Content area
This paper outlines the design and application of a didactic sequence aimed at facilitating primary students’ understanding of axial symmetry, utilizing a combination of digital artefacts and paper tasks. We wondered to what extent the designed didactic sequence is able to make primary school students formulate and validate effective programming strategies to construct symmetrical images with respect to an axis and identify the key properties of axial symmetry. Data analysis from a study carried out with fifth-grade students shows a link between the evolution of students’ programming strategies and the construction of mathematical knowledge related to the definition of axial symmetry. The digital artefact and the paper tasks were effective in bringing out programming strategies and some of the key properties of axial symmetry. However, the designed didactic sequence was not enough to allow students to identify all properties related to axial symmetry, and a subsequent intervention by the teacher was necessary. The results of the experimentation led us to expand the paper tasks with additional questions for students.
Introduction
In teaching and learning of geometry, the concept of symmetry is nowadays one of the most relevant in many countries and for all school levels (for example MIUR, 2018; NCTM, 2001). It has deep interdisciplinary links (Ellis-Davies, 1986; Leikin et al., 2000b; Weyl, 1952) and plays an extremely important role in problem-solving (Leikin et al., 2000a). Among the symmetries of the plane, axial symmetries, or reflections, play an important role. These are geometric transformations that fix the points of the axis of symmetry and map each point not on the axis to another point, in such a way that the segment joining them is perpendicular to the axis and the two points are equidistant from the axis. Axial symmetry involves a mirror-like reflection across an axis where points and their images are equidistant from and perpendicular to the axis. Because of its complexity, teaching axial symmetry presents unique challenges, addressing which requires innovative educational strategies. In teaching and learning of axial symmetries, students are often asked to identify which shapes (stars, hearts, etc.) are symmetrical. The aspect linked to the idea of transforming points into others or figures into others is often overlooked (Ng & Sinclair, 2015). According to Jagoda and Swoboda (2011), it is important to provide students with appropriate tasks that allow them to manipulate and to experiment, so that they can focus both on the action of geometric transformation and on the result of the transformation. In addition, students often receive ready-made content (definitions, theorems, proofs, and so on) that they simply have to assimilate and apply. Instead, it would be appropriate to let students actively work to construct and reconstruct a significant knowledge of geometric concepts, by involving them in the definition process too. However, difficulties may appear in students with respect to the mathematical definition of axial symmetry (Dello Grenier, 1985; Iacono & Ferrara Dentice, 2020; Swan, 2001; Zaslavsky, 1994). For instance, students often confuse the symmetry axis in irregular shapes, or struggle to visualize the axis when it does not align with familiar horizontal or vertical orientations. Moreover, students may have difficulty in identifying or in drawing the symmetry axis of a given figure, especially when the symmetry axis is slanted (Grenier, 1985; Hoyles & Healy, 1997; Jagoda, 2008; Küchemann, 1981; Leikin et al., 2000b).
To adequately address these difficulties related to teaching and learning symmetry, an important aspect must be considered. Active students’ involvement can be made explicit through visualization, manipulation, or through the use of software (for example, Sinclair et al., 2016). Baccaglini-Frank et al. (2018) describe a series of interactive and manipulative activities on linear symmetry for first- and second-grade classes: coloured circles arranged symmetrically on two sides of a line and the line itself can be dragged continuously on the screen while preserving symmetry. Faggiano et al. (2018) combined the potential of two different types of artefacts—a manipulatable artefact (paper and pins) and a digital artefact (Dynamic Geometry Environment)—in developing the concept of axial symmetry and its properties with fourth-grade students. The potential synergy between the combined use of these two artefacts is discussed by Mariotti and Montone (2020) through examples of experiments carried out at different school levels.
Computer programming could also be an effective tool to actively engage primary school students in building knowledge about symmetries and associating symmetry with movement. The educational use of computer programming is consistent with the recent directives (the Digital Education Action Plan) of the European Commission (n.d.) which promotes the introduction of the “computational thinking, coding and the creative and critical use of digital technologies” as early as the first grades.
This study is part of a wider research (Crisci et al., 2022, 2023) addressing difficulties related to symmetry in primary school using algorithmic and computer programming approaches. In this context, we designed a didactic sequence with the aim of supporting students in constructing mathematical knowledge related to axial symmetry. In particular, students should be able to identify key properties of an axial symmetry, such as the equidistance of corresponding points from the axis of symmetry, the perpendicularity between the axis and the segment connecting these points, the fact that the midpoint of this segment is on the axis and that the points on the axis are transformed into themselves. Our didactic sequence foresees that students work with slanted axes right away to make them approach a definition of axial symmetry in as general a condition as possible. Specifically, we designed a milieu (Brousseau, 1986), comprising both digital and tangible elements, that engages fifth-grade students in a dynamic learning process, where they actively manipulate geometric concepts.
The paper is organized as follows. Firstly, we describe the conceptual framework underlying our study. The “Method” section is divided into subsections, in which the design of the milieu, the participants in the study and the setting, the data collected and the analysis criteria are described. In the “Results” section the results of the study are presented. Finally, the paper concludes with the “Discussion and Conclusions” section.
Conceptual Background
In this section, we first report some research concerning the teaching/learning of symmetry in primary school in different countries around the world. Then, we focus on research concerning the use of computer programming and, in particular, visual programming (Scratch) for constructing mathematical and geometric concepts with primary school students. Next, we dwell on some elements of the Theory of Didactic Situations (TDS). Finally, we enunciate the principles that inspired the design of our didactic sequence and the research questions.
Symmetry in Primary School Around the World
The 1998 ICMI study highlights that geometry became an explicit topic in primary school only in the 1960s (Mammana & Villani, 1998). Almost all countries, including Great Britain, China, France, Germany, and Italy, based their primary school geometry curriculum on the study of two-dimensional shapes, such as circles and polygons, despite undisputedly important educators such as Froebel, Montessori, Pestalozzi, Steiner, Boole and Somervell built teaching programs on various aspects of geometry, including the geometry of transformations and, in particular, the symmetries (Sinclair & Bruce, 2015). Ng and Sinclair (2015) investigate primary school children’s learning of symmetries in a dynamic geometry environment and note that in the Western and Northern Canadian Protocol, symmetry is formally introduced in grade 4 (ages 9–10), where students are asked to identify and create symmetrical figures as well as draw one or more axes of symmetry. Xistouri and Pitta-Pantazi (2006) address the concept of symmetry in primary school in Cyprus, analyzing the performances of students in reflective symmetry tasks. Bartolini Bussi and Baccaglini-Frank (2015) use the Bee-Bot technology to mediate knowledge of geometric concepts through the symmetries of the movements of the robot in the Italian primary school.
Sinclair and Bruce (2015, p. 320) highlight the themes that best represent the current trends in the teaching and learning of geometry in the primary school curriculum. They include “the importance of transformational geometry in the curriculum (including symmetry as well as the isometries)” and “the affordances of digital technologies in geometric and spatial reasoning”.
Computer Programming for Constructing Mathematical and Geometric Knowledge
Several research projects in educational sciences have used programming since the 1980s. The seminal project in this sense is the LOGO programming language, conceived by Papert (1980). With the emergence of the LOGO language, research has focused on the potential of programming to foster mathematical learning through the writing of algorithms (Noss & Hoyles, 1996). Using the LOGO language, children have the opportunity to do mathematics in a meaningful way by linking concrete experiences with abstract mathematical concepts (Clements & Sarama, 1997). In this regard, LOGO was used by several authors, such as Bideault (1985), Salem (1988) and Clements et al. (2001) for learning geometry in primary school. More recently, new programming languages have been developed; these are more intuitive and thus accessible to learners of all ages without any advanced computer skills. As aforementioned, Bartolini Bussi and Baccaglini-Frank (2015) used the programmable robot Bee-Bot in order to mediate knowledge of geometric concepts from the early years of primary school.
Another example is the recent French project EXPIRE (Université Grenoble Alpes, n.d.), which aims to design teaching activities on specific mathematical concepts based on the visual programming environment Scratch (Scratch, n.d.) in primary school. Unlike textual programming, visual programming does not require knowledge of the syntax of a programming language. Users can intuitively create algorithms/programs (in the case of visual programming the two concepts are joined) by manipulating blocks of instructions, that is, by dragging and dropping them and encapsulating them in ordered sequences. Several authors (Benton et al., 2017, 2018; Crisci, 2020; Förster et al., 2018; Laurent et al., 2022; Zhang & Nouri, 2019) used Scratch for teaching and learning mathematical knowledge in primary school. Through the Scratch instructions, it is possible to make characters perform actions (e.g. moving and turning around) on a scene. By using Scratch, students can create stories and animations (Resnick et al., 2009) by making objects or characters move across a virtual scene by dragging visual components assembled as blocks (Maloney et al., 2010). Therefore, the action that students intend to perform necessarily occurs through programming. Students are thus forced to predict the outcome of their actions and to attribute meaning to it, and this can foster a metacognitive attitude in them (Rodríguez-Martínez et al., 2020). For these reasons, Scratch can promote the development of reasoning and problem-solving skills (Calao et al., 2015). Furthermore, since it allows to easily create lines and geometric shapes (Iskrenovic-Momcilovic, 2020), it can also be used effectively for learning geometry. Indeed, the geometric construction description is an algorithm that can be translated into a computer language by students (Schmidt-Thieme, 2009). Several articles deal with the use of Scratch in teaching and learning geometry, and in developing mathematical and geometrical skills. Dickson et al. (2022) use Scratch to improve students’ spatial reasoning skills. Ng et al. (2023) describe a geometry task based on programming with Scratch designed to facilitate students’ construction of the meaning of rotational symmetry. Iskrenovic-Momcilovic (2020) shows statistically significant differences in the learning of geometric shapes by third-grade students who studied conventionally compared to those who used a program implemented in Scratch. Students who used the program in Scratch performed 13% better than students who used conventional methods. Förster notes how Scratch is an excellent tool to enable students to accomplish all the geometric steps required to construct congruent triangles (Förster, 2015). Moreover, Förster describes an activity requiring programming with Scratch concerning polygons and tessellations (Förster, 2016). Gökdağ et al. (2023) also investigate tessellations through an activity with Scratch trying to understand students’ appreciation of aesthetics.
Chaachoua et al. (2018) identify key features for designing of effective tasks for learning mathematics using the visual programming environment Scratch: students are asked to create a computer program that allows a character to obtain a desired behavior on a scene, as required by the task; the blocks of instructions are designed to evoke mathematical meanings; the computer program running allows students to visualize the proposed procedure, and to recall concepts or properties useful for the construction of mathematical knowledge; the use of a visual programming language such as Scratch (which allows the implementation of ad-hoc blocks of instructions that hide more complex aspects of programming from students), does not require students to have advanced computer skills, and they can use programming as a tool for doing mathematics.
A milieu based on algorithms and computer programming could allow students to construct new mathematical knowledge. Algorithmics and computer programming are suitable for the development of three fundamental skills of mathematical thinking: decomposing a problem into easier sub-problems, reflecting on a task in terms of steps and actions, and describing problems and solutions at different levels of abstraction (Wing, 2006). Referring to axial symmetry, the request to implement symmetrical movements of a character with respect to an axis through programming could lead the student to decompose the problem into subproblems in search of effective programming strategies. Reflection on the task step by step and actions taken during the programming phase (e.g. choosing programming commands to implement movements of the character perpendicular to the axis to reach points equidistant from the axis) could lead her to reach different levels of abstraction related to the conceptualization of axial symmetry. Two different approaches can be adopted for promoting the introduction of algorithmics and computer programming from early school: (i) through an approach to computer programming within the specific domains of technology or computer science; (ii) through computer programming activities aimed at constructing knowledge related to a different domain (Tchounikine, 2016). The latter approach is a current subject of interest in the mathematics education research community, and it is the one we choose to adopt in this work.
Elements of TDS
The word milieu refers to everything that acts on the student or everything the student acts on. It may consist of physical, cultural, social, or human objects. Students interact with the milieu to modify their knowledge and accept temporary responsibility for their learning by actively engaging in the cognitive activity required by the situation. The milieu is a fundamental component of the Theory of Didactical Situations (TDS) of Brousseau (1986).
Brousseau distinguishes four intertwining types of situations, not always neatly: action, formulation, validation, and institutionalization. During the action situation, students act on the milieu, individually or in a group. The milieu provides feedback, allowing for empirical validation of the action by students. In the formulation situation, students make their strategies explicit (through oral and/or written productions) producing a description or representation of their actions. Moving on to the validation situation, students produce evidence to convince themselves and others (peers, teachers) of the validity and relevance of their proposed strategy. In the institutionalization situation, the teacher decontextualizes the knowledge acquired by students, making it available and reusable in different contexts.
A didactic situation may present a-didactic moments when students are left free to interact with the milieu and no explicit intervention by the teacher appears. An a-didactic situation is a situation in which students independently tackle a mathematical problem and the teacher lets them act even though she is present. The teacher does not intervene from the moment the students begin to address the problem until the moment they produce their answers (Brousseau, 2000).
The learner knows that the problem has been chosen to make her acquire new knowledge, but she also has to know that this knowledge is entirely justified by the internal logic of the situation and that she can construct it without appealing to didactic reasons. (Brousseau, 2000, p. 5, our translation).
Knowledge can be characterized by one or more a-didactic situations organized with didactic aims. The teacher should arrange such a-didactic situations so that the student is able to solve them. Such a-didactic situations determine the knowledge taught.
Design Principles and Research Questions
Our work fits into the context described so far. In particular, it fits into the most recent trends related to teaching and learning geometry highlighted by Sinclair and Bruce (2015) and into literature concerning computer programming for constructing geometric concepts. We designed a didactic sequence for primary school students with the aim that they could identify characteristic properties of axial symmetries. Our didactic sequence requires that students interact with a milieu consisting of the following:
A digital artefact, based on algorithmic and visual programming in the Scratch environment
Tasks on sheets of paper (in the following named paper tasks), which students have to carry out after dealing with the digital artefact.
Starting from the above, we formulated the following principles:
P1: According to Chaachoua et al. (2018), suitable educational tasks in visual programming environment that
Require students to create a computer program (without necessarily having advanced computer skills) through the manipulation of blocks of instructions that evoke mathematical meanings
Allow students to visualize the execution of the implemented procedure, can foster the construction of mathematical concepts and properties, and mathematical conceptualization.
P2: According to Brousseau (1986), a milieu could allow students, receiving feedback from it, to formulate and validate hypotheses, to evolve their strategies, and to autonomously construct their mathematical knowledge.
We designed our didactic sequence, and in particular, the digital artefact that constitutes a fundamental part of it, according to principles P1 and P2 with the aim of answering the following research question:
(RQ1) To what extent is the designed didactic sequence able to make primary school students formulate and validate effective programming strategies to construct symmetrical images with respect to an axis?
(RQ2) To what extent is the designed didactic sequence able to make primary school students identify the key properties of axial symmetry?
Method
As a methodological framework, we refer to Didactic Engineering (Artigue, 1988, 2008), which has an experimental scheme based on the conception, implementation, observation, and analysis of a didactic sequence. It consists of four phases. Firstly, there is a preliminary analysis which includes an analysis of the teaching content, an analysis of the habitual teaching and its effects (see previous sections). Secondly, there is the a priori analysis of the didactic sequence, in which researchers imagine some of all possible student behaviors (see the “The Design of the Milieu” section). The third phase is the classroom experimentation. Finally, there is the a posteriori analysis and evaluation, in which the researcher finds the distance between what was expected from the experimentation and what actually happened (see the “Results” section).
In the following, we describe the design of the milieu, the participants and the experiment setting, how data were collected, and the criteria used to analyze them.
The Design of the Milieu
Students interact with a milieu, which consists of two parts: a digital artefact and some paper tasks. Such interaction involves a sequence of a-didactic situations. Students are divided into pairs. First, they act on the digital artefact individually (each at her own pc) and communicate with her partner to formulate and validate orally the strategies that guided their action. Student pairs act on paper tasks after finishing working on the digital artefact.
The Digital Artefact
The digital artefact is implemented with the visual programming environment Scratch. Students, through visual programming, have to move 3 “sprites” (in the following named characters), Piero, Isabella and Giada, so that they can create a choreography for a dance show, in a symmetrical way with respect to a line. The request to create symmetrical choreography stems from the goal of tying the didactic situation to a real situation. Indeed, dancers often perform “symmetrical” movements in their choreographies. Therefore, this is a request linked to an authentic and not to an artificial situation. The Scratch interface shows a programming area on the left-hand side of the screen, and a scene on the right-hand side, under which the three characters appear with their names. By clicking on the small square representing a character, students can view the programming area associated with it on the left-hand side of the screen. All blocks of instructions provided to students in the programming area are non-standard, that is, they are not available in Scratch by default. They have been programmed by assembling several standard blocks and are made available to students, who can run them directly, without having to worry about technical aspects related to the complexity of the programming code they are working with (according to principle P1). In the scene, there are the three characters initially positioned at the bottom left; a slanted red line (representing the axis of symmetry) on which a blue dot is placed (the point from which the characters will start to draw the choreography); the unit of measurement (u), which corresponds to the length of each character’s step, at the top right (see Fig. 1).
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Fig. 1
The Scratch interface
The didactic sequence requires that students work on three consecutive Tasks (a-didactic moments).
In Task 1, students visualize Piero’s program already implemented, that is, the blocks of instructions associated with him are already set. Students are asked to “first observe Piero’s program, then run it (by clicking on the green flag) and, finally, observe Piero’s movements on the scene”. As soon as the green flag is clicked, Piero starts moving and leaves markers on the ground in the part of the scene to the left of the red line. He moves either along the red line, perpendicular to it or parallel to it. The following Table 1 shows the pre-established program for Piero, his behavior on the scene as a function of each block of instruction, and why each block of instruction was conceived, with respect to a particular choreography (Choreography C1, hereafter).
Table 1. The description of Piero’s blocks of instructions related to Choreography C1
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Task 1 was designed so that the student, simultaneously observing Piero’s program on the programming area and his movements on the scene, would be able to associate each block with the action/movement corresponding to it.
In Task 2, students are asked to “create a program for Isabella so that she makes the same movements as Piero, symmetrically on the other side of the red line”. By clicking on the small square representing Isabella’s character in the lower right corner of the Scratch interface (see Fig. 1), students visualize Isabella’s programming area on the left-hand side of the screen. The blocks of instructions for Isabella are the same as for Piero, but they are not encapsulated. Students are asked to drag and drop and encapsulate the blocks so that Isabella makes the same movements as Piero symmetrically on the other side of the red line.
After creating the program for Isabella, students click on the green flag. The markers previously dropped by Piero on the scene disappear and students simultaneously visualize Piero and Isabella’s movements. Since the blocks of instructions for Isabella are the same as for Piero, Isabella can also move either along the red line, perpendicular to it or parallel to it. The digital artefact provides feedback to the students’ actions (according to principle P2). The feedback is precisely the simultaneous movement of Piero and Isabella, which students can observe to verify if they have complied with the task. If Isabella’s movements are implemented “correctly” (i.e. in such a way that she “makes the same movements as Piero, symmetrically on the other side of the red line”), students visualize the two characters overlapping on the blue dot; Isabella’s character replicating Piero’s movements symmetrically on the other side of the line; the markers dropped by Isabella symmetrical to those dropped by Piero with respect to the red line. The symmetrical movements of Piero and Isabella ensure that the markers dropped on the scene by the two characters are symmetrical with respect to the line. We expect that students proceed initially by trial and error in implementing Isabella’s program. However, we expect that students are able to create Isabella’s program “correctly” due to the feedback. Once they have “correctly” implemented the program for Isabella, they could recognize that Piero and Isabella’s programs are “symmetrical” (i.e. the programs are identical, except for the direction of rotation of the characters). This may foster the development of programming strategies that allow students to create Isabella’s program from replication and adjustment of Piero’s one.
Task 2 was designed with the aim of allowing students to act on the digital artefact and, based on the feedback returned by the artefact itself, to formulate and validate hypotheses and evolve their own programming strategies (according to principle P2). The evolution of programming strategies could lead students to identify key properties of axial symmetry. Indeed, the blocks of instructions (and their text) are designed precisely with the aim of bringing out the mathematical meanings related to the concept of axial symmetry. For example, the blocks “move … units” and “turn ↺/↻ 90 degrees” are designed to invoke the concept of equidistance and perpendicularity from the axis, respectively. Furthermore, simultaneous Piero and Isabella’s movements should evoke mathematical meanings of symmetries. With regard to Choreography C1, the following Table 2 shows: Piero’s program; Isabella’s program (if implemented correctly); the respective movements on the scene of the two characters; the mathematical meanings concerning their simultaneous movement (according to principle P1).
Table 2. Piero and Isabella’s programs, their movements on the scene, and the mathematical meanings concerning their simultaneous movement, with regard to Choreography C1
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In Task 3, students are asked to “run Giada’s instruction ‘join the markers’ and visualize her movement on the scene”. It requires students to click on the small square representing Giada’s character in the lower right corner of the Scratch interface (see Fig. 1) and thereby display the corresponding programming area on the left side of the screen. Giada’s movements are pre-established like Piero’s ones. The instruction ‘join the markers’ allows the character to move and join the markers neatly with consecutive segments. Giada starts from the first marker dropped by Piero, joins all the markers dropped by Piero from the first to the last, joins the last one dropped by Piero with the last one dropped by Isabella, joins all the markers dropped by Isabella, from the last to the first and, finally, joins the first Isabella’s marker with the first Piero’s marker. Task 3 aims to link the concept of axial symmetry (as a correspondence between points in the plane) to that of a figure symmetrical with respect to a line. Giada draws a figure crossed by the red line that students visualize on the screen. Carrying out Task 3 provides additional feedback to students (according to Principle P2). Indeed, the resulting figure is symmetrical or not with respect to the red line, depending on how students programmed Isabella’s movements. It is symmetrical with respect to the red line if Isabella’s program is implemented “correctly”.
We have foreseen 5 choreographies to enable students to develop increasingly better solution strategies. The choreographies were designed so that students have a gradual approach to the difficulties of realization, both in terms of the symmetries of the figure and the program to be implemented. The gradualness consists, on the one hand, in the complexity of the figure that is created and of its symmetries (the number of points increases from one choreography to the next) and, on the other hand, in the length of the program that students are expected to implement. The following Table 3 shows, for each choreography C0, C1, C2, C3, C4, the Scratch scenes at the end of the execution of the programs of Piero (Task 1), Isabella (Task 2) and Giada (Task 3), respectively, in the case that Isabella’s program is “correctly” implemented. Choreographies C2 and C3, (in which Piero and Isabella leave markers also on the red line) are designed to evoke the property that the points belonging to the axis are symmetrical to themselves.1
Table 3. Tasks 1, 2 e 3 for each of the five choreographies C0, C1, C2, C3 and C4
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We expect that students, working on various increasingly complex choreographies, will be encouraged to experiment and refine their strategies based on the feedback provided by the digital artefact. This could lead them to find increasingly correct and time-efficient programming strategies. Consequently, they could identify the key properties of axial symmetry, as noted above, also thanks to the contribution of paper tasks, as described below.
The Paper Tasks
For each choreography, once students finish acting on the digital artefact, they work in pairs on some paper tasks (a-didactic moments). The aim of the paper tasks is twofold. On the one hand, the aim is to raise new, and more effective programming strategies. On the other hand, the aim is for students to formalize key properties of axial symmetry.
Regarding the first aim, the teacher (or the researcher) gives each pair of students a paper sheet with the following open question: “Look at the program you created for Isabella and compare it with Piero’s program. What do you observe?”. To answer this question, each student can visualize on her PC monitor the programming areas of Piero and Isabella. We expect each pair to recognize the “symmetry” between Piero and Isabella’s programs, and that this promotes the appearance (or consolidation) of a new programming strategy based on the replication and adaptation of Piero’s program for constructing Isabella’s one. More precisely, we expect that students, by observing Piero and Isabella’s programs, should be able to recognize the same instructions, and given in the same order, which differ only in the directions of rotation.
This new programming strategy could lead students to reflect on the mathematical meanings of the blocks (see Table 2). For example, observing Piero’s block with the instruction “move … units” and replicating it for Isabella, students could reflect on the equidistance from the axis of symmetrical points. Moreover, observing the “turn … 90 degrees” block in Piero’s program and replicating it for Isabella by reversing the direction of rotation, students could reflect on the perpendicularity to the axis of the segment joining two symmetrical points.
Regarding the second aim, the teacher (or the researcher) gives each pair of students a paper sheet showing a screenshot of the scene as visualized at the end of Task 1 (i.e. with the markers dropped on the scene by Piero). Students are asked to answer the following task: “Draw the markers dropped by Isabella on the other side of the line, then join all the markers to each other, neatly, and make a shared drawing. Answer the following question: What do you observe looking at the drawing?”. Students should reproduce with a pen the drawing of the figure they created by programming the movements of Isabella’s character and visualized on their PC monitor after running Giada’s block of instruction ‘join the markers’. At this stage, students might use rulers and squares. This could allow them to adopt specific strategies to reproduce as faithfully as possible what they visualize on the scene. For example, students could place the square perpendicular to the red line or take measurements to draw the points. We expect that drawing and subsequently observing the figure can foster the formulation of the key properties of axial symmetry (such as the equidistance of corresponding points from the axis or the perpendicularity of the segment connecting corresponding points with respect to the axis) from students.
Digital Artefact—Paper Tasks Match
The didactical sequence was designed so that the transition from digital artefact to paper tasks is repeated several times through carrying out the five planned choreographies. We expect this may foster the appearance of the situations of action, formulation and validation. In particular, students realize a situation of action every time they operate on the digital artefact and answer the paper tasks. Indeed, the choice of the number of steps and direction of rotation, and the manipulation of the instruction blocks by dragging and dropping, represent actions on the digital artefact. Drawing the markers as requested in the paper task, drawing the figure by connecting the markers, and writing the answers to the questions represent actions on the paper tasks. Since students work in pairs on both the digital artefact and the paper task, the formulation situation is guaranteed by the oral communication of the strategies between classmates; furthermore, the formulation appears in written form when students report their shared answers in the paper task. Validation is fostered by feedback from the milieu and occurs when students acquire awareness of the validity of the strategy (perhaps after having tested it several times) to the point of being able to justify it with their classmates and impose it as the most effective strategy.
Participants and Setting
The didactical sequence was experimented with 21 students of a fifth-grade class of a school of South Italy that is part of a network of schools that the University of Campania “L. Vanvitelli” (to which the authors of this paper belong) has contact with. However, so far no experimentation has been carried out with students from that school in the context of mathematics education. The class teacher had already carried out learning activities on symmetries with the students during the current and previous school years. Some of these activities, proposed in the school handbook, involved the construction of figures symmetrical with respect to vertical, horizontal and (in a more limited way) slanted axes placed at 45° with respect to a reference grid. The students involved in the experimentation had never tackled the activities on symmetries proposed in the textbook or by the teacher using digital technologies (e.g. dynamic geometry software). The students had never engaged in computer programming activities.
The study was carried out by a researcher, supported by the class teacher. It took place in the classroom in three curricular sessions of 90 min each. In each session, the students were randomly divided into pairs (only one group consisted of 3 members). The students’ desks were arranged close together so that each student could work on her own laptop and simultaneously look at her partner’s laptop and interact with her orally. In the first session, the researcher presented the characteristics of the Scratch environment via video projection and invited the pairs to interact with the digital artefact on the classroom computer to gain experience with it. In particular, the students learned how to switch from one character to another in the Scratch environment, to manipulate the blocks of instructions (i.e. how to drag, drop and encapsulate them to create a program), and to run the created program. In subsequent sessions, the students dealt with the choreographies in Fig. 3, using their laptop and communicating orally with their partner.
For each choreography, the students first dealt with the digital artefact and then with the paper tasks, as described in subsection “The Design of the Milieu”.
Human subject research approval was obtained as protocol RP-01–02-2023 of the Ethics Committee of the Department of Mathematics and Physics of the University of Campania “L. Vanvitelli”.
Data Collection and Analysis Criteria
We collected the following data:
The video recordings of the PC screens used by the students during the manipulation with the digital artefact
The paper sheets with the tasks related to each choreography, filled in by the pairs of students after working with the digital artefact
The audio recordings, group by group, of the whole activity with the digital artefact, as well as of the collaborative work moments related to the filling in of the paper tasks
The audio recordings of the collective discussions moderated by the researcher
The notes taken in class by the researcher and the teacher.
We analyzed the collected data in search of actions performed by students on the digital artefact or expressions or sentences orally declared or written by students that showed the appearance of action, formulation and validation situations foreseen by TDS. More specifically, to address the first research question RQ1 (“To what extent is the designed didactic sequence able to make primary school students formulate and validate effective programming strategies to construct symmetrical images with respect to an axis?”), we searched for the following:
Elements evidencing the emergence and consolidation of new programming strategies (we expect students to first create the program for Isabella just by observing Piero’s movements, but then search for different strategies to avoid errors and reduce working time, for example, “copying” the instructions from Piero’s program);
The reasons why such new programming strategies appeared (for example, due to a feedback provided by the digital artefact or due to the answers to the questions in the paper tasks).
To address the second research question RQ2 (“To what extent is the designed didactic sequence able to make primary school students identify the key properties of axial symmetry?”), our analysis focused on students’ oral and written references to axial symmetry’s key properties, specifically looking for mentions of direct indicators of conceptual understanding, that is.
“perpendicularity”, for example, an explicit use of the word “perpendicular” or the expression “90 degrees”;
“equidistance”, for example, an explicit use of expressions such as “same/equal distance/length”, or expressions that implicitly lead to the concept of equidistance, such as “the characters move the same number of steps on either side of the line”;
“coincidence of markers”, for example, expressions such as “Piero and Isabella overlap on the red line” or “markers overlap on the red line”;
“midpoints on the axis”, for example, expressions such as “the axis halves the segment joining two corresponding markers”.
Results
In this section, we show the results of a qualitative analysis aiming to highlight how students repeatedly activated situations of action, formulation and validation through the digital artefact and the paper tasks, managing to evolve their programming strategies.
Action, Formulation, and Validation Through the Digital Artefact
At the beginning of the activity with the digital artefact, students observed Piero’s program and clicked on the flag to visualize its running. Based on what the students visualized on the scene, they created Isabella’s program. Once the students thought they had created the correct program, they clicked again on the flag to verify the proper implementation. This action on the digital artefact entailed the appearance of a first programming strategy (movements-in-the-scene strategy), which was based exclusively on the visualization of Piero’s movements on the scene. This strategy was employed by most students in the initial phase of the activity with the artefact.
Some students observed the simultaneous movements of Piero and Isabella on the scene (a feedback of the milieu), and then realized that the movements-in-the-scene strategy was not effective, because the movements displayed on the scene were not symmetrical. So, they implemented a second strategy (program-instructions strategy), in which they focused exclusively on the instructions in Piero and Isabella’s programs, without worrying about observing their movements on the scene. In the notable case of Arianna and Sabrina, the shift from a visually dependent strategy to an instruction-based approach underscores the milieu’s capacity to foster adaptive learning strategies. In this case, the realization of the program-instructions strategy occurred following an oral formulation. Indeed, thanks to the feedback of the milieu, Sabrina realized that Isabella’s movements were not satisfactory for her, since the program created for Isabella did not make her perform the same movements as Piero symmetrically on the other side of the red line (see Fig. 2). Sabrina’s realization that Isabella’s movements were not satisfactory led to a pivotal strategy adjustment, illustrating the dynamic interaction between student cognition and digital feedback.
[See PDF for image]
Fig. 2
The Isabella’s program created by Sabrina
Arianna, after completing the task, helped her groupmate Sabrina in programming. She orally formulated a new strategy and proposed it to Sabrina: “copying” the instructions from Piero’s program, dragging and dropping them, in the same order, in the programming area of Isabella and reversing the direction of rotations, that is to replace instructions such as “turn … 90 degrees” with those that allow them to rotate in the opposite direction (“I copied from Piero, but… just I changed turn in that way, because she has to turn here, do you see?”; and also “This is Piero’s program… Now you have to do the same thing with Isabella…”). Sabrina switched from viewing Piero’s program to modifying Isabella’s program several times, until Piero and Isabella’s programs appear “symmetrical” to her.
Later in the activity, when working on choreographies C1 and C2, all the students adopted this programming strategy, which was considered more effective. This strategy became the goal for most of the students, who used it in the remaining choreographies, albeit with small variations. For example, some pairs of students organized their work as follows: one student visualized Piero’s program on her laptop and acted dictating to her partner the instructions (“go on the blue point”, “move 3 units”, “turn 90 degrees on the left”,…, “exit the scene”) to be included in Isabella’s program, one by one.
Only Arianna noticed a further option that would have made the execution of the task even faster. Working with the final choreography C4, Arianna acted in a new way by exploring some of the Scratch features and discovering the “duplicate” feature by right-clicking on Piero’s program (see Fig. 3). Arianna used Scratch’s “duplicate” function in Piero’s programming area, dragged and dropped the duplicate program into Isabella’s programming area, and modified it by reversing the directions of all 90-degrees rotations.
[See PDF for image]
Fig. 3
Arianna duplicates Piero’s program (in the box, the English translation of the features appearing by right-clicking)
Once she discovered this strategy (duplicate-block strategy), Arianna formulated it orally and shared it with her partner Sabrina: “Look: if I click here (right-clicking on the mouse when the cursor was on the instruction blocks of Piero’s program), ‘duplicate’ comes out and then I can drag this program directly onto Isabella. Do the same too!”. Then, she stated: “From now on, I no longer waste all that time copying the program by hand, because that way I do it much faster.”. In this statement, we recognize the validation situation, since Arianna imposed this strategy as the best from the point of view of effectiveness and correctness. In fact, by using this strategy, Arianna not only significantly reduced working times, but she was also sure of the correctness of Isabella’s program. Arianna’s discovery and application of the duplicate-block strategy represent a significant leap in operational efficiency and conceptual understanding. This strategy not only streamlined the programming task, but also cemented a deeper understanding of symmetry through direct manipulation of the digital tool, marking a key milestone in the learning journey.
In conclusion, using the first programming strategy (movements-in-the-scene strategy), students wrote Isabella’s program after observing Piero’s movements in the scene, but, without a visual reference such as the presence of a grid, they had difficulty establishing the number of Isabella’s steps. The second strategy (program-instructions strategy), as we expected, was not based on the visualization of Piero’s movements on the scene, but only on the visualization of Piero’s program, and on the block-by-block construction of Isabella’s program in a “symmetrical” way compared to Piero’s one. Finally, the third strategy (duplicate-block strategy) appeared to be the most effective in terms of view of the time necessary to carry out the Task and the correctness of the solution, since the only actions required were to use the feature “duplicate” (which does not cause any errors) and to reverse the rotation directions. In the transition from the first strategy to the second and then to the third, as we expected, the students’ focus shifted from observing the running of Piero’s program, and thus the character’s movements in the scene, to observing (and then duplicating) the program itself, which is then interpreted as a “trace” of the exact movements the character performs. However, students did not observe the mathematical meanings in the instruction blocks, but rather the “symmetry” of the programs. Referring directly to Piero’s program to create Isabella’s program allows the students to immediately identify not only the number of steps, but also the exact movements to be replicated, as well as the order in which they should be executed.
Paper Tasks Contribution
The use of paper tasks allowed formulations of both new programming strategies and some mathematical properties related to axial symmetry to appear in written form. Some students’ productions evoked the property of the axial symmetry that the axis cuts in half the segment that connects two corresponding points. For example, Sveva, when asked “Look at the program you created for Isabella and compare it with Piero’s program. What do you observe?”, answered: “An even line has come between the red line” (see Fig. 4a. Sveva). When the researcher asked for explanations, Sveva explained (miming with her fingers first the red line and then the two endpoints of the segment that converge on the line) that the line divided the segment in two equal parts. It seems that the student used the expression “even line” to indicate the segment cut in half by the red line (i.e. the axis of symmetry crosses the midpoint of the segment that connects the two corresponding points). Sveva’s description, “An even line has come between the red line”, subtly indicates her recognition of the axis of symmetry dividing a segment into equal parts. In this sense, the term “even line” used by Sveva could mean the student’s understanding of axial symmetry’s bisecting nature.
[See PDF for image]
Fig. 4
Written productions by Sveva and Cecilia referred to the paper task of the choreography C0 (literally translated from Italian)
The property of the symmetry axis of being the axis of the segment joining two symmetrical points seems to appear in Cecilia’s answer. Regarding choreography C0, she wrote: “It seems to me the base of a triangle” (see Fig. 4b. Cecilia). When the researcher asked for explanations, Cecilia specified: (pointing to a point on the red line) “If I take a point here on the red line and join it to the markers, a triangle comes out… The sides are all equal… (thinking about it, and pointing to another point on the red line) with only two equal sides!”. We observe how at first, Cecilia highlighted a particular point on the red line in such a way that the sides of the triangle are “all equal”. Subsequently, Cecilia realized that other triangles can be imagined with “only two equal sides”. Therefore, Cecilia, in the imagined figure, seems to focus on the points of the red line, recognizing that each one of its points is equidistant from the two endpoints of the segment. However, Sveva and Cecilia’s drawings (see Fig. 4) did not show clearly, neither the perpendicularity of the segment with respect to the axis nor the equidistance of the corresponding points. From the excerpts just analyzed, some properties of axial symmetry seem to emerge, albeit in a weak form.
Written formulations also occurred in Miriam, Enrico, and Arianna’s answers to the first paper task (“Look at the program you created for Isabella and compare it with Piero’s program. What do you observe?”). Miriam referred to “similarity” between the two programs, while Enrico stated that the two programs were “identical except for the direction of the 90 degrees turning”. Arianna also noticed that “in Piero’s program there is turn 90° to the right, in Isabella’s there is turn 90° to the left” (see Fig. 5). The observations of such students touch upon the essence of symmetry. Diving deeper into these perceptions could unveil more intricate programming strategies or mathematical properties of axial symmetry. Regarding programming strategies, answering the first paper task made students focus on Piero and Isabella’s program and not just on their movements in the scene. After answering that task, Arianna elaborated the program-instructions strategy while working on the new choreography, as described in the previous section. However, as far as the mathematical properties of axial symmetry are concerned, they did not emerge from the students’ written formulations.
[See PDF for image]
Fig. 5
The answers of Miriam, Enrico and Arianna to the paper task of the choreography C0 (literally translated from Italian)
Thanks to the evolving of the programming strategies and to the feedback of the milieu, in answering paper tasks, students were able to draw symmetrical figures. This allowed them to identify new properties related to axial symmetry, formulated in a written manner in the paper tasks and orally in conversations with the researcher in the classroom. For example, this involved three pairs of students who worked on choreography C4 and answered to the second paper task: “Draw the markers dropped by Isabella on the other side of the line, then join all the markers to each other, neatly, and make a shared drawing. Answer the following question: what do you observe looking at the drawing?” (see Fig. 6).
[See PDF for image]
Fig. 6
Answers to the second paper task on choreography C4 of three pairs of students (literally translated from Italian)
Students labeled (numerically or lexicographically) the markers dropped by Piero and Isabella and setted a correspondence among them. Simona and Miriam labeled the markers dropped by Piero with the numbers from 1 to 6 and those dropped by Isabella with the numbers from 7 to 12 (see Fig. 6a. Simona and Miriam). Then, in their answer, they formulated the correspondence among the markers by matching the pairs (i, i + 6), for i = 1,…, 6. The correspondence of points (2, 8), (3, 9), (4, 10) and (5, 11) is correctly stated, thus respecting the equidistance from the axis of the corresponding points. Lina and Enrico used an alphanumeric notation, indexing each pair of corresponding markers with a letter of the alphabet between A and F and assigning the number 1 to the markers dropped by Piero, and the number 2 to those dropped by Isabella, thus generating pairs such as (A1, A2) and (B1, B2) (see Fig. 6b. Lina and Enrico). This notation allowed them to indicate unequivocally the one-to-one correspondence between points. Furthermore, the two students (who stated that “the markers are perpendicular to each other”) seemed to recognize that corresponding points belong to a line that is perpendicular to the axis. Laura and Sveva also labeled the markers with the letters of the alphabet from A to N, ordering the letters starting from top to bottom and from left to right (see Fig. 6c. Laura and Sveva). Assigning letters to the markers could represent their intention to indicate a correspondence between them, even if such a correspondence is not explicitly stated. Referring to two matching markers dropped by Piero and Isabella, they stated that they are “on different but equal parts”. Therefore, it seems that the two students referred to the equidistance of corresponding points from the axis and that they evaluated the distance along the segment perpendicular to the axis, as shown in the drawing they produced.
We highlight that expressions such as “different but equal” or “contrary but equal” appeared in students’ written and oral formulations (for example, see Fig. 6c. Laura and Sveva). These expressions were recalled by the researcher, who asked the students to better explain what they meant by “equal” in a moment of interaction with the whole class. For example, Laura (who said: “Isabella and Piero do the same number of steps on both sides of the line”) referred to the “same number of steps”, recognizing the aspect of equidistance from the axis of corresponding points. With greater difficulties, the students tried to formulate the meaning they attributed to the expression “contrary”. Antonio stated that “contrary means that the markers are perpendicular to the line”. This statement is similar to that of Lina and Enrico in their answer to the second paper task (see Fig. 6b. Lina and Enrico) even if they stated that the markers are perpendicular to each other, not to the line. In these statements, we can recognize the property of axial symmetry that the segment joining corresponding points is perpendicular to the axis.
We conclude our analysis with the following remark. Most of the students, in working with the digital artefact at choreography C3, accepted the figure returned by the artefact as correct, thanks to the feedback of the milieu and to the recognized solidity and validity of the programming strategy used up to that moment. However, many students answered negatively to the researcher’s question “Did you expect this figure?”. We report, for example, the dialogue between the researcher (R) and Enrico (E):
E: “No, actually I was expecting some kind of pentagon …”
R: “A pentagon? Could you show me what this pentagon should have been like?”
E: “Thus” (with a finger placed on the screen in correspondence with the scene, Enrico draws a pentagon “joining” together all the markers except the last one as in Fig. 7)
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Fig. 7
The dotted line represents the pentagon drawn by Enrico on the screen with his finger
Enrico, in drawing the figure with his fingers, did not cross the last marker dropped by the characters on the line (a point belonging to the axis). Two other students also communicated to the researcher that they would have expected to see a pentagon. It seems that students’ trust in the developed programming strategy, consolidated up to that point, and the feedback from the digital artefact led students to accept the figure that fixes the points on the axis as correct, although not expected. However, during our study, the property of the axial symmetry of fixing the points on the axis did not emerge clearly, despite the choreographies C2 and C3 being designed for this purpose and despite the numerous attempts by the researcher to make the students reflect on this aspect.
Discussion and Conclusions
This work fits into recent research trends in the teaching and learning of geometry (in particular, symmetry) in primary school highlighted by Sinclair and Bruce (2015) (i.e. transformational geometry and the use of digital technologies in geometry) and the literature on computer programming for the construction of geometric concepts in primary school. In this paper, we described a didactic sequence designed according to the Theory of Didactical Situations (TDS) of Brousseau (1986) for primary school students, aimed at constructing mathematical knowledge on axial symmetry through algorithmics and computer programming. It includes a-didactic moments, during which students interact with a milieu (consisting of a digital artefact, based on visual programming, and paper tasks), and moments of interaction with the teacher with the aim of encouraging the emergence of the action, formulation and validation situations foreseen by the TDS.
In this paper, we wondered to what extent the designed didactic sequence was able.
To make primary school students formulate and validate effective programming strategies to construct symmetrical images with respect to an axis (RQ1)
To make primary school students able to identify key properties of axial symmetry (RQ2).
For the design of the didactic sequence and milieu, we used the methodological framework of Didactic Engineering (Artigue, 1988, 2008). We carried out a preliminary analysis of the teaching content, that is, axial symmetry, and its usual teaching in primary school. We did an a priori analysis of the didactic sequence, imagining possible student behaviors and solution strategies. We experimented with the didactic sequence in the classroom with students in a fifth-grade primary school class. We analyzed the results of the experimentation a posteriori, to detect the distance between what was expected from the experimentation and what actually happened.
Data analysis showed the link between the students’ evolution of programming strategies and the construction of mathematical knowledge related to the definition of axial symmetry. In investigating the extent to which our didactic sequence facilitated students’ understanding of axial symmetry (RQ1) and their ability to identify its key properties (RQ2), our analysis revealed a nuanced link between programming strategies and the construction of mathematical knowledge.
An early programming strategy (movements-in-the-scene strategy) involved the following steps: the students first observed Piero’s program; then they visualized the running of the program by clicking on the flag; based on what they visualized on the scene, they created Isabella’s program; they clicked again on the flag to verify its correct implementation. This programming strategy, which was based exclusively on the visualization of Piero’s movements on the scene, emerged in the initial phase of the activity, through the action on the digital artefact (see the section “Action, Formulation, and Validation Through the Digital Artefact” section). Applying the movements-in-the-scene strategy, many students failed to realize movements for Isabella that were symmetrical to Piero’s. They realized this through the milieu feedback, that is, by observing Piero and Isabella’s simultaneous movements in the scene. Moreover, after working with the digital artefact at the choreography C0, students responded to the first paper task (“Look at the program you created for Isabella and compare it with Piero’s program. What do you observe?”). Some students (e.g. Miriam) observed the “similarity” between the two programs, some (e.g. Enrico) noted that the two programs were “identical except for the direction of the 90 degrees of rotation”, others (e.g. Arianna) noted that “in Piero’s program there is turn 90° to the right, in Isabella’s there is turn 90° to the left”. Therefore, the students came up with new strategies (e.g. the case of Sabrina and Arianna) and formulated and validated these strategies through the feedback of the milieu (in particular, the simultaneous movement of the characters on the Scratch scene as well as the thought-provoking stimuli offered by the paper tasks), the different choreographies provided by the didactic sequence, and the collaborative work. A second strategy (program-instructions strategy) consisted of “copying” Piero’s program instructions, dragging and dropping them, in the same order, into Isabella’s programming area, and reversing the direction of the rotations for Isabella. This strategy emerged while working on choreographies C1 and C2 and was adopted by most students in the remaining choreography as well, albeit with small variations. Once the program-instructions strategy was consolidated, some students were concerned about finding the most effective way to apply it. The problem of making symmetrical movements was thus decomposed into simpler sub-problems (Wing, 2006). The question was no longer to write a program for Isabella but to figure out how to duplicate Piero’s program as effectively and efficiently as possible. In that attempt, a student (Arianna) discovered the “duplicate” function of Scratch and used it to duplicate Piero’s program in Isabella’s programming area. She finally reversed the directions of all rotations. This was the third strategy emerged (duplicate-block strategy). The program-instructions and duplicate-block strategies, which emerged through decomposing the task into simpler subtasks and reflecting in terms of steps and actions (Wing, 2006), fostered the arising of mathematical properties concerning axial symmetry. Some students (for example, Cecilia) recognized the axis of symmetry as the locus of points equidistant from the markers, while other students (for example, Sveva) identified the property of the axis of cutting in half the segment that joins two corresponding points. The point-to-point correspondence was highlighted by different pairs of students (e.g. Simona and Miriam, Lina and Enrico, Laura and Sveva) and formulated in writing in different ways (numerically or lexicographically). Thus, the decomposition of the problem into sub-problems and the synergy between the digital artefact and paper tasks (for example, see the contribution of paper tasks in program-instructions strategy) fostered the emergence of programming strategies and, consequently, the construction of students’ mathematical knowledge on axial symmetry. This result can be stated as a general principle P3 as follows:
P3: In accordance with Wing (2006), on the one hand, and Mariotti and Montone (2020), on the other hand, the decomposition of the problem into sub-problems and the synergy between digital and paper artefacts, respectively, are able to foster the emergence of programming strategies and promote students’ construction of mathematical knowledge.
Principle P3 is in addition to design principles P1 and P2 stated in the subsection “Design principles and research questions”. The three principles, P1, P2 and P3, represent an outcome of this study. The milieu and the didactic sequence are an example of the application of these principles. Such principles are to be considered general design principles that researchers and teachers could use to design visual programming environments aimed at constructing mathematical knowledge. From this point of view, we can state that this study has expanded the research on the use of computer and visual programming for the construction of mathematical and geometric concepts in primary school (see the “Computer Programming for Constructing Mathematical and Geometric Knowledge” section).
The didactic sequence favored the situations of action, formulation and validation. However, another aspect needs to be considered. The a-didactic moments (and therefore the milieu) were not sufficient to bring out the property of axial symmetry of fixing the points of the axis, although some features of the milieu were designed specifically with this goal (see the “Action, Formulation, and Validation Through the Digital Artefact” section, choreographies C2 and C3), and despite the interaction with the researcher and her numerous attempts to make students reflect on this aspect. While the didactic sequence stimulated action, formulation and validation situations, it fell short in fully elucidating the axial symmetry property of fixing points on the axis. This shortfall suggests the need for a more pronounced teacher intervention to bridge these conceptual gaps. Our analysis shows that the designed didactic sequence was not enough to allow students to identify all properties related to axial symmetry, but a subsequent targeted intervention by the teacher was necessary. The digital artefact was found to be effective in bringing out programming strategies, and the feedback provided by the artefact was so effective that students accepted as correct figures drawn on the Scratch scene even though they were not expected (see the last part of the “Paper Tasks Contribution” section). Paper tasks contributed to the emergence of new programming strategies. However, they only partly contributed to the emergence of key properties of axial symmetry (e.g. case of Miriam, Enrico, and Arianna).
Our findings suggest the designed didactic sequence, while effective in certain respects, requires augmentation through targeted teacher interventions and enhanced paper tasks. Future iterations of the sequence should incorporate additional probing questions to elicit deeper student engagement with the core mathematical properties of axial symmetry. For example, in the paper tasks, before the question “Look at the program you created for Isabella and compare it with Piero’s program. What do you observe?”, students could be asked:
“Look at the markers dropped on the stage by Piero and Isabella: what do you observe?”;
“How did you get Isabella to draw markers symmetrical to those of Piero?”.
Such questions could help students link programming aspects to mathematical ones. For example, the second question prompts the student to retrace what she did (using the blocks of instructions) to create the choreography for Isabella symmetrical to Piero’s. It is therefore a metacognitive question that could encourage reflection on aspects related to the equidistance of the markers from the axis and/or the perpendicularity of the segment joining them with respect to the axis. Moreover, at the end of all the choreographies, students could be shown a drawing similar to those in the column “Task 1” of Table 3 and first ask them, “Look at the markers Piero left on the stage and draw their symmetrical, just as Isabella would”, and then, “How did you draw Isabella’s markers?”. The request to draw symmetrical markers leads the student to detach herself from the digital artefact to make the drawing. At the same time, she might call to mind what she did previously in the programming environment. For example, she might ask, “How would I have done in Scratch to draw the symmetrical markers?”. This could encourage her deeper reflection on the connection between the meaning of the blocks of instructions and mathematical aspects related to symmetry. Further metacognitive reflections could be triggered by the subsequent request to describe how the symmetrical markers were drawn. The paper tasks revised as just described could promote the emergence of the mathematical properties of axial symmetry more clearly than observed in the experiment described in this paper. They would fit into a structured plan for teacher intervention based on the results of this study. In the future, we plan to carry out a new study with primary school students regarding the didactic sequence enriched with these new questions, and compare the results obtained.
Author Contribution
The authors (Rosamaria Crisci, Umberto Dello Iacono, Eva Ferrara Dentice) declare that they contributed equally to the research reported in and to the writing of this article.
Funding
Open access funding provided by Università degli Studi della Campania Luigi Vanvitelli within the CRUI-CARE Agreement.
Data Availability
The data that support the findings of this study are available from the corresponding author [UDI], upon request. Participants’ consent was obtained and documented with protocol RP-01–02-2023 of the Ethics Committee of the Department of Mathematics and Physics of the University of Campania “L. Vanvitelli”.
Declarations
Competing Interest
The authors declare no competing interests.
The video at the following link shows a use of the digital artefact relative to choreography C1: urly.it/3_fkc. For any difficulties viewing the video, please contact the corresponding author.
Publisher's Note
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