dynamic geometry software, the same problems were solved by students using the "straight line" tool. However, when these problems were repeated on pencil and paper, students were no longer successful. Thus, the straight line did not appear as the solution for aligning points based on the modeling of situations in space.

## Using a tool that is too short

From CP onward (6 years old and older), if pupils are to understand that straight lines can be extended and learn to do so, we feel that it is necessary for them to be able to produce tracings to solve problems. These drawings can then be checked and improved. Thus, in a study, pupils were asked to complete a network of (parallel) lines on an A3 sheet of paper (29.7 x 42 cm) with a geometrical instrument no longer than 20 cm. They were therefore required to switch the ruler, the instruction being: "You must extend the lines drawn" (cf. Figure 1).

<figure><img src="image_152.png" /><figcaption>Figure 1. The support</figcaption></figure>

We carried out experiments in approximately ten first-grade classes (6-year-old students) over several years, successively proposing several tracings of this type to the pupils in rectangular or round-cut sheet formats.

## Findings: A wide variety of products

In this problem, we observed: freehand tracing; lines partially drawn with a ruler but sometimes extended by hand; lines drawn with a ruler made up of broken lines; and correct tracings, approximately one-third of the total number, with possible inaccuracies in the continuity of the lines.
drawings; these practices do not always allow pupils' different initial conceptions to express themselves, whether they are erroneous or potentially useful. How can these conceptions be considered and taught in a way that challenges students if necessary? Should the concept of a straight line be approached on the basis of one of the spatial experiences that pupils may have encountered, e.g., a taut wire, the edge of a solid shape, a ray of light, a fold in a sheet of paper, the rectilinear trajectory of a moving object, all of which would be modeled by a line? Or is it better to define a straight line from two distinct points?

After conducting several experiments with 7-year-old students based on one or more of these choices, we found that the transition from solving spatial problems in which a straight line was the solution for aligning objects to modeling them on a sheet of paper presented several difficulties. On the one hand, this modeling was handled by the selected device and not by the student. In other words, it was the teacher who had to state that the e.g., taut wire, , was represented by a straight line. On the other hand, the students drew broken lines without realizing why their solutions were wrong. In computer simulations using 3D dynamic geometry software, the same problems were solved by students using the "straight line" tool. However, when these problems were repeated on pencil and paper, students were no longer successful. Thus, the straight line did not appear as the solution for aligning points based on the modeling of situations in space.

## Using a tool that is too short

From CP onward (6 years old and older), if pupils are to understand that straight lines can be extended and learn to do so, we feel that it is necessary for them to be able to produce tracings to solve problems. These drawings can then be checked and improved. Thus, in a study, pupils were asked to complete a network of (parallel) lines on an A3 sheet of paper (29.7 x 42 cm) with a geometrical instrument no longer than 20 cm. They were therefore required to switch the ruler, the instruction being: "You must extend the lines drawn" (cf. Figure 1).

<figure><img src="image_159.png" /><figcaption>Figure 1. The support</figcaption></figure>

We carried out experiments in approximately ten first-grade classes (6-year-old students) over several years, successively proposing several tracings of this type to the pupils in rectangular or round-cut sheet formats.

## Findings: A wide variety of products

In this problem, we observed: freehand tracing; lines partially drawn with a ruler but sometimes extended by hand; lines drawn with a ruler made up of broken lines; and correct tracings, approximately one-third of the total number, with possible inaccuracies in the continuity of the lines.
During the pooling of the results, both the analysis of the productions and the formulation by the
pupils of the conformity and the differences between the tracings enabled different judgment criteria
to be made explicit, which constituted one of the components of this learning process. Such judgments
were based on students' ability to perceive the difference between a curved and a straight line (Wu &
Ma, 2005). These criteria included understanding the task and the technical mastery of ruler use and
ruler transfer technology. To succeed, students had to be able to do the following:

* understand the goal to be achieved, namely, that a straight line must be drawn (the sight of either intersecting/diverging lines or lines drawn freehand was enough to disqualify some of the productions);

* master the drawing technique (position the ruler, hold it firmly in the middle without letting your fingers protrude, place the pencil, etc.);

* know how to switch the ruler when it is too short (i.e., place an intermediate point or slide the ruler over a sufficiently large part of the line), which presupposes that these techniques are formulated so that they can be shared and understood.

When successive problems were solved in the subsequent sessions, most pupils produced straight lines, often with technical inaccuracies concerning the switch of the ruler or the positioning of the pencil along it. Thus, such use of a ruler should be reinforced on numerous occasions, both mathematical and nonmathematical (underlining, framing, etc.).

THE RELATIONSHIP WITH SENSITIVE SPACE: A LONG-TERM EVOLUTION OF PROCEDURES AND EVIDENCE

Knowledge of a straight line and its properties continue in two directions in our proposals for
elementary school.

From CE2 (7 years old) onward, straight lines will be used to solve problems involving the alignment of points. Drawings on a sheet of paper serve as solutions to problems associating points and straight lines and representing spatial situations (e.g., determining the location of a fold hidden under a sheet of paper where only two of its points are known and marked by pins).

In the “Cours Moyen” (9-11-year-old students), drawing straight lines are part of the solutions to problems of parallelism, perpendicularity and symmetry. The question "How many points do you need to draw a straight line?" often arises after solving problems in which pupils must draw parallels. Some students are still content to draw lines in a guessed direction from one point, while others claim that two points are enough, and still others insist on using at least three points based on the argument that doing so "is more precise". The point of such an exchange is to enable students to distinguish between the properties of the straight line, which is defined by two distinct points, and the technical requirement for precision in drawing, for which many students recognize their own limitations.

For the study of geometric objects and modeling, we therefore give priority to problem-solving
involving the following:

* Engaging in actions (assemblies, comparisons...) regarding physical objects (shapes, solids) in grades GS to CE1 (from 5 to 7 years old), which enables pupils to apprehend the properties of such objects. Then, in CE1 and CE2 (7 and 8 years old), students start to construct figures.
During the pooling of the results, both the analysis of the productions and the formulation by the
pupils of the conformity and the differences between the tracings enabled different judgment criteria
to be made explicit, which constituted one of the components of this learning process. Such judgments
were based on students' ability to perceive the difference between a curved and a straight line (Wu &
Ma, 2005). These criteria included understanding the task and the technical mastery of ruler use and
ruler transfer technology. To succeed, students had to be able to do the following:

* understand the goal to be achieved, namely, that a straight line must be drawn (the sight of either intersecting/diverging lines or lines drawn freehand was enough to disqualify some of the productions);

* master the drawing technique (position the ruler, hold it firmly in the middle without letting your fingers protrude, place the pencil, etc.);

* know how to switch the ruler when it is too short (i.e., place an intermediate point or slide the ruler over a sufficiently large part of the line), which presupposes that these techniques are formulated so that they can be shared and understood.

When successive problems were solved in the subsequent sessions, most pupils produced straight lines, often with technical inaccuracies concerning the switch of the ruler or the positioning of the pencil along it. Thus, such use of a ruler should be reinforced on numerous occasions, both mathematical and nonmathematical (underlining, framing, etc.).

THE RELATIONSHIP WITH SENSITIVE SPACE: A LONG-TERM EVOLUTION OF PROCEDURES AND EVIDENCE

Knowledge of a straight line and its properties continue in two directions in our proposals for
elementary school.

From CE2 (7 years old) onward, straight lines will be used to solve problems involving the alignment of points. Drawings on a sheet of paper serve as solutions to problems associating points and straight lines and representing spatial situations (e.g., determining the location of a fold hidden under a sheet of paper where only two of its points are known and marked by pins).

In the “Cours Moyen” (9-11-year-old students), drawing straight lines are part of the solutions to problems of parallelism, perpendicularity and symmetry. The question "How many points do you need to draw a straight line?" often arises after solving problems in which pupils must draw parallels. Some students are still content to draw lines in a guessed direction from one point, while others claim that two points are enough, and still others insist on using at least three points based on the argument that doing so "is more precise". The point of such an exchange is to enable students to distinguish between the properties of the straight line, which is defined by two distinct points, and the technical requirement for precision in drawing, for which many students recognize their own limitations.

For the study of geometric objects and modeling, we therefore give priority to problem-solving
involving the following:

* Engaging in actions (assemblies, comparisons...) regarding physical objects (shapes, solids) in grades GS to CE1 (from 5 to 7 years old), which enables pupils to apprehend the properties of such objects. Then, in CE1 and CE2 (7 and 8 years old), students start to construct figures.
* The simple evocation of spatial contexts from CM1 (9 years old) onward and engaging in problem-solving by tracing on a sheet of paper. Solutions are validated by students describing how they were reached.

* Decontextualizing problems in CM2 (10 years old), such as "Can a triangle have two right angles?" Discussions between pupils help them understand that their various attempts at tracing do not enable them to answer such questions. Validation of their solutions are thus based on the statement of properties ("If there are two right angles, two sides are perpendicular to the third, then these two sides are therefore parallel; thus, it is not a triangle") and not on the accuracy of a drawing.

We previously mentioned a shift from parts assembly problems to drawing problems (graphical problems); to that extent, the sensitive space is changed.

## Research findings

The proofs produced by the pupils will therefore be constituted successively by the following:

* practical validation based on superposition;

* the formulation of the geometric characteristics of the shapes;

* the justification of their constructions made by describing each of their stages; and

* the production of reasoning based on known geometric properties.

This evolution is also expressed in language forms, as part of a language used by a student to control his or her actions or to communicate about a production (formulation of a procedure, checking the validation constraints of a solution, etc.).

## ABOUT THE RESOURCE

As mentioned at the beginning of this text, the second goal of our team is to make the results of our research available to teachers and trainers. The tools (e.g., progressions, situations) we have developed promote knowledge construction based on problem-solving. They have a certain "robustness" due in particular to the fact that the results and procedures that will be produced by students in a class are described in the situations, allowing teachers, who are generally not mathematics specialists, to anticipate their decisions based on their own class productions. This presupposes that the teacher's decisions are made explicit and that he or she has a progression plan in place that is backed by answers to the questions he or she may have after implementing them. This reliability seems partly due to, first, the level of coherence between the concepts of learning and the proposed situations and, second, to their experimentation in many classes for many years.

Our results relate to the following three levels:

* that of research, the construction of analytical tools and the testing of hypotheses;

* that of didactic engineering (Artigue, 2002) and the relationship between students' initial knowledge, the targeted knowledge and the proposed teaching devices; and

* that of the resources produced and their appropriation by teachers who wish to modify their teaching.

## CONCLUSION

The gestures made in relation to geometric objects in which we are interested are those used in purposeful, problem-solving actions, where a pupil can observe either success or failure; it is in this
* The simple evocation of spatial contexts from CM1 (9 years old) onward and engaging in problem-solving by tracing on a sheet of paper. Solutions are validated by students describing how they were reached.

* Decontextualizing problems in CM2 (10 years old), such as "Can a triangle have two right angles?" Discussions between pupils help them understand that their various attempts at tracing do not enable them to answer such questions. Validation of their solutions are thus based on the statement of properties ("If there are two right angles, two sides are perpendicular to the third, then these two sides are therefore parallel; thus, it is not a triangle") and not on the accuracy of a drawing.

We previously mentioned a shift from parts assembly problems to drawing problems (graphical problems); to that extent, the sensitive space is changed.

## Research findings

The proofs produced by the pupils will therefore be constituted successively by the following:

* practical validation based on superposition;

* the formulation of the geometric characteristics of the shapes;

* the justification of their constructions made by describing each of their stages; and

* the production of reasoning based on known geometric properties.

This evolution is also expressed in language forms, as part of a language used by a student to control his or her actions or to communicate about a production (formulation of a procedure, checking the validation constraints of a solution, etc.).

## ABOUT THE RESOURCE

As mentioned at the beginning of this text, the second goal of our team is to make the results of our research available to teachers and trainers. The tools (e.g., progressions, situations) we have developed promote knowledge construction based on problem-solving. They have a certain "robustness" due in particular to the fact that the results and procedures that will be produced by students in a class are described in the situations, allowing teachers, who are generally not mathematics specialists, to anticipate their decisions based on their own class productions. This presupposes that the teacher's decisions are made explicit and that he or she has a progression plan in place that is backed by answers to the questions he or she may have after implementing them. This reliability seems partly due to, first, the level of coherence between the concepts of learning and the proposed situations and, second, to their experimentation in many classes for many years.

Our results relate to the following three levels:

* that of research, the construction of analytical tools and the testing of hypotheses;

* that of didactic engineering (Artigue, 2002) and the relationship between students' initial knowledge, the targeted knowledge and the proposed teaching devices; and

* that of the resources produced and their appropriation by teachers who wish to modify their teaching.

## CONCLUSION

The gestures made in relation to geometric objects in which we are interested are those used in purposeful, problem-solving actions, where a pupil can observe either success or failure; it is in this
sense that we understand the use of the term "manipulation". In our view, this analysis of the procedures that have already developed by pupils in their experiences with objects from the physical world is an essential component of the study of the relationship between actions on materials and the acquisition of geometric notions. One of the functions of our learning proposals is to enable these procedures to be clarified and criticized. This empirical study, which is based on numerous experiments, shows that having students solve certain types of problems involving manipulation with objects or tracing can help them conceptualize mathematical objects through the progressive recognition of their properties. This engineering approach has been applied to other geometric objects and properties (parallelism, incidence relations in space, etc.) with the same results (Douaire et al., 2020).

## References

Artigue, M. (2002). Ingénierie didactique: quel rôle dans la recherche en didactique aujourd'hui [Didactic engineering: what role in didactic research today?], *Revue Internationale des Sciences de l'Éducation, 8*, 59-72.

Balacheff, N. (2020). The transition from mathematical argumentation to mathematical proof, a learning and teaching challenge (invited lecture). *In The 14th International Congress on Mathematical Education*.

Celi, V., Coutat-Gousseau, S., & Vendeira, C. (2019). Travailler avec les formes en maternelle: premiers pas vers des connaissances géométriques [Shape recognition in kindergarten, a didactic and psychological perspective]. *In Actes du 45ème Colloque de la COPIRELEM. Manipuler, Représenter, Communiquer: Quelle place pour les artefacts dans l'enseignement et l'apprentissage des mathématiques* (pp. 35-55).

Douaire, J., Argaud, H.-C., Emprin, F., et Frémin, M. (2020). *ERMEL géométrie CP-CE1*. Hatier.

Douaire, J., Argaud, H.-C., Emprin, F., et Frémin, M. (2023). *ERMEL Maternelle GS*. Hatier.

Douaire, J., Emprin, F., et Argaud, H.-C., (2023). Research and Production of a Resource for Geometric Learning in First and Second Grade in *Articulation between Tangible Space, Graphical Space and Geometrical Space*, C. Guille-Biel Winder & T. Assude (Eds), ISTE.

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

Houdement, C., & Kuzniak, A. (2003). Elementary geometry split into different geometrical paradigms. *In Proceedings of CERME* (Vol. 3, pp. 1-10). ERME.

Mazeau, M., et Pouhet, A. (2014). *Neuropsychologie et troubles des apprentissages chez l'enfant: du développement typique aux dys* [Neuropsychology and learning disorders in children: from typical development to dys.] Elsevier Masson.

Petitfour, E. (2017). Enseignement de la géométrie à la fin du cycle 3 - Proposition d'un dispositif de travail en dyade [Teaching geometry at the end of cycle 3 - Proposal for a dyad-based system.]. *Petit x, 103*, 5-31. hal-02268417

Sinclair, N., & Bruce, C. D. (2015). New opportunities in geometry education at the primary school. *ZDM, 47*, 319-329.

Wu, D.-B., & Ma, H.-L. (2005). A Study of the Geometric Concepts of Elementary School Students at van Hiele Level One. *International Group for the Psychology of Mathematics Education, 4*, 329-336.
sense that we understand the use of the term "manipulation". In our view, this analysis of the procedures that have already developed by pupils in their experiences with objects from the physical world is an essential component of the study of the relationship between actions on materials and the acquisition of geometric notions. One of the functions of our learning proposals is to enable these procedures to be clarified and criticized. This empirical study, which is based on numerous experiments, shows that having students solve certain types of problems involving manipulation with objects or tracing can help them conceptualize mathematical objects through the progressive recognition of their properties. This engineering approach has been applied to other geometric objects and properties (parallelism, incidence relations in space, etc.) with the same results (Douaire et al., 2020).

## References

Artigue, M. (2002). Ingénierie didactique: quel rôle dans la recherche en didactique aujourd'hui [Didactic engineering: what role in didactic research today?], *Revue Internationale des Sciences de l'Éducation, 8*, 59-72.

Balacheff, N. (2020). The transition from mathematical argumentation to mathematical proof, a learning and teaching challenge (invited lecture). *In The 14th International Congress on Mathematical Education*.

Celi, V., Coutat-Gousseau, S., & Vendeira, C. (2019). Travailler avec les formes en maternelle: premiers pas vers des connaissances géométriques [Shape recognition in kindergarten, a didactic and psychological perspective]. *In Actes du 45ème Colloque de la COPIRELEM. Manipuler, Représenter, Communiquer: Quelle place pour les artefacts dans l'enseignement et l'apprentissage des mathématiques* (pp. 35-55).

Douaire, J., Argaud, H.-C., Emprin, F., et Frémin, M. (2020). *ERMEL géométrie CP-CE1*. Hatier.

Douaire, J., Argaud, H.-C., Emprin, F., et Frémin, M. (2023). *ERMEL Maternelle GS*. Hatier.

Douaire, J., Emprin, F., et Argaud, H.-C., (2023). Research and Production of a Resource for Geometric Learning in First and Second Grade in *Articulation between Tangible Space, Graphical Space and Geometrical Space*, C. Guille-Biel Winder & T. Assude (Eds), ISTE.

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

Houdement, C., & Kuzniak, A. (2003). Elementary geometry split into different geometrical paradigms. *In Proceedings of CERME* (Vol. 3, pp. 1-10). ERME.

Mazeau, M., et Pouhet, A. (2014). *Neuropsychologie et troubles des apprentissages chez l'enfant: du développement typique aux dys* [Neuropsychology and learning disorders in children: from typical development to dys.] Elsevier Masson.

Petitfour, E. (2017). Enseignement de la géométrie à la fin du cycle 3 - Proposition d'un dispositif de travail en dyade [Teaching geometry at the end of cycle 3 - Proposal for a dyad-based system.]. *Petit x, 103*, 5-31. hal-02268417

Sinclair, N., & Bruce, C. D. (2015). New opportunities in geometry education at the primary school. *ZDM, 47*, 319-329.

Wu, D.-B., & Ma, H.-L. (2005). A Study of the Geometric Concepts of Elementary School Students at van Hiele Level One. *International Group for the Psychology of Mathematics Education, 4*, 329-336.
# CONNECTING SECONDARY AND COLLEGE GEOMETRY: RESOLUTION OF PROBLEMS OF CONSTRUCTION OR MEASURE OF CURVES USING DGS

Veronica Hoyos¹, Leonardo Robles-Pecina²

¹ National Pedagogical University-Mexico, ² Official Secondary School (EPO) 171

In this paper, we advance a schematic proposal on how to approach the resolution of problems of construction or measure of curves by intersections of straight lines or curves, a core topic in Cartesian Geometry, and whose solutions are based on geometric constructions of continuous movement, facilitated in this case using Dynamic Geometry Software (DGS). This proposal is based on some of the Cartesian geometric constructions (see, Jullien, 1996; Boyer, 1956; Bos, 1981) with articulated devices, which include the revision and study of conics. The empirical (and curricular) proposal for teaching the conic sections of one experienced teacher on the subject is reviewed and incorporated into the discussion, as well as some of the executions of his students.

## INTRODUCTION

In Mexico, in the Official Secondary Schools (located in the State of Mexico) the teaching and learning of geometry takes place in the second half of the 10th and the first half of the 11th grade in high school (SEP, 2017). In general terms, it can be said that the content of these two courses is reduced to learning some of the topics of trigonometry, the criteria of congruence and similarity of triangles (from a visual treatment of geometric properties), and analytic geometry basics. How are these topics related to the understanding and development of geometric thinking, which is one curricular axe in the mathematics education of the students at this educative level? (See Appendix 1).

In this paper, we advance a schematic proposal on how to approach the resolution of problems of finding the measure or construction of curves (by intersection of articulated straight lines or curves that slide in a plain, see Jullien, 1996, p.83), core topics in Cartesian Geometry, and whose solutions are based on geometric constructions of continuous movement, facilitated in this case using dynamic geometry software. And it is important to mention that the driving force that moved this proposal forward was a detailed review of the teaching experience at this educational level of the second author in this article.

The questions addressed in this paper are the following: (i) is there a connection between the topics proposed in the secondary geometry curriculum of SEMS (SEP, 2017), with the understanding and development of geometric thinking (an explicit study axis in this curriculum)? (ii) How are the topics proposed in this secondary geometry curriculum (see Appendix 1) related to the resolution of problems of finding the measure or construction of curves, core topics in Cartesian Geometry? (iii) How can the resolution of these problems be approached using dynamic geometry software? (iv) How does this approach contribute to the geometry teaching and learning proposed by the official study plans of SEMS (SEP, 2017)?
# CONNECTING SECONDARY AND COLLEGE GEOMETRY: RESOLUTION OF PROBLEMS OF CONSTRUCTION OR MEASURE OF CURVES USING DGS

Veronica Hoyos¹, Leonardo Robles-Pecina²

¹ National Pedagogical University-Mexico, ² Official Secondary School (EPO) 171

In this paper, we advance a schematic proposal on how to approach the resolution of problems of construction or measure of curves by intersections of straight lines or curves, a core topic in Cartesian Geometry, and whose solutions are based on geometric constructions of continuous movement, facilitated in this case using Dynamic Geometry Software (DGS). This proposal is based on some of the Cartesian geometric constructions (see, Jullien, 1996; Boyer, 1956; Bos, 1981) with articulated devices, which include the revision and study of conics. The empirical (and curricular) proposal for teaching the conic sections of one experienced teacher on the subject is reviewed and incorporated into the discussion, as well as some of the executions of his students.

## INTRODUCTION

In Mexico, in the Official Secondary Schools (located in the State of Mexico) the teaching and learning of geometry takes place in the second half of the 10th and the first half of the 11th grade in high school (SEP, 2017). In general terms, it can be said that the content of these two courses is reduced to learning some of the topics of trigonometry, the criteria of congruence and similarity of triangles (from a visual treatment of geometric properties), and analytic geometry basics. How are these topics related to the understanding and development of geometric thinking, which is one curricular axe in the mathematics education of the students at this educative level? (See Appendix 1).

In this paper, we advance a schematic proposal on how to approach the resolution of problems of finding the measure or construction of curves (by intersection of articulated straight lines or curves that slide in a plain, see Jullien, 1996, p.83), core topics in Cartesian Geometry, and whose solutions are based on geometric constructions of continuous movement, facilitated in this case using dynamic geometry software. And it is important to mention that the driving force that moved this proposal forward was a detailed review of the teaching experience at this educational level of the second author in this article.

The questions addressed in this paper are the following: (i) is there a connection between the topics proposed in the secondary geometry curriculum of SEMS (SEP, 2017), with the understanding and development of geometric thinking (an explicit study axis in this curriculum)? (ii) How are the topics proposed in this secondary geometry curriculum (see Appendix 1) related to the resolution of problems of finding the measure or construction of curves, core topics in Cartesian Geometry? (iii) How can the resolution of these problems be approached using dynamic geometry software? (iv) How does this approach contribute to the geometry teaching and learning proposed by the official study plans of SEMS (SEP, 2017)?
# THEORETICAL FRAMEWORK

In the work of Wasserman and colleagues (2023), the functional structure of the university courses offered in universities to future secondary-level mathematics teachers is reviewed, seeking to elucidate or find answers to the questions that exist everywhere regarding the usefulness of university mathematics for teaching school mathematics at the upper secondary level. According to these authors, this situation constitutes a challenge for the mathematics education of teachers at this educational level (p. 719).

In Mexico, though we are apparently disconnected from such educational problems, this is not the case, since to the question: who or what are the educational institutions in charge of training upper secondary level mathematics teachers in Mexico? The answer is that the mathematics training that high school mathematics teachers received is precisely what they have acquired during their undergraduate studies at the university. That is, in Mexico, it also happens that the mathematics knowledge of secondary school teachers goes through a double discontinuity, a term described by Félix Klein (2016) since the first third of the last century.

According to Wasserman et al. (2023, p. 721), the double discontinuity lies in the following: the first disconnection is that what the mathematics teacher learned at the university had little resemblance to what he knew when he left high school, and the second disconnection occurs again in the transition from studying mathematics at university and the subsequent return to teaching mathematics in the school curriculum. Moreover, Gueudet et al. (2017) had identified that “Klein's notion of a double discontinuity between university mathematics and school mathematics... can be seen to constitute the core of teacher mathematics education in both theoretical and practical aspects.” (Gueudet et al, 2017, p. 104).

By another hand, the mathematical topic addressed in this paper is the construction of curves by continuous movement that Descartes proposes in his *Géometrie*, it includes knowledge of basic geometric properties, such as the application of the Thales Theorem, and appears in the context of solving problems and theorems linked to the origins of analytical geometry, such as the resolution of the Pappus problem¹, and/or the algebraic representation of curves. The hypothesis advanced in this paper is that the historical context of the emergence of the algebraic representation of curves is a resource to work on establishing connections between university Geometry (and Calculus) and secondary Analytical Geometry. Moreover, the construction of curves by continuous movement suggested by Descartes could constitute an intuitive, or historical-evolutive approach to the study of conics from an evolutionary-historical perspective of the development of mathematical knowledge, as Moreno (2014) suggested by establishing that intuition and rigor in mathematics are coextensive.

Through an exploration of the construction of conics as continuous curves, as expressed by Descartes in his *Géometrie*, and the simulation – using dynamic geometry software – of Descartes' device, this teaching and learning proposal on conics will soon be implemented in one of the courses for the professional development of secondary teachers at the National Pedagogical University, in Mexico.
## THEORETICAL FRAMEWORK

In the work of Wasserman and colleagues (2023), the functional structure of the university courses offered in universities to future secondary-level mathematics teachers is reviewed, seeking to elucidate or find answers to the questions that exist everywhere regarding the usefulness of university mathematics for teaching school mathematics at the upper secondary level. According to these authors, this situation constitutes a challenge for the mathematics education of teachers at this educational level (p. 719).

In Mexico, though we are apparently disconnected from such educational problems, this is not the case, since to the question: who or what are the educational institutions in charge of training upper secondary level mathematics teachers in Mexico? The answer is that the mathematics training that high school mathematics teachers received is precisely what they have acquired during their undergraduate studies at the university. That is, in Mexico, it also happens that the mathematics knowledge of secondary school teachers goes through a double discontinuity, a term described by Félix Klein (2016) since the first third of the last century.

According to Wasserman et al. (2023, p. 721), the double discontinuity lies in the following: the first disconnection is that what the mathematics teacher learned at the university had little resemblance to what he knew when he left high school, and the second disconnection occurs again in the transition from studying mathematics at university and the subsequent return to teaching mathematics in the school curriculum. Moreover, Gueudet et al. (2017) had identified that “Klein's notion of a double discontinuity between university mathematics and school mathematics... can be seen to constitute the core of teacher mathematics education in both theoretical and practical aspects.” (Gueudet et al, 2017, p. 104).

By another hand, the mathematical topic addressed in this paper is the construction of curves by continuous movement that Descartes proposes in his *Géometrie*, it includes knowledge of basic geometric properties, such as the application of the Thales Theorem, and appears in the context of solving problems and theorems linked to the origins of analytical geometry, such as the resolution of the Pappus problem¹, and/or the algebraic representation of curves. The hypothesis advanced in this paper is that the historical context of the emergence of the algebraic representation of curves is a resource to work on establishing connections between university Geometry (and Calculus) and secondary Analytical Geometry. Moreover, the construction of curves by continuous movement suggested by Descartes could constitute an intuitive, or historical-evolutive approach to the study of conics from an evolutionary-historical perspective of the development of mathematical knowledge, as Moreno (2014) suggested by establishing that intuition and rigor in mathematics are coextensive.

Through an exploration of the construction of conics as continuous curves, as expressed by Descartes in his *Géometrie*, and the simulation – using dynamic geometry software – of Descartes' device, this teaching and learning proposal on conics will soon be implemented in one of the courses for the professional development of secondary teachers at the National Pedagogical University, in Mexico.
# METHODOLOGY AND DATA

## Construction of curves by continuous movement

In Descartes' *Géometrie*, a line is constructible if it is possible to draw it through continuous movement and/or through certain articulated mechanisms, where the points on the line are obtained from the intersection of other lines (curves, or even straight lines):

[Descartes] has in mind the justification of a determining procedure in his *Géometrie*: the recourse to lines generated by moving intersection... As in a deductive chain, no matter how long it is, it can lead to a condition exact provided that the rules of the method have been respected, thus the generation of a curved line can be extremely composite provided that the rules of composition are respected. These rules refer in fact to only one: that the movement that leads from one curve to the other is continuous and entirely determined... By this means one can always have exact knowledge of its measurement [of the curve's measurement]. The measurement here should not be taken as a numerical notion but as a constructible magnitude, as it has been presented at the beginning of Book I. (Jullien, 1996, pp. 77-78, our translation)

Descartes sought to establish a rule of general one-to-one correspondence between construction criteria and invariant considerations of proportionality in the articulated system in motion: "These are 'moving planes' where the principle is the following: a starting line will move [in a parallel way] along an axis and its intersection with an articulated ruler will produce a 'more composed line'..." (Jullien, 1996, p.83).

But the combinations of movements that Descartes had in mind (Bos, 1981, p. 310) involved only straight lines as moving parts, making it possible to generate curves in the plane, in particular, conics. For example, Descartes described the following construction artifact (see Figure 1).

A ruler GL pivots at G. It is linked at L with a device NKL which can be moved along the vertical axis while the direction of the line KN is kept constant. When L is moved along the vertical axis, the ruler turns around G and the line KN is moved downwards remaining parallel to itself. The intersection C of these two moving straight lines describes the curve GCE. (Bos, 1981, p. 311)

<figure><img src="image_153.png" /><figcaption>Figure 1: A diagram obtained through the use of DGS, shows the construction of the articulated system described by Descartes in his *Géometrie*, which, when sliding vertically, generates a hyperbola (see curve drawn in red)</figcaption></figure>
# METHODOLOGY AND DATA

## Construction of curves by continuous movement

In Descartes' *Géometrie*, a line is constructible if it is possible to draw it through continuous movement and/or through certain articulated mechanisms, where the points on the line are obtained from the intersection of other lines (curves, or even straight lines):

[Descartes] has in mind the justification of a determining procedure in his *Géometrie*: the recourse to lines generated by moving intersection... As in a deductive chain, no matter how long it is, it can lead to a condition exact provided that the rules of the method have been respected, thus the generation of a curved line can be extremely composite provided that the rules of composition are respected. These rules refer in fact to only one: that the movement that leads from one curve to the other is continuous and entirely determined... By this means one can always have exact knowledge of its measurement [of the curve's measurement]. The measurement here should not be taken as a numerical notion but as a constructible magnitude, as it has been presented at the beginning of Book I. (Jullien, 1996, pp. 77-78, our translation)

Descartes sought to establish a rule of general one-to-one correspondence between construction criteria and invariant considerations of proportionality in the articulated system in motion: "These are 'moving planes' where the principle is the following: a starting line will move [in a parallel way] along an axis and its intersection with an articulated ruler will produce a 'more composed line'..." (Jullien, 1996, p.83).

But the combinations of movements that Descartes had in mind (Bos, 1981, p. 310) involved only straight lines as moving parts, making it possible to generate curves in the plane, in particular, conics. For example, Descartes described the following construction artifact (see Figure 1).

A ruler GL pivots at G. It is linked at L with a device NKL which can be moved along the vertical axis while the direction of the line KN is kept constant. When L is moved along the vertical axis, the ruler turns around G and the line KN is moved downwards remaining parallel to itself. The intersection C of these two moving straight lines describes the curve GCE. (Bos, 1981, p. 311)

<figure><img src="image_160.png" /><figcaption>Figure 1: A diagram obtained through the use of DGS, shows the construction of the articulated system described by Descartes in his *Géometrie*, which, when sliding vertically, generates a hyperbola (see curve drawn in red)</figcaption></figure>
In this way, to obtain the equation of the hyperboloid, it is enough to apply the Thales Theorem to
the similar triangles formed by: (i) the vertices K, L, N & K, B, C, i.e., KLN ≅ KBC; (ii) the vertices
L, A, G & L, B, C, i.e., LAG ≅ LBC (see Figure 1). Considering now GA= a, NL= c, AB= x, KL= b,
CB= y (Bos, 1981, p. 311), the proportionality relationships between corresponding sides of the
triangles lead to obtaining the equation: cy + (c/b) yx + ay – ac = y2 (you can see an equivalent
diagram, perhaps clearer, in Figure 2).

<figure><img src="image_154.png" /><figcaption>Figure 2: An equivalent diagram (perhaps clearer) of the use of the SGD to draw the hyperboloid of Descartes</figcaption></figure>

The practice of teaching-learning conics, in grade 11 (students of 16 years old)

According to what was reported by Pecina (2023), the teaching of analytical geometry in the Official Preparatory Schools (EPO by their acronym in Spanish) of the State of Mexico (e.g., at the EPO 171) is carried out with an ordered planning of practical work in the classroom. In each of the sessions (see Table 1), and by the ‘55 methodology’ (see Hernández, 2023), the learning of the subject is guaranteed by the students' strict attention to the teacher's explanations, provided by the teacher for the students can follow and replicate the use of the formulae extracted from his planned presentation of curricular content in play. This sequence is based on a rote strategy of repetition and exercise, as shown by the planning (see Table 1) and the record of the student's activities and productions in each of the class sessions (see Figure 2).

The following Table 1 shows the planning prepared by the teacher responsible of class #21:
In this way, to obtain the equation of the hyperboloid, it is enough to apply the Thales Theorem to
the similar triangles formed by: (i) the vertices K, L, N & K, B, C, i.e., KLN ≅ KBC; (ii) the vertices
L, A, G & L, B, C, i.e., LAG ≅ LBC (see Figure 1). Considering now GA= a, NL= c, AB= x, KL= b,
CB= y (Bos, 1981, p. 311), the proportionality relationships between corresponding sides of the
triangles lead to obtaining the equation: cy + (c/b) yx + ay – ac = y2 (you can see an equivalent
diagram, perhaps clearer, in Figure 2).

<figure><img src="image_161.png" /><figcaption>Figure 2: An equivalent diagram (perhaps clearer) of the use of the SGD to draw the hyperboloid of Descartes</figcaption></figure>

The practice of teaching-learning conics, in grade 11 (students of 16 years old)

According to what was reported by Pecina (2023), the teaching of analytical geometry in the Official Preparatory Schools (EPO by their acronym in Spanish) of the State of Mexico (e.g., at the EPO 171) is carried out with an ordered planning of practical work in the classroom. In each of the sessions (see Table 1), and by the ‘55 methodology’ (see Hernández, 2023), the learning of the subject is guaranteed by the students' strict attention to the teacher's explanations, provided by the teacher for the students can follow and replicate the use of the formulae extracted from his planned presentation of curricular content in play. This sequence is based on a rote strategy of repetition and exercise, as shown by the planning (see Table 1) and the record of the student's activities and productions in each of the class sessions (see Figure 2).

The following Table 1 shows the planning prepared by the teacher responsible of class #21:
## Unit II. The Conics

<table><tr><td>Session planning #21</td><td>Ellipse's ordinary equation (the center out of the origin)</td><td>Estimated time: 210 min</td></tr></table>

Instrumental objective: Analyzing the ellipse with a center out of the origin, utilizing its ordinary equation to obtain references and magnitudes of its different elements in the Cartesian plane.

<table><caption>Table 1: Teacher planning to approach the study of the ellipse equation</caption><thead><tr><th></th><th>Task</th><th>Materials</th></tr></thead><tbody><tr><td></td><td>1. Revisiting previous session</td><td></td></tr><tr><td></td><td>2. The instrumental objective, the checklist, and the observation guide for the session are announced.</td><td></td></tr><tr><td></td><td>3. The teacher makes examples of graphing ellipses with the center outside the origin given the ordinary equation.</td><td>Blackboard and marker; wooden colors; pencil; eraser; pen; notebook; geometry set; projector;</td></tr><tr><td rowspan="5">Learning's Task</td><td>4. Groups are asked to perform exercises given the ordinary equation of the ellipse with the center outside the origin in real or simulated situations.</td><td>DVD; extended power cable.</td></tr><tr><td>5. Screening of the video "The Universe of History" to prepare a synthesis of the documentary.</td><td></td></tr><tr><td>6. Group conclusion.</td><td></td></tr><tr><td>7. Time to ask questions.</td><td></td></tr><tr><td>8. Account of the observation guide and checklist.</td><td></td></tr></tbody></table>

## CONCLUSIONS

This work sought to contribute to the line of research of establishing connections between mathematics at the university level and secondary mathematics (see Wasserman 2023; Gueudet et al., 2017), through the development of an intuitive, or historical-evolutive (see Moreno, 2014) proposal for teaching analytical geometry for the 11th grade (in high school) that looked to be complementary to what teachers have been implementing in practice in their Analytical Geometry courses at this educational level.

In his *Géometrie*, Descartes began addressing the problems of construction or measurement of curves based on the resolution of the Pappus problem. In his resolution, he arrived at one of the most celebrated mathematical discoveries, namely the algebraic representation of curves. However, another of the formidable ideas that underlie Descartes' resolution is the geometric idea of obtaining
## Unit II. The Conics

<table><tr><td>Session planning #21</td><td>Ellipse's ordinary equation (the center out of the origin)</td><td>Estimated time: 210 min</td></tr></table>

Instrumental objective: Analyzing the ellipse with a center out of the origin, utilizing its ordinary equation to obtain references and magnitudes of its different elements in the Cartesian plane.

<table><caption>Table 1: Teacher planning to approach the study of the ellipse equation</caption><thead><tr><th></th><th>Task</th><th>Materials</th></tr></thead><tbody><tr><td rowspan="8">Learning's Task</td><td>1. Revisiting previous session</td><td rowspan="8">Blackboard and marker; wooden colors; pencil; eraser; pen; notebook; geometry set; projector; DVD; extended power cable.</td></tr><tr><td>2. The instrumental objective, the checklist, and the observation guide for the session are announced.</td></tr><tr><td>3. The teacher makes examples of graphing ellipses with the center outside the origin given the ordinary equation.</td></tr><tr><td>4. Groups are asked to perform exercises given the ordinary equation of the ellipse with the center outside the origin in real or simulated situations.</td></tr><tr><td>5. Screening of the video "The Universe of History" to prepare a synthesis of the documentary.</td></tr><tr><td>6. Group conclusion.</td></tr><tr><td>7. Time to ask questions.</td></tr><tr><td>8. Account of the observation guide and checklist.</td></tr></tbody></table>

## CONCLUSIONS

This work sought to contribute to the line of research of establishing connections between mathematics at the university level and secondary mathematics (see Wasserman 2023; Gueudet et al., 2017), through the development of an intuitive, or historical-evolutive (see Moreno, 2014) proposal for teaching analytical geometry for the 11th grade (in high school) that looked to be complementary to what teachers have been implementing in practice in their Analytical Geometry courses at this educational level.

In his *Géometrie*, Descartes began addressing the problems of construction or measurement of curves based on the resolution of the Pappus problem. In his resolution, he arrived at one of the most celebrated mathematical discoveries, namely the algebraic representation of curves. However, another of the formidable ideas that underlie Descartes' resolution is the geometric idea of obtaining
or constructing curves through the continuous movement (or by sliding in the plane) of articulated mechanisms, based on the intersection between straight lines, or between straight lines and curves, or between curves and curves.

<figure><img src="image_155.png" /><figcaption>Figure 3. Students' products in the classroom, prepared after the teacher explained the topic, in this case, devoted to the study of the equation of the ellipse</figcaption></figure>

Such simulation, provided by the use of a DGS, probably enables student observation and application of invariant geometric properties of the moving device. Furthermore, discerning and applying the invariant geometric properties in play, such as Thales' Theorem, to finally arrive at the corresponding conic equation (or a quadratic equation in two variables), adds a different meaning than the commonly associated with conics when approached only through repetition and exercise of its rigorous definition. This proposal focuses on the utilization of basic geometric properties, such as Thales' theorem, as an argument for the construction of conics by continuous motion.

Algebraic curves in the plane are algebraic varieties of dimension one and can be thought of as the entry to the study of multidimensional algebraic varieties, a topic of Geometry in university mathematics (see Ramírez-Galarza, 2011). Or also as a complementary background to the study of the historical development of calculus (see Edwards, 1979).

The hypothesis that was advanced in this paper is that through the Cartesian approach to the construction or measurement of plane curves (see Jullien, 1996) it is possible to establish connections between the geometry at the university with the analytical geometry at the secondary level. Furthermore, this constitutes an intuitive, or historical-evolutionary approach (see Moreno, 2014) to the study of conics, and complements an approach of rote teaching by repetition and exercise of the definition of conics that is being implemented in practice (see Pecina, 2023).

Furthermore, this type of approach in the classroom likely promotes the recognition and establishment of invariant basic geometric properties that underlie the continuous construction of curves, and thus the generation of arguments for the development of geometric thinking in high school. Discerning
or constructing curves through the continuous movement (or by sliding in the plane) of articulated mechanisms, based on the intersection between straight lines, or between straight lines and curves, or between curves and curves.

<figure><img src="image_162.png" /><figcaption>Figure 3. Students' products in the classroom, prepared after the teacher explained the topic, in this case, devoted to the study of the equation of the ellipse</figcaption></figure>

Such simulation, provided by the use of a DGS, probably enables student observation and application of invariant geometric properties of the moving device. Furthermore, discerning and applying the invariant geometric properties in play, such as Thales' Theorem, to finally arrive at the corresponding conic equation (or a quadratic equation in two variables), adds a different meaning than the commonly associated with conics when approached only through repetition and exercise of its rigorous definition. This proposal focuses on the utilization of basic geometric properties, such as Thales' theorem, as an argument for the construction of conics by continuous motion.

Algebraic curves in the plane are algebraic varieties of dimension one and can be thought of as the entry to the study of multidimensional algebraic varieties, a topic of Geometry in university mathematics (see Ramírez-Galarza, 2011). Or also as a complementary background to the study of the historical development of calculus (see Edwards, 1979).

The hypothesis that was advanced in this paper is that through the Cartesian approach to the construction or measurement of plane curves (see Jullien, 1996) it is possible to establish connections between the geometry at the university with the analytical geometry at the secondary level. Furthermore, this constitutes an intuitive, or historical-evolutionary approach (see Moreno, 2014) to the study of conics, and complements an approach of rote teaching by repetition and exercise of the definition of conics that is being implemented in practice (see Pecina, 2023).

Furthermore, this type of approach in the classroom likely promotes the recognition and establishment of invariant basic geometric properties that underlie the continuous construction of curves, and thus the generation of arguments for the development of geometric thinking in high school. Discerning
and applying invariant geometric properties, such as Thales' Theorem, to finally arrive at the
corresponding conic equation (or a quadratic equation in two variables), adds a different meaning
than the commonly associated with conics from an approach of rote teaching through repetition and
exercise of its rigorous definition.

Footnotes

[1] In Descartes' *Géometrie*, the Pappus problem in the case of 3 or 4 straight lines, leads to obtaining the general equation of the second degree in two unknowns, i.e. is an algebraic description of conics. A version of the Pappus problem from Jullien (1996, p. 72), is the following:

Given four coplanar lines D₁, D₂, D₃, D₄ given in position, consider a point C, [and] denoted as d₁, d₂, d₃, d₄ the lengths of the segments that join C with D₁ (respectively 2, 3, 4), under a given angle. The relationship (d₁.d₂) is then fixed compared to (d₃.d₄) ... One of the lines is taken as the 'abscissae axis', one point A as the origin, and a direction determines the ordinates. One point C, solution of the problem, is associated with the two coordinates AB=x & BC=y (BC is the first line involved in the analysis of the problem). Descartes then shows, with the help of simple considerations, mobilizing the given lines and angles (those that join C with one of the given lines, under a given angle) ... [that] Pappus's problem, up to five lines, produces an equation [with two unknowns] that does not go beyond the second degree... [and] its solutions can be constructed with a ruler and compass. (Jullien, 1996, pp. 72-74)

<figure><img src="image_156.png" /><figcaption>Figure 4: Image of the Pappus' Problem in the case of four straight lines (Jullien, 1996, p.72)</figcaption></figure>

References

Bos, H. (1981). On the representation of curves in Descartes' *Géométrie*. *Archive for History of Exact Sciences*, 295-338.

Boyer, C. (1956). *History of Analytic Geometry*. Dover.

Edwards, Ch. (1979). *The historical development of the calculus*. Springer.

Gueudet, G., Bosh, M., DiSessa, A., Kwon, O., and Verschaffel, L. (2017). Transitions in mathematics education: The panel debate. In G. Kaiser (Ed.), *Proceedings of the 13th International Congress in Mathematics Education*, ICME-13 Monographs.

Hernández, O. (2003). *Metodología* 55. Retrieved, on 01/12/024 at
http://www.prepa224.edu.mx/Metodologia.html

Jullien, V. (1996). *Descartes. La “Géométrie” de 1637*. PUF.

Moreno, L. (2014). Intuir y formalizar: Procesos coextensivos. *Educación Matemática, 25 Años*, pp. 185-206.

Pecina, L. (2023). Enseñanza de la Geometría Analítica en la Escuela Preparatoria Oficial 171. Reporte sintético de la práctica en el salón de clases. Documento de Trabajo. Sin publicar.
and applying invariant geometric properties, such as Thales' Theorem, to finally arrive at the
corresponding conic equation (or a quadratic equation in two variables), adds a different meaning
than the commonly associated with conics from an approach of rote teaching through repetition and
exercise of its rigorous definition.

Footnotes

[1] In Descartes' *Géometrie*, the Pappus problem in the case of 3 or 4 straight lines, leads to obtaining the general equation of the second degree in two unknowns, i.e. is an algebraic description of conics. A version of the Pappus problem from Jullien (1996, p. 72), is the following:

Given four coplanar lines D₁, D₂, D₃, D₄ given in position, consider a point C, [and] denoted as d₁, d₂, d₃, d₄ the lengths of the segments that join C with D₁ (respectively 2, 3, 4), under a given angle. The relationship (d₁.d₂) is then fixed compared to (d₃.d₄) ... One of the lines is taken as the 'abscissae axis', one point A as the origin, and a direction determines the ordinates. One point C, solution of the problem, is associated with the two coordinates AB=x & BC=y (BC is the first line involved in the analysis of the problem). Descartes then shows, with the help of simple considerations, mobilizing the given lines and angles (those that join C with one of the given lines, under a given angle) ... [that] Pappus's problem, up to five lines, produces an equation [with two unknowns] that does not go beyond the second degree... [and] its solutions can be constructed with a ruler and compass. (Jullien, 1996, pp. 72-74)

<figure><img src="image_163.png" /><figcaption>Figure 4: Image of the Pappus' Problem in the case of four straight lines (Jullien, 1996, p.72)</figcaption></figure>

References

Bos, H. (1981). On the representation of curves in Descartes' *Géométrie*. *Archive for History of Exact Sciences*, 295-338.

Boyer, C. (1956). *History of Analytic Geometry*. Dover.

Edwards, Ch. (1979). *The historical development of the calculus*. Springer.

Gueudet, G., Bosh, M., DiSessa, A., Kwon, O., and Verschaffel, L. (2017). Transitions in mathematics education: The panel debate. In G. Kaiser (Ed.), *Proceedings of the 13th International Congress in Mathematics Education*, ICME-13 Monographs.

Hernández, O. (2003). *Metodología* 55. Retrieved, on 01/12/024 at
http://www.prepa224.edu.mx/Metodologia.html

Jullien, V. (1996). *Descartes. La “Géométrie” de 1637*. PUF.

Moreno, L. (2014). Intuir y formalizar: Procesos coextensivos. *Educación Matemática, 25 Años*, pp. 185-206.

Pecina, L. (2023). Enseñanza de la Geometría Analítica en la Escuela Preparatoria Oficial 171. Reporte sintético de la práctica en el salón de clases. Documento de Trabajo. Sin publicar.
Ramírez-Galarza, A. (2011). *Geometría Analítica. Una introducción a la Geometría* [nivel universitario]. México: Fac. de Ciencias-UNAM.

SEP [Public Education Secretariat] (2017). *Planes de estudio SEMS* [Sistema de Educación Media-Superior]. Mexico: SEP [See at https://dgb.sep.gob.mx/storage/recursos/2023/08/WiVoqQZSMA-Matematicas-III.pdf]

Uribe, L. (2005). *Geometría analítica [nivel medio-superior]*. Grupo Editorial Uribe GC.

Wasserman, N., Buchbinder, O., and Buchholtz, N. (2023). Making university mathematics matter for secondary teacher preparation. *ZDM*, 55, 719-736.

Appendix 1. Geometry curriculum for Secondary School in Mexico (students of 15-18 years old)

<table><thead><tr><th>Axis</th><th>Components</th><th>Central content</th></tr></thead><tbody><tr><td rowspan="2">From the treatment of space, form, and measurement to the geometric and trigonometric thinking</td><td>Structure and transformation: Geometry basic elements</td><td>Fundamental concepts of space and form, “the geometric”<br>The study of geometric figures and their properties<br>Treatment of geometric formulas for areas and volumes<br>Visual treatment of geometric properties, the criteria of congruence and similarity of triangles</td></tr><tr><td>Angularity and constructions: Elements of the Plane Trigonometry</td><td>Basic concepts of the trigonometry<br>Uses and functions of trigonometric relationships in the triangle<br>Trigonometric functions and its properties<br>Measurements of angles and trigonometric relationships<br>From the unit circle to the Cartesian plane. An introduction to ratios of magnitudes to real functions. Visualizing trigonometric formulas and identities</td></tr><tr><td>Geometric loci and reference systems. From geometric to analytical thinking</td><td>System of reference and location: Elements of Analytic Geometry</td><td>Analytical Geometry as an algebraic method for solving geometric tasks. Treatment in various coordinate systems<br>Basic concepts of the system rectangular coordinates, orientation and position in the plane. The role of the origin of coordinates in reference systems.<br>Recognition and construction of geometric loci: line, circle, ellipse, parabola and hyperbola<br>Visual treatment and representations multiples of the geometric loci: rectangular and parametric coordinates, singular points, roots and asymptotic behavior</td></tr></tbody></table>
Ramírez-Galarza, A. (2011). *Geometría Analítica. Una introducción a la Geometría* [nivel universitario]. México: Fac. de Ciencias-UNAM.

SEP [Public Education Secretariat] (2017). *Planes de estudio SEMS* [Sistema de Educación Media-Superior]. Mexico: SEP [See at https://dgb.sep.gob.mx/storage/recursos/2023/08/WiVoqQZSMA-Matematicas-III.pdf]

Uribe, L. (2005). *Geometría analítica [nivel medio-superior]*. Grupo Editorial Uribe GC.

Wasserman, N., Buchbinder, O., and Buchholtz, N. (2023). Making university mathematics matter for secondary teacher preparation. *ZDM*, 55, 719-736.

Appendix 1. Geometry curriculum for Secondary School in Mexico (students of 15-18 years old)

<table><thead><tr><th>Axis</th><th>Components</th><th>Central content</th></tr></thead><tbody><tr><td rowspan="2">From the treatment of space, form, and measurement to the geometric and trigonometric thinking</td><td>Structure and transformation: Geometry basic elements</td><td>Fundamental concepts of space and form, "the geometric"<br>The study of geometric figures and their properties<br>Treatment of geometric formulas for areas and volumes<br>Visual treatment of geometric properties, the criteria of congruence and similarity of triangles</td></tr><tr><td>Angularity and constructions: Elements of the Plane Trigonometry</td><td>Basic concepts of the trigonometry<br>Uses and functions of trigonometric relationships in the triangle<br>Trigonometric functions and its properties<br>Measurements of angles and trigonometric relationships<br>From the unit circle to the Cartesian plane. An introduction to ratios of magnitudes to real functions. Visualizing trigonometric formulas and identities</td></tr><tr><td>Geometric loci and reference systems. From geometric to analytical thinking</td><td>System of reference and location: Elements of Analytic Geometry</td><td>Analytical Geometry as an algebraic method for solving geometric tasks. Treatment in various coordinate systems<br>Basic concepts of the system rectangular coordinates, orientation and position in the plane. The role of the origin of coordinates in reference systems.<br>Recognition and construction of geometric loci: line, circle, ellipse, parabola and hyperbola<br>Visual treatment and representations multiples of the geometric loci: rectangular and parametric coordinates, singular points, roots and asymptotic behavior</td></tr></tbody></table>
# HOW CAN VIRTUAL GEOMETRY MANIPULATIVES BE USED IN WAYS THAT MITIGATE THEIR ONTOLOGICAL, TECHNOLOGICAL AND PEDAGOGICAL LIMITATIONS?

Ulrich Kortenkamp¹, Kevin Larkin²

¹ Potsdam University ² Griffith University

The use of virtual geometry manipulatives in mathematics and mathematics education is increasingly commonplace. In this paper we identify three sets of concerns regarding their use. Firstly, ontological ones whereby it is perhaps impossible to faithfully represent the three-dimensional world in two-dimensions. Secondly, technological ones in attempting to represent geometrical objects using digital tools. Finally, pedagogical ones relating to the impact of these two concerns on decisions regarding the use of virtual geometrical manipulatives in educationally appropriate ways. We then suggest the use of trust certificates as a way forward in responding to these three concerns and argue that trust certificates provide students with a procedure for developing confidence in evaluating the accuracy of digital representations. Finally, we propose that this procedure is useful in evaluating problem solutions generated by Artificial Intelligence software. Given the rapid proliferation of digital representations, in the form of virtual geometry manipulatives and artificial intelligence, in our view a process for determining the accuracy of information is a skill that students require.

## INTRODUCTION

Manipulatives have long played a critical role in the teaching and learning of geometry (Moyer-Packenham & Bolyard, 2016) and activities using manipulatives form the basis for enhancing a range of geometrical learning processes. Operations using manipulatives, both physical and virtual (Ladel, 2009) are purposeful, and it is essential that students are aware of the relations between the objects and the concepts they are learning. There is a growing body of evidence indicating that quality virtual manipulatives (VM) support mathematical learning (Ladel & Kortenkamp 2016; Larkin, 2016). Given that there has been a rapid increase in the use of virtual geometry manipulatives (VGM) in mathematics education since the last ICMI Geometry Study in 1995, it is opportune to reflect on issues raised by their use.

Moyer-Packenham and Bolyard (2016) define a VM as “an interactive, technology-enabled visual representation of a dynamic mathematical object, including all of the programmable features that allow it to be manipulated, that presents opportunities for constructing mathematical knowledge” (p. 13). At their core, VM provide students with opportunities to interweave iconic and symbolic representations, with the actions that they perform on them, to emphasize the underlying mathematical concepts. In this paper, VM include both comprehensive geometry packages such as *Cabri, Geometers' Sketchpad, Cinderella*, and *GeoGebra*, as well as stand-alone geometry apps such as *Coordinate Geometry* or *Cubeling*.

Critiques of VM (Larkin, 2016; Namukasa et al., 2016) have consistently found that most VGM are very unlikely to support the development of geometrical understanding. For example, Larkin (2016)
# HOW CAN VIRTUAL GEOMETRY MANIPULATIVES BE USED IN WAYS THAT MITIGATE THEIR ONTOLOGICAL, TECHNOLOGICAL AND PEDAGOGICAL LIMITATIONS?

Ulrich Kortenkamp¹, Kevin Larkin²

¹ Potsdam University ² Griffith University

The use of virtual geometry manipulatives in mathematics and mathematics education is increasingly commonplace. In this paper we identify three sets of concerns regarding their use. Firstly, ontological ones whereby it is perhaps impossible to faithfully represent the three-dimensional world in two-dimensions. Secondly, technological ones in attempting to represent geometrical objects using digital tools. Finally, pedagogical ones relating to the impact of these two concerns on decisions regarding the use of virtual geometrical manipulatives in educationally appropriate ways. We then suggest the use of trust certificates as a way forward in responding to these three concerns and argue that trust certificates provide students with a procedure for developing confidence in evaluating the accuracy of digital representations. Finally, we propose that this procedure is useful in evaluating problem solutions generated by Artificial Intelligence software. Given the rapid proliferation of digital representations, in the form of virtual geometry manipulatives and artificial intelligence, in our view a process for determining the accuracy of information is a skill that students require.

## INTRODUCTION

Manipulatives have long played a critical role in the teaching and learning of geometry (Moyer-Packenham & Bolyard, 2016) and activities using manipulatives form the basis for enhancing a range of geometrical learning processes. Operations using manipulatives, both physical and virtual (Ladel, 2009) are purposeful, and it is essential that students are aware of the relations between the objects and the concepts they are learning. There is a growing body of evidence indicating that quality virtual manipulatives (VM) support mathematical learning (Ladel & Kortenkamp 2016; Larkin, 2016). Given that there has been a rapid increase in the use of virtual geometry manipulatives (VGM) in mathematics education since the last ICMI Geometry Study in 1995, it is opportune to reflect on issues raised by their use.

Moyer-Packenham and Bolyard (2016) define a VM as “an interactive, technology-enabled visual representation of a dynamic mathematical object, including all of the programmable features that allow it to be manipulated, that presents opportunities for constructing mathematical knowledge” (p. 13). At their core, VM provide students with opportunities to interweave iconic and symbolic representations, with the actions that they perform on them, to emphasize the underlying mathematical concepts. In this paper, VM include both comprehensive geometry packages such as *Cabri, Geometers' Sketchpad, Cinderella*, and *GeoGebra*, as well as stand-alone geometry apps such as *Coordinate Geometry* or *Cubeling*.

Critiques of VM (Larkin, 2016; Namukasa et al., 2016) have consistently found that most VGM are very unlikely to support the development of geometrical understanding. For example, Larkin (2016)
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.