losing track of it in the following years when new numbers are introduced. All this does not allow for
an in-depth critical analysis and discovery of the students guided by the teacher. To all this we must
add that important historical knowledge is lost, i.e. the idea of a static mathematics is consolidated.
On the contrary, the Number Line brings together millennia of theory much of which, like the
identification of a geometric point with a pair of ordered real numbers by R. Descartes (1596-1650)
and the rigorous definition of real numbers by R. Dedekind (1831-1916) and G. Cantor (1845-1918),
relatively close to us. Obviously, in the case of the secondary school, the connection with the
philosophical and epistemological themes is lost.

The geometric Number Line has the characteristic of one-dimensional space to satisfy the total ordering of the various numerical sets. For the Euclidean construction of the Number Line, the *Elements* make available, in addition to the two aforementioned tools, the following fundamental geometric entities (Sbaragli, 2005): a) the point (Deff. I,I; I,III; Post. I,I); b) the 'straight line', the segment in modern parlance (Deff. I,II; I,IV; I,VI; Post. I,II); c) the circumference (Def. I,XV; Post. I,III); d) the 'surface', the plane in modern parlance (Def. I,VII).

Below is the schematic description of the construction of the Number Line for numerical points that
can be constructed starting from the set $\mathbb{N}$. The restructuring phase of the Number Line as virtual
geometric artifact will also be an opportunity to think about numbers' features and their history.

**Construction of $\mathbb{N}$**

To construct $\mathbb{N}$ it is necessary to choose a point to identify it with 0. Since we want to construct a
straight line through points it is necessary to choose one in a one-dimensional space (geodetic line).
But you also have to consider the straightedge and compass construction types that require at least a
two-dimensional space or plane $\Pi$ (Barbieri, Maschietto, 2012).

Having defined the geometric locus where to locate the point 0, we must ask ourselves whether by
choosing it at random we are sure of finding that point. This requires that the plane consists of points
that are complete and this property, in general, is not explicit in the *Elements* but it seems that it can
be deduced from the Post. I,III. However, this certainty comes with the introduction of the Cartesian
plane and the axiomatic analysis carried out by Hilbert at the end of the 19th century.

The placement of the 1 is very important as it
identifies the unit referred to in Def. VII, I and for the
Def. VII,III this unity, both arithmetic and geometric,
will be the measure of all our construction (Figure 3).
The choice of the point corresponding to 1 is
completely arbitrary among those which are distant
from 0 of the distance taken as unit. To do this, we use
the compass to draw a circumference. At this point the
basic structure of the Number Line for $\mathbb{N}$ has been
built following the Post. I,I. The segment $\overline{01}$ has
deliberately not been drawn because we want to
highlight the existing numerical void between two
consecutive naturals since $\mathbb{N}$ is non-dense ($\mathbb{N} \not\subset \mathbb{R}$ and $\mathbb{N} \not\subset \mathbb{Q}$).

<figure><img src="image_105.png" /><figcaption>Figure 3</figcaption></figure>
losing track of it in the following years when new numbers are introduced. All this does not allow for
an in-depth critical analysis and discovery of the students guided by the teacher. To all this we must
add that important historical knowledge is lost, i.e. the idea of a static mathematics is consolidated.
On the contrary, the Number Line brings together millennia of theory much of which, like the
identification of a geometric point with a pair of ordered real numbers by R. Descartes (1596-1650)
and the rigorous definition of real numbers by R. Dedekind (1831-1916) and G. Cantor (1845-1918),
relatively close to us. Obviously, in the case of the secondary school, the connection with the
philosophical and epistemological themes is lost.

The geometric Number Line has the characteristic of one-dimensional space to satisfy the total ordering of the various numerical sets. For the Euclidean construction of the Number Line, the *Elements* make available, in addition to the two aforementioned tools, the following fundamental geometric entities (Sbaragli, 2005): a) the point (Deff. I,I; I,III; Post. I,I); b) the 'straight line', the segment in modern parlance (Deff. I,II; I,IV; I,VI; Post. I,II); c) the circumference (Def. I,XV; Post. I,III); d) the 'surface', the plane in modern parlance (Def. I,VII).

Below is the schematic description of the construction of the Number Line for numerical points that
can be constructed starting from the set $\mathbb{N}$. The restructuring phase of the Number Line as virtual
geometric artifact will also be an opportunity to think about numbers' features and their history.

**Construction of $\mathbb{N}$**

To construct $\mathbb{N}$ it is necessary to choose a point to identify it with 0. Since we want to construct a
straight line through points it is necessary to choose one in a one-dimensional space (geodetic line).
But you also have to consider the straightedge and compass construction types that require at least a
two-dimensional space or plane $\Pi$ (Barbieri, Maschietto, 2012).

Having defined the geometric locus where to locate the point 0, we must ask ourselves whether by
choosing it at random we are sure of finding that point. This requires that the plane consists of points
that are complete and this property, in general, is not explicit in the *Elements* but it seems that it can
be deduced from the Post. I,III. However, this certainty comes with the introduction of the Cartesian
plane and the axiomatic analysis carried out by Hilbert at the end of the 19th century.

The placement of the 1 is very important as it
identifies the unit referred to in Def. VII, I and for the
Def. VII,III this unity, both arithmetic and geometric,
will be the measure of all our construction (Figure 3).
The choice of the point corresponding to 1 is
completely arbitrary among those which are distant
from 0 of the distance taken as unit. To do this, we use
the compass to draw a circumference. At this point the
basic structure of the Number Line for $\mathbb{N}$ has been
built following the Post. I,I. The segment $\overline{01}$ has
deliberately not been drawn because we want to
highlight the existing numerical void between two
consecutive naturals since $\mathbb{N}$ is non-dense ($\mathbb{N} \not\subset \mathbb{R}$ and $\mathbb{N} \not\subset \mathbb{Q}$).

<figure><img src="image_111.png" /><figcaption>Figure 3</figcaption></figure>
As for the construction by induction of the natural number consecutive to $n$, a geometric construction must be found to identify the point corresponding to 2 which is $\overline{01}$ from 1 and is on the straight line being constructed. Not wanting to enforce the Post. I,II the assurance that the construction of successive naturals is possible without the use of the straightedge is based on the Mascheroni-Mohr theorem (1672/1797). To satisfy the unit 'distance' from 1, draw the circumference with center at 1 and radius $\overline{01}$ by identifying the two points A and B on the plane (Figure 4). The constructability problem is to identify the point 2 on the circle with center 1 aligned to 0 and 1. This point can be identified as the opposite to 0 of the diameter $\overline{02}$ with respect to 1, viewed as midpoint. Therefore one must construct the vertex opposite 0 of a

polygon inscribed in the circle with center 1. The candidate polygons are all those with an even number of sides (square, hexagon, etc.) and with a vertex at 0. Among these polygons it is preferable to choose the hexagon since its side is congruent with the radius of the circumscribing circumference. Ultimately, therefore, point 2 is constructed by pointing the compass at A or at B with amplitude $\overline{A1}$ or $\overline{B1}$ to identify, respectively, either point C or point D intersection with the circumference with center 1. Again, centering in C or D with the same amplitude identifies the point 2 aligned with 0 and 1. The construction of all subsequent natural numbers is carried out with the same procedure in sequence.

<figure><img src="image_106.png" /><figcaption>Figure 4</figcaption></figure>

## Construction of $\mathbb{Z}$

The relative integers $\mathbb{Z}$ will be constructed with the same procedure described for $\mathbb{N}$ but along the negative direction (on the left by Western convention).

## Construction of $\mathbb{Q}$

In this case the set is dense i.e. $\frac{p}{q}, \frac{s}{t} \in \mathbb{Q}$, $\frac{p}{q} < \frac{s}{t}$, $\exists \frac{k}{m} \in \mathbb{Q} \mid \frac{p}{q} < \frac{k}{m} < \frac{s}{t}$ but not complete, i.e. $\frac{p}{q}, \frac{s}{t} \in \mathbb{Q}$, $\frac{p}{q} < \frac{s}{t}$, $\exists x \notin \mathbb{Q} \mid \frac{p}{q} < x < \frac{s}{t}$. If the subdivision of the interval between $n$ and $n+1$ is equidistant, the number of rational points with denominator $m$ to be identified will be $m-1$, unless there are any overlaps with previous points as the equivalent rationals (fractions reduced to lowest terms or not).

<figure><img src="image_107.png" /><figcaption>Figure 5</figcaption></figure>
As for the construction by induction of the natural number consecutive to $n$, a geometric construction must be found to identify the point corresponding to 2 which is $\overline{01}$ from 1 and is on the straight line being constructed. Not wanting to enforce the Post. I,II the assurance that the construction of successive naturals is possible without the use of the straightedge is based on the Mascheroni-Mohr theorem (1672/1797). To satisfy the unit 'distance' from 1, draw the circumference with center at 1 and radius $\overline{01}$ by identifying the two points A and B on the plane (Figure 4). The constructability problem is to identify the point 2 on the circle with center 1 aligned to 0 and 1. This point can be identified as the opposite to 0 of the diameter $\overline{02}$ with respect to 1, viewed as midpoint. Therefore one must construct the vertex opposite 0 of a

polygon inscribed in the circle with center 1. The candidate polygons are all those with an even number of sides (square, hexagon, etc.) and with a vertex at 0. Among these polygons it is preferable to choose the hexagon since its side is congruent with the radius of the circumscribing circumference. Ultimately, therefore, point 2 is constructed by pointing the compass at A or at B with amplitude $\overline{A1}$ or $\overline{B1}$ to identify, respectively, either point C or point D intersection with the circumference with center 1. Again, centering in C or D with the same amplitude identifies the point 2 aligned with 0 and 1. The construction of all subsequent natural numbers is carried out with the same procedure in sequence.

## Construction of $\mathbb{Z}$

The relative integers $\mathbb{Z}$ will be constructed with the same procedure described for $\mathbb{N}$ but along the negative direction (on the left by Western convention).

## Construction of $\mathbb{Q}$

In this case the set is dense i.e. $\frac{p}{q}, \frac{s}{t} \in \mathbb{Q}$, $\frac{p}{q} < \frac{s}{t}$, $\exists \frac{k}{m} \in \mathbb{Q} \mid \frac{p}{q} < \frac{k}{m} < \frac{s}{t}$ but not complete, i.e. $\frac{p}{q}, \frac{s}{t} \in \mathbb{Q}, \frac{p}{q} < \frac{s}{t}, \exists x \notin \mathbb{Q} \mid \frac{p}{q} < x < \frac{s}{t}$. If the subdivision of the interval between $n$ and $n+1$ is equidistant, the number of rational points with denominator $m$ to be identified will be $m-1$, unless there are any overlaps with previous points as the equivalent rationals (fractions reduced to lowest terms or not).

<figure><img src="image_112.png" /><figcaption>Figure 4</figcaption></figure>

<figure><img src="image_113.png" /><figcaption>Figure 5</figcaption></figure>
Due to the lack of completeness, we will continue not to take in account the segment that joins two consecutive naturals and we will use Thales' theorem, i.e. intercept theorem, with the following construction.

For the points corresponding to the two consecutive natural numbers, the straight lines $\overline{SM}$ and $\overline{WZ}$ are drawn with reference to Prop. I,11 and with the protocol of Prop. I,1 (Figure 5). For the identification of the *m*-1 rationals along the aforesaid passing lines, in our case between 0 and 1, the $k_1, ..., k_m$ points equidistant from each other are identified, starting from 0 and 1, along the four half-lines. Subsequently the point 0 joins the $k_m$ on the straight line passing through 1. The same is done by joining the point 1 with the $k_m$ on the straight line passing through 0. The drawing continues by joining the remaining points of one perpendicular with those of the other perpendicular which they find in the half-plane opposite to the one containing them with a segment parallel to the outgoing segment, respectively from 0 or 1 and joining the opposite $k_m$. The intersection of the segments joining the points on the perpendiculars will identify the rationals sought.

## Construction of $\mathbb{R}$

Among the reals, the simplest numbers to construct there are the radicals $\sqrt{n}$ with $n \in \mathbb{N}$. The design of these rationals is possible by applying the Pythagorean theorem (Prop. I,47) considering the formula

<figure><img src="image_108.png" /><figcaption>Figure 6</figcaption></figure>

$$ \sqrt{n} = \sqrt{(\sqrt{n}-1)^2 + 1^2} $$

that is, constructing a right triangle with one side congruent to unity and the other congruent with the immediately preceding natural one: the number sought will be that of the hypotenuse. Also in this case, not wanting to materialize the Number Line, we must consider that the construction with the Pythagorean theorem will have to be carried out in the two semi-planes identified by the unit segments constructed on the perpendicular passing through the point corresponding to $\sqrt{n-1}$.

The drawing of the perpendicular passing through an algebraic irrational not coinciding with a natural one poses the problem, if one uses the procedure seen for

the construction of the rationals, of identifying two points along the diameter of the circumference with center $\sqrt{n-1}$. In reality, the problem is not in identifying both: it is possible to identify the distance with the immediately lower natural $k$ as radius but it becomes impossible to identify, along the diameter of the circumference, the antipodal point at the distance $\sqrt{n-1-k}$, if not coinciding with a natural. To overcome this, one can make use of Prop. I,31 which allows the drawing of a parallel for a point external to a given straight line, i.e. the perpendicular drawn for the point identifying the natural immediately lower than the irrational considered. For example, $\sqrt{2}$ is identified by drawing the perpendicular for point 1 and on it the unit distance $\overline{10}$ (point AA) is reported (Figure 6). Then the circumference with center 0 and radius AA is drawn which is equal to the length $\sqrt{2}$.
Due to the lack of completeness, we will continue not to take in account the segment that joins two consecutive naturals and we will use Thales' theorem, i.e. intercept theorem, with the following construction.

For the points corresponding to the two consecutive natural numbers, the straight lines $\overline{SM}$ and $\overline{WZ}$ are drawn with reference to Prop. I,11 and with the protocol of Prop. I,1 (Figure 5). For the identification of the *m*-1 rationals along the aforesaid passing lines, in our case between 0 and 1, the $k_1, ..., k_m$ points equidistant from each other are identified, starting from 0 and 1, along the four half-lines. Subsequently the point 0 joins the $k_m$ on the straight line passing through 1. The same is done by joining the point 1 with the $k_m$ on the straight line passing through 0. The drawing continues by joining the remaining points of one perpendicular with those of the other perpendicular which they find in the half-plane opposite to the one containing them with a segment parallel to the outgoing segment, respectively from 0 or 1 and joining the opposite $k_m$. The intersection of the segments joining the points on the perpendiculars will identify the rationals sought.

### Construction of $\mathbb{R}$

Among the reals, the simplest numbers to construct there are the radicals $\sqrt{n}$ with $n \in \mathbb{N}$. The design of these rationals is possible by applying the Pythagorean theorem (Prop. I,47) considering the formula

<figure><img src="image_114.png" /><figcaption>Figure 6</figcaption></figure>

$$ \sqrt{n} = \sqrt{(\sqrt{n}-1)^2 + 1^2} $$

that is, constructing a right triangle with one side congruent to unity and the other congruent with the immediately preceding natural one: the number sought will be that of the hypotenuse. Also in this case, not wanting to materialize the Number Line, we must consider that the construction with the Pythagorean theorem will have to be carried out in the two semi-planes identified by the unit segments constructed on the perpendicular passing through the point corresponding to $\sqrt{n-1}$.

The drawing of the perpendicular passing through an algebraic irrational not coinciding with a natural one poses the problem, if one uses the procedure seen for

the construction of the rationals, of identifying two points along the diameter of the circumference with center $\sqrt{n-1}$. In reality, the problem is not in identifying both: it is possible to identify the distance with the immediately lower natural $k$ as radius but it becomes impossible to identify, along the diameter of the circumference, the antipodal point at the distance $\sqrt{n-1-k}$, if not coinciding with a natural. To overcome this, one can make use of Prop. I,31 which allows the drawing of a parallel for a point external to a given straight line, i.e. the perpendicular drawn for the point identifying the natural immediately lower than the irrational considered. For example, $\sqrt{2}$ is identified by drawing the perpendicular for point 1 and on it the unit distance $\overline{10}$ (point AA) is reported (Figure 6). Then the circumference with center 0 and radius AA is drawn which is equal to the length $\sqrt{2}$.
This circumference crosses the numerical void between 1 and 2, so we need to draw an arc that identifies the point $\sqrt{2}$ aligned with the naturals. Considering that the triangle 01AA is isosceles and right-angled with a central angle equal to $\frac{\pi}{4}$, we have that AA is the vertex of the octagon inscribed in the circumference with center 0. Therefore, the point $\sqrt{2}$ is identified by intersecting the circumference with center 0 and the circumference with center in AA and radius $\overline{AAC}$.

Up to now we have identified the following points (for rationals there are only fractions with a denominator of 3) of a Number Line by points which gives a better idea than what is deduced in school textbooks with the presence of 'empty spaces' of non-constructible numbers, i.e. the place of the transcendentals (Figure 7).

<figure><img src="image_109.png" /><figcaption>Figure 7: Some constructable points along the Number Line</figcaption></figure>

To locate the geometric point corresponding to $\sqrt{3}$ you can repeat the procedure seen before but now the angle at 0 is 35.2° i.e. between 32.7° and 36° corresponding, respectively, to the angles at the center of an 11-regular agon and a 10-regular agon. This tells us that it is not possible to identify $\sqrt{3}$ starting from a vertex of the inscribed polygon as was done for $\sqrt{2}$. The construction with the Pythagorean theorem is similar to the one carried out for $\sqrt{2}$. In order to identify the point corresponding to $\sqrt{n+1}$ without prolonging the segment $\overline{01}$ we can apply both the drawing of a straight line parallel to a given point through an external point (Prop. I,31) and the 2nd Euclid's theorem (Prop. VI,8).

From what we have seen we are able to identify all the geometric points of the irrationals of even index. Those of odd index, such as $\sqrt[3]{2}$, cannot be drawn with straightedge and compass according to the theory of Galois groups. Even less constructively identifiable are the non-algebraic irrationals, i.e. the transcendentals such as $e$ and $\pi$. At the end it is possible to draw the straight line according to the Post. I,II resulting in the only way to draw but not identify those numbers that form the continuum.

## CONCLUSIONS

This proposal aims to highlight the intimate connection, both theoretical and didactic, between geometry and number theory mediated with DGS.

The aspect that is brought to light with the deconstructuring and restructuring of the geometric Number Line is that of a virtual artifact that is educationally useful and susceptible to an internal dynamic that allows us to grasp the fundamental elements of mathematics: numbers and geometrical points. In particular, it allows the study of constructible numbers both from a theoretical and historical point of view offering a geometric view to understand the different features of numbers.

About this didactic proposal, only the theoretical part concerning numerical sets was carried out in a secondary school in past years while the historical study and the constructability of numbers are activities that will be a subject of experimentation, always in secondary school, for the purposes of doctoral work. As regards the theoretical part carried out, obviously, in the secondary school the approach was more qualitative with, however, some operational considerations with the natural and
This circumference crosses the numerical void between 1 and 2, so we need to draw an arc that identifies the point $\sqrt{2}$ aligned with the naturals. Considering that the triangle 01AA is isosceles and right-angled with a central angle equal to $\frac{\pi}{4}$, we have that AA is the vertex of the octagon inscribed in the circumference with center 0. Therefore, the point $\sqrt{2}$ is identified by intersecting the circumference with center 0 and the circumference with center in AA and radius $\overline{AAC}$.

Up to now we have identified the following points (for rationals there are only fractions with a denominator of 3) of a Number Line by points which gives a better idea than what is deduced in school textbooks with the presence of 'empty spaces' of non-constructible numbers, i.e. the place of the transcendentals (Figure 7).

<figure><img src="image_115.png" /><figcaption>Figure 7: Some constructable points along the Number Line</figcaption></figure>

To locate the geometric point corresponding to $\sqrt{3}$ you can repeat the procedure seen before but now the angle at 0 is 35.2° i.e. between 32.7° and 36° corresponding, respectively, to the angles at the center of an 11-regular agon and a 10-regular agon. This tells us that it is not possible to identify $\sqrt{3}$ starting from a vertex of the inscribed polygon as was done for $\sqrt{2}$. The construction with the Pythagorean theorem is similar to the one carried out for $\sqrt{2}$. In order to identify the point corresponding to $\sqrt{n+1}$ without prolonging the segment $\overline{01}$ we can apply both the drawing of a straight line parallel to a given point through an external point (Prop. I,31) and the 2nd Euclid's theorem (Prop. VI,8).

From what we have seen we are able to identify all the geometric points of the irrationals of even index. Those of odd index, such as $\sqrt[3]{2}$, cannot be drawn with straightedge and compass according to the theory of Galois groups. Even less constructively identifiable are the non-algebraic irrationals, i.e. the transcendentals such as $e$ and $\pi$. At the end it is possible to draw the straight line according to the Post. I,II resulting in the only way to draw but not identify those numbers that form the continuum.

## CONCLUSIONS

This proposal aims to highlight the intimate connection, both theoretical and didactic, between geometry and number theory mediated with DGS.

The aspect that is brought to light with the deconstructuring and restructuring of the geometric Number Line is that of a virtual artifact that is educationally useful and susceptible to an internal dynamic that allows us to grasp the fundamental elements of mathematics: numbers and geometrical points. In particular, it allows the study of constructible numbers both from a theoretical and historical point of view offering a geometric view to understand the different features of numbers.

About this didactic proposal, only the theoretical part concerning numerical sets was carried out in a secondary school in past years while the historical study and the constructability of numbers are activities that will be a subject of experimentation, always in secondary school, for the purposes of doctoral work. As regards the theoretical part carried out, obviously, in the secondary school the approach was more qualitative with, however, some operational considerations with the natural and
the rational numbers up to the radical irrational ones in connection with the Pythagorean theorem and
the relative design.

By highlighting the geometric aspect, the euclidean construction of the Number Line proves to be a
useful tool for understanding the important and fruitful historical-scientific period known as the 'crisis
of the foundations' which had in D. Hilbert (1862-1943) the protagonist of rigor of Euclidean
geometry thus opening up other avenues of research (Hartshorne, 2000). For this reason, it is
important to highlight the theoretical and historical aspects of geometry presented must also be seen
as cultural enrichment of each teacher to modulate her/his didactic transposition.

Acknowledgements

This study is part of the doctoral research topic.

References

Barbieri, S., & Maschietto, M. (2012). Attività nel laboratorio di matematica: costruzioni con riga e compasso. In O. Robutti, & M. Mosca (Eds.), *Atti del V Convegno Nazionale DiFiMa2011* (pp.181-192). Kim Williams Books.

Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), *Handbook of International Research in Mathematics Education* (2nd ed., pp. 746-783), Routledge.

D'Amore, B. (2003). *Le basi filosofiche, pedagogiche, epistemologiche e concettuali della Didattica della Matematica*. Pitagora.

Frykholm, J. (2010). *Learning to think mathematically with the number line. a resource for teachers, a tool for young children*. The Math Learning Center.

Hartshorne, R. (2000). *Geometry: Euclid and beyond*. Springer.

Mariotti, M. A., & Maffia, A. (2018). Dall'utilizzo degli artefatti ai significati matematici: il ruolo dell'insegnante nel processo di mediazione semiotica. *Didattica della matematica. Dalla Ricerca alle Pratiche d'Aula*, 4, 50-64.

Rizzo, A. (2000). La natura degli artefatti e la loro progettazione. *Sistemi Intelligenti*, XII, 3, 437-452.

Sbaragli, S. (2005). L'importanza delle diverse rappresentazioni semiotiche. Il caso degli enti primitivi della geometria. *Bollettino dei Docenti di Matematica*, 50, May, 69-76.

Wallis, J. (1685). *Treatise of algebra*. London.
the rational numbers up to the radical irrational ones in connection with the Pythagorean theorem and
the relative design.

By highlighting the geometric aspect, the euclidean construction of the Number Line proves to be a
useful tool for understanding the important and fruitful historical-scientific period known as the 'crisis
of the foundations' which had in D. Hilbert (1862-1943) the protagonist of rigor of Euclidean
geometry thus opening up other avenues of research (Hartshorne, 2000). For this reason, it is
important to highlight the theoretical and historical aspects of geometry presented must also be seen
as cultural enrichment of each teacher to modulate her/his didactic transposition.

Acknowledgements

This study is part of the doctoral research topic.

References

Barbieri, S., & Maschietto, M. (2012). Attività nel laboratorio di matematica: costruzioni con riga e compasso. In O. Robutti, & M. Mosca (Eds.), *Atti del V Convegno Nazionale DiFiMa2011* (pp.181-192). Kim Williams Books.

Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), *Handbook of International Research in Mathematics Education* (2nd ed., pp. 746-783), Routledge.

D'Amore, B. (2003). *Le basi filosofiche, pedagogiche, epistemologiche e concettuali della Didattica della Matematica*. Pitagora.

Frykholm, J. (2010). *Learning to think mathematically with the number line. a resource for teachers, a tool for young children*. The Math Learning Center.

Hartshorne, R. (2000). *Geometry: Euclid and beyond*. Springer.

Mariotti, M. A., & Maffia, A. (2018). Dall'utilizzo degli artefatti ai significati matematici: il ruolo dell'insegnante nel processo di mediazione semiotica. *Didattica della matematica. Dalla Ricerca alle Pratiche d'Aula*, 4, 50-64.

Rizzo, A. (2000). La natura degli artefatti e la loro progettazione. *Sistemi Intelligenti*, XII, 3, 437-452.

Sbaragli, S. (2005). L'importanza delle diverse rappresentazioni semiotiche. Il caso degli enti primitivi della geometria. *Bollettino dei Docenti di Matematica*, 50, May, 69-76.

Wallis, J. (1685). *Treatise of algebra*. London.
# DYNAMIC GEOMETRY IN PRIMARY SCHOOL

Francine Athias

University of Franche-Comte, ELLIADD, France

In this paper we want to investigate Dynamic geometry Environment (DGE) in primary school. DGE is a way to develop powerful mathematics knowledge thanks to its ability to visually make explicit geometrical concepts. This theoretical possibility to better understanding geometry is not so easily to be shared by students in the classroom. The aim of this paper is to expound the epistemic value of DGE when reproducing figures in geometry in primary school, as a didactic milieu. Situations are designed by teachers and researchers in a cooperative engineering, based on habits and geometrical knowledge (didactic contract). Then they are implemented in the classroom. The shared understanding is a way to highlight geometrical properties. Doing so we want to show how the limits during teaching class when reproducing figures may be exceeded thanks to the DGE and a deep understanding by the collective of teachers and researchers.

## INTRODUCTION

In this paper we want to investigate Dynamic Geometry Environment (DGE) in primary school. DGE is a way to develop powerful mathematics knowledge acquisition in a micro world (Leung, 2008). As we know, geometry is a kind of knowledge area based on two registers: the visualization of shapes in order to represent the space and the language for stating some properties (Duval, 2005, 2007). Furthermore the DGE's ability leads students to visually make explicit the implicit through the use of properties and the dragging shapes. The dragging mode is usually used to help students for proof and explanation (Hanna, 2000; Jones, 2000). But this research is up to students in high school (for example 9th or 10th grade). A few research concerns young students (4th or 5th grade, Assude 2005) or disabled students (Athias, 2021). We want to highlight the dragging mode to help these young students to develop their geometrical skills to better understand the properties in stake.

This work is grounded in a cooperative engineering (Sensevy & Bloor, 2020). Cooperative engineering refers to a methodological process in which a collective of teachers and researchers work together to design, implement, and re-implement a teaching sequence. In our case, it means to design a sequence about geometry by using a dynamic geometry software (DGS), GeoGebra (https://www.geogebra.org/classic?lang=fr), at primary school (4-5th grades). The cooperative engineering are built throughout different phases, which are not necessarily chronological. The first phase is a moment to work on the aim of the geometry teaching. It is necessary to focus on the links between geometric knowledge and the drawing. Each teacher member of the collective becomes familiar with the reproduction of figures with usual tools (ruler, compass, square set) and with the software in order to progress towards the conceptualization of geometric objects. The second phase focuses on the design of a sequence in geometry. Based on the knowledge acquired in the first phase, a teaching sequence is designed and implemented by the teachers, based on this shared understanding. Responsibility for implementation is shared by the whole team, teachers and researchers. The third phase involves actual implementation by one or more teachers. Most of the time, this implementation
# DYNAMIC GEOMETRY IN PRIMARY SCHOOL

Francine Athias

University of Franche-Comte, ELLIADD, France

In this paper we want to investigate Dynamic geometry Environment (DGE) in primary school. DGE is a way to develop powerful mathematics knowledge thanks to its ability to visually make explicit geometrical concepts. This theoretical possibility to better understanding geometry is not so easily to be shared by students in the classroom. The aim of this paper is to expound the epistemic value of DGE when reproducing figures in geometry in primary school, as a didactic milieu. Situations are designed by teachers and researchers in a cooperative engineering, based on habits and geometrical knowledge (didactic contract). Then they are implemented in the classroom. The shared understanding is a way to highlight geometrical properties. Doing so we want to show how the limits during teaching class when reproducing figures may be exceeded thanks to the DGE and a deep understanding by the collective of teachers and researchers.

## INTRODUCTION

In this paper we want to investigate Dynamic Geometry Environment (DGE) in primary school. DGE is a way to develop powerful mathematics knowledge acquisition in a micro world (Leung, 2008). As we know, geometry is a kind of knowledge area based on two registers: the visualization of shapes in order to represent the space and the language for stating some properties (Duval, 2005, 2007). Furthermore the DGE's ability leads students to visually make explicit the implicit through the use of properties and the dragging shapes. The dragging mode is usually used to help students for proof and explanation (Hanna, 2000; Jones, 2000). But this research is up to students in high school (for example 9th or 10th grade). A few research concerns young students (4th or 5th grade, Assude 2005) or disabled students (Athias, 2021). We want to highlight the dragging mode to help these young students to develop their geometrical skills to better understand the properties in stake.

This work is grounded in a cooperative engineering (Sensevy & Bloor, 2020). Cooperative engineering refers to a methodological process in which a collective of teachers and researchers work together to design, implement, and re-implement a teaching sequence. In our case, it means to design a sequence about geometry by using a dynamic geometry software (DGS), GeoGebra (https://www.geogebra.org/classic?lang=fr), at primary school (4-5th grades). The cooperative engineering are built throughout different phases, which are not necessarily chronological. The first phase is a moment to work on the aim of the geometry teaching. It is necessary to focus on the links between geometric knowledge and the drawing. Each teacher member of the collective becomes familiar with the reproduction of figures with usual tools (ruler, compass, square set) and with the software in order to progress towards the conceptualization of geometric objects. The second phase focuses on the design of a sequence in geometry. Based on the knowledge acquired in the first phase, a teaching sequence is designed and implemented by the teachers, based on this shared understanding. Responsibility for implementation is shared by the whole team, teachers and researchers. The third phase involves actual implementation by one or more teachers. Most of the time, this implementation
is followed by a session report, organized around the students' productions, screen copies of the blackboard or videos of the practice. Each video is commented on by the teacher, who becomes increasingly familiar with the sequence as the implementation progresses. The fourth stage involves an analysis of the implementation, by the group of teachers and researchers. This highlights the sequence's interests, limitations and prospects. The fifth phase involves iterative implementation, using the same approach as above. Finally, to conclude on these five phases, it's clear that the implementation and its analysis (phases 3 and 4) allow us to delve deeper into the geometrical knowledge involved (phase 1) and the improvement of the teaching sequence (phase 2). In addition, the new iteration (phase 5) enables the sequence to be reworked. In another way, the goals of designing teaching sequences and developing theories of teaching and learning are intertwined in a cooperative engineering. Moreover, cooperative engineering could be seen as fundamental research within an anthropological approach (Chevallard & Sensevy 2014). It aims to elaborate a theory of practice, and in the same time, it assumes an engineering function such as to build better educational designs (devices and practice). In this paper we develop this engineering function throughout a case study, which is remarkable: it represents what could be done in a sequence of geometry in primary school.

In a first part we will present some theoretical concepts. In a second part we will show the possible limits when reproducing a figure with geometrical instruments (ruler, compass, square) in the paper and pencil environment during a specific lesson (at the end of the year), written by the team of the teachers and the researcher. In a third part we will try to expose how students use the dragging mode to represent a new shape in the DGE (at the end of the year), written by the team of the teachers and the researcher. In a fourth part we will engage a discussion before concluding.

# THEORETICAL FRAMEWORK AND METHODOLOGY

## Part 1: Figure versus drawing

I will explain the relationship between drawing, figures and theoretical geometry. Laborde and Capponi (1992) propose to reinterpret this distinction between drawing and figure by using the triplet referent, signifier, signified. As a material entity, the drawing (the visible trace on a material support) can be considered as a signifier of a theoretical referent (that object of theoretical geometry). The geometric figure thus consists of the pairing of a given referent with all its drawings. We can see it as a pair (referent/drawing, one of the drawings taken from the set of all possible drawings). In this approach, the relationship between a drawing and its referent is constructed by a subject, the signified of the geometric figure. A graphic representation gives access to the analogical dimensions of perception, while the text enables digital processing of the same theoretical object. It's precisely this point that interests us at primary school level. In the following I use the term of "figure" (Laborde & Capponi, 1992) to take into account that it's not only a drawing (for example a circle), some geometrical properties are embodied thanks the use of tools.

In primary school doing geometry could be characterized by the use of instruments (such as ruler, compass and square set) to draw shapes in the paper and pencil environment (Mathe et al., 2022). Most of the geometry's knowledge in primary school is expressed in the same way as that of theoretical geometry, for example the circle. But a circle (in geometry) has to be characterized by a center and a radius (or a point of the circle). The geometric use of instruments (ruler, compass and
is followed by a session report, organized around the students' productions, screen copies of the blackboard or videos of the practice. Each video is commented on by the teacher, who becomes increasingly familiar with the sequence as the implementation progresses. The fourth stage involves an analysis of the implementation, by the group of teachers and researchers. This highlights the sequence's interests, limitations and prospects. The fifth phase involves iterative implementation, using the same approach as above. Finally, to conclude on these five phases, it's clear that the implementation and its analysis (phases 3 and 4) allow us to delve deeper into the geometrical knowledge involved (phase 1) and the improvement of the teaching sequence (phase 2). In addition, the new iteration (phase 5) enables the sequence to be reworked. In another way, the goals of designing teaching sequences and developing theories of teaching and learning are intertwined in a cooperative engineering. Moreover, cooperative engineering could be seen as fundamental research within an anthropological approach (Chevallard & Sensevy 2014). It aims to elaborate a theory of practice, and in the same time, it assumes an engineering function such as to build better educational designs (devices and practice). In this paper we develop this engineering function throughout a case study, which is remarkable: it represents what could be done in a sequence of geometry in primary school.

In a first part we will present some theoretical concepts. In a second part we will show the possible limits when reproducing a figure with geometrical instruments (ruler, compass, square) in the paper and pencil environment during a specific lesson (at the end of the year), written by the team of the teachers and the researcher. In a third part we will try to expose how students use the dragging mode to represent a new shape in the DGE (at the end of the year), written by the team of the teachers and the researcher. In a fourth part we will engage a discussion before concluding.

# THEORETICAL FRAMEWORK AND METHODOLOGY

## Part 1: Figure versus drawing

I will explain the relationship between drawing, figures and theoretical geometry. Laborde and Capponi (1992) propose to reinterpret this distinction between drawing and figure by using the triplet referent, signifier, signified. As a material entity, the drawing (the visible trace on a material support) can be considered as a signifier of a theoretical referent (that object of theoretical geometry). The geometric figure thus consists of the pairing of a given referent with all its drawings. We can see it as a pair (referent/drawing, one of the drawings taken from the set of all possible drawings). In this approach, the relationship between a drawing and its referent is constructed by a subject, the signified of the geometric figure. A graphic representation gives access to the analogical dimensions of perception, while the text enables digital processing of the same theoretical object. It's precisely this point that interests us at primary school level. In the following I use the term of "figure" (Laborde & Capponi, 1992) to take into account that it's not only a drawing (for example a circle), some geometrical properties are embodied thanks the use of tools.

In primary school doing geometry could be characterized by the use of instruments (such as ruler, compass and square set) to draw shapes in the paper and pencil environment (Mathe et al., 2022). Most of the geometry's knowledge in primary school is expressed in the same way as that of theoretical geometry, for example the circle. But a circle (in geometry) has to be characterized by a center and a radius (or a point of the circle). The geometric use of instruments (ruler, compass and
set square) could be a way for conceptualizing the objects of theoretical geometry. Petitfour (2015) is interested in the potential of a geometric technical language "relating to the manipulation of instruments in connection with geometric properties, without mentioning these properties". We consider this technical language in conjunction with the DGE. Selecting the "circle" button is like "taking the compass". If you can set the compass point in the paper and pencil environment, without having to explain it, you can't set a point anywhere on the dynamic figure. This is because the point moves with the mouse, as it is a "free" point.

I wonder how the change of instrument could encourage students to act on geometric properties.

## Part 2: Joint action theory in didactic

I want to investigate what it happens in the classroom when the students use these different instruments (ruler, compass, set square and DGS). I will analyze it throughout both concepts of didactic contract and the didactic milieu. The didactic contract (Brousseau, 1997; Sensevy, 2014) can only be understood in relation to the didactic milieu. That is to say, when students work on a problem (any situation that confronts the student with a difficulty action, here reproducing a figure), they will look for ways to solve the problem within their existing knowledge and habits (the use of the geometrical instruments). The didactic milieu could be seen as the problem the students have to deal with. Recognizing this clarifies how the already-there capacities of the didactic contract only become meaningful in relation to the problem that the student must work on. These capacities are necessary but not sufficient to successfully undertake this work, otherwise the student would not learn anything new. There is thus a kind of initial imbalance between the contract (the already-there, which mainly results from the previous joint action) and the milieu (that which is to be known, and which is embedded in a specific symbolic structure). A new aspect of the didactical contract due to the DGE must be stressed (Laborde, 1998). Some pupils try systematically some menu items on every object: they hope to discover the solution or a hint. The effects of the dragging test could lead the students to have some clues to reproduce the figure.

The question which is addressed here is the following: how reproducing figures in the paper and pencil environment and the dynamic geometry environment could lead young students to a better understanding of geometric properties?

To try to answer this question we rely on various data. These data come from the implementations of a sequence in geometry based on the DGE and in particular on classroom videos and students’ drawings. Moreover I add a lot of exchanges during the cooperative engineering meetings. In this paper I have chosen to describe and analyze what happens in the classroom of one of the teachers who is part of the team of two teachers and one researcher. This class (4 and 5th grade in France), while being specific, is part of the team’s culture and the culture of dynamic geometry. The teacher and his students worked with the software early in the year. For example, they reproduce figures in the paper and pencil environment and in the dynamic environment. The students are on their own to reproduce the figure. The teacher is there but she doesn’t help. The specific session was video-recorded and transcribed. I selected two moments in particular since the geometric properties are tangible, but not achievable. The tasks are co-designed by the team as mentioned before. Their implementation led us to a better understanding of the link between both environments and their strengths and limits.
set square) could be a way for conceptualizing the objects of theoretical geometry. Petitfour (2015) is interested in the potential of a geometric technical language "relating to the manipulation of instruments in connection with geometric properties, without mentioning these properties". We consider this technical language in conjunction with the DGE. Selecting the "circle" button is like "taking the compass". If you can set the compass point in the paper and pencil environment, without having to explain it, you can't set a point anywhere on the dynamic figure. This is because the point moves with the mouse, as it is a "free" point.

I wonder how the change of instrument could encourage students to act on geometric properties.

## Part 2: Joint action theory in didactic

I want to investigate what it happens in the classroom when the students use these different instruments (ruler, compass, set square and DGS). I will analyze it throughout both concepts of didactic contract and the didactic milieu. The didactic contract (Brousseau, 1997; Sensevy, 2014) can only be understood in relation to the didactic milieu. That is to say, when students work on a problem (any situation that confronts the student with a difficulty action, here reproducing a figure), they will look for ways to solve the problem within their existing knowledge and habits (the use of the geometrical instruments). The didactic milieu could be seen as the problem the students have to deal with. Recognizing this clarifies how the already-there capacities of the didactic contract only become meaningful in relation to the problem that the student must work on. These capacities are necessary but not sufficient to successfully undertake this work, otherwise the student would not learn anything new. There is thus a kind of initial imbalance between the contract (the already-there, which mainly results from the previous joint action) and the milieu (that which is to be known, and which is embedded in a specific symbolic structure). A new aspect of the didactical contract due to the DGE must be stressed (Laborde, 1998). Some pupils try systematically some menu items on every object: they hope to discover the solution or a hint. The effects of the dragging test could lead the students to have some clues to reproduce the figure.

The question which is addressed here is the following: how reproducing figures in the paper and pencil environment and the dynamic geometry environment could lead young students to a better understanding of geometric properties?

To try to answer this question we rely on various data. These data come from the implementations of a sequence in geometry based on the DGE and in particular on classroom videos and students' drawings. Moreover I add a lot of exchanges during the cooperative engineering meetings. In this paper I have chosen to describe and analyze what happens in the classroom of one of the teachers who is part of the team of two teachers and one researcher. This class (4 and 5th grade in France), while being specific, is part of the team's culture and the culture of dynamic geometry. The teacher and his students worked with the software early in the year. For example, they reproduce figures in the paper and pencil environment and in the dynamic environment. The students are on their own to reproduce the figure. The teacher is there but she doesn't help. The specific session was video-recorded and transcribed. I selected two moments in particular since the geometric properties are tangible, but not achievable. The tasks are co-designed by the team as mentioned before. Their implementation led us to a better understanding of the link between both environments and their strengths and limits.
# RESULTS

## Result 1

The compass doesn't allow to take into account some geometric properties of a specific figure. The student can't become aware of the properties.

I will describe and analyze the situation in the following. It was designed by the team of the teachers and the researcher. We decided to let students acting alone. The specific figure I want to present now is the circle into the square. The teacher S gives students this figure on the white board (cf fig. 1).

<figure><img src="image_110.png" /><figcaption>Figure 3: the circle into the square</figcaption></figure>

The first task given by the teacher is to let students to describe the figure. The aim is to perceptually recognize a circle, a square and the link between the two geometric objects: the circle is internally tangent to the square. For elementary school pupils, the circle is inside the square, the circle "borders" the square. Only perceptual recognition of geometric objects is expected. For students from 4th or 5th grade, there's no problem.

The second task given by the teacher is to let students to reproduce the figure using geometrical instruments. They have to begin by drawing the square. I explain the choice we made of drawing the square first. Drawing the circle and then the square means using the tangents to the circle, drawing the radii first. Discussions within the engineering team have ruled out this possibility.

In this paper I am only interested in the construction of the circle. During the description phase, nothing was said about the characteristic elements of the circle. This is up to the students.

The teacher wanted the students use geometrical knowledge. One can notice that the center of the circle is not written. The students have to analyze a geometric figure with a view to reproducing it using instruments. This implies that students can make hypotheses (more or less explicit) about geometric properties and check them. The prior knowledge (the didactic contract) that pupils need to understand the problem and draw up a procedure is: (recognize a square and construct it with a ruler, compass and square) recognize and draw a circle with a compass, knowing its center and radius or a point. Pupils must reproduce the figure, without measuring.

Now I present a student in the classroom.

A student was drawing the figure. After drawing the square he took his compass. He hesitated where to put the tip of the compass. He decided to draw the diagonal of the square thanks to his ruler. He took the compass again, pushed one leg of the compasses into the paper with the spike, put the pencil
# RESULTS

## Result 1

The compass doesn't allow to take into account some geometric properties of a specific figure. The student can't become aware of the properties.

I will describe and analyze the situation in the following. It was designed by the team of the teachers and the researcher. We decided to let students acting alone. The specific figure I want to present now is the circle into the square. The teacher S gives students this figure on the white board (cf fig. 1).

<figure><img src="image_116.png" /><figcaption>Figure 3: the circle into the square</figcaption></figure>

The first task given by the teacher is to let students to describe the figure. The aim is to perceptually recognize a circle, a square and the link between the two geometric objects: the circle is internally tangent to the square. For elementary school pupils, the circle is inside the square, the circle "borders" the square. Only perceptual recognition of geometric objects is expected. For students from 4th or 5th grade, there's no problem.

The second task given by the teacher is to let students to reproduce the figure using geometrical instruments. They have to begin by drawing the square. I explain the choice we made of drawing the square first. Drawing the circle and then the square means using the tangents to the circle, drawing the radii first. Discussions within the engineering team have ruled out this possibility.

In this paper I am only interested in the construction of the circle. During the description phase, nothing was said about the characteristic elements of the circle. This is up to the students.

The teacher wanted the students use geometrical knowledge. One can notice that the center of the circle is not written. The students have to analyze a geometric figure with a view to reproducing it using instruments. This implies that students can make hypotheses (more or less explicit) about geometric properties and check them. The prior knowledge (the didactic contract) that pupils need to understand the problem and draw up a procedure is: (recognize a square and construct it with a ruler, compass and square) recognize and draw a circle with a compass, knowing its center and radius or a point. Pupils must reproduce the figure, without measuring.

Now I present a student in the classroom.

A student was drawing the figure. After drawing the square he took his compass. He hesitated where to put the tip of the compass. He decided to draw the diagonal of the square thanks to his ruler. He took the compass again, pushed one leg of the compasses into the paper with the spike, put the pencil
on the paper approximately on the square and moved the pencil around while keeping the legs at the same angle. At the end he puts the model and his figure next to each other (c.f. Figure 2).

<figure><img src="image_111.png" /><figcaption>Figure 4: Both figures</figcaption></figure>

The center of the circle is the point of intersection of the diagonals. At first the feedback of the milieu throughout the use of the compass in the paper and pencil environment is useful: the student is able to deal with it (you can see this extract here: https://mediacenter.univ-fcomte.fr/permalink/v1263dd7dfe545uq3gcy/iframe/). A kind of initial imbalance between the contract and the milieu is achieved. Then to finish the circle a point of the circle must be determined, here the midpoint of side of the square. The student's habits thus enable him to act in such a way as to solve the reproduction question. But they don't allow him to look any further into geometric relationships.

## Result 2

In the dynamic environment, the dragging test highlights the “invisible” point (the midpoint of side) and allows the student to take into account the geometric properties of the figure.

I will describe and analyze the situation in the following. This time the teacher S asks the students to reproduce the same figure thanks to the DGS GeoGebra. As before, I describe a student reproducing the circle (c.f. Table 1).

The student is familiar with the software and the dragging test. He knows that he has to drag any point. The student is able to say that the reproduction is not right. Habits and geometric knowledge in the dynamic environment enable the student to begin appropriately, to understand the feedback of the milieu. After a few attempts the student is to visualize that point G must be declared as the midpoint of [AB]. As a lot of students, this student uses the well-known button “midpoint of a segment”. But, as Laborde (1998) explained, some others tried some other buttons, for example, the unknown perpendicular bisector. After all, in the dynamic environment, the dragging test highlights the point, the midpoint of a segment, “invisible” in the paper and pencil environment.
on the paper approximately on the square and moved the pencil around while keeping the legs at the same angle. At the end he puts the model and his figure next to each other (c.f. Figure 2).

<figure><img src="image_117.png" /><figcaption>Figure 4: Both figures</figcaption></figure>

The center of the circle is the point of intersection of the diagonals. At first the feedback of the milieu throughout the use of the compass in the paper and pencil environment is useful: the student is able to deal with it (you can see this extract here: https://mediacenter.univ-fcomte.fr/permalink/v1263dd7dfe545uq3gcy/iframe/). A kind of initial imbalance between the contract and the milieu is achieved. Then to finish the circle a point of the circle must be determined, here the midpoint of side of the square. The student's habits thus enable him to act in such a way as to solve the reproduction question. But they don't allow him to look any further into geometric relationships.

## Result 2

In the dynamic environment, the dragging test highlights the “invisible” point (the midpoint of side) and allows the student to take into account the geometric properties of the figure.

I will describe and analyze the situation in the following. This time the teacher S asks the students to reproduce the same figure thanks to the DGS GeoGebra. As before, I describe a student reproducing the circle (c.f. Table 1).

The student is familiar with the software and the dragging test. He knows that he has to drag any point. The student is able to say that the reproduction is not right. Habits and geometric knowledge in the dynamic environment enable the student to begin appropriately, to understand the feedback of the milieu. After a few attempts the student is to visualize that point G must be declared as the midpoint of [AB]. As a lot of students, this student uses the well-known button “midpoint of a segment”. But, as Laborde (1998) explained, some others tried some other buttons, for example, the unknown perpendicular bisector. After all, in the dynamic environment, the dragging test highlights the point, the midpoint of a segment, “invisible” in the paper and pencil environment.
The square is already done
The circle is done

<figure><img src="image_112.png" /></figure>

A and B are moved

<figure><img src="image_113.png" /></figure>

The circle is outside the square.

<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /></figure>

We don't look before The student drew the The student used the The student move the that moment: we want two diagonals of the drag-test, by dragging point G. This time, the to inquiry how the square. He named F the the points A, B and C. circle is out of the student finds the point of intersection of The circle seemed to be square. characteristic elements the two diagonals. He fine. of the circle, the center chose the “circle” and a point of the circle. button. The circle moves with the mouse. A point G is created by a simple click on the segment [AB].

<caption>Table 1: What the student does</caption>

## Result 3

The teacher plays a decisive role in clarifying the geometric properties involved in a reproduction task.

I will describe and analyze the situation in the following. After that moment, the teacher summarizes the work done: "when you make a circle, you need to know its center and one of its points". Doing so she is able to point out knowledge issues to students. As we explained, the situation was written by the team of the teachers and the researcher. Even if the responsibility of the implementation of this situation is shared by the collective of teachers and researchers, the teacher has to challenge this situation in the classroom. She has to share her understanding of geometrical concepts. During a meeting, she explained to the collective that she had found it difficult to help students in the dynamic environment. She suggested changing the moment of description of the figure (description phase). The following year, when she proposed the figure (the circle in the square), she distributed a sheet to each student, asking them to use their instruments and code it. Only after this individual moment of analysis did, the students described the figure (c.f. Figure 3).
The square is already done
The circle is done

<figure><img src="image_118.png" /></figure>

A and B are moved

<figure><img src="image_119.png" /></figure>

The circle is outside the square.

<figure><img src="image_120.png" /></figure>

<figure><img src="image_121.png" /></figure>

We don't look before The student drew the The student used the The student move the that moment: we want two diagonals of the drag-test, by dragging point G. This time, the to inquiry how the square. He named F the the points A, B and C. circle is out of the student finds the point of intersection of The circle seemed to be square. characteristic elements the two diagonals. He fine. of the circle, the center chose the “circle” and a point of the circle. button. The circle moves with the mouse. A point G is created by a simple click on the segment [AB].

<caption>Table 1: What the student does</caption>

## Result 3

The teacher plays a decisive role in clarifying the geometric properties involved in a reproduction task.

I will describe and analyze the situation in the following. After that moment, the teacher summarizes the work done: "when you make a circle, you need to know its center and one of its points". Doing so she is able to point out knowledge issues to students. As we explained, the situation was written by the team of the teachers and the researcher. Even if the responsibility of the implementation of this situation is shared by the collective of teachers and researchers, the teacher has to challenge this situation in the classroom. She has to share her understanding of geometrical concepts. During a meeting, she explained to the collective that she had found it difficult to help students in the dynamic environment. She suggested changing the moment of description of the figure (description phase). The following year, when she proposed the figure (the circle in the square), she distributed a sheet to each student, asking them to use their instruments and code it. Only after this individual moment of analysis did, the students described the figure (c.f. Figure 3).
<figure><img src="image_116.png" /><figcaption>Figure 5: An example</figcaption></figure>

I chose this example to show how the students become aware of geometrical properties. This student named the points (around 20 students over 23 in the classroom). This student showed that she takes into account the midpoint of the segments (around 18 students over 23). He pointed the center of the circle (around 15 students over 23).

Based on this first analysis, the inquiry is easier. When students do not define the midpoint of the segment in the dynamic environment, the teacher can then refer them to the preliminary analysis of the figure. The distance between the already-there and the problem is adequate. It is thus a kind of balance between the contract and the milieu. What's more, these moments highlight the need for the two characteristic elements of the circle, in the paper-and-pencil environment and in the dynamic environment. As a result, recognition of the work done "when you make a circle, you need to know its center and one of its points" is more effective.

## DISCUSSION AND CONCLUSION

Our research question is about reproducing figures in the paper and pencil environment, and in the dynamic geometry environment, to lead young students to a better understanding of geometric properties. This situation illustrates the need to declare geometrical properties in the dynamic environment, whereas the need for this property is invisible in the paper and pencil environment. However, at elementary school level, it is not possible to justify this property: we need to show the midpoint of the segment, but it's not possible to explain why. It is based on the visualization of shapes (Duval, 2005). Furthermore, the teacher summarized the geometric work done. She was able to understand the geometrical properties at stake. Substantial international research into the use of digital technologies in the teaching and learning of mathematics has yielded powerful results. Despite the massive investment of resources, impact on the reality of school practices has been limited (Lagrange & Kynigos, 2014). I am searching a way to reconsider the research results by working together with the teachers. The situation we want to provide in the engineering work leads us to a better understanding of the link between both environments by the teachers and the researchers. The visualization enabled by dynamic geometry increases the power of action for both students and teacher. Students who are able to use geometric concepts in their everyday lives: "I see the circle with center A that passes through B" (Athias, 2019). Teachers and researchers are taking into account the importance of planning appropriate situations collectively. Based on these results, we have developed a program for pupils with long-term learning difficulties (Athias, 2021).
<figure><img src="image_122.png" /><figcaption>Figure 5: An example</figcaption></figure>

I chose this example to show how the students become aware of geometrical properties. This student named the points (around 20 students over 23 in the classroom). This student showed that she takes into account the midpoint of the segments (around 18 students over 23). He pointed the center of the circle (around 15 students over 23).

Based on this first analysis, the inquiry is easier. When students do not define the midpoint of the segment in the dynamic environment, the teacher can then refer them to the preliminary analysis of the figure. The distance between the already-there and the problem is adequate. It is thus a kind of balance between the contract and the milieu. What's more, these moments highlight the need for the two characteristic elements of the circle, in the paper-and-pencil environment and in the dynamic environment. As a result, recognition of the work done "when you make a circle, you need to know its center and one of its points" is more effective.

## DISCUSSION AND CONCLUSION

Our research question is about reproducing figures in the paper and pencil environment, and in the dynamic geometry environment, to lead young students to a better understanding of geometric properties. This situation illustrates the need to declare geometrical properties in the dynamic environment, whereas the need for this property is invisible in the paper and pencil environment. However, at elementary school level, it is not possible to justify this property: we need to show the midpoint of the segment, but it's not possible to explain why. It is based on the visualization of shapes (Duval, 2005). Furthermore, the teacher summarized the geometric work done. She was able to understand the geometrical properties at stake. Substantial international research into the use of digital technologies in the teaching and learning of mathematics has yielded powerful results. Despite the massive investment of resources, impact on the reality of school practices has been limited (Lagrange & Kynigos, 2014). I am searching a way to reconsider the research results by working together with the teachers. The situation we want to provide in the engineering work leads us to a better understanding of the link between both environments by the teachers and the researchers. The visualization enabled by dynamic geometry increases the power of action for both students and teacher. Students who are able to use geometric concepts in their everyday lives: "I see the circle with center A that passes through B" (Athias, 2019). Teachers and researchers are taking into account the importance of planning appropriate situations collectively. Based on these results, we have developed a program for pupils with long-term learning difficulties (Athias, 2021).
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Chevallard, Y., & Sensevy, G. (2014). Anthropological approaches in mathematics education, French perspectives. In *Encyclopedia of mathematics education* (pp. 38-43). Springer. https://hal.science/hal-01139352

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

Duval, R. (2007). Cognitive functioning and the understanding of mathematical processes of proof. In P. Boero (ed.), *Theorems in school: From history, epistemology and cognition to classroom practice* (p. 135-161). Brill.

Hanna, G. (2000). Proof, explanation and exploration: An overview. *Educational Studies in Mathematics*, 44(1), 5-23. https://doi.org/10.1023/A:1012737223465

Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. *Educational Studies in Mathematics*, 44(1), 55-85. https://doi.org/10.1023/A:1012789201736

Laborde, C. (1998). Computer technology and the teaching of geometry. In C. Mammana & V. Villani (Eds.), *Perspectives on the teaching of geometry for the 21st Century* (Vol. 5, pp. 109-158). Springer Netherlands.
https://doi.org/10.1007/978-94-011-5226-6_5

Laborde, C., & Capponi, B. (1994). Cabri-géomètre constituant d'un milieu pour l'apprentissage de la notion de figure géométrique. Recherches en Didactique des Mathématiques, 14(1.2), 165-210.

Lagrange, J.-B., & Kynigos, C. (2014). Digital technologies to teach and learn mathematics: Context and re-contextualization. *Educational Studies in Mathematics*, 85(3), 381-403. https://doi.org/10.1007/s10649-013-9525-z

Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. *International Journal of Computers for Mathematical Learning*, 13(2), 135-157. https://doi.org/10.1007/s10758-008-9130-x

Mathé, A.-C., Barrier, T., & Perrin-Glorian, M.-J. (2020). Enseigner la géométrie élémentaire: enjeux, ruptures et continuités. Academia-l'Harmattan.

Petitfour, E. (2015). *Enseignement de la géométrie à des élèves en difficulté d'apprentissage: étude du processus d'accès à la géométrie d'élèves dyspraxiques visuo-spatiaux lors de la transition CM2-6ème* [PhD Thesis]. Université Paris Diderot-Paris 7.

Sensevy, G. (2014). Characterizing teaching effectiveness in the Joint Action Theory in Didactics: An exploratory study in primary school. *Journal of Curriculum Studies*, 46(5), 577-610.
https://doi.org/10.1080/00220272.2014.931466

Sensevy, G., & Bloor, T. (2020). Cooperative didactic engineering. In S. Lerman (Éd.), *Encyclopedia of mathematics education* (p. 141-145). Springer. https://doi.org/10.1007/978-3-030-15789-0_100037
References

Assude, T. (2005). Time management in the work economy of a class: A case study: Integration of Cabri in primary school mathematics teaching. *Educational Studies in Mathematics*, 59(1), 183-203.
https://doi.org/10.1007/s10649-005-5888-0

Athias, F. (2019). Un exemple d'usage de la de la géométrie dynamique. *Grand N*, 103, 57-70.

Athias, F. (2021). Une ingénierie coopérative pour partager des connaissances. *Éducation & Didactique*, 15, 85-101. https://doi.org/10.4000/educationdidactique.9245

Brousseau, G. (1997). *The theory of didactic situations in mathematics*. Kluwer.

Chevallard, Y., & Sensevy, G. (2014). Anthropological approaches in mathematics education, French perspectives. In *Encyclopedia of mathematics education* (pp. 38-43). Springer. https://hal.science/hal-01139352

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

Duval, R. (2007). Cognitive functioning and the understanding of mathematical processes of proof. In P. Boero (ed.), *Theorems in school: From history, epistemology and cognition to classroom practice* (p. 135-161). Brill.

Hanna, G. (2000). Proof, explanation and exploration: An overview. *Educational Studies in Mathematics*, 44(1), 5-23. https://doi.org/10.1023/A:1012737223465

Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. *Educational Studies in Mathematics*, 44(1), 55-85. https://doi.org/10.1023/A:1012789201736

Laborde, C. (1998). Computer technology and the teaching of geometry. In C. Mammana & V. Villani (Eds.), *Perspectives on the teaching of geometry for the 21st Century* (Vol. 5, pp. 109-158). Springer Netherlands.
https://doi.org/10.1007/978-94-011-5226-6_5

Laborde, C., & Capponi, B. (1994). Cabri-géomètre constituant d'un milieu pour l'apprentissage de la notion de figure géométrique. Recherches en Didactique des Mathématiques, 14(1.2), 165-210.

Lagrange, J.-B., & Kynigos, C. (2014). Digital technologies to teach and learn mathematics: Context and re-contextualization. *Educational Studies in Mathematics*, 85(3), 381-403. https://doi.org/10.1007/s10649-013-9525-z

Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. *International Journal of Computers for Mathematical Learning*, 13(2), 135-157. https://doi.org/10.1007/s10758-008-9130-x

Mathé, A.-C., Barrier, T., & Perrin-Glorian, M.-J. (2020). Enseigner la géométrie élémentaire: enjeux, ruptures et continuités. Academia-l'Harmattan.

Petitfour, E. (2015). *Enseignement de la géométrie à des élèves en difficulté d'apprentissage: étude du processus d'accès à la géométrie d'élèves dyspraxiques visuo-spatiaux lors de la transition CM2-6ème* [PhD Thesis]. Université Paris Diderot-Paris 7.

Sensevy, G. (2014). Characterizing teaching effectiveness in the Joint Action Theory in Didactics: An exploratory study in primary school. *Journal of Curriculum Studies*, 46(5), 577-610.
https://doi.org/10.1080/00220272.2014.931466

Sensevy, G., & Bloor, T. (2020). Cooperative didactic engineering. In S. Lerman (Éd.), *Encyclopedia of mathematics education* (p. 141-145). Springer. https://doi.org/10.1007/978-3-030-15789-0_100037
# THE NOTION OF ANGLE AND THE GGBOT AS A TOOL-TO-THINK-WITH... OR WITHOUT

Anna Baccaglini-Frank¹, Silvia Funghi², Elisa Miragliotta³

¹Università di Pisa, ²Università di Genova, ³Università di Pavia

This study is about how geometrical reasoning can be supported through experiences with artifacts. Specifically, taking a semiotic perspective, we focus on the potential of primary school activities exploiting a drawing robot, the GGBot, a descendant of Papert's robotic turtle, in relation to the concept of angle. We analyze the interview of a 5th grader who participated in such activities to highlight his way of seeing the contour of a figure, that, we claim, showcases the extent to which GGBot's functionalities can become a "tool-to-think-with" (or, actually, without).

## INTRODUCTION

Managing the theoretical nature of mathematical objects and the spatial nature of their (physical or mental) representations is a complex process with a long development (e.g., Fischbein, 1993). Geometrical reasoning is strongly connected with developing various ways of seeing and managing representations of geometrical objects, also making use of analytical reasoning (Duval, 1995). Here we will consider analytical reasoning as a specific kind of geometrical reasoning, that is related to recognizing subunits and composing and decomposing representations of figures (Duval, 1995).

It is widely recognized that the use of artifacts can provide experiences promoting multiple and dynamic representations of geometrical objects, that with appropriate tasks can be highly effective in fostering analytical and geometrical reasoning (e.g., Leung et al., 2023). This study provides an example of how during such experiences the artifact can support human cognition, becoming a tool-to-think-with, and it can proceed to become internalized, giving birth to a psychological tool (Vygotsky, 1978). More specifically, we focus on the geometrical notion of angle, because it is crucial in primary school geometry and it is one of the most challenging and misleading geometrical objects (e.g., Fischbein, 1993; Devichi & Munier, 2013), although it is experienced quite early in school (e.g., Bartolini Bussi & Baccaglini-Frank, 2015). Indeed, such notion is epistemologically challenging: for example, as described by Freudenthal (1973), when considering a pair of half lines with a common origin, in *Elementary Geometry* an angle is viewed as a static part of a plane comprised between such an unordered pair, while in *Goniometry* it is viewed as a dynamic turn between an ordered pair.

The protagonist of our study is Sam, a 5th grade student who learned about angles in grades 4 and 5 through a sequence of activities with GGBot, a descendant of Papert's drawing turtle (Baccaglini-Frank & Mariotti, 2022) that we will introduce in the following sections. Here we analyze part of a task-based interview about a polygon and its angles. The analyses focus on the signs Sam produced through geometrical and analytical reasoning. We will argue that such results showcase the extent to which GGBot's functionalities can become a "tool-to-think-with" (or, actually, without).
# THE NOTION OF ANGLE AND THE GGBOT AS A TOOL-TO-THINK-WITH... OR WITHOUT

Anna Baccaglini-Frank¹, Silvia Funghi², Elisa Miragliotta³

¹Università di Pisa, ²Università di Genova, ³Università di Pavia

This study is about how geometrical reasoning can be supported through experiences with artifacts. Specifically, taking a semiotic perspective, we focus on the potential of primary school activities exploiting a drawing robot, the GGBot, a descendant of Papert's robotic turtle, in relation to the concept of angle. We analyze the interview of a 5th grader who participated in such activities to highlight his way of seeing the contour of a figure, that, we claim, showcases the extent to which GGBot's functionalities can become a "tool-to-think-with" (or, actually, without).

## INTRODUCTION

Managing the theoretical nature of mathematical objects and the spatial nature of their (physical or mental) representations is a complex process with a long development (e.g., Fischbein, 1993). Geometrical reasoning is strongly connected with developing various ways of seeing and managing representations of geometrical objects, also making use of analytical reasoning (Duval, 1995). Here we will consider analytical reasoning as a specific kind of geometrical reasoning, that is related to recognizing subunits and composing and decomposing representations of figures (Duval, 1995).

It is widely recognized that the use of artifacts can provide experiences promoting multiple and dynamic representations of geometrical objects, that with appropriate tasks can be highly effective in fostering analytical and geometrical reasoning (e.g., Leung et al., 2023). This study provides an example of how during such experiences the artifact can support human cognition, becoming a tool-to-think-with, and it can proceed to become internalized, giving birth to a psychological tool (Vygotsky, 1978). More specifically, we focus on the geometrical notion of angle, because it is crucial in primary school geometry and it is one of the most challenging and misleading geometrical objects (e.g., Fischbein, 1993; Devichi & Munier, 2013), although it is experienced quite early in school (e.g., Bartolini Bussi & Baccaglini-Frank, 2015). Indeed, such notion is epistemologically challenging: for example, as described by Freudenthal (1973), when considering a pair of half lines with a common origin, in *Elementary Geometry* an angle is viewed as a static part of a plane comprised between such an unordered pair, while in *Goniometry* it is viewed as a dynamic turn between an ordered pair.

The protagonist of our study is Sam, a 5th grade student who learned about angles in grades 4 and 5 through a sequence of activities with GGBot, a descendant of Papert's drawing turtle (Baccaglini-Frank & Mariotti, 2022) that we will introduce in the following sections. Here we analyze part of a task-based interview about a polygon and its angles. The analyses focus on the signs Sam produced through geometrical and analytical reasoning. We will argue that such results showcase the extent to which GGBot's functionalities can become a "tool-to-think-with" (or, actually, without).
# THEORETICAL FRAMEWORK AND RESEARCH QUESTION

This study is grounded in *Theory of Semiotic Mediation* (Bartolini Bussi & Mariotti, 2008), a Vygotskian socio-constructivist theory, according to which students are guided by their teacher to construct mathematical knowledge by solving appropriately designed tasks with the use of appropriately chosen artifacts – that is, artifacts and mathematical tasks that are considered helpful for unfolding the target mathematical meanings. During a first phase of the *didactic cycle* students use the *artifact* (e.g., a physical compass, the dragging tool in a Dynamic Geometry Environment) to solve a task and develop *utilization schemes*. The artifact together with such schemes constitutes an *instrument* in the sense of Rabardel (1995), through which the students produce *artifact signs*, closely related to the task and to the artifact. We can now look at two developmental processes that the activity of working with an artifact can foster.

One process concerns the unfolding of the artifact's *semiotic potential*, the artifact's double relationship with personal meanings (related to artifact signs) and with mathematical meanings (related to *mathematical signs*), that the teacher aims to construct through the orchestration of mathematical discussions during the didactic cycles.

A second process concerns the student's *internalization* (e.g., Bartolini Bussi & Mariotti, 2008), the "transformation" of what has been a mainly external process into an internal process. At this phase the student will produce artifact signs and possibly make references to the artifact, even in absence of the artifact. Moreover, this process can go a step further: the instrument can be used as a *psychological tool* (Kozulin, 1998) to reason about and solve a problem. At this phase it is possible to observe the emergence of artifact signs, and possibly mathematical signs, without any observable physical action or presence of the artifact. In other words, it will have become a tool-to-think-with(out).

## The semiotic potential of the GGBot with respect to the notion of angle

How does this apply to our study? Our artifact is the GGBot (short for “GREATGeometryBot”), an object moving on wheels (see Figure 1), similar to Papert’s turtle, that can be programmed using a graphical block-based coding language similar to Scratch (it is realized in SNAP!). The two back wheels and a metal sphere in the front allow it to move on the plane (Figure 1).

<figure><img src="image_117.png" /><figcaption>Figure 1: a) top view of GGBot with nose and trunk marker holders; b) translation movement direction; c) direction of departure and arrival after a rotation with vertex at the “trunk” marker, and trace mark left by a red marker in the “nose”; d) trace marks left if the GGBot is programmed to “do the path” of the black triangle.</figcaption></figure>

GGBot has two marker holders, one in front (“nose”) and one between the wheels (“trunk”) so that it can “provide situated signs that can be elaborated into geometrical notions – such as segment, vertex, angle, rotation, polygon – while still carrying the situatedness given by the real movement of the
# THEORETICAL FRAMEWORK AND RESEARCH QUESTION

This study is grounded in *Theory of Semiotic Mediation* (Bartolini Bussi & Mariotti, 2008), a Vygotskian socio-constructivist theory, according to which students are guided by their teacher to construct mathematical knowledge by solving appropriately designed tasks with the use of appropriately chosen artifacts – that is, artifacts and mathematical tasks that are considered helpful for unfolding the target mathematical meanings. During a first phase of the *didactic cycle* students use the *artifact* (e.g., a physical compass, the dragging tool in a Dynamic Geometry Environment) to solve a task and develop *utilization schemes*. The artifact together with such schemes constitutes an *instrument* in the sense of Rabardel (1995), through which the students produce *artifact signs*, closely related to the task and to the artifact. We can now look at two developmental processes that the activity of working with an artifact can foster.

One process concerns the unfolding of the artifact's *semiotic potential*, the artifact's double relationship with personal meanings (related to artifact signs) and with mathematical meanings (related to *mathematical signs*), that the teacher aims to construct through the orchestration of mathematical discussions during the didactic cycles.

A second process concerns the student's *internalization* (e.g., Bartolini Bussi & Mariotti, 2008), the "transformation" of what has been a mainly external process into an internal process. At this phase the student will produce artifact signs and possibly make references to the artifact, even in absence of the artifact. Moreover, this process can go a step further: the instrument can be used as a *psychological tool* (Kozulin, 1998) to reason about and solve a problem. At this phase it is possible to observe the emergence of artifact signs, and possibly mathematical signs, without any observable physical action or presence of the artifact. In other words, it will have become a tool-to-think-with(out).

## The semiotic potential of the GGBot with respect to the notion of angle

How does this apply to our study? Our artifact is the GGBot (short for “GREATGeometryBot”), an object moving on wheels (see Figure 1), similar to Papert’s turtle, that can be programmed using a graphical block-based coding language similar to Scratch (it is realized in SNAP!). The two back wheels and a metal sphere in the front allow it to move on the plane (Figure 1).

<figure><img src="image_123.png" /><figcaption>Figure 1: a) top view of GGBot with nose and trunk marker holders; b) translation movement direction; c) direction of departure and arrival after a rotation with vertex at the “trunk” marker, and trace mark left by a red marker in the “nose”; d) trace marks left if the GGBot is programmed to “do the path” of the black triangle.</figcaption></figure>

GGBot has two marker holders, one in front (“nose”) and one between the wheels (“trunk”) so that it can “provide situated signs that can be elaborated into geometrical notions – such as segment, vertex, angle, rotation, polygon – while still carrying the situatedness given by the real movement of the
physical artefact” (Baccaglini-Frank & Mariotti, 2022, p. 2660). The possible movements (translations and rotations) and trace marks are shown in Figure 1.

GGBot’s semiotic potential with respect to the notion of angle consists in the relationships between the signs it can produce as it “turns changing direction” and seeing an angle as the rotation that takes one side of the angle onto the other (see Figures 1b and 1c). Concerning angles of polygons, GGBot’s greatest semiotic potential is respect to “exterior angle(s)”, seen as the change(s) in direction of a person walking along the contour of the figure (Figure 1d).

Such potential can only be realized through well-designed tasks. A dual pair of tasks that were used frequently during the didactic cycle in Sam’s class, were the following: the *planning task*, or *figure-to-code task* (students are given the name of a figure and asked to produce a code with the blocks, working in pairs, to make GGBot draw the required figure on the paper); the *prediction task*, or *code-to-figure task* (students are given a code and asked to predict the trace mark GGBot will leave on the paper when executing such a code).

As Baccaglini-Frank & Mariotti (2022) have highlighted,

> “an essential feature of the semiotic potential of this artefact is its building on the relationship between the global movement and its breaking up into steps and turning points and the geometrical meaning of a polygon at a global and an analytical level. From a cognitive point of view [...], the task consists in breaking down a path that is imagined to be generated through physical continuous motion, into geometrical elements [...] of a different nature: they are static and discrete.” (p. 2662).

Specifically, concerning the notion of angle, the artifact GGBot can support the development of geometrical reasoning stemming from a “temporal” and dynamic conception, similar to the one described by Freudenthal in his Goniometry. Moreover, the dual tasks described above have the potential of eliciting analytical reasoning, since in both cases a “whole” figure needs to be decomposed into (or recomposed from) smaller components: in the case of the angle these are single (parts of) sides, with a vertex in common and with a specific reciprocal inclination, given by “how much” GGBot needs to turn to face first one direction and then the other.

## Research questions

Our questions are grounded in the theoretical framework above, and we ask specifically: Is it possible for GGBot to act as a psychological tool for solving primary school geometry tasks? If so, how can such a tool support students’ analytical (recognizing components and using them to compose/decompose a figure) and geometric reasoning (recognizing and using specific representations of) concerning angle?

## METHODS

We analyze here part of a task-based interview with Sam, a 5th grade student, whose class had worked with GGBot since the beginning of 4th grade, following the proposals of the *PerContare* project (teacher guides designed by many researchers including the first two authors, see Bartolini Bussi & Baccaglini-Frank, 2015; Baccaglini-Frank et al., 2023). Initially the activities only made use of the marker in the trunk and of right-angle rotations, for both planning tasks and prediction tasks; subsequently, also the marker in the nose was inserted, and through open problems the students were encouraged to discover new commands with parameters for controlling the amplitude of the rotations (turns) and the length of the translations (steps).
physical artefact” (Baccaglini-Frank & Mariotti, 2022, p. 2660). The possible movements (translations and rotations) and trace marks are shown in Figure 1.

GGBot’s semiotic potential with respect to the notion of angle consists in the relationships between the signs it can produce as it “turns changing direction” and seeing an angle as the rotation that takes one side of the angle onto the other (see Figures 1b and 1c). Concerning angles of polygons, GGBot’s greatest semiotic potential is respect to “exterior angle(s)”, seen as the change(s) in direction of a person walking along the contour of the figure (Figure 1d).

Such potential can only be realized through well-designed tasks. A dual pair of tasks that were used frequently during the didactic cycle in Sam’s class, were the following: the *planning task*, or *figure-to-code task* (students are given the name of a figure and asked to produce a code with the blocks, working in pairs, to make GGBot draw the required figure on the paper); the *prediction task*, or *code-to-figure task* (students are given a code and asked to predict the trace mark GGBot will leave on the paper when executing such a code).

As Baccaglini-Frank & Mariotti (2022) have highlighted,

> “an essential feature of the semiotic potential of this artefact is its building on the relationship between the global movement and its breaking up into steps and turning points and the geometrical meaning of a polygon at a global and an analytical level. From a cognitive point of view [...], the task consists in breaking down a path that is imagined to be generated through physical continuous motion, into geometrical elements [...] of a different nature: they are static and discrete.” (p. 2662).

Specifically, concerning the notion of angle, the artifact GGBot can support the development of geometrical reasoning stemming from a “temporal” and dynamic conception, similar to the one described by Freudenthal in his Goniometry. Moreover, the dual tasks described above have the potential of eliciting analytical reasoning, since in both cases a “whole” figure needs to be decomposed into (or recomposed from) smaller components: in the case of the angle these are single (parts of) sides, with a vertex in common and with a specific reciprocal inclination, given by “how much” GGBot needs to turn to face first one direction and then the other.

## Research questions

Our questions are grounded in the theoretical framework above, and we ask specifically: Is it possible for GGBot to act as a psychological tool for solving primary school geometry tasks? If so, how can such a tool support students’ analytical (recognizing components and using them to compose/decompose a figure) and geometric reasoning (recognizing and using specific representations of) concerning angle?

## METHODS

We analyze here part of a task-based interview with Sam, a 5th grade student, whose class had worked with GGBot since the beginning of 4th grade, following the proposals of the *PerContare* project (teacher guides designed by many researchers including the first two authors, see Bartolini Bussi & Baccaglini-Frank, 2015; Baccaglini-Frank et al., 2023). Initially the activities only made use of the marker in the trunk and of right-angle rotations, for both planning tasks and prediction tasks; subsequently, also the marker in the nose was inserted, and through open problems the students were encouraged to discover new commands with parameters for controlling the amplitude of the rotations (turns) and the length of the translations (steps).
Sam's interview was part of a larger data collection aimed at assessing students' learning at the end of a 3-year cycle of teachers' use of the *PerContare* materials. After assigning and collecting a written set of tasks to 112 5th graders in 5 classes, two researchers (the second author is one of them) visited each classroom and individually interviewed selected students in a quiet room outside the students' regular classroom. Sam was one of the students selected because of the mathematical depth of some of their answers; others were selected because of their mistakes or reference to specific artifacts that helped them think. During the interview, the students were shown their previous written work and asked to further explain their thinking. Moreover, the researchers would ask additional questions when they noted clues of internalization of a particular artifact. This was very clear in the case of Sam. Figure 2 shows: (2a) the original question ("How many interior angles does this figure have?"), followed by the request to explain own reasoning; (2b) Sam's first and (2c) second drawings in response to researchers' questions.

<figure><img src="image_118.png" /><figcaption>Figure 2: a) original task; b) first figure Sam used to explain his reasoning with numbers indicating the order of his counting; c) Sam's second figure, showing what "the nose would draw"</figcaption></figure>

We selected this task and Sam’s answers corresponding to figures 2b and 2c because of Sam’s explanation of the relationship between angles drawn with markers in the trunk and in the nose of GGBot, and because the original question introduces the slippery term “interior angles” that frequently induces incorrect answers (this was an item of Italian national assessment test for grade 5 in 2009: in the national student sample, the percentage of A answers was 66%). GGBot was not available in the physical or virtual form during the interview.

To perform the analyses we did the following. We transcribed the interview and searched in the text and video for artifact signs and mathematical signs (verbal utterances, written signs on the paper used, gestures). Since the GGBot was never available during the interview, we interpreted the artifact signs as hints that the internalization of GGBot as a psychological tool. We then related such artifact signs to the mathematical signs produced by Sam, and we proceeded to identify instances of analytical thinking (when the signs are used to distinguish components or possible decompositions or re-compositions of angle) and geometrical thinking (more in general, when the signs are used to talk and reason about representations of angle), till we reached agreement.

## ANALYSIS

We selected 3 excerpts that are particularly illuminating with respect to Sam’s internalization of the GGBot relative to the notion of angle. We provide summaries of skipped parts around each excerpt.

### Excerpt 1: How would GGBot do it?

In the written task, asking for the number of interior angles in the star (Fig. 2a), Sam had answered “10” and explained on the sheet of paper: “I calculated all the interior angles (without falling in the
Sam’s interview was part of a larger data collection aimed at assessing students’ learning at the end of a 3-year cycle of teachers’ use of the *PerContare* materials. After assigning and collecting a written set of tasks to 112 5^th^ graders in 5 classes, two researchers (the second author is one of them) visited each classroom and individually interviewed selected students in a quiet room outside the students’ regular classroom. Sam was one of the students selected because of the mathematical depth of some of their answers; others were selected because of their mistakes or reference to specific artifacts that helped them think. During the interview, the students were shown their previous written work and asked to further explain their thinking. Moreover, the researchers would ask additional questions when they noted clues of internalization of a particular artifact. This was very clear in the case of Sam. Figure 2 shows: (2a) the original question (“How many interior angles does this figure have?”, followed by the request to explain own reasoning; (2b) Sam’s first and (2c) second drawings in response to researchers’ questions.

<figure><img src="image_124.png" /><figcaption>Figure 2: a) original task; b) first figure Sam used to explain his reasoning with numbers indicating the order of his counting; c) Sam’s second figure, showing what “the nose would draw”</figcaption></figure>

We selected this task and Sam’s answers corresponding to figures 2b and 2c because of Sam’s explanation of the relationship between angles drawn with markers in the trunk and in the nose of GGBot, and because the original question introduces the slippery term “interior angles” that frequently induces incorrect answers (this was an item of Italian national assessment test for grade 5 in 2009: in the national student sample, the percentage of A answers was 66%). GGBot was not available in the physical or virtual form during the interview.

To perform the analyses we did the following. We transcribed the interview and searched in the text and video for artifact signs and mathematical signs (verbal utterances, written signs on the paper used, gestures). Since the GGBot was never available during the interview, we interpreted the artifact signs as hints that the internalization of GGBot as a psychological tool. We then related such artifact signs to the mathematical signs produced by Sam, and we proceeded to identify instances of analytical thinking (when the signs are used to distinguish components or possible decompositions or re-compositions of angle) and geometrical thinking (more in general, when the signs are used to talk and reason about representations of angle), till we reached agreement.

## ANALYSIS

We selected 3 excerpts that are particularly illuminating with respect to Sam’s internalization of the GGBot relative to the notion of angle. We provide summaries of skipped parts around each excerpt.

### Excerpt 1: How would GGBot do it?

In the written task, asking for the number of interior angles in the star (Fig. 2a), Sam had answered “10” and explained on the sheet of paper: “I calculated all the interior angles (without falling in the
trap)". When one of the interviewers asked to say something more about the "trap", he explained the way he counted the angles producing the Figure 2b. Referring to such angles, to better explain how he reasoned, Sam mentioned "turns" (Italian: "svolte"): the interviewers immediately asked him more about what these referred to, and he spoke of the movements of GGBot (original excerpt here).

Sam: So let's assume it starts from here: it has to go straight and turn, so this is already an angle. Then it has to go straight and turn, and it's 2 angles. [He repeats the refrain "go straight and turn" four times] Straight, turn, 7. Then it has to go straight again and then turn, and it's 8. It has to come, uh, it has to go straight and then turn, that's 9. Go straight and turn, and it's 10... and then come back to where it started from.

Table 1 shows correspondences between GGBot's movements, Sam's verbal utterances and his gestures associated to the figural units he identifies in the star (segments and angles).

<table><caption>Table 1: Overview of GGBot's predicated movements, Sam's verbal utterances and his gestures</caption><thead><tr><td>Movement</td><td>Verbal utterances</td><td>Hand gestures</td></tr></thead><tbody><tr><td>Step forward</td><td>"go straight" or only "straight"</td><td>He sweeps a segment with the right index finger</td></tr><tr><td>Rotation</td><td>"turn"</td><td>He points the index finger at a vertex and rotates it; after counting angle number 5, the finger rotation is less emphatic</td></tr></tbody></table>

Sam temporalizes the figure by looking at its contour as a path to be followed – in this attempt we see an instance of internalization: Sam talks as if an imagined GGBot is tracing the star. Internalization is also evident in the turning gestures: they change during the excerpt, becoming less emphasized as if the movement is just imagined or unessential. The small arcs Sam had previously drawn (see Figure 2b) are different from those that the GGBot would draw, but this confirms our point: internalization is more than a replication of a past experience. Rather, Sam uses GGBot's movement as a (psychological) tool for supporting analytical reasoning and detecting angles in the spatial and dynamic essence of its figural units: just a turn preceded and followed by a straight path.

## Excerpt 2: What about the GGBot's nose?

The interviewers, curious about the marks that Sam uses to count the times GGBot turns, and so the angles of the star, and their correspondence to marks that either of GGBot's markers would leave, ask Sam what GGBot would draw if it had markers both in the trunk and in the nose (original excerpt here). Sam produced Figure 2c and the signs analyzed in Table 2.

Again we can see how GGBot supports analytical reasoning to decompose the path into a combination of translations and rotations, but there is more. The predicted movements of the GGBot's nose are used as psychological means to solve a new task and not just to reproduce a known procedure. Indeed, the deixis (see "it makes this" when Sam is drawing the arc, Table 2) manifests a deeper identification between Sam and the robot: Sam's gestures are no longer just a reference to the movement of the GGBot, but gestures and GGbot's motion overlap intimately. Sam talks as if the hand holding the pencil was the GGBot. Moreover, Sam does not trace the actual angle that GGBot "makes with the nose", but his verbal and gestural productions suggest that it is just imagined. When Sam claims that GGBot "lies here on the tip", the gesture points to the lengthening (just mimicked) of a side (see
trap)". When one of the interviewers asked to say something more about the "trap", he explained the way he counted the angles producing the Figure 2b. Referring to such angles, to better explain how he reasoned, Sam mentioned "turns" (Italian: "svolte"): the interviewers immediately asked him more about what these referred to, and he spoke of the movements of GGBot (original excerpt here).

Sam: So let's assume it starts from here: it has to go straight and turn, so this is already an angle. Then it has to go straight and turn, and it's 2 angles. [He repeats the refrain "go straight and turn" four times] Straight, turn, 7. Then it has to go straight again and then turn, and it's 8. It has to come, uh, it has to go straight and then turn, that's 9. Go straight and turn, and it's 10... and then come back to where it started from.

Table 1 shows correspondences between GGBot's movements, Sam's verbal utterances and his gestures associated to the figural units he identifies in the star (segments and angles).

<table><caption>Table 1: Overview of GGBot's predicated movements, Sam's verbal utterances and his gestures</caption><thead><tr><th>Movement</th><th>Verbal utterances</th><th>Hand gestures</th></tr></thead><tbody><tr><td>Step forward</td><td>"go straight" or<br>only "straight"</td><td>He sweeps a segment with the right index finger</td></tr><tr><td>Rotation</td><td>"turn"</td><td>He points the index finger at a vertex and rotates it; after<br>counting angle number 5, the finger rotation is less<br>emphatic</td></tr></tbody></table>

Sam temporalizes the figure by looking at its contour as a path to be followed – in this attempt we see an instance of internalization: Sam talks as if an imagined GGBot is tracing the star. Internalization is also evident in the turning gestures: they change during the excerpt, becoming less emphasized as if the movement is just imagined or unessential. The small arcs Sam had previously drawn (see Figure 2b) are different from those that the GGBot would draw, but this confirms our point: internalization is more than a replication of a past experience. Rather, Sam uses GGBot's movement as a (psychological) tool for supporting analytical reasoning and detecting angles in the spatial and dynamic essence of its figural units: just a turn preceded and followed by a straight path.

## Excerpt 2: What about the GGBot's nose?

The interviewers, curious about the marks that Sam uses to count the times GGBot turns, and so the angles of the star, and their correspondence to marks that either of GGBot's markers would leave, ask Sam what GGBot would draw if it had markers both in the trunk and in the nose (original excerpt here). Sam produced Figure 2c and the signs analyzed in Table 2.

Again we can see how GGBot supports analytical reasoning to decompose the path into a combination of translations and rotations, but there is more. The predicted movements of the GGBot's nose are used as psychological means to solve a new task and not just to reproduce a known procedure. Indeed, the deixis (see "it makes this" when Sam is drawing the arc, Table 2) manifests a deeper identification between Sam and the robot: Sam's gestures are no longer just a reference to the movement of the GGBot, but gestures and GGbot's motion overlap intimately. Sam talks as if the hand holding the pencil was the GGBot. Moreover, Sam does not trace the actual angle that GGBot "makes with the nose", but his verbal and gestural productions suggest that it is just imagined. When Sam claims that GGBot "lies here on the tip", the gesture points to the lengthening (just mimicked) of a side (see
Figure 4) which corresponds to the direction taken by a fictional GGBot. In absence of a physical artefact, Sam's gestures are tools that realize the body sytonicity with the imagined GGBot.

<table><thead><tr><th>Predicted nose trace mark</th><th>Verbal utterances</th><th>Pencil gestures</th></tr></thead><tbody><tr><td>Figure 3</td><td>"go straight" or only "straight"</td><td>The pencil runs along one side</td></tr><tr><td>Figure 4</td><td>"it makes this" or "it lies here on the tip and it makes this with the nose"</td><td>The pencil draws an arc from this vertex to an inner point on the consecutive side; it moves back and forth along the side around the vertex, and then along the drawn arc.</td></tr></tbody></table>

<figure><img src="image_119.png" /></figure>

<figure><img src="image_120.png" /><figcaption>Table 2: Predicated nose trace mark, Sam's verbal utterances, and pencil gestures from excerpt 2</figcaption></figure>

Excerpt 3 – Comparing different signs for angles

Intrigued, the researchers ask Sam to better explain the difference between the angles he drew in Figures 2b and 2c (original excerpt here). He explains:

Sam: Because GGBot instead of making the angles as they are on the protractor, it works with the supplementary angles, if I am not mistaken, which are on the basis of 180. Because if we made a 90-degree angle [he draws a right angle]... oh god, maybe with 90 you don't understand so much....

and, invited by a researcher to show a drawing, he produces the signs in Table 3.

<table><thead><tr><th>Predicted trace marks</th><th>Verbal utterances</th><th>Pencil gestures</th></tr></thead><tbody><tr><td>Figure 5</td><td>"an obtuse [acute] angle"</td><td>He draws an acute angle</td></tr><tr><td>Figure 6</td><td>"instead of going straight", "it will go here on the edge"</td><td>He traces the extension of one side of the angle; points at the acute angle; runs along one of its sides and stops at its vertex</td></tr><tr><td>Figure 7</td><td>"with the nose, to turn, it will do this", "on the basis of the supplementary angles", "let's say this is 30 [...] and this 50 [degrees]"</td><td>He draws an arc from the vertex of the acute angle to its other side; then he draws more lightly also an arc for the obtuse angle</td></tr></tbody></table>

<figure><img src="image_121.png" /></figure>

<figure><img src="image_122.png" /></figure>

<figure><img src="image_123.png" /><figcaption>Table 3: Predicated nose trace mark, Sam's verbal utterances, and pencil gestures from excerpt 3</figcaption></figure>

Sam is now using formal mathematical signs and the GGBot's movement to focus on the features of different representations of angle, demonstrating sophisticated geometrical reasoning. He speaks of "supplementary angles" to distinguish angles made by GGBot from those of the protractor (another artifact that he had the opportunity to experience). Interestingly, he uses dynamic expressions (*do*, *make*, *works*) to indicate movements when referring to the former, and static ones (*are on*) when speaking about the latter. His control over the mathematical concept of supplementary angles seems quite strong: he starts with an example (right angle) but quickly changes his mind, acknowledging
Figure 4) which corresponds to the direction taken by a fictional GGBot. In absence of a physical artefact, Sam's gestures are tools that realize the body sytonicity with the imagined GGBot.

<table><thead><tr><th>Predicted nose trace mark</th><th>Verbal utterances</th><th>Pencil gestures</th></tr></thead><tbody><tr><td>Figure 3</td><td>"go straight" or only "straight"</td><td>The pencil runs along one side</td></tr><tr><td>Figure 4</td><td>"it makes this" or "it lies here on the tip and it makes this with the nose"</td><td>The pencil draws an arc from this vertex to an inner point on the consecutive side; it moves back and forth along the side around the vertex, and then along the drawn arc.</td></tr></tbody></table>

<figure><img src="image_125.png" /></figure>

<figure><img src="image_126.png" /><figcaption>Table 2: Predicated nose trace mark, Sam's verbal utterances, and pencil gestures from excerpt 2</figcaption></figure>

Excerpt 3 – Comparing different signs for angles

Intrigued, the researchers ask Sam to better explain the difference between the angles he drew in Figures 2b and 2c (original excerpt here). He explains:

Sam: Because GGBot instead of making the angles as they are on the protractor, it works with the supplementary angles, if I am not mistaken, which are on the basis of 180. Because if we made a 90-degree angle [he draws a right angle]... oh god, maybe with 90 you don't understand so much....

and, invited by a researcher to show a drawing, he produces the signs in Table 3.

<table><thead><tr><th>Predicted trace marks</th><th>Verbal utterances</th><th>Pencil gestures</th></tr></thead><tbody><tr><td>Figure 5</td><td>"an obtuse [acute] angle"</td><td>He draws an acute angle</td></tr><tr><td>Figure 6</td><td>"instead of going straight", "it will go here on the edge"</td><td>He traces the extension of one side of the angle; points at the acute angle; runs along one of its sides and stops at its vertex</td></tr><tr><td>Figure 7</td><td>"with the nose, to turn, it will do this", "on the basis of the supplementary angles", "let's say this is 30 [...] and this 50 [degrees]"</td><td>He draws an arc from the vertex of the acute angle to its other side; then he draws more lightly also an arc for the obtuse angle</td></tr></tbody></table>

<figure><img src="image_127.png" /></figure>

<figure><img src="image_128.png" /></figure>

<figure><img src="image_129.png" /><figcaption>Table 3: Predicated nose trace mark, Sam's verbal utterances, and pencil gestures from excerpt 3</figcaption></figure>

Sam is now using formal mathematical signs and the GGBot's movement to focus on the features of different representations of angle, demonstrating sophisticated geometrical reasoning. He speaks of "supplementary angles" to distinguish angles made by GGBot from those of the protractor (another artifact that he had the opportunity to experience). Interestingly, he uses dynamic expressions (*do*, *make*, *works*) to indicate movements when referring to the former, and static ones (*are on*) when speaking about the latter. His control over the mathematical concept of supplementary angles seems quite strong: he starts with an example (right angle) but quickly changes his mind, acknowledging
that in such a special case, his example would have been less understandable, since both the exterior
angle and the interior one would have both been right angles. Sam then continues to construct his
example, mixing artifact signs and mathematical signs (Table 3): the verbal utterances about GGBot’s
movement are intertwined with more general and atemporal forms (“obtuse angle”, “on the basis of
the supplementary angles”, “let's say this is 30”). Moreover, this example allows him to better explain
what he was referring to in excerpt 2 (Table 2), since after that he re-draws the arc associated to the
action of “turning” starting from a side of the angle, not its vertex: this is coherent with the “nose”
marker’s trace, i.e. with the actual angle of rotation of the GGBot.

Shortly after, Sam identifies a pair of supplementary angles, and he says:

Sam: ...ah no, sorry. This one is 30 and this one is 150. He is turning, he makes an angle of 150 degrees [he retraces the small arc]. Internally there is 30 [he retraces the small arc] and together they make 180 [he marks the two horizontal sides a whole]. Because maybe we see the initial angle like this [he retraces the sides of the acute angle], but the angle he sees is this [he retraces the sides of the obtuse angle].

Sam talks about the two angles drawn earlier (Figure 7) by mixing different signs: the obtuse angle is drawn turning (It: “svoltando”), while the acute angle *is* inside. The verbal expression “there is” addresses a static dimension that is less related to the process of drawing the angle, while the gesture of retracing the small arc in the acute angle recalls the trace mark GGBot would make when rotating. Possibly, these different takes on angles mirror two way of looking at – and reasoning about – angles: one way is static in nature – angles *are somewhere* and, for example, “they are on the protractor” – and the other is dynamic and recalls the angle that the GGBot “makes turning”. The dynamism is partially overcome by the whole figure representing supplementary angles that Sam now sees as two adjacent angles, as the gesture reveals. In the last utterance the dynamism has been completely overcome, as if the original angle that “we see” and the one that “he [GGBot] sees” have blended together, giving rise to something new, indicated through a single repeated gesture with the pencil along their sides. This interpretation is strengthened by Sam’s use of “see”, instead of “do”.

CONCLUSION AND DISCUSSION

We set out to explore whether GGBot could act as a psychological tool solving primary school geometry tasks, and found compelling evidence that, yes, it is indeed possible. The case of Sam allowed us to gain insight into how such a tool supported his analytical and geometric reasoning on a specific task. Digging deeper into what Sam had internalized, we discovered how GGBot had supported his conceptualization of the notion of angle, and in particular of supplementary angles, proving that it plays the role of tool to recall and productively use geometrical notions, developing ever more effective objects-to-think-with (or, in our case, without).

Our analyses revealed how working with GGBot led Sam to re-thinking a figure, decomposing it and
using analytic reasoning to relate the commands, movements, trace marks and figural units in a global
whole. This is possible not only through the use of GGBot, but thanks to the dyad composed of the
artifacit and the dual tasks we have designed, which elicit processes of (de)composition and
anticipation. Specifically, we discovered that the sequence of activities with GGBot that was
proposed to Sam led him to conceive angles as rotations, but also as geometrical figures comprising
two sides and an “inclination” between them, recognizing different kinds of angles (right, acute,
obtuse), their mutual relationships (two supplementary angles) and angles of closed figures (interior
that in such a special case, his example would have been less understandable, since both the exterior
angle and the interior one would have both been right angles. Sam then continues to construct his
example, mixing artifact signs and mathematical signs (Table 3): the verbal utterances about GGBot’s
movement are intertwined with more general and atemporal forms (“obtuse angle”, “on the basis of
the supplementary angles”, “let's say this is 30”). Moreover, this example allows him to better explain
what he was referring to in excerpt 2 (Table 2), since after that he re-draws the arc associated to the
action of “turning” starting from a side of the angle, not its vertex: this is coherent with the “nose”
marker’s trace, i.e. with the actual angle of rotation of the GGBot.

Shortly after, Sam identifies a pair of supplementary angles, and he says:

Sam: ...ah no, sorry. This one is 30 and this one is 150. He is turning, he makes an angle of 150 degrees [he retraces the small arc]. Internally there is 30 [he retraces the small arc] and together they make 180 [he marks the two horizontal sides a whole]. Because maybe we see the initial angle like this [he retraces the sides of the acute angle], but the angle he sees is this [he retraces the sides of the obtuse angle].

Sam talks about the two angles drawn earlier (Figure 7) by mixing different signs: the obtuse angle is drawn turning (It: “svoltando”), while the acute angle *is* inside. The verbal expression “there is” addresses a static dimension that is less related to the process of drawing the angle, while the gesture of retracing the small arc in the acute angle recalls the trace mark GGBot would make when rotating. Possibly, these different takes on angles mirror two way of looking at – and reasoning about – angles: one way is static in nature – angles *are somewhere* and, for example, “they are on the protractor” – and the other is dynamic and recalls the angle that the GGBot “makes turning”. The dynamism is partially overcome by the whole figure representing supplementary angles that Sam now sees as two adjacent angles, as the gesture reveals. In the last utterance the dynamism has been completely overcome, as if the original angle that “we see” and the one that “he [GGBot] sees” have blended together, giving rise to something new, indicated through a single repeated gesture with the pencil along their sides. This interpretation is strengthened by Sam’s use of “see”, instead of “do”.

CONCLUSION AND DISCUSSION

We set out to explore whether GGBot could act as a psychological tool solving primary school geometry tasks, and found compelling evidence that, yes, it is indeed possible. The case of Sam allowed us to gain insight into how such a tool supported his analytical and geometric reasoning on a specific task. Digging deeper into what Sam had internalized, we discovered how GGBot had supported his conceptualization of the notion of angle, and in particular of supplementary angles, proving that it plays the role of tool to recall and productively use geometrical notions, developing ever more effective objects-to-think-with (or, in our case, without).

Our analyses revealed how working with GGBot led Sam to re-thinking a figure, decomposing it and
using analytic reasoning to relate the commands, movements, trace marks and figural units in a global
whole. This is possible not only through the use of GGBot, but thanks to the dyad composed of the
artifect and the dual tasks we have designed, which elicit processes of (de)composition and
anticipation. Specifically, we discovered that the sequence of activities with GGBot that was
proposed to Sam led him to conceive angles as rotations, but also as geometrical figures comprising
two sides and an “inclination” between them, recognizing different kinds of angles (right, acute,
obtuse), their mutual relationships (two supplementary angles) and angles of closed figures (interior
and exterior). In particular, Sam constructed a theoretical lens that let him interpret the expression "interior angles" in the initial task coherently with formal geometry and correctly solve it, differently from many other Italian students in the national assessment. Moreover, thanks to synergies with other artifacts such as the protractor, Sam merges dynamic and static experiences, reaching a generalized static and atemporal (but with dynamic features encapsulated within) conception of angle.

In this perspective, the learning path through the *PerContare* activities with GGBot fully exploited the artifact's semiotic potential, leading Sam to develop a psychological tool that he can use in different ways, unpacking as necessary the internalized experiences when necessary, for example, to "see" a geometric figure as a part of the plane enclosed by consecutive segments, or supplementary angles as the "pair" of angles, one "seen from the figure" and the other "seen by GGBot" as it traces the contour. Of course there remain many open questions and issues to study in more depth. Just to mention one in which we are particularly interested, we are curious to explore how the dual tasks presented can be harbingers of similar activities in dynamic geometry (with step-by-step construction open problems) useful for enhancing key geometric skills (Leung et al., 2023).

## References

Baccaglini-Frank, A., Funghi, S., Maracci, M., & Ramploud, A. (2023). Learning about multiplication by comparing algorithms: "One times one, but actually they are ten times ten". *The Journal of Mathematical Behavior, 70*, 101024.

Baccaglini-Frank, A., & Mariotti, M. A. (2022). "Doing well" in the Teaching for Robust Understanding approach revealed by the lens of the semiotic potential of tasks with the GGBot. In *Proc. of the 12th Congress of the European Society for Research in Mathematics Education (CERME12)* (pp. 2658–2665). ERME / Free University of Bozen-Bolzano.

Bartolini Bussi, M. G., & Baccaglini-Frank, A. (2015). Geometry in early years: sowing seeds for a mathematical definition of squares and rectangles. *ZDM, 47*(3), 391–405.

Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic Mediation in the Mathematics Classroom: Artefacts and Signs after a Vygotskian Perspective. In *Handbook of International research in Mathematics education* (2nd edition) (pp. 746–783). Routledge Taylor & Francis Group.

Devichi, C., & Munier, V. (2013). About the concept of angle in elementary school: Misconceptions and teaching sequences. *The Journal of Mathematical Behavior, 32*(1), 1–19.

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

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

Freudenthal, H. (1973). *Mathematics as an educational task*. Reidel.

Kozulin, A. (1998). *Psychological tools: a sociocultural approach to education*. Harvard University Press.

Leung, A., Baccaglini-Frank, A., Mariotti, M.A., Miragliotta, E. (2023). Enhancing geometric skills with digital technology: the case of dynamic geometry. In B. Pepin, G. Gueudet, & J. Choppin (Eds.), *Handbook of digital resources in mathematics education*. Springer.

Rabardel, P. (1995). *Les hommes et les technologies – Approche cognitive des instruments contemporains*. Armand Colin.

Vygotsky, L. S. (1978). *Mind in society: the development of higher psychological processes*. Harvard University Press.
and exterior). In particular, Sam constructed a theoretical lens that let him interpret the expression "interior angles" in the initial task coherently with formal geometry and correctly solve it, differently from many other Italian students in the national assessment. Moreover, thanks to synergies with other artifacts such as the protractor, Sam merges dynamic and static experiences, reaching a generalized static and atemporal (but with dynamic features encapsulated within) conception of angle.

In this perspective, the learning path through the *PerContare* activities with GGBot fully exploited the artifact's semiotic potential, leading Sam to develop a psychological tool that he can use in different ways, unpacking as necessary the internalized experiences when necessary, for example, to "see" a geometric figure as a part of the plane enclosed by consecutive segments, or supplementary angles as the "pair" of angles, one "seen from the figure" and the other "seen by GGBot" as it traces the contour. Of course there remain many open questions and issues to study in more depth. Just to mention one in which we are particularly interested, we are curious to explore how the dual tasks presented can be harbingers of similar activities in dynamic geometry (with step-by-step construction open problems) useful for enhancing key geometric skills (Leung et al., 2023).

## References

Baccaglini-Frank, A., Funghi, S., Maracci, M., & Ramploud, A. (2023). Learning about multiplication by comparing algorithms: "One times one, but actually they are ten times ten". *The Journal of Mathematical Behavior, 70*, 101024.

Baccaglini-Frank, A., & Mariotti, M. A. (2022). "Doing well" in the Teaching for Robust Understanding approach revealed by the lens of the semiotic potential of tasks with the GGBot. In *Proc. of the 12th Congress of the European Society for Research in Mathematics Education (CERME12)* (pp. 2658–2665). ERME / Free University of Bozen-Bolzano.

Bartolini Bussi, M. G., & Baccaglini-Frank, A. (2015). Geometry in early years: sowing seeds for a mathematical definition of squares and rectangles. *ZDM, 47*(3), 391–405.

Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic Mediation in the Mathematics Classroom: Artefacts and Signs after a Vygotskian Perspective. In *Handbook of International research in Mathematics education* (2nd edition) (pp. 746–783). Routledge Taylor & Francis Group.

Devichi, C., & Munier, V. (2013). About the concept of angle in elementary school: Misconceptions and teaching sequences. *The Journal of Mathematical Behavior, 32*(1), 1–19.

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

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

Freudenthal, H. (1973). *Mathematics as an educational task*. Reidel.

Kozulin, A. (1998). *Psychological tools: a sociocultural approach to education*. Harvard University Press.

Leung, A., Baccaglini-Frank, A., Mariotti, M.A., Miragliotta, E. (2023). Enhancing geometric skills with digital technology: the case of dynamic geometry. In B. Pepin, G. Gueudet, & J. Choppin (Eds.), *Handbook of digital resources in mathematics education*. Springer.

Rabardel, P. (1995). *Les hommes et les technologies – Approche cognitive des instruments contemporains*. Armand Colin.

Vygotsky, L. S. (1978). *Mind in society: the development of higher psychological processes*. Harvard University Press.
# A STRUCTURED PRACTICAL ACTIVITY TO ENHANCE
UNDERSTANDING OF CIRCLE CIRCUMFERENCE AND $π$
APPROXIMATION

Masoud Bahramibidkalme¹, Zahra Gooya², Soheila Gholamazad³

¹ National Organization for Development of Exceptional Talents (Tehran Allameh Helli 1 Branch),

² Shahid Beheshti University, ³ Mathematics Education Organization for Research & Educational Planning, Ministry of Education, Iran

This paper presents a structured practical activity designed to enhance middle school students' understanding of circles in geometry. The activity focuses on promoting "relational understanding" and deepening students' engagement with concepts such as the number $π$ and the formula for calculating the circumference of circle. The findings showed the potential of practical structured activities in enhancing students' conceptual understanding of mathematics. This research contributes to the teaching practices in geometry education, emphasizing the importance of experiential learning. To conclude, the structured practical activity demonstrates its effectiveness in advancing students' understanding of circle-related concepts.

## INTRODUCTION

The circle, a simple closed curve, has held its place in human knowledge since long before recorded history. As far back as 1700 BC, the Rhind Papyrus provided methods for calculating the area of circular fields. As well, the history of mathematics reveals the profound influence of the circle, inspiring the development of geometric and calculus concepts (Koestler, 1990). The fundamental nature of a circle has made it as a cornerstone for mathematical understanding and has become as essential part of school mathematics curricula worldwide. This paper reports on a segment of a larger study that aimed to explore the ways for enhancing middle grades students' comprehension of geometrical concepts. Specifically, this study focused on a structured practical activity designed to promote "relational understanding" in the context of calculating the circumference of a circle. Traditionally, students were introduced to the approximation of $π$ as 3.14, a key constant in mathematics that allows for the calculation of a circle's circumference as $π$ times its diameter. While this formula has often taught by rote learning, sometimes to justify its correctness, students are asked to measure the diameter and circumference of various circular objects such as coins and rings and calculate their ratio. The findings will help to design structured practical activities for improving middle grade students' understanding of circle and its properties. The focus of this study was to explore how these activities might enhance students' conceptual knowledge and understanding related to circle. By implementing these structured practical activities and assessing their effectiveness, the research seeks to provide insights to take instructional approaches for teaching circle-related concepts. The theoretical foundation for the development of these activities is rooted in the role of structured practical activities in Skemp's framework for "relational understanding" in mathematics (1989).
# A STRUCTURED PRACTICAL ACTIVITY TO ENHANCE
UNDERSTANDING OF CIRCLE CIRCUMFERENCE AND $π$
APPROXIMATION

Masoud Bahramibidkalme¹, Zahra Gooya², Soheila Gholamazad³

¹ National Organization for Development of Exceptional Talents (Tehran Allameh Helli 1 Branch),

² Shahid Beheshti University, ³ Mathematics Education Organization for Research & Educational Planning, Ministry of Education, Iran

This paper presents a structured practical activity designed to enhance middle school students' understanding of circles in geometry. The activity focuses on promoting "relational understanding" and deepening students' engagement with concepts such as the number $π$ and the formula for calculating the circumference of circle. The findings showed the potential of practical structured activities in enhancing students' conceptual understanding of mathematics. This research contributes to the teaching practices in geometry education, emphasizing the importance of experiential learning. To conclude, the structured practical activity demonstrates its effectiveness in advancing students' understanding of circle-related concepts.

## INTRODUCTION

The circle, a simple closed curve, has held its place in human knowledge since long before recorded history. As far back as 1700 BC, the Rhind Papyrus provided methods for calculating the area of circular fields. As well, the history of mathematics reveals the profound influence of the circle, inspiring the development of geometric and calculus concepts (Koestler, 1990). The fundamental nature of a circle has made it as a cornerstone for mathematical understanding and has become as essential part of school mathematics curricula worldwide. This paper reports on a segment of a larger study that aimed to explore the ways for enhancing middle grades students' comprehension of geometrical concepts. Specifically, this study focused on a structured practical activity designed to promote "relational understanding" in the context of calculating the circumference of a circle. Traditionally, students were introduced to the approximation of $π$ as 3.14, a key constant in mathematics that allows for the calculation of a circle's circumference as $π$ times its diameter. While this formula has often taught by rote learning, sometimes to justify its correctness, students are asked to measure the diameter and circumference of various circular objects such as coins and rings and calculate their ratio. The findings will help to design structured practical activities for improving middle grade students' understanding of circle and its properties. The focus of this study was to explore how these activities might enhance students' conceptual knowledge and understanding related to circle. By implementing these structured practical activities and assessing their effectiveness, the research seeks to provide insights to take instructional approaches for teaching circle-related concepts. The theoretical foundation for the development of these activities is rooted in the role of structured practical activities in Skemp's framework for "relational understanding" in mathematics (1989).
STRUCTURED PRACTICAL ACTIVITIES

Previous studies have emphasized the benefits of incorporating structured practical activities in mathematics education. Such activities provide students with hands-on experiences, allowing them to actively engage with mathematical concepts and develop a deeper understanding (Lesh, 2003; Hiebert et al., 2007). The teaching of circle-related concepts, including circumference and the approximation of π, has often relied on rote learning approaches. However, recent research has advocated for more meaningful and experiential learning experiences to promote conceptual understanding (Battista & Clements, 1996; Lehrer & Chazan, 1998). By engaging students in measuring the diameter and circumference of real-world objects, they can explore the relationship between these measures and discover the role of π in calculating circumference (Gravemeijer, 1994.)

Relational understanding framework

Structured practical activities presented in the model of intelligent learning (Skemp, 1989), allow students to experience mathematics at various levels increasing their power of prediction and to find suitable solutions for their real-life problems. Structured practical activities are designed to help learners to do the followings:

* to build structured mathematical knowledge and to make predictable predictions.

* to take pleasure from finding their predictions confirmed by events.

* to put students in a position to immediately correct their own thinking rather than wait for a teacher to tell them where they went wrong.

* to get control in directing their own learning process.

* to provide shared sensory experiences which ensure that there is a common ground for co-operative learning by exchanging ideas and discussion to extend learning into abstract areas of thinking.

* to make it possible for the natural creativity of the learners to come into action (paraphrase Skemp, 1989).

Skemp's framework of “relational understanding” emphasizes the importance of connecting mathematical concepts and procedures to develop a deeper comprehension of mathematics (1989). In the context of circle-related concepts, this framework suggests that structured practical activities can facilitate students' construction of mental schemas, enabling them to relate the formula for the circumference of circle to underlying geometric properties of circle. The importance of these kinds of experiences has been recognized by Polya (2004, p. 113) who asserted that “before obtaining certainty, we must be satisfied with a more or less plausible guess”. These experiences give learners a greater control in directing their own learning processes as opposed to situations in which, they depend on their teachers or other knowledge authorities to tell them whether they have answered the questions correctly or what they did wrong. These guesses “can be a tool to help learner to transmission from ‘external conviction proof scheme’ to other desirable proof schemas” (Harel, 2008, p. 492). Overall, literature supports the use of structured practical activities to enhance students’ understanding of circle-related concepts. These activities provide students with opportunities to explore, measure, and reason about circle, fostering relational understanding. By incorporating Skemp's framework this study aims to contribute to the understanding of effective instructional approaches for teaching circle-related concepts in middle school mathematics education.
# STRUCTURED PRACTICAL ACTIVITIES

Previous studies have emphasized the benefits of incorporating structured practical activities in mathematics education. Such activities provide students with hands-on experiences, allowing them to actively engage with mathematical concepts and develop a deeper understanding (Lesh, 2003; Hiebert et al., 2007). The teaching of circle-related concepts, including circumference and the approximation of $\pi$, has often relied on rote learning approaches. However, recent research has advocated for more meaningful and experiential learning experiences to promote conceptual understanding (Battista & Clements, 1996; Lehrer & Chazan, 1998). By engaging students in measuring the diameter and circumference of real-world objects, they can explore the relationship between these measures and discover the role of $\pi$ in calculating circumference (Gravemeijer, 1994.)

## Relational understanding framework

Structured practical activities presented in the model of intelligent learning (Skemp, 1989), allow students to experience mathematics at various levels increasing their power of prediction and to find suitable solutions for their real-life problems. Structured practical activities are designed to help learners to do the followings:

* to build structured mathematical knowledge and to make predictable predictions.

* to take pleasure from finding their predictions confirmed by events.

* to put students in a position to immediately correct their own thinking rather than wait for a teacher to tell them where they went wrong.

* to get control in directing their own learning process.

* to provide shared sensory experiences which ensure that there is a common ground for co-operative learning by exchanging ideas and discussion to extend learning into abstract areas of thinking.

* to make it possible for the natural creativity of the learners to come into action (paraphrase Skemp, 1989).

Skemp's framework of “relational understanding” emphasizes the importance of connecting mathematical concepts and procedures to develop a deeper comprehension of mathematics (1989). In the context of circle-related concepts, this framework suggests that structured practical activities can facilitate students' construction of mental schemas, enabling them to relate the formula for the circumference of circle to underlying geometric properties of circle. The importance of these kinds of experiences has been recognized by Polya (2004, p. 113) who asserted that “before obtaining certainty, we must be satisfied with a more or less plausible guess”. These experiences give learners a greater control in directing their own learning processes as opposed to situations in which, they depend on their teachers or other knowledge authorities to tell them whether they have answered the questions correctly or what they did wrong. These guesses “can be a tool to help learner to transmission from ‘external conviction proof scheme’ to other desirable proof schemas” (Harel, 2008, p. 492). Overall, literature supports the use of structured practical activities to enhance students’ understanding of circle-related concepts. These activities provide students with opportunities to explore, measure, and reason about circle, fostering relational understanding. By incorporating Skemp's framework this study aims to contribute to the understanding of effective instructional approaches for teaching circle-related concepts in middle school mathematics education.
## THE STUDY

The purpose of the study was to investigate the effect of “structured practical activities” on Grade 8 students’ understanding of the circumference of circle in Iran. To conduct the study, a seven-step activity was designed and developed as an extra-curricular activity in a middle school for gifted students. The activity provided a problematic situation to stimulate “intellectual need” of students in a way that Harel (2013) has described as the following:

* To establish common definitions, notations and conventions, and to describe mathematical objects unambiguously categorized as “need to communication.”

* “Need to quantify and to calculate values of quantities and relations among them”, categorized as “need for computation.”

## METHODOLOGY

14 Grade 8 students voluntarily participated in the study. The study was conducted in 2016 school year and the duration of instruction was 80 minutes. The first author was the mathematics teacher of that class. He divided the 14 students into four groups of three to four members with no specific order or criteria for the formation of groups. To carry out the activity, enough number of plastic strips with different colors where distributed among the groups, and students could form them to circles of various sizes (Figure 1). The teacher/researcher acted as facilitator and guide in class, while each group was involved in doing the activity, and the data were collected through students’ work.

## DESCRIPTION OF THE ACTIVITY

The length of the circumference of a circle *C* is related to its radius *r* and diameter *d*, by the formula

$$C = 2\pi r = \pi d.$$

* The proportion of circumference of circle to its diameter is constant.

* Knowing this constant, one can calculate the circumference of any given circle when the length of its diameter is known.

The activity for finding the circumference of circle consisted of seven stages to bring about an opportunity for students to observe the relation between quantities; in this example, the circumference of circle with different diameters. This is what Harel (2013) calls “need for computation” as students go forward in various computations to ensure that the concept is internalized, organized and consequently, retainable for them.

## DATA COLLECTION

To implement the activity, a set of circles made from flexible plastic strips in various colors was used, and data was collected by utilizing these plastic rings.
## THE STUDY

The purpose of the study was to investigate the effect of “structured practical activities” on Grade 8 students’ understanding of the circumference of circle in Iran. To conduct the study, a seven-step activity was designed and developed as an extra-curricular activity in a middle school for gifted students. The activity provided a problematic situation to stimulate “intellectual need” of students in a way that Harel (2013) has described as the following:

* To establish common definitions, notations and conventions, and to describe mathematical objects unambiguously categorized as “need to communication.”

* “Need to quantify and to calculate values of quantities and relations among them”, categorized as “need for computation.”

## METHODOLOGY

14 Grade 8 students voluntarily participated in the study. The study was conducted in 2016 school year and the duration of instruction was 80 minutes. The first author was the mathematics teacher of that class. He divided the 14 students into four groups of three to four members with no specific order or criteria for the formation of groups. To carry out the activity, enough number of plastic strips with different colors where distributed among the groups, and students could form them to circles of various sizes (Figure 1). The teacher/researcher acted as facilitator and guide in class, while each group was involved in doing the activity, and the data were collected through students’ work.

## DESCRIPTION OF THE ACTIVITY

The length of the circumference of a circle *C* is related to its radius *r* and diameter *d*, by the formula

$$C = 2\pi r = \pi d.$$

* The proportion of circumference of circle to its diameter is constant.

* Knowing this constant, one can calculate the circumference of any given circle when the length of its diameter is known.

The activity for finding the circumference of circle consisted of seven stages to bring about an opportunity for students to observe the relation between quantities; in this example, the circumference of circle with different diameters. This is what Harel (2013) calls “need for computation” as students go forward in various computations to ensure that the concept is internalized, organized and consequently, retainable for them.

## DATA COLLECTION

To implement the activity, a set of circles made from flexible plastic strips in various colors was used, and data was collected by utilizing these plastic rings.
<figure><img src="image_124.png" /><figcaption>Figure 1</figcaption></figure>

The relation between the diameters of colored rings to the big black ring as referent circle, was as follows:

* The diameter of every red circle is 1/2 of the diameter of black one.

* The diameter of every green circle is 1/3 of the diameter of black one.

* The diameter of every blue circle is 1/4 of the diameter of black one.

* The diameter of every brown circle is 1/5 of the diameter of black one.

## DATA ANALYSIS

The data collected for this study was analyzed according to the seven stages designed for the activity and the findings presented accordingly.

### Stage 1

At this stage, students were encouraged to cut the biggest (black) circle and straighten it out and make it into a strip as figure 2. The circumference of the biggest circle is equal to the length of this strip. Then they should cut the green circles, straighten them out and put them besides of black strip. The purpose of this stage was to draw students’ attention to the relation between circumference of the black circle and each green circle. The expectation was that students could see the length of each green strip is half of the length of the black strip and all students realized and observed it.

<caption>Figure 2</caption>

### Stage 2

At this stage, the teacher asked students to cut the green circles, straighten them out and put them under the red strip as Figure 3. The purpose was to draw students’ attention to the relation between circumference of the black, red, and green circles. As the teacher expected, all students noticed that the length of each green strip is 1/3 of the length of the red one.
<figure><img src="image_130.png" /><figcaption>Figure 1</figcaption></figure>

The relation between the diameters of colored rings to the big black ring as referent circle, was as follows:

* The diameter of every red circle is 1/2 of the diameter of black one.

* The diameter of every green circle is 1/3 of the diameter of black one.

* The diameter of every blue circle is 1/4 of the diameter of black one.

* The diameter of every brown circle is 1/5 of the diameter of black one.

## DATA ANALYSIS

The data collected for this study was analyzed according to the seven stages designed for the activity and the findings presented accordingly.

### Stage 1

At this stage, students were encouraged to cut the biggest (black) circle and straighten it out and make it into a strip as figure 2. The circumference of the biggest circle is equal to the length of this strip. Then they should cut the green circles, straighten them out and put them besides of black strip. The purpose of this stage was to draw students’ attention to the relation between circumference of the black circle and each green circle. The expectation was that students could see the length of each green strip is half of the length of the black strip and all students realized and observed it.

<caption>Figure 2</caption>

### Stage 2

At this stage, the teacher asked students to cut the green circles, straighten them out and put them under the red strip as Figure 3. The purpose was to draw students’ attention to the relation between circumference of the black, red, and green circles. As the teacher expected, all students noticed that the length of each green strip is 1/3 of the length of the red one.
<figure><img src="image_125.png" /><figcaption>Figure 3</figcaption></figure>

Stage 3

At this stage, students were asked to answer the following question:

**Question 1:** *How many blue circles are needed to obtain strip lengths equal to black, red and green ones?*

All students successfully noticed the relation between the circumferences of colored circles with black color circle as reference circle. After wards they had to examine their guesses by cutting the blue circles, straightening them out, and putting them under the green strips as Figure 4. In this case, there was a situation for students to make predictions, examine them, and find out whether their predictions could be confirmed or not.

<figure><img src="image_126.png" /><figcaption>Figure 4</figcaption></figure>

Stage 4

At Stage4, students were expected to answer the second question:

**Question 2:** *How many brown circles are needed to obtain a strip that its length is equal to black, red, green, and brown ones?*

To answer this question, students needed to examine their guesses by cutting the brown circles, set them straight and putting them under the previous strips as in Figure 5. Similar to the previous stage, students made predictions, and investigated them to approve or reject their predictions.

<figure><img src="image_127.png" /><figcaption>Figure 5</figcaption></figure>
<figure><img src="image_131.png" /><figcaption>Figure 3</figcaption></figure>

Stage 3

At this stage, students were asked to answer the following question:

**Question 1:** *How many blue circles are needed to obtain strip lengths equal to black, red and green ones?*

All students successfully noticed the relation between the circumferences of colored circles with black color circle as reference circle. After wards they had to examine their guesses by cutting the blue circles, straightening them out, and putting them under the green strips as Figure 4. In this case, there was a situation for students to make predictions, examine them, and find out whether their predictions could be confirmed or not.

<figure><img src="image_132.png" /><figcaption>Figure 4</figcaption></figure>

Stage 4

At Stage4, students were expected to answer the second question:

**Question 2:** *How many brown circles are needed to obtain a strip that its length is equal to black, red, green, and brown ones?*

To answer this question, students needed to examine their guesses by cutting the brown circles, set them straight and putting them under the previous strips as in Figure 5. Similar to the previous stage, students made predictions, and investigated them to approve or reject their predictions.

<figure><img src="image_133.png" /><figcaption>Figure 5</figcaption></figure>
Stage 5

The goal of stage 5 was to provide an opportunity for students to explore that when the diameter of a
circle is multiplied by any natural number, its circumference will be multiplied by that number
accordingly. The visible observation was that almost all students successfully generalized the findings
from the last four stages and expressed their findings as the subsequent conjecture:

"If the diameter of circle A is multiplied by m, its circumference will be multiplied by m as well". The following formula is representing this conjecture as "if diameter of circle A = m × (diameter of circle B), then circumference of circle will be A = m × (circumference of circle B)."

Stage 6

At this stage, there was a chance to find a large group of circles as:

* Circles with diameter 1/2, 1/3, 1/4, and so on.

* Circles with diameter *m/n* multiplies of diameter of a given circle where *m/n* is a positive rational number.

By reflecting on the results gained from stage 6, students were able to see the emergence of the key
idea as:

If the circumference of only one circle is known, the circumference of any circle could be found by knowing the ratio of its diameter to the diameter of that circle.

Furthermore, the students asked interesting and deep questions including the followings:

* Is there any circle that other people (for example mathematicians) had tried to calculate its circumference?

* If there is such a circle that mathematicians calculated the circumference of it, how could we calculate it as well?

Stage 7

At this stage, students were asked to draw a line on a sheet of paper and divide it into seven equal segments that the length of each is the same as the diameter of brown circles. Then cut some brown circles, flatten them out and put them over a line as is shown in figure following:

<figure><img src="image_128.png" /><figcaption>Figure 6</figcaption></figure>

The goal was to attract the students' attention to the relation between circumference and diameter of circles. At this stage, they saw that the sum of the lengths of seven brown circles was equivalent to the sum of the lengths of the 22 brown circles.

FINDINGS

The analysis of the data showed that “structured practical activities” are suitable means to enhance
students’ understanding of mathematical concepts and specifically, the circumference of circle. The
data collected from14 Grade 8 students arranged in four groups. The analysis of the data revealed that
all four groups found the number of blue circles needed for a strip with the same length as black strip,
and they verified this number by doing a structured activity. At the fourth stage, all groups guessed
## Stage 5

The goal of stage 5 was to provide an opportunity for students to explore that when the diameter of a circle is multiplied by any natural number, its circumference will be multiplied by that number accordingly. The visible observation was that almost all students successfully generalized the findings from the last four stages and expressed their findings as the subsequent conjecture:

"If the diameter of circle A is multiplied by m, its circumference will be multiplied by m as well". The following formula is representing this conjecture as "if diameter of circle A = m × (diameter of circle B), then circumference of circle will be A = m × (circumference of circle B)."

## Stage 6

At this stage, there was a chance to find a large group of circles as:

* Circles with diameter 1/2, 1/3, 1/4, and so on.

* Circles with diameter m/n multiplies of diameter of a given circle where m/n is a positive rational number.

By reflecting on the results gained from stage 6, students were able to see the emergence of the key idea as:

*If the circumference of only one circle is known, the circumference of any circle could be found by knowing the ratio of its diameter to the diameter of that circle.*

Furthermore, the students asked interesting and deep questions including the followings:

* Is there any circle that other people (for example mathematicians) had tried to calculate its circumference?

* If there is such a circle that mathematicians calculated the circumference of it, how could we calculate it as well?

## Stage 7

At this stage, students were asked to draw a line on a sheet of paper and divide it into seven equal segments that the length of each is the same as the diameter of brown circles. Then cut some brown circles, flatten them out and put them over a line as is shown in figure following:

<figure><img src="image_134.png" /><figcaption>Figure 6</figcaption></figure>

The goal was to attract the students' attention to the relation between circumference and diameter of circles. At this stage, they saw that the sum of the lengths of seven brown circles was equivalent to the sum of the lengths of the 22 brown circles.

## FINDINGS

The analysis of the data showed that “structured practical activities” are suitable means to enhance students’ understanding of mathematical concepts and specifically, the circumference of circle. The data collected from 14 Grade 8 students arranged in four groups. The analysis of the data revealed that all four groups found the number of blue circles needed for a strip with the same length as black strip, and they verified this number by doing a structured activity. At the fourth stage, all groups guessed
the number of brown circles for constructing a strip with equal length as earlier strips, examined their guesses and verified the numbers. At the end of this stage, they could generalize their findings from previous stages and concluded that if the diameter of circle is multiplied by any number, its circumference will be multiplied with that number as well. At the end of stage 6, students were able to infer that for calculating the circumference of any circle, it is sufficient to find the circumference of a specific circle such as a circle with unit diameter. The students came up with this important outcome by reflecting on the results obtained from earlier stages. At this point, the teacher/researcher, as facilitator, challenged the students by his enquiries to become assured that it is a useful decision to give a name for the circumference of such a circle and prepared a situation to introduce number $π$ as a constant ratio of the circumference of circle to its diameter. Finally, by going through stage 7, students realized that when they folded a sheet of paper and divided it into seven equal segments, they made a strip that its approximate length was 22 times the diameter of the same circle. In fact, at the end of stage 7, they came up with a good approximation for $π$ as 22/7.

## DISCUSSION

Conventionally, students are told that the ratio of circumference of every circle to its diameter is constant. However, in this study, a structured activity was designed with seven stages and carried out with 14 volunteer Grade8th students, to make a guess regarding the relation between circumference and diameter of circle. The findings indicated that the students examined their guess and came up with a more generalized version of it at the final stage of the activity. Thus, the conclusion is that:

* Students found out that for calculating the circumference of every circle, finding the circumference of one circle is sufficient. Therefore, there is a suitable situation to inform them historically that the circle with radius 1 has been chosen to calculate its circumference and the length of circumference of this circle is represented by symbol $π$. In fact, the structured activities are potentially appropriate to arise the students’ intellectual needs for establishing common definitions, notations, and conventions to fulfil the “need to communication” in mathematics learning.

* Structured activities could create serious need for computation as the students involving in this activity did. They tried to find a way of calculating the circumference of circle with radius 1 or as Harel (2013) puts it, to fulfil the “need for computation.”

Finally, the structured activity designed for this study, created a suitable and challenging situation for Grade 8 students to acquire conceptual knowledge of the concept of $π$, construct the formula of circumference of circle, and find the relation between diameter and circumference of circle. This activity gave students a chance to enhance their conceptual understanding and enabled them to apply their knowledge “autonomously” and “spontaneously” and to “reorganize” the knowledge that they are able to construct them as required by situations created by structured activities. Harel (2013) emphasizes, these activities could create learning situations for students to meet their intellectual needs.

## References

* Battista, M. T., & Clements, D. H. (1996). Students' understanding of three-dimensional rectangular arrays of cubes. *Journal for Research in Mathematics Education* 27(3), 258-292.

* Gravemeijer, K. P. E. (1994). *Developing realistic mathematics education.*
the number of brown circles for constructing a strip with equal length as earlier strips, examined their guesses and verified the numbers. At the end of this stage, they could generalize their findings from previous stages and concluded that if the diameter of circle is multiplied by any number, its circumference will be multiplied with that number as well. At the end of stage 6, students were able to infer that for calculating the circumference of any circle, it is sufficient to find the circumference of a specific circle such as a circle with unit diameter. The students came up with this important outcome by reflecting on the results obtained from earlier stages. At this point, the teacher/researcher, as facilitator, challenged the students by his enquiries to become assured that it is a useful decision to give a name for the circumference of such a circle and prepared a situation to introduce number $π$ as a constant ratio of the circumference of circle to its diameter. Finally, by going through stage 7, students realized that when they folded a sheet of paper and divided it into seven equal segments, they made a strip that its approximate length was 22 times the diameter of the same circle. In fact, at the end of stage 7, they came up with a good approximation for $π$ as 22/7.

## DISCUSSION

Conventionally, students are told that the ratio of circumference of every circle to its diameter is constant. However, in this study, a structured activity was designed with seven stages and carried out with 14 volunteer Grade8th students, to make a guess regarding the relation between circumference and diameter of circle. The findings indicated that the students examined their guess and came up with a more generalized version of it at the final stage of the activity. Thus, the conclusion is that:

* Students found out that for calculating the circumference of every circle, finding the circumference of one circle is sufficient. Therefore, there is a suitable situation to inform them historically that the circle with radius 1 has been chosen to calculate its circumference and the length of circumference of this circle is represented by symbol $π$. In fact, the structured activities are potentially appropriate to arise the students’ intellectual needs for establishing common definitions, notations, and conventions to fulfil the “need to communication” in mathematics learning.

* Structured activities could create serious need for computation as the students involving in this activity did. They tried to find a way of calculating the circumference of circle with radius 1 or as Harel (2013) puts it, to fulfil the “need for computation.”

Finally, the structured activity designed for this study, created a suitable and challenging situation for Grade 8 students to acquire conceptual knowledge of the concept of $π$, construct the formula of circumference of circle, and find the relation between diameter and circumference of circle. This activity gave students a chance to enhance their conceptual understanding and enabled them to apply their knowledge “autonomously” and “spontaneously” and to “reorganize” the knowledge that they are able to construct them as required by situations created by structured activities. Harel (2013) emphasizes, these activities could create learning situations for students to meet their intellectual needs.

## References

Battista, M. T., & Clements, D. H. (1996). Students' understanding of three-dimensional rectangular arrays of cubes. *Journal for Research in Mathematics Education* 27(3), 258-292.

Gravemeijer, K. P. E. (1994). *Developing realistic mathematics education*.
Harel, G. (2008). DNR perspective on mathematics curriculum and instruction, Part I: focus on proving. *ZDM, 40*, 487-500.

Harel, G. (2008). DNR Perspective on mathematics curriculum and instruction, Part II: with reference to teacher's knowledge. *ZDM. 40*, 893-907.

Harel, G. (2013). Intellectual need. In *Vital directions for mathematics education research* (pp. 119-151). Springer.

Koestler, A. (2017). *The sleepwalkers: A history of man's changing vision of the universe*. Penguin.

Lehrer, R., & Chazan, D. (Eds.) (2012). *Designing learning environments for developing understanding of geometry and space*. Routledge.

Mousoulides, N., Sriraman, B., & Christou, C. (2007). From problem solving to modelling. *Education, 12*(1), 23-47.

Polya, G. (2004). *How to solve it: A new aspect of mathematical method* (Vol. 85). Princeton University Press.

Skemp, R. (1989). *Mathematics in the primary schools*. Routledge.

Skemp, R. (1986). *The psychology of mathematics learning*. Penguin Books.
Harel, G. (2008). DNR perspective on mathematics curriculum and instruction, Part I: focus on proving. *ZDM, 40*, 487-500.

Harel, G. (2008). DNR Perspective on mathematics curriculum and instruction, Part II: with reference to teacher's knowledge. *ZDM. 40*, 893-907.

Harel, G. (2013). Intellectual need. In *Vital directions for mathematics education research* (pp. 119-151). Springer.

Koestler, A. (2017). *The sleepwalkers: A history of man's changing vision of the universe*. Penguin.

Lehrer, R., & Chazan, D. (Eds.) (2012). *Designing learning environments for developing understanding of geometry and space*. Routledge.

Mousoulides, N., Sriraman, B., & Christou, C. (2007). From problem solving to modelling. *Education, 12*(1), 23-47.

Polya, G. (2004). *How to solve it: A new aspect of mathematical method* (Vol. 85). Princeton University Press.

Skemp, R. (1989). *Mathematics in the primary schools*. Routledge.

Skemp, R. (1986). *The psychology of mathematics learning*. Penguin Books.
# THE DEVELOPMENT OF SEQUENTIAL AND DISCURSIVE APPREHENSION IN KINDERGARTEN STUDENTS WHEN THEY BUILD POLYGONS

Melania Bernabeu, Àngela Buforn, Salvador Castillo

University of Alicante

To develop the geometrical thinking is important to carry out relevant interactive learning tasks that connect the representation of the figures (perceptual) with the geometrical concept (conceptual). Two aspects to develop the geometrical thinking are the sequential and discursive apprehensions. A good practice to develop these apprehensions are building tasks with didactic material. The aim of this study is to generate information about the sequential and discursive apprehension of kindergarten students when they build polygons with manipulative material. The participants were 17 kindergarten students of 3-year-olds that performed a task of build polygons with manipulative material after an instruction designed ad hoc. The results obtained show that students began to develop the dimensional deconstruction after this instruction. That is, students began to be able to make mathematical sense of the built figures through both apprehensions, anticipating and building figures (sequential apprehensions) and justifying the properties of built figures according to the geometric concepts (discursive apprehension).

## INTRODUCTION

The geometrical thinking is an important aspect of the mathematical development in children (Clements, 2004). This type of thinking is developed before children enter school, and when they enter, in some cases, is not explored in appropriate ways (Jatisunda et al., 2021). Most of the tasks in kindergarten are related to circle or name certain figures, which are usually prototypical figures (the prototypical figures are the representations of a concept most commonly used, e.g. a square in prototypical position as an example of quadrilateral), without using the attributes that define them (Bernabeu, 2022; Clements et al., 1999). Due to the few examples of geometric figures shown in class (Verdine et al., 2016), the low cognitive level tasks (Bernabeu, 2022; Clements et al., 1999) and the lack of knowledge of many teachers or pre-service teachers about geometry (Bernabeu et al., 2019), many students have a limited concept image. The concept image (Tall & Vinner, 1981) is considered as the overall cognitive structure associated with a concept, which includes the mental representations and associated properties. So, the concept image can be limited or rigid including only certain prototypical examples and examples that contradict the definition (Maier & Benz, 2020). But the concept image' cognitive structure is dynamic, being constructed and changing over time through the different experiences that children encounter (Bernabeu, 2022). For this reason, the teachers recognize the need for relevant interactive learning tasks that connect the representation of the figures with the geometrical concept (Kinzer et al., 2016), that is, tasks that develop the figural concept (symbiosis between the perceptual and the conceptual, (Fischbein, 1993)). A good practice to develop figural concept, and amplify the concept image, of children is through building tasks. For building, didactic material offers a wide combination of geometrical figures' representations (not only
# THE DEVELOPMENT OF SEQUENTIAL AND DISCURSIVE APPREHENSION IN KINDERGARTEN STUDENTS WHEN THEY BUILD POLYGONS

Melania Bernabeu, Àngela Buforn, Salvador Castillo

University of Alicante

To develop the geometrical thinking is important to carry out relevant interactive learning tasks that connect the representation of the figures (perceptual) with the geometrical concept (conceptual). Two aspects to develop the geometrical thinking are the sequential and discursive apprehensions. A good practice to develop these apprehensions are building tasks with didactic material. The aim of this study is to generate information about the sequential and discursive apprehension of kindergarten students when they build polygons with manipulative material. The participants were 17 kindergarten students of 3-year-olds that performed a task of build polygons with manipulative material after an instruction designed ad hoc. The results obtained show that students began to develop the dimensional deconstruction after this instruction. That is, students began to be able to make mathematical sense of the built figures through both apprehensions, anticipating and building figures (sequential apprehensions) and justifying the properties of built figures according to the geometric concepts (discursive apprehension).

## INTRODUCTION

The geometrical thinking is an important aspect of the mathematical development in children (Clements, 2004). This type of thinking is developed before children enter school, and when they enter, in some cases, is not explored in appropriate ways (Jatisunda et al., 2021). Most of the tasks in kindergarten are related to circle or name certain figures, which are usually prototypical figures (the prototypical figures are the representations of a concept most commonly used, e.g. a square in prototypical position as an example of quadrilateral), without using the attributes that define them (Bernabeu, 2022; Clements et al., 1999). Due to the few examples of geometric figures shown in class (Verdine et al., 2016), the low cognitive level tasks (Bernabeu, 2022; Clements et al., 1999) and the lack of knowledge of many teachers or pre-service teachers about geometry (Bernabeu et al., 2019), many students have a limited concept image. The concept image (Tall & Vinner, 1981) is considered as the overall cognitive structure associated with a concept, which includes the mental representations and associated properties. So, the concept image can be limited or rigid including only certain prototypical examples and examples that contradict the definition (Maier & Benz, 2020). But the concept image' cognitive structure is dynamic, being constructed and changing over time through the different experiences that children encounter (Bernabeu, 2022). For this reason, the teachers recognize the need for relevant interactive learning tasks that connect the representation of the figures with the geometrical concept (Kinzer et al., 2016), that is, tasks that develop the figural concept (symbiosis between the perceptual and the conceptual, (Fischbein, 1993)). A good practice to develop figural concept, and amplify the concept image, of children is through building tasks. For building, didactic material offers a wide combination of geometrical figures' representations (not only
prototypical examples), which is very important in geometry teaching and learning (Herbst et al.,
2017). However, most of the research developed about kindergarten' geometry is based on the
recognition of geometrical figures (Clements et al., 1999; Maier & Benz, 2020; Verdine et al., 2016),
building of 3D shapes, normally with blocks (Casey et al., 2008; Kinzer et al., 2016), and few ones
are based on characterizing the ability of pre-school students to build geometric figures (2D) with
manipulative materials. For this reason, in this study, we will characterize the way kindergarten
students build polygons with manipulative material as an ability to connect the representation of the
figures (perceptual) with the geometrical concept (conceptual).

FRAMEWORK

To build a geometric figure with manipulative material it is necessary to give mathematical meaning to the pieces that constitute the material, assuming that these are the parts of the figure to be represented. This manner of seeing the material in a mathematical way by making use of geometric concepts to give mathematical meaning to their parts is what is meant by *dimensional deconstruction* (Duval, 2017). The dimensional deconstruction can be evidenced through the *sequential apprehension* which is the ability to construct a configuration or to describe its construction (Duval, 1995). Thus, for example, in tasks of building geometric figures with manipulative material, students need to make effective use of the material anticipating the parts that compose the figure to be obtained. That is, students need to anticipate the deconstruction of the 2D configuration to be constructed into 1D figural units (Duval, 2017). For example, to build a quadrilateral (2D configuration), a student must anticipate that four segments joined (1D figural units) are needed. In this sense, students need to endow mathematical meaning to the material, for example, the sticks of Meccano. Therefore, the student will consider the Meccano sticks as if they were sides of the quadrilateral (1D figural units), so the student will have to join the sticks to build a four-sided closed plane figure. As sequential apprehension is also describing the building process, the child can anticipate through the words what material is needed to build a certain figure. For example, to make a triangle "I need three Meccano sticks". In this way, dimensional deconstruction is also evident through the student's discourse, as the children give mathematical meaning to the parts that will compose the figure to be represented.

Furthermore, when a child is asked what a partner has built, providing the properties of the building, the dimensional deconstruction is evidenced through *discursive apprehension*, which is the ability to relate the configuration with mathematical definitions or properties to solve a geometric task (Duval, 1995). Thus, for example, a child, when faced with the building of a triangle with Meccano, may justify that it is a triangle because it has three sides or three sticks.

Considering these theoretical references, we aim to generate information about the manifestation of
dimensional deconstruction through sequential and discursive apprehension when kindergarten
students build polygons with manipulative material, promoting the expansion of the concept image
and the development of the figural concept of the students.

METHOD

Participants and context

The participants in this study were 17 kindergarten students (3-year-olds). In relation to the curricular context, the geometry expectations of the NCTM (2000) to grades Pre-K-2 (from prekindergarten to
prototypical examples), which is very important in geometry teaching and learning (Herbst et al.,
2017). However, most of the research developed about kindergarten' geometry is based on the
recognition of geometrical figures (Clements et al., 1999; Maier & Benz, 2020; Verdine et al., 2016),
building of 3D shapes, normally with blocks (Casey et al., 2008; Kinzer et al., 2016), and few ones
are based on characterizing the ability of pre-school students to build geometric figures (2D) with
manipulative materials. For this reason, in this study, we will characterize the way kindergarten
students build polygons with manipulative material as an ability to connect the representation of the
figures (perceptual) with the geometrical concept (conceptual).

FRAMEWORK

To build a geometric figure with manipulative material it is necessary to give mathematical meaning to the pieces that constitute the material, assuming that these are the parts of the figure to be represented. This manner of seeing the material in a mathematical way by making use of geometric concepts to give mathematical meaning to their parts is what is meant by *dimensional deconstruction* (Duval, 2017). The dimensional deconstruction can be evidenced through the *sequential apprehension* which is the ability to construct a configuration or to describe its construction (Duval, 1995). Thus, for example, in tasks of building geometric figures with manipulative material, students need to make effective use of the material anticipating the parts that compose the figure to be obtained. That is, students need to anticipate the deconstruction of the 2D configuration to be constructed into 1D figural units (Duval, 2017). For example, to build a quadrilateral (2D configuration), a student must anticipate that four segments joined (1D figural units) are needed. In this sense, students need to endow mathematical meaning to the material, for example, the sticks of Meccano. Therefore, the student will consider the Meccano sticks as if they were sides of the quadrilateral (1D figural units), so the student will have to join the sticks to build a four-sided closed plane figure. As sequential apprehension is also describing the building process, the child can anticipate through the words what material is needed to build a certain figure. For example, to make a triangle "I need three Meccano sticks". In this way, dimensional deconstruction is also evident through the student's discourse, as the children give mathematical meaning to the parts that will compose the figure to be represented.

Furthermore, when a child is asked what a partner has built, providing the properties of the building, the dimensional deconstruction is evidenced through *discursive apprehension*, which is the ability to relate the configuration with mathematical definitions or properties to solve a geometric task (Duval, 1995). Thus, for example, a child, when faced with the building of a triangle with Meccano, may justify that it is a triangle because it has three sides or three sticks.

Considering these theoretical references, we aim to generate information about the manifestation of
dimensional deconstruction through sequential and discursive apprehension when kindergarten
students build polygons with manipulative material, promoting the expansion of the concept image
and the development of the figural concept of the students.

METHOD

Participants and context

The participants in this study were 17 kindergarten students (3-year-olds). In relation to the curricular context, the geometry expectations of the NCTM (2000) to grades Pre-K-2 (from prekindergarten to
grade 2), all students should (i) [...] build [...] two-dimensional shapes and (ii) describe attributes and parts of two-dimensional shapes. Therefore, in this research, we will focus on making sense of the concept of the polygon (a closed plane geometric figure with straight, non-crossed sides) and the class of polygon according to the number of sides by building figures with manipulative material and describe the attributes of built geometric figures. The students who participated in this research had not received any instruction on geometry during the 2021-2022 school year except for what they knew from their own extracurricular experience.

## Instruction and data collection instrument

The instruction was carried out by a member of the research group. The instruction consisted of 4 sessions to address the polygon concept and polygons according to the number of sides, and a final test through individual interviews. In this study we will focus on session 3 (polygons according to the number of sides) (instruction), to explain how the concept of this class of polygon was constructed, and on the task of building from the final test (data collection instrument), which will show our results.

In session 3, once the polygon concept had been constructed (sessions 1 and 2), we carried out tasks to know the classes of polygons according to the number of sides, specifically, triangles, quadrilaterals, and pentagons. For this, we performed a task to recognize differences between different pairs of cards containing polygons with different numbers of sides (Task 1 of Figure 1). In this task, students had to recognize that, for example, one was a polygon with three sides and the other with four sides. Following this task, we performed another task in which we presented a set of polygons with a common attribute, e.g., having four sides, and asked the students what the showed polygons had in common (Task 2 of Figure 1). Once they had identified the common attribute (having four sides), the teacher-researcher would say what term the four-sided polygons were called, i.e., quadrilaterals. The same applies to triangles and pentagons. After this, a task of classifying polygons according to the number of sides was carried out (Task 3 of Figure 1). Lastly, a task of building polygons according to the number of sides, using the manipulative material Meccano, was performed (Task 4 of Figure 1). First, the students were left free to build their polygons (sequential apprehension) and if they had not built any of the polygons learnt in class, they were instructed to build a pentagon, for example. After the building of each polygon, another classmate was asked which polygon had been built and why (discursive apprehension).

<table><caption>Figure 1: Tasks of the session 3</caption><tr><td>Task 1: recognize the differences (e.g., three sides vs four sides)</td><td>Task 2: recognize the common attribute (e.g., polygons of four sides)</td></tr><tr><td></td><td></td></tr><tr><td>Task 3: classify polygon according to number of sides</td><td>Task 4: build a polygon with Meccano (e.g., pentagon)</td></tr></table>
grade 2), all students should (i) [...] build [...] two-dimensional shapes and (ii) describe attributes and parts of two-dimensional shapes. Therefore, in this research, we will focus on making sense of the concept of the polygon (a closed plane geometric figure with straight, non-crossed sides) and the class of polygon according to the number of sides by building figures with manipulative material and describe the attributes of built geometric figures. The students who participated in this research had not received any instruction on geometry during the 2021-2022 school year except for what they knew from their own extracurricular experience.

## Instruction and data collection instrument

The instruction was carried out by a member of the research group. The instruction consisted of 4 sessions to address the polygon concept and polygons according to the number of sides, and a final test through individual interviews. In this study we will focus on session 3 (polygons according to the number of sides) (instruction), to explain how the concept of this class of polygon was constructed, and on the task of building from the final test (data collection instrument), which will show our results.

In session 3, once the polygon concept had been constructed (sessions 1 and 2), we carried out tasks to know the classes of polygons according to the number of sides, specifically, triangles, quadrilaterals, and pentagons. For this, we performed a task to recognize differences between different pairs of cards containing polygons with different numbers of sides (Task 1 of Figure 1). In this task, students had to recognize that, for example, one was a polygon with three sides and the other with four sides. Following this task, we performed another task in which we presented a set of polygons with a common attribute, e.g., having four sides, and asked the students what the showed polygons had in common (Task 2 of Figure 1). Once they had identified the common attribute (having four sides), the teacher-researcher would say what term the four-sided polygons were called, i.e., quadrilaterals. The same applies to triangles and pentagons. After this, a task of classifying polygons according to the number of sides was carried out (Task 3 of Figure 1). Lastly, a task of building polygons according to the number of sides, using the manipulative material Meccano, was performed (Task 4 of Figure 1). First, the students were left free to build their polygons (sequential apprehension) and if they had not built any of the polygons learnt in class, they were instructed to build a pentagon, for example. After the building of each polygon, another classmate was asked which polygon had been built and why (discursive apprehension).

<figure><img src="image_135.png" /><figcaption>Figure 1: Tasks of the session 3</figcaption></figure>
In the interview of the final questionnaire, one of the tasks was focused on sequential and discursive apprehension. Specifically, the task asked: "Build the following figures with Meccano: (a) polygon, (b) triangle, and (c) pentagon". In the interview, the teacher-researcher asked the students what they needed to build these polygons and the students anticipated the Meccano sticks they would need (whether the students anticipated through oral discourse what material they needed or whether they took the necessary material to build the requested polygon) (sequential apprehension). Then, the students built the demanded polygon (sequential apprehension) and finally, the teacher-researcher asked students by the attributes of the built polygons (discursive apprehension).

## Analysis

The segments of the videos from the building task of the final interview were viewed by the members of the research group. For this, it was analyzed for each of the students whether they had solved the different commands of the task correctly. That is, if the student (i) had built the geometric figure requested, (ii) had anticipated what material they needed (sequential apprehension), (iii) built the figure requested (sequential apprehension) and (iv) justified the attributes of the figure they had built (discursive apprehension). At the end of the heading line, we added a column to note some observations about the students' way of solving the task. In the case of building a "polygon", we considered as non-polygons the constructions that remained open or had crossed sides (since in class it was reinforced that to be a polygon, the sticks of the Meccano had to be the ends of the sticks together and not cut each other), which we will consider as almost correct. In addition, in the building of the polygon, if they built a triangle, they were asked to build it again after the pentagon.

Once the fragment of a student's interview was viewed, the annotations of the members of the research group were discussed and if there were discrepancies, the video was watched again to reach an agreement. In this way, we obtained the characteristics of geometric thinking that students showed when building polygons with manipulative material.

## RESULTS

The results are divided into two sections. First, we will show the ability of students to build with manipulative material examples of polygons, pentagon, and triangle. Second, we will show the evidence of the sequential and discursive apprehension when students built triangles and pentagons.

### The success levels when building polygons

The Table 1 shows the frequency of correct, almost correct, and incorrect buildings of polygon, triangles and pentagons made by children.

<table><caption>Table 1: Frequency of correct, almost correct, and incorrect of the buildings</caption><thead><tr><th>n=17</th><th>Polygon</th><th>Pentagon</th><th>Triangle</th></tr></thead><tbody><tr><th>Correct</th><td>6</td><td>4</td><td>9</td></tr><tr><th>Almost correct</th><td>7</td><td>0</td><td>2</td></tr><tr><th>Incorrect</th><td>4</td><td>13</td><td>6</td></tr></tbody></table>

According to polygons, 6 students built a polygon correctly, joining and avoiding the Meccano's sticks being crossed or cutting each other. From these 6 students, 4 built triangles, of which three of
In the interview of the final questionnaire, one of the tasks was focused on sequential and discursive apprehension. Specifically, the task asked: "Build the following figures with Meccano: (a) polygon, (b) triangle, and (c) pentagon". In the interview, the teacher-researcher asked the students what they needed to build these polygons and the students anticipated the Meccano sticks they would need (whether the students anticipated through oral discourse what material they needed or whether they took the necessary material to build the requested polygon) (sequential apprehension). Then, the students built the demanded polygon (sequential apprehension) and finally, the teacher-researcher asked students by the attributes of the built polygons (discursive apprehension).

## Analysis

The segments of the videos from the building task of the final interview were viewed by the members of the research group. For this, it was analyzed for each of the students whether they had solved the different commands of the task correctly. That is, if the student (i) had built the geometric figure requested, (ii) had anticipated what material they needed (sequential apprehension), (iii) built the figure requested (sequential apprehension) and (iv) justified the attributes of the figure they had built (discursive apprehension). At the end of the heading line, we added a column to note some observations about the students' way of solving the task. In the case of building a "polygon", we considered as non-polygons the constructions that remained open or had crossed sides (since in class it was reinforced that to be a polygon, the sticks of the Meccano had to be the ends of the sticks together and not cut each other), which we will consider as almost correct. In addition, in the building of the polygon, if they built a triangle, they were asked to build it again after the pentagon.

Once the fragment of a student's interview was viewed, the annotations of the members of the research group were discussed and if there were discrepancies, the video was watched again to reach an agreement. In this way, we obtained the characteristics of geometric thinking that students showed when building polygons with manipulative material.

## RESULTS

The results are divided into two sections. First, we will show the ability of students to build with manipulative material examples of polygons, pentagon, and triangle. Second, we will show the evidence of the sequential and discursive apprehension when students built triangles and pentagons.

### The success levels when building polygons

The Table 1 shows the frequency of correct, almost correct, and incorrect buildings of polygon, triangles and pentagons made by children.

<table><caption>Table 1: Frequency of correct, almost correct, and incorrect of the buildings</caption><thead><tr><th>n=17</th><th>Polygon</th><th>Pentagon</th><th>Triangle</th></tr></thead><tbody><tr><th>Correct</th><td>6</td><td>4</td><td>9</td></tr><tr><th>Almost correct</th><td>7</td><td>0</td><td>2</td></tr><tr><th>Incorrect</th><td>4</td><td>13</td><td>6</td></tr></tbody></table>

According to polygons, 6 students built a polygon correctly, joining and avoiding the Meccano's sticks being crossed or cutting each other. From these 6 students, 4 built triangles, of which three of