## Class sizes (back to 2022)

Public and private primary education

* 22.4 pupils per class, average number of pupils in pre-elementary classes

* 21.6 pupils per class, average number of pupils in elementary schools

Secondary public and private

* 25.9 students per class, average number of students in college training (excluding Segpa)

* 17.9 students per class, average number of students in vocational high school courses

* 30.3 students per class, average number of students in general and technological high school courses

## Diplomas (session 223)

* 89.1 % Diplôme national du brevet (DNB) pass rate (after 3e)

* 90.9 % Baccalauréat pass rate (after terminale)

* 79.3% Proportion of baccalaureate holders in a generation

Average expenditure per pupil per year (calendar year 2018, provisional data)

* €7,440 per primary school pupil

* €9,150 per middle school student

* €11,570 per general and technological high school student

* €13,220 per vocational high school student

## TEACHER'S PRESENTATION

The plenary session will begin with a presentation of a few schools with successive zoom in:

* Elements (photos, short video clips, data, information) about the teacher's town, school and classroom.

* Place of geometry in the cursus.

* Examples of geometric activities proposed in the classroom, in the form of resource documents, materials used, video extracts, typical errors, and specific characteristics of the audience.

* The special-needs audiences they cater to.

* The technological tools they use in their classrooms.

* Their equipment.

* Problems and difficulties associated with teaching geometry.

The presentations will be followed by a discussion with the researchers of the ICMI study.

## Participant teachers:

* Kindergarten: classes of colleagues presented by Christine Trouillet

* Primary school: Bernadette Da Motta (CE2), Aurélie Marche (CP-CE1) and Mélanie Binet: teachers in Reims

* Secondary school: Isabelle Audra (Paul Langevin's Middle school in Romilly-sur-Seine) accompanied by Marie-Paul Foy

* University: to be determined
## Class sizes (back to 2022)

Public and private primary education

* 22.4 pupils per class, average number of pupils in pre-elementary classes

* 21.6 pupils per class, average number of pupils in elementary schools

Secondary public and private

* 25.9 students per class, average number of students in college training (excluding Segpa)

* 17.9 students per class, average number of students in vocational high school courses

* 30.3 students per class, average number of students in general and technological high school courses

## Diplomas (session 223)

* 89.1 % Diplôme national du brevet (DNB) pass rate (after 3e)

* 90.9 % Baccalauréat pass rate (after terminale)

* 79.3% Proportion of baccalaureate holders in a generation

Average expenditure per pupil per year (calendar year 2018, provisional data)

* €7,440 per primary school pupil

* €9,150 per middle school student

* €11,570 per general and technological high school student

* €13,220 per vocational high school student

## TEACHER'S PRESENTATION

The plenary session will begin with a presentation of a few schools with successive zoom in:

* Elements (photos, short video clips, data, information) about the teacher's town, school and classroom.

* Place of geometry in the cursus.

* Examples of geometric activities proposed in the classroom, in the form of resource documents, materials used, video extracts, typical errors, and specific characteristics of the audience.

* The special-needs audiences they cater to.

* The technological tools they use in their classrooms.

* Their equipment.

* Problems and difficulties associated with teaching geometry.

The presentations will be followed by a discussion with the researchers of the ICMI study.

## Participant teachers:

* Kindergarten: classes of colleagues presented by Christine Trouillet

* Primary school: Bernadette Da Motta (CE2), Aurélie Marche (CP-CE1) and Mélanie Binet: teachers in Reims

* Secondary school: Isabelle Audra (Paul Langevin's Middle school in Romilly-sur-Seine) accompanied by Marie-Paul Foy

* University: to be determined
DISCUSSION TOPICS
DISCUSSION TOPICS
TOPIC A

Theoretical perspectives
# TOPIC A

## Theoretical perspectives
# UNDERGRADUATE MATHEMATICS STUDENTS' REASONING AND ARGUMENTATION IN PROBLEM-SOLVING GEOMETRIC ACTIVITIES

Tania Azucena Chicalote-Jiménez¹, Daniel José Ortiz-May², Adrián Gómez-Árciga³

¹,² CINVESTAV, ³ Autonomous University of Baja California

The aim of this study is to analyze the extent to which a problem-solving approach in geometry activities engages undergraduate students in mathematical discussions and tasks to formulate conjectures, seek different ways to support and validate those conjectures and communicate their findings to peers and the instructor. To this end, the participants were encouraged to use a Dynamic Geometry System (GeoGebra) to model the tasks dynamically and identify and explore the relationships between objects in terms of empirical and analytic arguments. As a result, the participants initially interpreted concepts involved in tasks statements in terms of their geometric meaning to represent them. Subsequently, they proceeded to explore the model by moving select elements within it. After conducting this exploration, the participants formulated conjectures regarding objects' behaviors or relationships that were explained and supported through empirical and formal arguments. In other words, students extended their ways of reasoning transitioning from empirical to formal arguments to support and validate their conjectures.

## INTRODUCTION

Over the last three decades, research in mathematics education has been interested in identifying, explaining, and understanding the difficulties that first-year students face during their integration into university-level education. For instance, Leviatan (2008) argues that these difficulties are related to the contrast between how mathematics is taught at a high school level versus at a university level. While the former tends to focus on developing algorithmic skills aimed at solving concrete and routine exercises, the latter demands abstraction skills and inquiry questioning and emphasizes non-routine problem-solving and mathematical rigor. The data obtained from diagnostic admission exams for bachelor's degrees in mathematics in Mexico indicate that the majority of students lack a basic understanding of fundamental mathematical concepts and are primarily focused on reproducing problem-solving algorithms. This means that most students begin their college education with limited knowledge of mathematical concepts and problem-solving strategies.

This highlights how crucial it is to shift the focus in mathematics education towards perspectives that prioritize the development of problem-solving and sense-making skills about mathematical concepts. That is, activities and instructions ought to be designed in such a way that they foster the development of students' mathematical thinking. Such instructional frameworks should empower students to articulate and address diverse problem types across the spectrum of their academic and social education. Santos-Trigo (2019) argues that, in a problem-solving based instruction, students can engage in mathematical knowledge by conceptualizing the discipline as a set of dilemmas that need to be represented, explored, analyzed, explained and justified, i.e., learners need to develop and value an inquiring approach to understand concepts and to solve problems. Additionally, the use of digital
# UNDERGRADUATE MATHEMATICS STUDENTS' REASONING AND ARGUMENTATION IN PROBLEM-SOLVING GEOMETRIC ACTIVITIES

Tania Azucena Chicalote-Jiménez¹, Daniel José Ortiz-May², Adrián Gómez-Árciga³

¹,² CINVESTAV, ³ Autonomous University of Baja California

The aim of this study is to analyze the extent to which a problem-solving approach in geometry activities engages undergraduate students in mathematical discussions and tasks to formulate conjectures, seek different ways to support and validate those conjectures and communicate their findings to peers and the instructor. To this end, the participants were encouraged to use a Dynamic Geometry System (GeoGebra) to model the tasks dynamically and identify and explore the relationships between objects in terms of empirical and analytic arguments. As a result, the participants initially interpreted concepts involved in tasks statements in terms of their geometric meaning to represent them. Subsequently, they proceeded to explore the model by moving select elements within it. After conducting this exploration, the participants formulated conjectures regarding objects' behaviors or relationships that were explained and supported through empirical and formal arguments. In other words, students extended their ways of reasoning transitioning from empirical to formal arguments to support and validate their conjectures.

## INTRODUCTION

Over the last three decades, research in mathematics education has been interested in identifying, explaining, and understanding the difficulties that first-year students face during their integration into university-level education. For instance, Leviatan (2008) argues that these difficulties are related to the contrast between how mathematics is taught at a high school level versus at a university level. While the former tends to focus on developing algorithmic skills aimed at solving concrete and routine exercises, the latter demands abstraction skills and inquiry questioning and emphasizes non-routine problem-solving and mathematical rigor. The data obtained from diagnostic admission exams for bachelor's degrees in mathematics in Mexico indicate that the majority of students lack a basic understanding of fundamental mathematical concepts and are primarily focused on reproducing problem-solving algorithms. This means that most students begin their college education with limited knowledge of mathematical concepts and problem-solving strategies.

This highlights how crucial it is to shift the focus in mathematics education towards perspectives that prioritize the development of problem-solving and sense-making skills about mathematical concepts. That is, activities and instructions ought to be designed in such a way that they foster the development of students' mathematical thinking. Such instructional frameworks should empower students to articulate and address diverse problem types across the spectrum of their academic and social education. Santos-Trigo (2019) argues that, in a problem-solving based instruction, students can engage in mathematical knowledge by conceptualizing the discipline as a set of dilemmas that need to be represented, explored, analyzed, explained and justified, i.e., learners need to develop and value an inquiring approach to understand concepts and to solve problems. Additionally, the use of digital
technologies as a problem-solving tool allows students to explore mathematical tasks in different ways. In this sense, we argue that exposing students to a problem-solving approach where students can rely on digital technologies can foster crucial skills to elaborate conjectures, justify and communicate results, which promote their integration to university education.

Therefore, in this research, we aimed to characterize, through geometry activities based on a problem-solving approach (Polya, 1965; Schoenfeld, 1985) and mediated by the coordinated use of GeoGebra, the tools, skills and difficulties exhibited by undergraduate students as they work on geometric tasks that encouraged them to formulate conjectures, explore different ways to support and validate those conjectures and to communicate their results. The research seeks to address the question: To what extent do these geometric activities, mediated by the use of GeoGebra, promote the development of mathematical processes, such as formulating, supporting and validating conjectures, and elaborating mathematical arguments among undergraduate students?

## CONCEPTUAL FRAMEWORK

When students begin their university education, they face difficulties related to their lack of abstraction skills, inquiry reasoning, general sense of mathematical rigor, and non-routine problem-solving skills (Leviatan, 2008). This indicates that moving from a secondary-level of education to a university-level entails a process of transition and integration. Research on the secondary-tertiary transition in mathematics, has considered undergoing perspectives to explain and address the problems and stages that students experience during this period. In this matter, it has been found that some aspects influencing students during the secondary-tertiary transition are related to their previous experiences in learning mathematics, the quality of study strategies, the way of thinking in/for mathematics, and conceptual background about mathematics ideas (Rach & Heinze, 2016; Di Martino & Gregorio, 2019). In this sense, Schoenfeld (2022) highlights the importance of providing challenging activities to students, i.e., non-routine problems as well as generating educational environments that promote "learning to think in and with the discipline, internalizing the discipline's habits of mind and practices of doing mathematics". Thus, students need to have close encounters with processes linked to mathematical reasoning as the formulation of conjectures, examples and counterexamples, and arguments that allows validation of mathematical states.

On the subject of argumentation and proof in university-level mathematics education, Meyer (2020) expresses that several aspects need to be considered when analyzing how students formulate their arguments, such as the structure, the mathematical content involved, and the recipient-orientation. Furthermore, De Villiers (2010) states that it is essential for teachers to explore authentic and meaningful ways of incorporating experimentation and proof into mathematics education, to provide students with a deeper, more holistic insight into the nature of mathematics. In respect of this matter, as mentioned by De Villiers (2010), we understand that experimentation involves any intuitive, inductive, or analogical reasoning employed when:

a) Mathematical conjectures and/or statements are evaluated numerically, visually, graphically, diagrammatically, physically, kinesthetically, analogically, etc.;

b) Conjectures, generalizations or conclusions are made based on intuition or experience obtained through any of the above methods. (p.205)

In this regard, Liljedahl (2016) and Schoenfeld (2022, et al., 2016) state that it is fundamental to
technologies as a problem-solving tool allows students to explore mathematical tasks in different ways. In this sense, we argue that exposing students to a problem-solving approach where students can rely on digital technologies can foster crucial skills to elaborate conjectures, justify and communicate results, which promote their integration to university education.

Therefore, in this research, we aimed to characterize, through geometry activities based on a problem-solving approach (Polya, 1965; Schoenfeld, 1985) and mediated by the coordinated use of GeoGebra, the tools, skills and difficulties exhibited by undergraduate students as they work on geometric tasks that encouraged them to formulate conjectures, explore different ways to support and validate those conjectures and to communicate their results. The research seeks to address the question: To what extent do these geometric activities, mediated by the use of GeoGebra, promote the development of mathematical processes, such as formulating, supporting and validating conjectures, and elaborating mathematical arguments among undergraduate students?

## CONCEPTUAL FRAMEWORK

When students begin their university education, they face difficulties related to their lack of abstraction skills, inquiry reasoning, general sense of mathematical rigor, and non-routine problem-solving skills (Leviatan, 2008). This indicates that moving from a secondary-level of education to a university-level entails a process of transition and integration. Research on the secondary-tertiary transition in mathematics, has considered undergoing perspectives to explain and address the problems and stages that students experience during this period. In this matter, it has been found that some aspects influencing students during the secondary-tertiary transition are related to their previous experiences in learning mathematics, the quality of study strategies, the way of thinking in/for mathematics, and conceptual background about mathematics ideas (Rach & Heinze, 2016; Di Martino & Gregorio, 2019). In this sense, Schoenfeld (2022) highlights the importance of providing challenging activities to students, i.e., non-routine problems as well as generating educational environments that promote "learning to think in and with the discipline, internalizing the discipline's habits of mind and practices of doing mathematics". Thus, students need to have close encounters with processes linked to mathematical reasoning as the formulation of conjectures, examples and counterexamples, and arguments that allows validation of mathematical states.

On the subject of argumentation and proof in university-level mathematics education, Meyer (2020) expresses that several aspects need to be considered when analyzing how students formulate their arguments, such as the structure, the mathematical content involved, and the recipient-orientation. Furthermore, De Villiers (2010) states that it is essential for teachers to explore authentic and meaningful ways of incorporating experimentation and proof into mathematics education, to provide students with a deeper, more holistic insight into the nature of mathematics. In respect of this matter, as mentioned by De Villiers (2010), we understand that experimentation involves any intuitive, inductive, or analogical reasoning employed when:

a) Mathematical conjectures and/or statements are evaluated numerically, visually, graphically, diagrammatically, physically, kinesthetically, analogically, etc.;

b) Conjectures, generalizations or conclusions are made based on intuition or experience obtained through any of the above methods. (p.205)

In this regard, Liljedahl (2016) and Schoenfeld (2022, et al., 2016) state that it is fundamental to
create robust learning environments that support students in not only developing the qualified knowledge and processes underlying mathematical thinking but also fostering the development of a sense of agency and authority to make sense of mathematical objects and practices. Thus, we considered aspects related to TRU framework (Schoenfeld, 2014) to create powerful mathematics classrooms in which exist 1) Mathematical content; 2) Cognitive demand; 3) Access to mathematical content; 4) Agency, authority and identity and 5) Use of assessments. Moreover, Santos-Trigo (2019) highlights the importance of incorporating technologies such as Dynamic Geometry Systems as the foundation to represent, explore and extend mathematical tasks when designing a powerful mathematical classroom.

Thus, first, we provide a general description of the teaching practices carried out during our research and the task design implemented; then, we present a qualitative analysis of the processes set in motion by the participants. Regarding the data analysis, we identified the resources and heuristics (Schoenfeld, 1985), as well as the conceptual and procedural tools (Melhuish et al., 2022) that students activated during their problem-solving work. For this purpose, we considered the Authentic Mathematical Proof Activity (AMPA) theoretical framework proposed by Melhuish et al. (2022). Out of the ten procedural tools expressed by those authors, we sought to identify: 1) the refinement or analysis of a proof, a statement, or definition by focusing on the attainment of assumptions; 2) the process of translating informal ideas into formal or symbolic rhetorical forms; 3) the elaboration of analogies, i.e., the process of importing proofs, statements or concepts across different domains adapting them to new schemas; 4) the use of examples, such as using a specific and concrete representation of a statement, concept or proof that represents a class of objects; and 5) the elaboration of diagrams and visual representations of mathematical objects (statements, concepts or proofs) that capture structural properties.

## METHODOLOGY AND TASK DESIGN

The data gathered from this research resulted from the implementation of a geometry course with 30 first-year undergraduate students. The main objective of the course was to lead students to develop deductive reasoning through problem-solving activities through the use of a DGS (GeoGebra). This allowed them to make sense of the problem statements and explore, represent, and identify geometric relations between the mathematical objects involved through direct manipulation, namely dragging. Due to space constraints in this paper, we will describe the analysis of one team's work (E1) regarding one of the four implemented activities (T3). However, we will provide a brief discussion of all 15 teams. We summarize the methods followed during the tasks down below:

1. All tasks were presented as problems, whether they had a situational context or not. Tasks were adapted from rutinary textbook problems into open-ended problems that required the formulation of conjectures.

2. Students were then asked to validate their conjectures by generating an argument or giving a counterexample. Both the elaboration of the conjecture and its validation or refutation were expected to be done with the support of GeoGebra.

3. The activities were conducted by a teacher, one of the researchers, by guiding the participants' work through questions and prompts. These had the dual purpose of fostering reasoning and argumentation skills and encouraging exploration and understanding of mathematical concepts by manipulating dynamic models.
create robust learning environments that support students in not only developing the qualified knowledge and processes underlying mathematical thinking but also fostering the development of a sense of agency and authority to make sense of mathematical objects and practices. Thus, we considered aspects related to TRU framework (Schoenfeld, 2014) to create powerful mathematics classrooms in which exist 1) Mathematical content; 2) Cognitive demand; 3) Access to mathematical content; 4) Agency, authority and identity and 5) Use of assessments. Moreover, Santos-Trigo (2019) highlights the importance of incorporating technologies such as Dynamic Geometry Systems as the foundation to represent, explore and extend mathematical tasks when designing a powerful mathematical classroom.

Thus, first, we provide a general description of the teaching practices carried out during our research and the task design implemented; then, we present a qualitative analysis of the processes set in motion by the participants. Regarding the data analysis, we identified the resources and heuristics (Schoenfeld, 1985), as well as the conceptual and procedural tools (Melhuish et al., 2022) that students activated during their problem-solving work. For this purpose, we considered the Authentic Mathematical Proof Activity (AMPA) theoretical framework proposed by Melhuish et al. (2022). Out of the ten procedural tools expressed by those authors, we sought to identify: 1) the refinement or analysis of a proof, a statement, or definition by focusing on the attainment of assumptions; 2) the process of translating informal ideas into formal or symbolic rhetorical forms; 3) the elaboration of analogies, i.e., the process of importing proofs, statements or concepts across different domains adapting them to new schemas; 4) the use of examples, such as using a specific and concrete representation of a statement, concept or proof that represents a class of objects; and 5) the elaboration of diagrams and visual representations of mathematical objects (statements, concepts or proofs) that capture structural properties.

## METHODOLOGY AND TASK DESIGN

The data gathered from this research resulted from the implementation of a geometry course with 30 first-year undergraduate students. The main objective of the course was to lead students to develop deductive reasoning through problem-solving activities through the use of a DGS (GeoGebra). This allowed them to make sense of the problem statements and explore, represent, and identify geometric relations between the mathematical objects involved through direct manipulation, namely dragging. Due to space constraints in this paper, we will describe the analysis of one team's work (E1) regarding one of the four implemented activities (T3). However, we will provide a brief discussion of all 15 teams. We summarize the methods followed during the tasks down below:

1. All tasks were presented as problems, whether they had a situational context or not. Tasks were adapted from rutinary textbook problems into open-ended problems that required the formulation of conjectures.

2. Students were then asked to validate their conjectures by generating an argument or giving a counterexample. Both the elaboration of the conjecture and its validation or refutation were expected to be done with the support of GeoGebra.

3. The activities were conducted by a teacher, one of the researchers, by guiding the participants' work through questions and prompts. These had the dual purpose of fostering reasoning and argumentation skills and encouraging exploration and understanding of mathematical concepts by manipulating dynamic models.
4. Students had an average of three days to tackle the task and were permitted to work in pairs or individually, depending on the task; this allowed students to socialize and share their initial ideas.

5. After exploring each task, a face-to-face session was held to discuss questions, difficulties, conjectures, and possible initial arguments expressed by students. They were encouraged to share their work with the rest of the class to gain insight into their ideas and receive feedback from their peers and the teacher. The objectives of this type of session were to lead the students 1) to identify the concepts involved in the problem as well as weak resources or cognitive obstacles; 2) to identify the conditions given by the problem's statement and its relation to the problem's representation, 3) to decide and raise an initial statement of the proposed conjecture, and 4) to exchange possible strategies to approach the argumentation of the conjecture.

6. Students were asked to register their problem-solving processes, as well as questions, ideas, or actions that arose while tackling the task and during face-to-face sessions. This constituted the main source of data for this research. For this purpose, the teacher asked students to include in their registries the following elements:

* Description of all explorations oriented to understanding the problem statement and generating conjectures. For both data analysis and the statement of a conjecture, we sought to determine what concepts, strategies, processes and/or digital tools were activated by the students when exploring and understanding the problem.

* Description of the process followed for the proposal of a plan or strategy that leads to solving the problem. We sought to identify what resources (concepts, mathematical content, evidence, or previous results) students recognized as helpful when tackling the problem, how they activated them when they sought a solution strategy, and what type of inquiries they made.

* Description of the problem-solving process, as well as all types of algebraic or empirical procedures, diagrams, etc. We analyzed the types of arguments students used when they proved or validated their conjectures, the use of mathematical notation they exhibited, and the logical structure exhibited in their proofs.

* Problem extensions. We analyzed whether students could pose new or similar questions or problems to be solved based on what they had learned, explored innovative solutions and/or generalized their strategies. Through these questions, we aimed to encourage students to engage in the activity of problem-posing.

## Task T3

Now, we present the instructions and questions given to the students to guide their reasoning process in Task 3 (it is worth noting that the original language of the instructions was Spanish). Figure 1 shows the problem statement.

Instructions: The solution process for this problem must be thoroughly registered on a logbook. Section 1. Exploring and obtaining a conjecture: a) Create a model of the embedded figure in the problem using GeoGebra; then, explore the problem by manipulating the elements of your model using the available tools in GeoGebra; b) Register any questions you asked yourself during the construction of the model. Establish the assumptions given in the problem and those that you have
4. Students had an average of three days to tackle the task and were permitted to work in pairs or individually, depending on the task; this allowed students to socialize and share their initial ideas.

5. After exploring each task, a face-to-face session was held to discuss questions, difficulties, conjectures, and possible initial arguments expressed by students. They were encouraged to share their work with the rest of the class to gain insight into their ideas and receive feedback from their peers and the teacher. The objectives of this type of session were to lead the students 1) to identify the concepts involved in the problem as well as weak resources or cognitive obstacles; 2) to identify the conditions given by the problem's statement and its relation to the problem's representation, 3) to decide and raise an initial statement of the proposed conjecture, and 4) to exchange possible strategies to approach the argumentation of the conjecture.

6. Students were asked to register their problem-solving processes, as well as questions, ideas, or actions that arose while tackling the task and during face-to-face sessions. This constituted the main source of data for this research. For this purpose, the teacher asked students to include in their registries the following elements:

* *Description of all explorations oriented to understanding the problem statement and generating conjectures.* For both data analysis and the statement of a conjecture, we sought to determine what concepts, strategies, processes and/or digital tools were activated by the students when exploring and understanding the problem.

* *Description of the process followed for the proposal of a plan or strategy that leads to solving the problem.* We sought to identify what resources (concepts, mathematical content, evidence, or previous results) students recognized as helpful when tackling the problem, how they activated them when they sought a solution strategy, and what type of inquiries they made.

* *Description of the problem-solving process*, as well as all types of algebraic or empirical procedures, diagrams, etc. We analyzed the types of arguments students used when they proved or validated their conjectures, the use of mathematical notation they exhibited, and the logical structure exhibited in their proofs.

* *Problem extensions.* We analyzed whether students could pose new or similar questions or problems to be solved based on what they had learned, explored innovative solutions and/or generalized their strategies. Through these questions, we aimed to encourage students to engage in the activity of problem-posing.

## Task T3

Now, we present the instructions and questions given to the students to guide their reasoning process in Task 3 (it is worth noting that the original language of the instructions was Spanish). Figure 1 shows the problem statement.

Instructions: The solution process for this problem must be thoroughly registered on a logbook. *Section 1. Exploring and obtaining a conjecture:* a) Create a model of the embedded figure in the problem using GeoGebra; then, explore the problem by manipulating the elements of your model using the available tools in GeoGebra; b) Register any questions you asked yourself during the construction of the model. Establish the assumptions given in the problem and those that you have
considered for the construction. Write an initial guess (conjecture) about the answers to the questions stated in the problem; c) If you move the sides of the quadrilateral, does your guess still work? If you move point F along segment CD, is your conjecture still true? d) Considering the initial conditions that you think are necessary to solve the problem, determine a possible conjecture about the ratio of the shaded area to the non-shaded area. **Section 2. Elaboration of a solution plan:** a) Describe the mathematical concepts you consider are underlying in the formulation of your conjecture. Which of them do you master, and which don't? b) Use online platforms, videos, or books to do some research on those concepts. Add your query sources. c) Describe how these concepts or results could help you to solve the problem.

En la siguiente figura se muestra el cuadrilátero ABCD con BC paralelo a AD. Se traza el segmento EF donde E y F son puntos medios de AB y CD respectivamente. ¿Será posible establecer una relación entre el área sombreada y el área no sombreada? ¿Cuál sería esa relación?

<figure><img src="image_13.png" /></figure>

*Translation:* The following figure shows the quadrilateral ABCD with BC parallel to AD. Segment EF is drawn, with E and F being the midpoints of AB and CD, respectively. Is it possible to establish a relationship between the shaded and the non-shaded areas? What would that relationship be?

<caption>Figure 1: Problem corresponding to Task T3</caption>

*Section 3. Development of an argument and solution:* a) Describe an argument that allows you to validate the conjecture you have established. If you realize that your conjecture is false, present a counterexample and then seek to establish a new conjecture and validate it. Repeat this process until you can find an argument (or proof) that shows that your conjecture is true. *Section 4. Extension of the problem:* a) Pose at least another way to solve the problem, a new similar problem, or question that you could solve based on what you found out during this problem.

Analyzing these elements, we derive several results regarding the reasoning methods and approaches students employed to solve problems and support statements.

## ANALYSIS AND FINDINGS ON TASK 3

We now show the analysis of the types of reasoning and problem-solving processes followed by the team E1 when arguing the validation of their statements. We also provide some insight on the difficulties or obstacles faced by students and how they overcame them.

Team E1 initially approached the problem by employing paper drawings despite the problem indicating the use of GeoGebra. Realizing that this method complicated their exploration, the students subsequently shifted to using GeoGebra to construct the figure and represent the given problem and its conditions. Then, students manipulated (dragged) points along parallel sides, leading to the identification of various triangles with similar properties. They further posed the question "What happens when the points are evenly distributed on the parallel lines?" and explored potential changes to the figure by dragging its vertices, examining how their initial ideas were influenced. In this sense, manipulating the model was important for students to identify the existence of similar triangles. Then, students constructed rectangles by focusing on the points along the AD, BC and EF segments,
considered for the construction. Write an initial guess (conjecture) about the answers to the questions stated in the problem; c) If you move the sides of the quadrilateral, does your guess still work? If you move point F along segment CD, is your conjecture still true? d) Considering the initial conditions that you think are necessary to solve the problem, determine a possible conjecture about the ratio of the shaded area to the non-shaded area. **Section 2. Elaboration of a solution plan:** a) Describe the mathematical concepts you consider are underlying in the formulation of your conjecture. Which of them do you master, and which don't? b) Use online platforms, videos, or books to do some research on those concepts. Add your query sources. c) Describe how these concepts or results could help you to solve the problem.

<figure><img src="image_12.png" /><figcaption>Figure 1: Problem corresponding to Task T3</figcaption></figure>

*Section 3. Development of an argument and solution:* a) Describe an argument that allows you to validate the conjecture you have established. If you realize that your conjecture is false, present a counterexample and then seek to establish a new conjecture and validate it. Repeat this process until you can find an argument (or proof) that shows that your conjecture is true. *Section 4. Extension of the problem:* a) Pose at least another way to solve the problem, a new similar problem, or question that you could solve based on what you found out during this problem.

Analyzing these elements, we derive several results regarding the reasoning methods and approaches students employed to solve problems and support statements.

## ANALYSIS AND FINDINGS ON TASK 3

We now show the analysis of the types of reasoning and problem-solving processes followed by the team E1 when arguing the validation of their statements. We also provide some insight on the difficulties or obstacles faced by students and how they overcame them.

Team E1 initially approached the problem by employing paper drawings despite the problem indicating the use of GeoGebra. Realizing that this method complicated their exploration, the students subsequently shifted to using GeoGebra to construct the figure and represent the given problem and its conditions. Then, students manipulated (dragged) points along parallel sides, leading to the identification of various triangles with similar properties. They further posed the question “What happens when the points are evenly distributed on the parallel lines?” and explored potential changes to the figure by dragging its vertices, examining how their initial ideas were influenced. In this sense, manipulating the model was important for students to identify the existence of similar triangles. Then, students constructed rectangles by focusing on the points along the AD, BC and EF segments,
viewing it as a helpful approach to reaching conclusions for the given problem (Figure 2a). Their initial state was: "The shaded area is 1/4 of the total area". Afterward, they utilized the measure area tool in GeoGebra to compare the areas of the shaded and unshaded triangles, along with the entire rectangle ABCD (Figure 2b) as they formulated the conjecture: "[In rectangle ABCD] The shaded area is 1/4 of the total area, and the unshaded area is 3/4 of the total. The unshaded area is three times the shaded area." This approach allowed them to visually confirm the validity of their conjecture and state verbally that the shaded area is invariant to the position of the points on AD and BC as long as the shaded regions do not intersect each other. To support the validity of their statement, students employed pivotal concepts from the exploration and understanding phase, such as parallel lines and angle relationships, midpoint of a line segment and similarity. They considered angles formed by transversal lines (BG, GH, HI, IC) intersecting parallel lines (AD, EF, BC) (Figure 2b). This approach led to the establishment of triangle similarity. Consequently, they applied Thales's first theorem along with the area formulas for triangles and rectangles. Using these methods, they conducted a detailed analysis, comparing the areas of the shaded and unshaded triangles to the overall area of the rectangle, which allowed them to support their conjecture (Figure 2c). Although the students successfully argued their conjecture, it is evident that there is room for improvement in the use of mathematical notation, particularly regarding the establishment of similar triangles and the ratio of areas. Enhancing this notation could lead to a more concise and effective argument. Finally, team E1 did not explicitly pose a new problem or additional questions related to that posed initially.

<figure><img src="image_14.png" /><figcaption>Figure 2: Illustrations of the exploration processes followed by team E1</figcaption></figure>

In general terms, the students used GeoGebra to construct the figure and represent the problem and its conditions. They established initial conditions and made plausible conjectures about the type of quadrilateral involved and its area. During exploration, common questions were raised about point distribution on parallel lines and the impact on the quadrilateral when it is a parallelogram without restrictions, for example: "What happens when the points are evenly distributed on the parallel lines? What happens if there are no restrictions on the points but it [the quadrilateral] is a parallelogram?" While 80% of teams shared similar conjectures, variations on hypotheses existed; for example, Team E2 generalized to the parallelogram ABCD instead of the rectangle. Following a feedback session,
viewing it as a helpful approach to reaching conclusions for the given problem (Figure 2a). Their initial state was: “The shaded area is 1/4 of the total area”. Afterward, they utilized the measure area tool in GeoGebra to compare the areas of the shaded and unshaded triangles, along with the entire rectangle ABCD (Figure 2b) as they formulated the conjecture: “[In rectangle ABCD] The shaded area is 1/4 of the total area, and the unshaded area is 3/4 of the total. ∴ The unshaded area is three times the shaded area.” This approach allowed them to visually confirm the validity of their conjecture and state verbally that the shaded area is invariant to the position of the points on AD and BC as long as the shaded regions do not intersect each other. To support the validity of their statement, students employed pivotal concepts from the exploration and understanding phase, such as parallel lines and angle relationships, midpoint of a line segment and similarity. They considered angles formed by transversal lines (BG, GH, HI, IC) intersecting parallel lines (AD, EF, BC) (Figure 2b). This approach led to the establishment of triangle similarity. Consequently, they applied Thales’s first theorem along with the area formulas for triangles and rectangles. Using these methods, they conducted a detailed analysis, comparing the areas of the shaded and unshaded triangles to the overall area of the rectangle, which allowed them to support their conjecture (Figure 2c). Although the students successfully argued their conjecture, it is evident that there is room for improvement in the use of mathematical notation, particularly regarding the establishment of similar triangles and the ratio of areas. Enhancing this notation could lead to a more concise and effective argument. Finally, team E1 did not explicitly pose a new problem or additional questions related to that posed initially.

<figure><img src="image_13.png" /><figcaption>Figure 2: Illustrations of the exploration processes followed by team E1</figcaption></figure>

In general terms, the students used GeoGebra to construct the figure and represent the problem and its conditions. They established initial conditions and made plausible conjectures about the type of quadrilateral involved and its area. During exploration, common questions were raised about point distribution on parallel lines and the impact on the quadrilateral when it is a parallelogram without restrictions, for example: “What happens when the points are evenly distributed on the parallel lines? What happens if there are no restrictions on the points but it [the quadrilateral] is a parallelogram?” While 80% of teams shared similar conjectures, variations on hypotheses existed; for example, Team E2 generalized to the parallelogram ABCD instead of the rectangle. Following a feedback session,
students collaboratively, along with the teacher, used GeoGebra to represent and discuss the
generalization or particularization of the conjecture, as well as visual principles for argumentation.
The manipulation of the model played a crucial role, enabling students to articulate a more structured,
albeit not entirely rigorous, reasoning.

Summarizing findings, we assert that: 1) students developed skills for analyzing statements by identifying assumptions; 2) the process of translating informal ideas into formal or symbolic rhetorical forms requires more activities and guidance from teachers to develop this particular skill; 3) students applied previous concepts and statements to this new problem; 4) they constructed and manipulated a dynamic and concrete representation to explore the problem and concepts related to stating a conjecture, allowing them to actively participate into the processes of argumentation; and 5) they elaborated diagrams and visual representations of the proposed statement and related concepts, capturing their structural properties.

CONCLUSIONS

The teaching dynamics aimed to provide students with equal access to geometric content through
course notes, suggested bibliography, interactive GeoGebra applets, and the opportunity for online
research. The problems and questions posed prompted students to generate conjectures, explore
different ways to validate them, and consequently develop the ability to argue rather than solely rely
on predetermined solutions. Group discussions fostered individual and collaborative engagement
with the activities. While it is observed that students improved their reasoning and argumentation
skills through the activities, it is also evident that this development is a gradual process that may
evoke feelings of confusion and frustration, particularly due to their familiarity with routine problems.
Furthermore, it has been noted that there is a lack of motivation for students to formulate their
problems, whether derived from those suggested by the teacher or independently conceived.
Therefore, the role of the teacher is considered crucial in guiding and providing feedback throughout
the monitoring and control processes.

Given this, the dynamics in mathematics courses must provide activities that bring students closer to
the realization of activities of a high mathematical level and empower students to recognize their
mathematical resources and to make decisions on applying them in problem-solving. This entails
manipulating auxiliary tools, namely technologies, to explore and construct mathematical objects and
formulate conjectures. Creating spaces that cultivate oral and written arguments, using mathematical
notation and symbology, and refining proofs are essential aspects of developing geometrical
mathematical thinking guided by a logical sequence that progressively leads students toward the
processes of abstraction and demonstration. Such practices strengthen mathematical reasoning,
particularly during the transition from high school to university.

Finally, developing course content based on non-routine problem-solving or initial questions posed by the teacher, requiring students to conduct preliminary exploration mediated by the use of a GDS like GeoGebra during class, promotes: 1) the exploration and manipulation of geometrical models, as well as the understanding of the problem and mathematical content; 2) the formulation of conjectures; 3) the visual validation or generation of visual geometrical counterexamples to initial conjectures; and 4) the development of many possible solution strategies. By engaging in these processes, the structuring of arguments and proofs for the conjectures is facilitated, contributing to
students collaboratively, along with the teacher, used GeoGebra to represent and discuss the
generalization or particularization of the conjecture, as well as visual principles for argumentation.
The manipulation of the model played a crucial role, enabling students to articulate a more structured,
albeit not entirely rigorous, reasoning.

Summarizing findings, we assert that: 1) students developed skills for analyzing statements by identifying assumptions; 2) the process of translating informal ideas into formal or symbolic rhetorical forms requires more activities and guidance from teachers to develop this particular skill; 3) students applied previous concepts and statements to this new problem; 4) they constructed and manipulated a dynamic and concrete representation to explore the problem and concepts related to stating a conjecture, allowing them to actively participate into the processes of argumentation; and 5) they elaborated diagrams and visual representations of the proposed statement and related concepts, capturing their structural properties.

CONCLUSIONS

The teaching dynamics aimed to provide students with equal access to geometric content through course notes, suggested bibliography, interactive GeoGebra applets, and the opportunity for online research. The problems and questions posed prompted students to generate conjectures, explore different ways to validate them, and consequently develop the ability to argue rather than solely rely on predetermined solutions. Group discussions fostered individual and collaborative engagement with the activities. While it is observed that students improved their reasoning and argumentation skills through the activities, it is also evident that this development is a gradual process that may evoke feelings of confusion and frustration, particularly due to their familiarity with routine problems. Furthermore, it has been noted that there is a lack of motivation for students to formulate their problems, whether derived from those suggested by the teacher or independently conceived. Therefore, the role of the teacher is considered crucial in guiding and providing feedback throughout the monitoring and control processes.

Given this, the dynamics in mathematics courses must provide activities that bring students closer to
the realization of activities of a high mathematical level and empower students to recognize their
mathematical resources and to make decisions on applying them in problem-solving. This entails
manipulating auxiliary tools, namely technologies, to explore and construct mathematical objects and
formulate conjectures. Creating spaces that cultivate oral and written arguments, using mathematical
notation and symbology, and refining proofs are essential aspects of developing geometrical
mathematical thinking guided by a logical sequence that progressively leads students toward the
processes of abstraction and demonstration. Such practices strengthen mathematical reasoning,
particularly during the transition from high school to university.

Finally, developing course content based on non-routine problem-solving or initial questions posed by the teacher, requiring students to conduct preliminary exploration mediated by the use of a GDS like GeoGebra during class, promotes: 1) the exploration and manipulation of geometrical models, as well as the understanding of the problem and mathematical content; 2) the formulation of conjectures; 3) the visual validation or generation of visual geometrical counterexamples to initial conjectures; and 4) the development of many possible solution strategies. By engaging in these processes, the structuring of arguments and proofs for the conjectures is facilitated, contributing to
the development of robust reasoning within the mathematics field. It was observed that students were afforded opportunities to actively use diverse mathematical skills and explore various approaches to construct compelling arguments, substantiating the relevance and validity of their conjectures. The results indicate that students extended their ways of reasoning, enabling them to transition from empirical to formal arguments in presenting the solutions to the problems.

## Acknowledgements

The authors wish to thank the Department of Mathematics, Faculty of Sciences, National Autonomous University of Mexico (UNAM), and their colleagues for making this study possible by providing funding and the space for data collection.

## References

De Villiers, M. (2010). Experimentation and proof in mathematics. In G. Hanna, H.N. Jahnke, & H. Pulte (Eds.). *Explanation and proof in mathematics: Philosophical and educational perspectives*, 205-221.
https://doi.org/10.1007/978-1-4419-0576-5_14

Di Martino, P., Gregorio, F. (2019). The mathematical crisis in secondary-tertiary transition. *International Journal of Science and Mathematics Education*, 17, 825-843.
https://doi.org/10.1007/s10763-018-9894-y

Leviatan, T. (2008). Bridging a cultural gap. *Mathematics Education Research Journal*, 20, 105-116.
https://doi.org/10.1007/BF03217480

Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick y E. Pehkonen (Eds.). *Posing and solving mathematical problems: Advances and new perspectives*, 361-386.
https://doi.org/10.1007/978-3-319-28023-3_21

Melhuish, K., Vroom, K., Lew, K., & Ellis, B. (2022). Operationalizing authentic mathematical proof activity using disciplinary tools. *Journal of Mathematical Behavior*, 68.
https://doi.org/10.1016/j.jmathb.2022.101009

Meyer, M., Schnell, S. (2020). What counts as a “good” argument in school? How teachers grade students’ mathematical arguments. *Educational Studies in Mathematics*, 105, 35-51.
https://doi.org/10.1007/s10649-020-09974-z

Polya, G. (1965). *Cómo plantear y resolver problemas*. (J. Zugazagoitia, Trans.). Trillas.

Rach, S., & Heinze, A. (2016). The transition from school to university in mathematics: Which influence do school-related variables have? *International Journal of Science and Mathematics Education*, 15, 1343-1363.
https://doi.org/10.1007/s10763-016-9744-8

Santos-Trigo, M. (2019). Mathematical problem solving and the use of digital technologies. In: Liljedahl, P., Santos-Trigo, M. (Eds.). *Mathematical Problem Solving. ICME-13 Monographs*. Springer.
https://doi.org/10.1007/978-3-030-10472-6_4

Schoenfeld, A. H. (1985). *Mathematical Problem Solving*. Academic Press.

Schoenfeld, A. H. (2014). What makes for powerful classrooms, and how can we support teachers in creating them? A story of research and practice, productively intertwined. *Educational Researcher*, 43(8), 404-412.
https://doi.org/10.3102/0013189X14554450

Schoenfeld, A. H., Thomas, M., & Barton, B. (2016). On understanding and improving the teaching of university mathematics. *International Journal of STEM Education* 3(4).
https://doi.org/10.1186/s40594-016-0038-z

Schoenfeld, A. H. (2022). Why are learning and teaching mathematics so difficult? *In Handbook of Cognitive Mathematics*, 1-35. Springer.
https://doi.org/10.1007/978-3-030-44982-7_10-1
the development of robust reasoning within the mathematics field. It was observed that students were afforded opportunities to actively use diverse mathematical skills and explore various approaches to construct compelling arguments, substantiating the relevance and validity of their conjectures. The results indicate that students extended their ways of reasoning, enabling them to transition from empirical to formal arguments in presenting the solutions to the problems.

## Acknowledgements

The authors wish to thank the Department of Mathematics, Faculty of Sciences, National Autonomous University of Mexico (UNAM), and their colleagues for making this study possible by providing funding and the space for data collection.

## References

De Villiers, M. (2010). Experimentation and proof in mathematics. In G. Hanna, H.N. Jahnke, & H. Pulte (Eds.). *Explanation and proof in mathematics: Philosophical and educational perspectives*, 205-221.
https://doi.org/10.1007/978-1-4419-0576-5_14

Di Martino, P., Gregorio, F. (2019). The mathematical crisis in secondary-tertiary transition. *International Journal of Science and Mathematics Education*, 17, 825-843.
https://doi.org/10.1007/s10763-018-9894-y

Leviatan, T. (2008). Bridging a cultural gap. *Mathematics Education Research Journal*, 20, 105-116.
https://doi.org/10.1007/BF03217480

Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick y E. Pehkonen (Eds.). *Posing and solving mathematical problems: Advances and new perspectives*, 361-386.
https://doi.org/10.1007/978-3-319-28023-3_21

Melhuish, K., Vroom, K., Lew, K., & Ellis, B. (2022). Operationalizing authentic mathematical proof activity using disciplinary tools. *Journal of Mathematical Behavior*, 68.
https://doi.org/10.1016/j.jmathb.2022.101009

Meyer, M., Schnell, S. (2020). What counts as a “good” argument in school? How teachers grade students’ mathematical arguments. *Educational Studies in Mathematics*, 105, 35-51.
https://doi.org/10.1007/s10649-020-09974-z

Polya, G. (1965). *Cómo plantear y resolver problemas*. (J. Zugazagoitia, Trans.). Trillas.

Rach, S., & Heinze, A. (2016). The transition from school to university in mathematics: Which influence do school-related variables have? *International Journal of Science and Mathematics Education*, 15, 1343-1363.
https://doi.org/10.1007/s10763-016-9744-8

Santos-Trigo, M. (2019). Mathematical problem solving and the use of digital technologies. In: Liljedahl, P., Santos-Trigo, M. (Eds.). *Mathematical Problem Solving. ICME-13 Monographs*. Springer.
https://doi.org/10.1007/978-3-030-10472-6_4

Schoenfeld, A. H. (1985). *Mathematical Problem Solving*. Academic Press.

Schoenfeld, A. H. (2014). What makes for powerful classrooms, and how can we support teachers in creating them? A story of research and practice, productively intertwined. *Educational Researcher*, 43(8), 404-412.
https://doi.org/10.3102/0013189X14554450

Schoenfeld, A. H., Thomas, M., & Barton, B. (2016). On understanding and improving the teaching of university mathematics. *International Journal of STEM Education* 3(4).
https://doi.org/10.1186/s40594-016-0038-z

Schoenfeld, A. H. (2022). Why are learning and teaching mathematics so difficult? *In Handbook of Cognitive Mathematics*, 1-35. Springer.
https://doi.org/10.1007/978-3-030-44982-7_10-1
# SPATIAL REASONING IN AUTHENTIC CONTEXTS OF AN ENGINEERING CHALLENGE: TAPPING INTO CHILDREN'S INTUITIVE UNDERSTANDING OF RELATIONSHIPS BETWEEN OBJECTS AND SELF

Viktor Freiman¹, Olga Fellus²

¹ Université de Moncton ² Brock University

We report on results of a study featuring kindergarten students solving an engineering challenge of constructing a shelter for a stuffed animal using carton materials and tools in a makerspace. Our analysis focuses on spatial reasoning (SR) skills that were manifested in the children's intuitive exploration of space and shapes as they were planning, realizing, testing, and adjusting the shelters they constructed. The construction context allowed the children to enact creativity and flexible thinking through spatial reasoning. We suggest that makerspaces contexts carry promising ways to surface SR. We underscore the need to make explicit the mathematics employed so that this can be recognized by teachers, students, and parents. We offer a four-part model as a working mechanism. Questions remain, however, pertaining to connections of SR in engineering tasks with aspects such as affect, materiality, agency, authorial identity, accessibility to high-ceiling mathematical concepts and the socio-ecological turn in mathematics education.

## CONTEXT AND RESEARCH FOCUS

Along with other places around the world, school makerspaces are taking ground in New Brunswick (NB), Canada, K-12, emphasizing STEM education (NBED, 2016). Several non-profit organizations and charity associations in NB provide much-needed support and assistance to teachers in developing a variety of making activities to foster students' creativity, innovation, computational skills, coding capacity, and an entrepreneurial spirit across grade levels and educational curricula, both in formal and informal contexts. Among these organizations, Brilliant Labs, had initiated a research collaboration with the CompeTI.CA (Compétences en TIC en Atlantique/ICT competences in the Atlantic Canada) and had set up a partnership network team to conduct case studies of school makerspaces (Freiman, 2020).

Overall, CompeTI.CA's main objectives are three pronged: (1) to identify information and communication technological competencies continuum; (2) to research exemplary practices (case studies) in school makerspaces; and (3) to develop new practices for teachers to use. To respond to the call of the 26th ICMI Study on Advances in Geometry Education, we aim to zoom in and out on kindergarten students' intuitive SR during their work on an engineering challenge at the school's STEM Lab (makerspace). In the next section, we present key concepts that frame our work.

## CONCEPTUAL FRAMEWORK

### An engineering challenge and design in making activities

Learning activities that provide students with engineering challenges and design opportunities are considered as beneficial in supporting and further developing modeling, discourse, and natural
# SPATIAL REASONING IN AUTHENTIC CONTEXTS OF AN ENGINEERING CHALLENGE: TAPPING INTO CHILDREN'S INTUITIVE UNDERSTANDING OF RELATIONSHIPS BETWEEN OBJECTS AND SELF

Viktor Freiman¹, Olga Fellus²

¹ Université de Moncton ² Brock University

We report on results of a study featuring kindergarten students solving an engineering challenge of constructing a shelter for a stuffed animal using carton materials and tools in a makerspace. Our analysis focuses on spatial reasoning (SR) skills that were manifested in the children's intuitive exploration of space and shapes as they were planning, realizing, testing, and adjusting the shelters they constructed. The construction context allowed the children to enact creativity and flexible thinking through spatial reasoning. We suggest that makerspaces contexts carry promising ways to surface SR. We underscore the need to make explicit the mathematics employed so that this can be recognized by teachers, students, and parents. We offer a four-part model as a working mechanism. Questions remain, however, pertaining to connections of SR in engineering tasks with aspects such as affect, materiality, agency, authorial identity, accessibility to high-ceiling mathematical concepts and the socio-ecological turn in mathematics education.

## CONTEXT AND RESEARCH FOCUS

Along with other places around the world, school makerspaces are taking ground in New Brunswick (NB), Canada, K-12, emphasizing STEM education (NBED, 2016). Several non-profit organizations and charity associations in NB provide much-needed support and assistance to teachers in developing a variety of making activities to foster students' creativity, innovation, computational skills, coding capacity, and an entrepreneurial spirit across grade levels and educational curricula, both in formal and informal contexts. Among these organizations, Brilliant Labs, had initiated a research collaboration with the CompeTI.CA (Compétences en TIC en Atlantique/ICT competences in the Atlantic Canada) and had set up a partnership network team to conduct case studies of school makerspaces (Freiman, 2020).

Overall, CompeTI.CA's main objectives are three pronged: (1) to identify information and communication technological competencies continuum; (2) to research exemplary practices (case studies) in school makerspaces; and (3) to develop new practices for teachers to use. To respond to the call of the 26th ICMI Study on Advances in Geometry Education, we aim to zoom in and out on kindergarten students' intuitive SR during their work on an engineering challenge at the school's STEM Lab (makerspace). In the next section, we present key concepts that frame our work.

## CONCEPTUAL FRAMEWORK

### An engineering challenge and design in making activities

Learning activities that provide students with engineering challenges and design opportunities are considered as beneficial in supporting and further developing modeling, discourse, and natural
scaffolding processes in which students collaborate with their peers to solve real-life problems (DeJarnette, 2012). In these activities, students develop understanding through interaction and observation of their environment (Vygotsky, 1978). Design projects help students to see connections between science concepts and solutions to real-world problems (Sadler et al., 2014). In mathematics, LeBlanc et al. (2022) documented rich mathematical connections in students’ work in makerspaces. Similarly, Bush et al. (2022) suggest that teaching through design is learner-driven and goal-oriented, provides authentic contexts, incorporates constraints, and allows for use of a variety of materials, resources, and tools. Teaching through design involves teamwork and also requires the use of tangible artifacts and discrete processes to suggest varied solutions (Sluis-Thiescheffer et al., 2016).

Among several types of materials and technologies available to support design activities, Deed et al. (2022) provide several arguments for the *use of cardboard* as it supports embodied experiences; creative and imaginative play; students’ wellbeing—social emotional and physical safety; a process of idea generation; and reconfiguring classroom spaces to support engagement and productivity through construction processes to resolve an engineering challenge. These are qualitatively and fundamentally connected to the development of spatial reasoning (Omundsen, 2014).

Hatzigianni et al. (2021) conceptualize makerspaces as workspaces where open-ended resources are used as enablers of learning through integration of the elements of design, creation, and innovation that position young children as equally capable makers. The maker mindset is promoted in Papert’s theory of constructionism, which emphasizes child-led activities, where thinking processes are more valuable and critical than end-products, and where exploration and experimentation are positioned as core to the learning process. Constructionist thinking draws on constructivism, another learning theory that considers how learners construct knowledge through interaction with experience and ideas (Hatzigianni et al., 2021). Heroman (2017) emphasizes the open-ended tinkering in making experiences. That is, as children mature, their ability to use tools, collaborate with others, experiment, observe, make discoveries, tap into prior knowledge, communicate, and persevere continue to develop and flourish. These are key developmental experiences that are made available to learners working in contexts of makerspaces. Hughes et al. (2019) offer a model for a design thinking cycle (ask, imagine, plan, create, improve; Figure 1), which we used to frame manifestations of SR (Ramey et al., 2018)—a construct that we will examine in the next sections.

<figure><img src="image_15.png" /><figcaption>Figure 1. A design thinking cycle (Hughes et al., 2019)</figcaption></figure>
scaffolding processes in which students collaborate with their peers to solve real-life problems (DeJarnette, 2012). In these activities, students develop understanding through interaction and observation of their environment (Vygotsky, 1978). Design projects help students to see connections between science concepts and solutions to real-world problems (Sadler et al., 2014). In mathematics, LeBlanc et al. (2022) documented rich mathematical connections in students’ work in makerspaces. Similarly, Bush et al. (2022) suggest that teaching through design is learner-driven and goal-oriented, provides authentic contexts, incorporates constraints, and allows for use of a variety of materials, resources, and tools. Teaching through design involves teamwork and also requires the use of tangible artifacts and discrete processes to suggest varied solutions (Sluis-Thiescheffer et al., 2016).

Among several types of materials and technologies available to support design activities, Deed et al. (2022) provide several arguments for the *use of cardboard* as it supports embodied experiences; creative and imaginative play; students’ wellbeing—social emotional and physical safety; a process of idea generation; and reconfiguring classroom spaces to support engagement and productivity through construction processes to resolve an engineering challenge. These are qualitatively and fundamentally connected to the development of spatial reasoning (Omundsen, 2014).

Hatzigianni et al. (2021) conceptualize makerspaces as workspaces where open-ended resources are used as enablers of learning through integration of the elements of design, creation, and innovation that position young children as equally capable makers. The maker mindset is promoted in Papert’s theory of constructionism, which emphasizes child-led activities, where thinking processes are more valuable and critical than end-products, and where exploration and experimentation are positioned as core to the learning process. Constructionist thinking draws on constructivism, another learning theory that considers how learners construct knowledge through interaction with experience and ideas (Hatzigianni et al., 2021). Heroman (2017) emphasizes the open-ended tinkering in making experiences. That is, as children mature, their ability to use tools, collaborate with others, experiment, observe, make discoveries, tap into prior knowledge, communicate, and persevere continue to develop and flourish. These are key developmental experiences that are made available to learners working in contexts of makerspaces. Hughes et al. (2019) offer a model for a design thinking cycle (ask, imagine, plan, create, improve; Figure 1), which we used to frame manifestations of SR (Ramey et al., 2018)—a construct that we will examine in the next sections.

<figure><img src="image_14.png" /><figcaption>Figure 1. A design thinking cycle (Hughes et al., 2019)</figcaption></figure>
Spatial reasoning: What is it? And why should it be developed from early grades?

Being considered as a critical aspect of intelligence (Smith, 1964), SR involves everything from how we perceive the world, to locating and navigating within it, to imagining, manipulating, and transforming two- and three-dimensional figures. SR is malleable and can be refined and improved upon regardless of age or gender (Stepankova et al., 2014) while tapping into “the ability to recognize and (mentally) manipulate the spatial properties of objects and the spatial relations among objects” (Bruce et al., 2017, p. 147). In mathematics curricula, SR is, more often than not, assigned to the geometry strand. The issue with this is that in spite of its critical role in the development of the learner, teachers find themselves less equipped to teach geometry (Moss et al., 2015)—and by extension SR.

SR grounds “relations among two or more objects or between one’s moving body and objects or landmarks in the environment” (Davis & the Spatial Reasoning Study Group, 2015, p. 5). It includes transformations in 2D and 3D shapes, angles, and direction (Mulligan et al., 2018) and takes place through interactive processes by locating, orienting, comparing, scaling, re/de/composing objects, finding/creating symmetry, rotating, classifying, balancing, diagramming, transforming, navigating, and sensing (Mulligan et al., 2018). SR can also “be practiced with limited or no use of the eyes—with the hands, with the moving body and gestures” (Whiteley et al., 2015, p. 11).

But how can SR be noticed and identified and why should we care about SR? Several studies shed light on important aspects of SR that can be noticed in a variety of out-of-school situations, and further enhanced in school situations (McCluskey et al., 2018). For instance, at a very young age (K-2), children can already develop very robust understandings of parallel lines (Sinclair & Bruce, 2014), which can be built upon in teaching/learning geometrical properties. In terms of learning outcomes, SR can be associated with mathematics performance (Hawes et al., 2022). In addition, those with developed SR are more likely to enter and succeed in STEM (Wai et al., 2009).

Pulling all these threads together, we recognize the compatibility of and interdependence between different conditions for SR to be noticed and worked on. We follow Latour’s (1990) idea of objects as actants to suggest that as much as humans enact manipulations and transformations on objects, objects as well enact their affordances to direct the ways they are used. In this regard, we see discourse patterns, instructional approaches, and teaching strategies as non-reified actants. To us, the entry point to noticing manifestations of SR is located at the context-specific, situated arrangement of objects both reified and non-reified. The idea of the situatedness of SR is then a manifestation of the allowable, reproducible, imitable, transferable, yet contextually bounded spatial-related actions and reactions. Understanding SR as context dependent, relational, goal oriented, and interactive (Figure 2) helps us identify and potentially expand context-dependent affordances (Das & Winter, 2016).

<figure><img src="image_16.png" /><figcaption>Figure 2: Elements of spatial reasoning</figcaption></figure>
Spatial reasoning: What is it? And why should it be developed from early grades?

Being considered as a critical aspect of intelligence (Smith, 1964), SR involves everything from how we perceive the world, to locating and navigating within it, to imagining, manipulating, and transforming two- and three-dimensional figures. SR is malleable and can be refined and improved upon regardless of age or gender (Stepankova et al., 2014) while tapping into “the ability to recognize and (mentally) manipulate the spatial properties of objects and the spatial relations among objects” (Bruce et al., 2017, p. 147). In mathematics curricula, SR is, more often than not, assigned to the geometry strand. The issue with this is that in spite of its critical role in the development of the learner, teachers find themselves less equipped to teach geometry (Moss et al., 2015)—and by extension SR.

SR grounds “relations among two or more objects or between one’s moving body and objects or landmarks in the environment” (Davis & the Spatial Reasoning Study Group, 2015, p. 5). It includes transformations in 2D and 3D shapes, angles, and direction (Mulligan et al., 2018) and takes place through interactive processes by locating, orienting, comparing, scaling, re/de/composing objects, finding/creating symmetry, rotating, classifying, balancing, diagramming, transforming, navigating, and sensing (Mulligan et al., 2018). SR can also “be practiced with limited or no use of the eyes—with the hands, with the moving body and gestures” (Whiteley et al., 2015, p. 11).

But how can SR be noticed and identified and why should we care about SR? Several studies shed light on important aspects of SR that can be noticed in a variety of out-of-school situations, and further enhanced in school situations (McCluskey et al., 2018). For instance, at a very young age (K-2), children can already develop very robust understandings of parallel lines (Sinclair & Bruce, 2014), which can be built upon in teaching/learning geometrical properties. In terms of learning outcomes, SR can be associated with mathematics performance (Hawes et al., 2022). In addition, those with developed SR are more likely to enter and succeed in STEM (Wai et al., 2009).

Pulling all these threads together, we recognize the compatibility of and interdependence between different conditions for SR to be noticed and worked on. We follow Latour’s (1990) idea of objects as actants to suggest that as much as humans enact manipulations and transformations on objects, objects as well enact their affordances to direct the ways they are used. In this regard, we see discourse patterns, instructional approaches, and teaching strategies as non-reified actants. To us, the entry point to noticing manifestations of SR is located at the context-specific, situated arrangement of objects both reified and non-reified. The idea of the situatedness of SR is then a manifestation of the allowable, reproducible, imitable, transferable, yet contextually bounded spatial-related actions and reactions. Understanding SR as context dependent, relational, goal oriented, and interactive (Figure 2) helps us identify and potentially expand context-dependent affordances (Das & Winter, 2016).

<figure><img src="image_15.png" /><figcaption>Figure 2: Elements of spatial reasoning</figcaption></figure>
SR is an important part of the NB K-12 mathematics curriculum and is included in the *Geometry Strand* (MEDPENB, 2016). Over the past decades, the STEM movement pushed center stage new opportunities for noticing and fostering spatial reasoning through more informal contexts of school makerspaces (Freiman, 2020). The question we investigate in this context is What aspects of SR emerge when kindergarten students work on an engineering challenge design task?

## RESEARCH SETTING AND METHODS

As part of a bigger case study of school makerspaces, we observed two groups of 20 students and their classroom teachers from a K-5 elementary school working in a STEM Lab (makerspace) in an urban area in NB, Canada. After obtaining ethics approval and parents' consent, the first author entered the Lab to video record students' work and conduct post-project interviews with some students. Our analysis focuses on one group working on *building a shelter for a stuffed animal*. The shelter needed *to be waterproof and protect the animal from high winds*. Twelve video-segments were first analyzed on the presence of spatial reasoning as per the actions listed above. Then initial codes were assigned to each video-segment. We then identified aspects across all segments.

Observing the children as they were working on constructing their shelter, we realized that we needed to adjust Hughes et al.'s (2019) design thinking cycle to reflect the design phases the children manifested in their collaborative work. We created a three-phase process: Planning → Realization → Testing and Adjusting (Figure 3).

<figure><img src="image_17.png" /><figcaption>Figure 3: Interconnected cogs in makerspaces (Fellus, Freiman, & Lurette, forthcoming)</figcaption></figure>

To better understand how the children engaged with SR, the following overarching questions guided our analysis that align with the three stages of planning, realization, and testing and adjusting: 1) What aspects of SR do children manifest in planning their work? 2) What elements of SR are observable during prototyping? 3) How do they modify their geometric structures in testing and adjusting their engineered product? In particular, we looked into the ways children articulated their ideas of how to approach the task (planning stage). We also looked into how children organized and managed their space to realize their construction (realization stage). Finally, we learned how children test, readjust, and improve the prototype through SR.

## FINDINGS AND DISCUSSION

The process of solving the challenge is highly complex. Students were mostly working in small groups on their own with little or no intervention from the teacher. As the children were planning,
SR is an important part of the NB K-12 mathematics curriculum and is included in the *Geometry Strand* (MEDPENB, 2016). Over the past decades, the STEM movement pushed center stage new opportunities for noticing and fostering spatial reasoning through more informal contexts of school makerspaces (Freiman, 2020). The question we investigate in this context is What aspects of SR emerge when kindergarten students work on an engineering challenge design task?

## RESEARCH SETTING AND METHODS

As part of a bigger case study of school makerspaces, we observed two groups of 20 students and their classroom teachers from a K-5 elementary school working in a STEM Lab (makerspace) in an urban area in NB, Canada. After obtaining ethics approval and parents' consent, the first author entered the Lab to video record students' work and conduct post-project interviews with some students. Our analysis focuses on one group working on *building a shelter for a stuffed animal*. The shelter needed *to be waterproof and protect the animal from high winds*. Twelve video-segments were first analyzed on the presence of spatial reasoning as per the actions listed above. Then initial codes were assigned to each video-segment. We then identified aspects across all segments.

Observing the children as they were working on constructing their shelter, we realized that we needed to adjust Hughes et al.'s (2019) design thinking cycle to reflect the design phases the children manifested in their collaborative work. We created a three-phase process: Planning → Realization → Testing and Adjusting (Figure 3).

<figure><img src="image_16.png" /><figcaption>Figure 3: Interconnected cogs in makerspaces (Fellus, Freiman, & Lurette, forthcoming)</figcaption></figure>

To better understand how the children engaged with SR, the following overarching questions guided our analysis that align with the three stages of planning, realization, and testing and adjusting: 1) What aspects of SR do children manifest in planning their work? 2) What elements of SR are observable during prototyping? 3) How do they modify their geometric structures in testing and adjusting their engineered product? In particular, we looked into the ways children articulated their ideas of how to approach the task (planning stage). We also looked into how children organized and managed their space to realize their construction (realization stage). Finally, we learned how children test, readjust, and improve the prototype through SR.

## FINDINGS AND DISCUSSION

The process of solving the challenge is highly complex. Students were mostly working in small groups on their own with little or no intervention from the teacher. As the children were planning,
realizing, and testing and adjusting, we noticed multiple instances of SR, which were built on the situatedness of the context (i.e., minimal teacher intervention, available tools and materials, the space) that allowed children to work with relation to others and to their environment towards the goal of constructing the shelter through actions of locating, orienting, comparing, scaling, composing, decomposing and recomposing objects (Mulligan et al., 2018).

In the *planning stage*, which was the initial phase of the project, students were discussing the problem and exchanging ideas. They used gestures to *estimate* the size of their shelter and organized their space as they *were getting ready to start their* construction of their imagined shapes (Figure 4).

<figure><img src="image_18.png" /><figcaption>Figure 4: Children estimate and compare lengths and sizes of stuffed animals and projected designs</figcaption></figure>

The second phase consisted of *building a prototype*. Students tried to construct walls using some spatial relationships. At first, students opted for a rectangular shape by synchronizing their movements to produce the desired shape. They also had to decide how to fasten the joints of the faces of the structure to make their construction more solid. They adjusted for size (pushing a toy through a “door”) and trying to make the structure more sturdy by adding screws. This was accompanied by a vivant discussion and coordinated actions (Figure 5; Clements, 2004).

<figure><img src="image_19.png" /><figcaption>Figure 5: Children discussing their construction and coordinating their actions</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Figure 6: Children orienting, rotating, balancing, transforming, and synchronizing their movements</figcaption></figure>

Some of the students decided to follow the line of the folds in the cardboard (intuitively grasping a concept of a straight line). We also see two children doing their cuts simultaneously holding their knives perpendicular to the cardboard and trying to synchronize their movements to create parallel sides of the rectangle (Figure 6; Clements & Battista, 1990). Then they pull all pieces together to set up their shelters in a shape of rectangular prisms (Figure 7).
realizing, and testing and adjusting, we noticed multiple instances of SR, which were built on the situatedness of the context (i.e., minimal teacher intervention, available tools and materials, the space) that allowed children to work with relation to others and to their environment towards the goal of constructing the shelter through actions of locating, orienting, comparing, scaling, composing, decomposing and recomposing objects (Mulligan et al., 2018).

In the *planning stage*, which was the initial phase of the project, students were discussing the problem and exchanging ideas. They used gestures to *estimate* the size of their shelter and organized their space as they *were getting ready to start their* construction of their imagined shapes (Figure 4).

<figure><img src="image_17.png" /><figcaption>Figure 4: Children estimate and compare lengths and sizes of stuffed animals and projected designs</figcaption></figure>

The second phase consisted of *building a prototype*. Students tried to construct walls using some spatial relationships. At first, students opted for a rectangular shape by synchronizing their movements to produce the desired shape. They also had to decide how to fasten the joints of the faces of the structure to make their construction more solid. They adjusted for size (pushing a toy through a “door”) and trying to make the structure more sturdy by adding screws. This was accompanied by a vivant discussion and coordinated actions (Figure 5; Clements, 2004).

<figure><img src="image_18.png" /><figcaption>Figure 5: Children discussing their construction and coordinating their actions</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 6: Children orienting, rotating, balancing, transforming, and synchronizing their movements</figcaption></figure>

Some of the students decided to follow the line of the folds in the cardboard (intuitively grasping a concept of a straight line). We also see two children doing their cuts simultaneously holding their knives perpendicular to the cardboard and trying to synchronize their movements to create parallel sides of the rectangle (Figure 6; Clements & Battista, 1990). Then they pull all pieces together to set up their shelters in a shape of rectangular prisms (Figure 7).
<figure><img src="image_21.png" /><figcaption>Figure 7: Children complete their 3D prototype</figcaption></figure>

The final phase was testing and making their construction more sturdy: “waterproof” and
withstanding against the “wind.” Many had to readjust their construction by switching to a triangular
shape, for example. Indeed, when testing their constructions, students realized that their construction
needs to be more solid and well sealed to protect their animal from winds and rain. They undertook
a process of **transformation** of their initial constructions (Figure 8; Pruden et al., 2011).

<figure><img src="image_22.png" /><figcaption>Figure 8: Children testing, rebuilding, and transforming their constructions</figcaption></figure>

Children transitioned shapes of structures as they were navigating toward their goal of constructing the shelter for their stuffed animal. Figure 9 showcases an example of decomposing a rectangular prism and recomposing it as a triangular prism by bending the two side walls to create symmetrically centered, balanced walls for the shelter (Mulligan et al., 2018).

<figure><img src="image_23.png" /><figcaption>Figure 9: Children composing, de/re/composing and transforming structures</figcaption></figure>

CONCLUSIONS

Overall, we found that engineering challenges enrich young children’s spatial explorations by providing them with a sense of spatial insight into complex mathematical structures and relationships between objects and self (Ramey et al., 2018), at least at the intuitive level. The concept of SR is enmeshed with the expression and experience of movement (embodiment) allowing students to design, build, test, and adjust sophisticated geometric constructions within situated contexts that are relational, goal oriented, and interactive. While our findings seem to corroborate with existing literature on SR showing the importance of early STEM learning opportunities, the context of makerspaces lets us direct the discussion towards theoretical concepts that emerge when we reflect on students’ intuitive understanding of geometry. Hence, the construct of agency seems to be a promising entry point into the realm of the socio-ecological turn in mathematics education (Bush et
<figure><img src="image_20.png" /><figcaption>Figure 7: Children complete their 3D prototype</figcaption></figure>

The final phase was testing and making their construction more sturdy: “waterproof” and
withstanding against the “wind.” Many had to readjust their construction by switching to a triangular
shape, for example. Indeed, when testing their constructions, students realized that their construction
needs to be more solid and well sealed to protect their animal from winds and rain. They undertook
a process of **transformation** of their initial constructions (Figure 8; Pruden et al., 2011).

<figure><img src="image_21.png" /><figcaption>Figure 8: Children testing, rebuilding, and transforming their constructions</figcaption></figure>

Children transitioned shapes of structures as they were navigating toward their goal of constructing the shelter for their stuffed animal. Figure 9 showcases an example of decomposing a rectangular prism and recomposing it as a triangular prism by bending the two side walls to create symmetrically centered, balanced walls for the shelter (Mulligan et al., 2018).

<figure><img src="image_22.png" /><figcaption>Figure 9: Children composing, de/re/composing and transforming structures</figcaption></figure>

CONCLUSIONS

Overall, we found that engineering challenges enrich young children’s spatial explorations by providing them with a sense of spatial insight into complex mathematical structures and relationships between objects and self (Ramey et al., 2018), at least at the intuitive level. The concept of SR is enmeshed with the expression and experience of movement (embodiment) allowing students to design, build, test, and adjust sophisticated geometric constructions within situated contexts that are relational, goal oriented, and interactive. While our findings seem to corroborate with existing literature on SR showing the importance of early STEM learning opportunities, the context of makerspaces lets us direct the discussion towards theoretical concepts that emerge when we reflect on students’ intuitive understanding of geometry. Hence, the construct of agency seems to be a promising entry point into the realm of the socio-ecological turn in mathematics education (Bush et
al., 2022). This is a necessary step that allows us to reflect on the multidirectional relationship
between one’s embodied self and their environment. This reorientation of mathematics education
through the lens of agency is how we navigate public discourse in the volatile, uncertain, complex,
and ambiguous world. These ideas require further exploration and discussion.

Acknowledgements

This project was supported by Brilliant Labs/Labos créatifs. We also thank the children and their
teachers for sharing their enthusiasm working on their projects.

References

Bruce, C. D., Davis, B., Sinclair, N., McGarvey, L., Hallowell, D., Drefs, M., ... & Woolcott, G. (2017). Understanding gaps in research networks: using “spatial reasoning” as a window into the importance of networked educational research. *Educational Studies in Mathematics*, *95*(2), 143–161.

Bush, S., Edelen, D., Roberts, T., Maiorca, C., Ivy, J. T., Cook, K. L., ... & Cox Jr, R. (2022). Humanistic STE (A) M instruction through empathy: Leveraging design thinking to improve society. *Pedagogies: An International Journal*, 1–20.

Clements, D. H. (2004). Geometric and spatial thinking in early childhood education. In Clements, D. H., Sarama, J., & DiBiase, A. M. (Eds.), *Engaging young children in mathematics: Standards for early childhood mathematics education*. (pp. 267–297). Routledge.

Clements, D. H., & Battista, M. T. (1990). The effects of Logo on children's conceptualizations of angle and polygons. *Journal for Research in Mathematics Education, 21*(5), 356–371.

Das, R. D., & Winter, S. (2016). A context-sensitive conceptual framework for activity modeling. *Journal of Spatial Information Science*, 2016(12), 45–85.

Davis, B., & The Spatial Reasoning Study Group, (Eds.), (2015). *Spatial reasoning in the early years: Principles, assertions, and speculations*. Routledge.

Deed, C., Cardellino, P., Matthews, E., & Southall, A. (2022). A qualitative exploration of cardboard architecture in post-pandemic schools. *International Journal of Educational Research Open*, 3, 100186.

DeJarnette, N. (2012). America’s children: Providing early exposure to STEM (science, technology, engineering and math) initiatives. *Education, 133*(1), 77–84.

Fellus, O., Freiman, V., & Lurette, O. (forthcoming). Rapturousness in makerspaces: Delight in construction. In X. Robichaud, O. Fellus, & V. Freiman (Eds.), *Mathematics and Its Connections to the Arts and Sciences (MACAS)*.

Freiman, V. (2020). Issues of teaching in a new technology-rich environment: Investigating the case of New Brunswick (Canada) School makerspaces. In Y. B.-D. Kolikant, D. Martinovic, & M. Milner-Bolotin (Eds.), *STEM teachers and teaching in the era of change: Professional expectations and advancement in 21st Century Schools*. (pp. 273–292). Springer. doi:10.1007/978-3-030-29396-3_15.

Hatzigianni, M., Stevenson, M., Falloon, G., Bower, M., & Forbes, A. (2021). Young children’s design thinking skills in makerspaces. *International Journal of Child-Computer Interaction*, 27, 100216.

Hawes, Z., Gilligan-Lee, K., & Mix, K. (2022). Effects of spatial training on mathematics performance: A meta-analysis. *Developmental Psychology*, 58(1), 112–137.

Heroman, C. (2017). Preschool through Grade 3: Making and tinkering: Bringing design challenges to the classroom. *Young Children*, 72(2), 72–79.

Hughes, J., Morrison, L., Kajamaa, A., & Kumpulainen, K. (2019). Makerspaces promoting students’ design thinking and collective knowledge creation: Examples from Canada and Finland. In A. L. Brooks, E. Brooks, & C. Sylla (Eds.), *Interactivity, game creation, design, learning, and innovation* (pp. 343–352). Springer.
al., 2022). This is a necessary step that allows us to reflect on the multidirectional relationship
between one’s embodied self and their environment. This reorientation of mathematics education
through the lens of agency is how we navigate public discourse in the volatile, uncertain, complex,
and ambiguous world. These ideas require further exploration and discussion.

Acknowledgements

This project was supported by Brilliant Labs/Labos créatifs. We also thank the children and their
teachers for sharing their enthusiasm working on their projects.

References

Bruce, C. D., Davis, B., Sinclair, N., McGarvey, L., Hallowell, D., Drefs, M., ... & Woolcott, G. (2017). Understanding gaps in research networks: using “spatial reasoning” as a window into the importance of networked educational research. *Educational Studies in Mathematics*, *95*(2), 143–161.

Bush, S., Edelen, D., Roberts, T., Maiorca, C., Ivy, J. T., Cook, K. L., ... & Cox Jr, R. (2022). Humanistic STE (A) M instruction through empathy: Leveraging design thinking to improve society. *Pedagogies: An International Journal*, 1–20.

Clements, D. H. (2004). Geometric and spatial thinking in early childhood education. In Clements, D. H., Sarama, J., & DiBiase, A. M. (Eds.), *Engaging young children in mathematics: Standards for early childhood mathematics education*. (pp. 267–297). Routledge.

Clements, D. H., & Battista, M. T. (1990). The effects of Logo on children's conceptualizations of angle and polygons. *Journal for Research in Mathematics Education, 21*(5), 356–371.

Das, R. D., & Winter, S. (2016). A context-sensitive conceptual framework for activity modeling. *Journal of Spatial Information Science*, 2016(12), 45–85.

Davis, B., & The Spatial Reasoning Study Group, (Eds.), (2015). *Spatial reasoning in the early years: Principles, assertions, and speculations*. Routledge.

Deed, C., Cardellino, P., Matthews, E., & Southall, A. (2022). A qualitative exploration of cardboard architecture in post-pandemic schools. *International Journal of Educational Research Open*, 3, 100186.

DeJarnette, N. (2012). America’s children: Providing early exposure to STEM (science, technology, engineering and math) initiatives. *Education, 133*(1), 77–84.

Fellus, O., Freiman, V., & Lurette, O. (forthcoming). Rapturousness in makerspaces: Delight in construction. In X. Robichaud, O. Fellus, & V. Freiman (Eds.), *Mathematics and Its Connections to the Arts and Sciences (MACAS)*.

Freiman, V. (2020). Issues of teaching in a new technology-rich environment: Investigating the case of New Brunswick (Canada) School makerspaces. In Y. B.-D. Kolikant, D. Martinovic, & M. Milner-Bolotin (Eds.), *STEM teachers and teaching in the era of change: Professional expectations and advancement in 21st Century Schools*. (pp. 273–292). Springer. doi:10.1007/978-3-030-29396-3_15.

Hatzigianni, M., Stevenson, M., Falloon, G., Bower, M., & Forbes, A. (2021). Young children’s design thinking skills in makerspaces. *International Journal of Child-Computer Interaction*, 27, 100216.

Hawes, Z., Gilligan-Lee, K., & Mix, K. (2022). Effects of spatial training on mathematics performance: A meta-analysis. *Developmental Psychology*, 58(1), 112–137.

Heroman, C. (2017). Preschool through Grade 3: Making and tinkering: Bringing design challenges to the classroom. *Young Children*, 72(2), 72–79.

Hughes, J., Morrison, L., Kajamaa, A., & Kumpulainen, K. (2019). Makerspaces promoting students’ design thinking and collective knowledge creation: Examples from Canada and Finland. In A. L. Brooks, E. Brooks, & C. Sylla (Eds.), *Interactivity, game creation, design, learning, and innovation* (pp. 343–352). Springer.
Latour, B. (1990). Technology is society made durable. *The Sociological Review*, 38(S1), 103–131.

LeBlanc, M., Freiman, V., & Furlong, C. (2022). From STEm to STEM: Learning from Students Working in School Makerspaces. In C. Michelsen, A. Beckmann, V. Freiman, U. T. Jankvist, & A. Savard (Eds.), *Mathematics and Its Connections to the Arts and Sciences (MACAS) 15 Years of Interdisciplinary Mathematics Education* (pp. 179–203). Springer.

McCluskey, C., Mulligan, J., & Van Bergen, P. (2018). Noticing mathematical pattern and structure embodied in young children’s play. In J. Hunter, L. Darragh, & P. Perger (Eds.), *Making waves, opening spaces: proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia* (pp. 535–542). MERGA.

MEDPENB (2016). *Programme de mathématiques. Première année*. GNB.

Moss, J., Hawes, Z., Naqvi, S., & Caswell, B. (2015). Adapting Japanese Lesson Study to enhance the teaching and learning of geometry and spatial reasoning in early years classrooms: a case study. *ZDM*, 47(3), 377–390.

Mulligan, J., Woolcott, G., Mitchelmore, M., & Davis, B. (2018). Connecting mathematics learning through spatial reasoning. *Mathematics Education Research Journal*, 30, 77–87.

NBED (2016). *Everyone at their best. 10 Year Education Plan*. GNB.

Omundsen, J. (2014). Cardboard Boat Building in Math Class: This Article Examines How the Implementation of an Integrative Learning Experience Encouraged Middle School Students to Work Collaboratively, and Apply Their Knowledge in Relevant and Meaningful Ways. *Middle School Journal*, 46(2), 3–9.

Pruden, S., Levine, S., & Huttenlocher, J. (2011). Children’s spatial thinking: Does talk about the spatial world matter? *Developmental Science*, 14(6), 1417–1430.

Ramey, K. E., Stevens, R., & Uttal, D. H. (2018, January). STEAM learning in an in-school makerspace: The role of distributed spatial sensemaking. In *Proceedings of the 13th International Conference of the Learning Sciences, London, UK* (Vol. 1).

Sadler, P. M., Coyle, H. P., & Schwartz, M. (2014). Engineering competitions in the middle school classroom: Key elements in developing effective design challenges. In *Design Education* (pp. 299–327). Routledge.

Sluis-Thiescheffer, W., Bekker, T., Eggen, B., Vermeeren, A., & De Ridder, H. (2016). Measuring and comparing novelty for design solutions generated by young children through different design methods. *Design Studies*, 43, 48–73.

Sinclair, N., & Bruce, C. D. (2014). Research forum: Spatial reasoning for young learners. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), *Proceedings of the joint meeting of PME 38 and PME-NA 36* (Vol. 1, pp. 173–203). Vancouver: PME.

Smith, I. A. (1964). *Spatial ability: Its educational and social significance*. London: University Press.

Stepankova, H., Lukavsky, J., Buschkuehl, M., Kopecek, M., Ripova, D., & Jaeggi, S. M. (2014). The malleability of working memory and visuospatial skills: A randomized controlled study in older adults. *Developmental Psychology*, 50(4), 1049–1059.

Vygotsky, L. S. (1978). *Mind in society: Development of higher psychological processes*. Harvard university press.

Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. *Journal of Educational Psychology*, 101(4), 817–835.

Whiteley, W., Sinclair, N., & Davis, B. (2015). What is spatial reasoning? In B. Davis and the Spatial Reasoning Study Group (Eds.), *Spatial reasoning in the early years: Principles, assertions and speculations* (pp. 3–14). Taylor and Francis.
Latour, B. (1990). Technology is society made durable. *The Sociological Review*, 38(S1), 103–131.

LeBlanc, M., Freiman, V., & Furlong, C. (2022). From STEm to STEM: Learning from Students Working in School Makerspaces. In C. Michelsen, A. Beckmann, V. Freiman, U. T. Jankvist, & A. Savard (Eds.), *Mathematics and Its Connections to the Arts and Sciences (MACAS) 15 Years of Interdisciplinary Mathematics Education* (pp. 179–203). Springer.

McCluskey, C., Mulligan, J., & Van Bergen, P. (2018). Noticing mathematical pattern and structure embodied in young children’s play. In J. Hunter, L. Darragh, & P. Perger (Eds.), *Making waves, opening spaces: proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia* (pp. 535–542). MERGA.

MEDPENB (2016). *Programme de mathématiques. Première année*. GNB.

Moss, J., Hawes, Z., Naqvi, S., & Caswell, B. (2015). Adapting Japanese Lesson Study to enhance the teaching and learning of geometry and spatial reasoning in early years classrooms: a case study. *ZDM*, 47(3), 377–390.

Mulligan, J., Woolcott, G., Mitchelmore, M., & Davis, B. (2018). Connecting mathematics learning through spatial reasoning. *Mathematics Education Research Journal*, 30, 77–87.

NBED (2016). *Everyone at their best. 10 Year Education Plan*. GNB.

Omundsen, J. (2014). Cardboard Boat Building in Math Class: This Article Examines How the Implementation of an Integrative Learning Experience Encouraged Middle School Students to Work Collaboratively, and Apply Their Knowledge in Relevant and Meaningful Ways. *Middle School Journal*, 46(2), 3–9.

Pruden, S., Levine, S., & Huttenlocher, J. (2011). Children’s spatial thinking: Does talk about the spatial world matter? *Developmental Science*, 14(6), 1417–1430.

Ramey, K. E., Stevens, R., & Uttal, D. H. (2018, January). STEAM learning in an in-school makerspace: The role of distributed spatial sensemaking. In *Proceedings of the 13th International Conference of the Learning Sciences, London, UK* (Vol. 1).

Sadler, P. M., Coyle, H. P., & Schwartz, M. (2014). Engineering competitions in the middle school classroom: Key elements in developing effective design challenges. In *Design Education* (pp. 299–327). Routledge.

Sluis-Thiescheffer, W., Bekker, T., Eggen, B., Vermeeren, A., & De Ridder, H. (2016). Measuring and comparing novelty for design solutions generated by young children through different design methods. *Design Studies*, 43, 48–73.

Sinclair, N., & Bruce, C. D. (2014). Research forum: Spatial reasoning for young learners. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), *Proceedings of the joint meeting of PME 38 and PME-NA 36* (Vol. 1, pp. 173–203). Vancouver: PME.

Smith, I. A. (1964). *Spatial ability: Its educational and social significance*. London: University Press.

Stepankova, H., Lukavsky, J., Buschkuehl, M., Kopecek, M., Ripova, D., & Jaeggi, S. M. (2014). The malleability of working memory and visuospatial skills: A randomized controlled study in older adults. *Developmental Psychology*, 50(4), 1049–1059.

Vygotsky, L. S. (1978). *Mind in society: Development of higher psychological processes*. Harvard university press.

Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. *Journal of Educational Psychology*, 101(4), 817–835.

Whiteley, W., Sinclair, N., & Davis, B. (2015). What is spatial reasoning? In B. Davis and the Spatial Reasoning Study Group (Eds.), *Spatial reasoning in the early years: Principles, assertions and speculations* (pp. 3–14). Taylor and Francis.
# IDENTIFYING A SEQUENCE OF CORE SKILLS FOR DEDUCTIVE PROVING IN SECONDARY SCHOOL GEOMETRY

Taro Fujita¹, Yutaka Kondo², Hiroyuki Kumakura³, Shinichi Miyawaki⁴, Susumu Kunimune³, Kojiro Shojima⁵, and Keith Jones⁶

¹ University of Exeter, ² Nara University of Education, ³ Shizuoka University, ⁴ Ozu elementary school, ⁵ National Centre for University Entrance Examination, ⁶ University of Southampton

Developing an effective teaching sequence for deductive proving in geometry still faces many challenges. In this paper, we aim to contribute to this issue by identifying 'core skills' for deductive geometry. By analysing the survey data from 238 G8, and 208 G9 students, we identified 7 items related to the generality of proofs, the structure of proofs, the economical nature of hierarchical relationships, and spatial reasoning. Our findings have implications for the teaching of deductive proofs in geometry as our results suggest guidance in considering a sequence of geometry teaching for deductive proving.

## INTRODUCTION

Mathematics education research documents how the teaching of geometrical reasoning still faces challenges, and how students continue to have difficulties to write and read geometrical proofs, to solve problems in both 2D/3D contexts, are heavily influenced by visual aspects of geometrical figures, and so on (e.g., Sinclair et al., 2016).

Given that proof is one of the forms of geometrical reasoning that is important to be taught and learnt in school geometry, and while there can be many different forms of proof (e.g., visual proofs, formal proofs, etc.), in this paper we focus on deductive geometrical proofs in lower secondary schools because a) deductive proofs are often one of the main topics in geometry curricular in secondary schools and b) it is still recognised one of the most challenging areas for both teachers and students. In particular, our goal is to explore what types of teaching sequences might be helpful in promoting the learning of geometric proofs in lower secondary schools, a theme identified by the Discussion Document for ICMI Study 26 on geometry education. In order to achieve our goal, we analyse data from 238 G(rade) 8 and 208 G9 students to address the following research questions (RQ): What skills are particularly important for students to solve deductive proofs and reasoning problems in geometry, what question items might be used to characterise these skills, and how can this information inform a teaching sequence for the teaching of deductive proofs in geometry.

We define 'core skills' as 'those that may be particularly related to students' successful problem solving' (Fujita et al., 2022), and can be used to suggest a teaching sequence for particular topics. For example, Fujita et al. (2022) identified specific core skills in 3D geometry by analysing a total of 2303 G4-9 Japanese students, and use the result to inform a teaching sequence for developing spatial reasoning. In what follows, we first present our theoretical framework in relation to our research questions. This informs what types of skills are necessary for deductive proving in geometry. Second, we explain our research methodology - survey items, participants, and analytical approaches. We then
# IDENTIFYING A SEQUENCE OF CORE SKILLS FOR DEDUCTIVE PROVING IN SECONDARY SCHOOL GEOMETRY

Taro Fujita¹, Yutaka Kondo², Hiroyuki Kumakura³, Shinichi Miyawaki⁴, Susumu Kunimune³, Kojiro Shojima⁵, and Keith Jones⁶

¹University of Exeter, ²Nara University of Education, ³Shizuoka University, ⁴Ozu elementary school, ⁵National Centre for University Entrance Examination, ⁶University of Southampton

Developing an effective teaching sequence for deductive proving in geometry still faces many challenges. In this paper, we aim to contribute to this issue by identifying 'core skills' for deductive geometry. By analysing the survey data from 238 G8, and 208 G9 students, we identified 7 items related to the generality of proofs, the structure of proofs, the economical nature of hierarchical relationships, and spatial reasoning. Our findings have implications for the teaching of deductive proofs in geometry as our results suggest guidance in considering a sequence of geometry teaching for deductive proving.

## INTRODUCTION

Mathematics education research documents how the teaching of geometrical reasoning still faces challenges, and how students continue to have difficulties to write and read geometrical proofs, to solve problems in both 2D/3D contexts, are heavily influenced by visual aspects of geometrical figures, and so on (e.g., Sinclair et al., 2016).

Given that proof is one of the forms of geometrical reasoning that is important to be taught and learnt in school geometry, and while there can be many different forms of proof (e.g., visual proofs, formal proofs, etc.), in this paper we focus on deductive geometrical proofs in lower secondary schools because a) deductive proofs are often one of the main topics in geometry curricular in secondary schools and b) it is still recognised one of the most challenging areas for both teachers and students. In particular, our goal is to explore what types of teaching sequences might be helpful in promoting the learning of geometric proofs in lower secondary schools, a theme identified by the Discussion Document for ICMI Study 26 on geometry education. In order to achieve our goal, we analyse data from 238 G(rade) 8 and 208 G9 students to address the following research questions (RQ): What skills are particularly important for students to solve deductive proofs and reasoning problems in geometry, what question items might be used to characterise these skills, and how can this information inform a teaching sequence for the teaching of deductive proofs in geometry.

We define 'core skills' as 'those that may be particularly related to students' successful problem solving' (Fujita et al., 2022), and can be used to suggest a teaching sequence for particular topics. For example, Fujita et al. (2022) identified specific core skills in 3D geometry by analysing a total of 2303 G4-9 Japanese students, and use the result to inform a teaching sequence for developing spatial reasoning. In what follows, we first present our theoretical framework in relation to our research questions. This informs what types of skills are necessary for deductive proving in geometry. Second, we explain our research methodology - survey items, participants, and analytical approaches. We then
present the results of our analysis, identifying ‘core skills’ and critically discuss how our findings can
be used to consider a teaching sequence to support the learning of geometric proofs in lower
secondary schools.

THEORETICAL FRAMEWORK OF THIS STUDY

In this study we take Miyazaki et al.’s definition for a deductive proof – “a deductive proof to consist of the following components: singular propositions (premises, conclusions, and intermediate propositions between them), universal propositions (theorems, definitions, etc.), and the appropriate connectives between singular propositions and universal propositions” (p. 255). Understanding deductive proofs require certain skills and understanding of the various aspects. One of them is the understanding of the structure of the proofs (Koseki, 1987; Moore, 1994). For example, Miyazaki et al. (2017) propose the three levels of the understanding of the structure of proofs from seeing a proofs as a cluster of unrelated elements such as assumptions, conclusions, theorems used etc. to a logical network of these elements as well as logical inferences such as syllogism. Also, in addition to these structural/logical aspects of proofs, it is essential to understand the generality of deductive proofs (Koseki, 1987; Kunimune et al., 2010). For example, once a statement is proved deductively, then it is not necessary to use a few examples or measurement to verify the statement. However, this is known to be quite difficult for students to develop the skills to understand this generality of deductive proofs (e.g., Hoyles & Healy, 2007; Marco et al., 2020).

Another important set of skills relate to the nature of geometrical figures. Geometry is a domain with
a dual nature – the practical and visual element of a geometrical figure can, at the same time, be
studied theoretically and conceptually (Fujita & Jones, 2003). Geometrical figures are figural
concepts (Fischbein, 1993), having both visual and conceptual aspects. This notion is useful when
considering, for example, students’ reasoning of hierarchical relationships between geometrical
figures. As studies have shown (e.g., Fujita, 2012), if students rely on visual aspects of figures, then
their reasoning might be restricted by such visual information, and they may not accept why squares
or rectangles are also parallelograms. The skills in understanding hierarchical relationships help
develop the economical nature of geometrical reasoning and proofs (de Villiers, 1994), which is also
related to the generality of proofs described above.

Finally, in geometry various skills related to visualising and manipulating geometrical figures are essential. In recent years the importance of spatial reasoning skills is increasingly recognised (e.g. Lowrie et al., 2021), such as mental rotations of geometrical figures (e.g. Bruce & Hawes, 2015), using domain specific knowledge to reason about 3D shapes (e.g. Fujita et al., 2020; Fujita et al., 2022) and so on.

In summary, we consider that students’ understanding of deductive proving in geometry can be
studied through identifying skills relating to the structure of proofs, the generality of proofs, the nature
of geometrical figures, and spatial reasoning.

METHODOLOGY

Context and survey items

In this study, we used a survey with a total of 33 items. The survey items, which were also based on
our theoretical framework, were taken from our previous studies (e.g., Koseki, 1987; Kunimune, et
present the results of our analysis, identifying ‘core skills’ and critically discuss how our findings can
be used to consider a teaching sequence to support the learning of geometric proofs in lower
secondary schools.

THEORETICAL FRAMEWORK OF THIS STUDY

In this study we take Miyazaki et al.’s definition for a deductive proof – “a deductive proof to consist of the following components: singular propositions (premises, conclusions, and intermediate propositions between them), universal propositions (theorems, definitions, etc.), and the appropriate connectives between singular propositions and universal propositions” (p. 255). Understanding deductive proofs require certain skills and understanding of the various aspects. One of them is the understanding of the structure of the proofs (Koseki, 1987; Moore, 1994). For example, Miyazaki et al. (2017) propose the three levels of the understanding of the structure of proofs from seeing a proofs as a cluster of unrelated elements such as assumptions, conclusions, theorems used etc. to a logical network of these elements as well as logical inferences such as syllogism. Also, in addition to these structural/logical aspects of proofs, it is essential to understand the generality of deductive proofs (Koseki, 1987; Kunimune et al., 2010). For example, once a statement is proved deductively, then it is not necessary to use a few examples or measurement to verify the statement. However, this is known to be quite difficult for students to develop the skills to understand this generality of deductive proofs (e.g., Hoyles & Healy, 2007; Marco et al., 2020).

Another important set of skills relate to the nature of geometrical figures. Geometry is a domain with
a dual nature – the practical and visual element of a geometrical figure can, at the same time, be
studied theoretically and conceptually (Fujita & Jones, 2003). Geometrical figures are figural
concepts (Fischbein, 1993), having both visual and conceptual aspects. This notion is useful when
considering, for example, students’ reasoning of hierarchical relationships between geometrical
figures. As studies have shown (e.g., Fujita, 2012), if students rely on visual aspects of figures, then
their reasoning might be restricted by such visual information, and they may not accept why squares
or rectangles are also parallelograms. The skills in understanding hierarchical relationships help
develop the economical nature of geometrical reasoning and proofs (de Villiers, 1994), which is also
related to the generality of proofs described above.

Finally, in geometry various skills related to visualising and manipulating geometrical figures are essential. In recent years the importance of spatial reasoning skills is increasingly recognised (e.g. Lowrie et al., 2021), such as mental rotations of geometrical figures (e.g. Bruce & Hawes, 2015), using domain specific knowledge to reason about 3D shapes (e.g. Fujita et al., 2020; Fujita et al., 2022) and so on.

In summary, we consider that students’ understanding of deductive proving in geometry can be
studied through identifying skills relating to the structure of proofs, the generality of proofs, the nature
of geometrical figures, and spatial reasoning.

METHODOLOGY

Context and survey items

In this study, we used a survey with a total of 33 items. The survey items, which were also based on
our theoretical framework, were taken from our previous studies (e.g., Koseki, 1987; Kunimune, et
al., 2010; Fujita, 2012; Fujita et al., 2020, Fujita et al., 2022) and have the following three main
components, namely 1) Deductive proof, 2) Geometric figures, and 3) 3D geometry (see Appendix).
In the first section 1) Deductive proof, students were asked to judge the generality of a deductive
proof (Q1), write a simple proof (Q2), identify the definition, assumptions, conclusions and theorems
used, and point out a circularity (see also Kunimune, et al., 2010). The second section 2) Geometric
figures tests students’ knowledge and understanding of geometric figures in the context of
hierarchical relationships between quadrilaterals. The questions are adopted from Koseki (1987) and
Fujita (2012) In this section, students were asked to select parallelograms (Q5), write a definition of
a parallelogram (Q6) and solve problems using their knowledge and understanding of the
relationships (Q7-10). The third section tests students' geometrical thinking in 3D geometry contexts.
The items are almost identical to those used in Fujita et al. (2022).

Participants

It is expected that G8 students who have studied the required Japanese curriculum content would be able to answer all the questions. From February to March 2022, a survey was conducted with 238 G8, and 208 G9 students aged from 14 to 15 years from three state, nonselective lower secondary schools in two different cities in Japan. The recruitment process was exactly the same as described in Fujita et al. (2022). The schools were recruited through the authors’ contacts and the study aims and survey procedures were explained to teachers, and then they agreed to participate in the survey. The period from February to March was chosen as students in each grade have experienced the prescribed mathematical curriculum (the Japanese school academic year is from April to March). The survey question items were directly distributed to the students by their class teachers with paper forms, and in general, students completed answering them within 30 min.

Analysis procedure

In this study, we attempt to identify which survey items might be particularly related to the core skills
in deductive proofs in geometry. In order to identify a few items from 33 questions, we used the
method developed and explored in Fujita et al. (2022), which utilises 2PLM analysis with the
following procedure, in which “we divide the question items (set C) into two groups, set A from the
extracted items and set B from the others (i.e., C=A+B), and calculate the Pearson correlation
coefficient r between the sum of the scores of sets A and B” (p. 446). Also, this step employs the
following principles (ibids., p. 446):

1. Consistency with theory: The extracted question items are closely related to our theoretical framework.

2. Consistency with past research evidence: The extracted question items are also identified as important in past/other related surveys.

3. Less is more (or Occam's razor): Consider the whole set of question items as C. Suppose from C we extract five items (set A) but four items are also suggested (set A'). The rest of the items are B (and B'). It is better if the cardinal number of set A is less than set B(= C-A), that is, |A|<|B| (and |A'|<|B'|). Then we calculate correlations between (set A, set B) and (set A', set B'), and if the correlation value r' between sets A' and B' (r'A'B') is higher than r between sets A and B (rAB) (i.e., |A'|>|A'| but r'A'B' > rAB) then set A' will be more likely to be
al., 2010; Fujita, 2012; Fujita et al., 2020, Fujita et al., 2022) and have the following three main
components, namely 1) Deductive proof, 2) Geometric figures, and 3) 3D geometry (see Appendix).
In the first section 1) Deductive proof, students were asked to judge the generality of a deductive
proof (Q1), write a simple proof (Q2), identify the definition, assumptions, conclusions and theorems
used, and point out a circularity (see also Kunimune, et al., 2010). The second section 2) Geometric
figures tests students' knowledge and understanding of geometric figures in the context of
hierarchical relationships between quadrilaterals. The questions are adopted from Koseki (1987) and
Fujita (2012) In this section, students were asked to select parallelograms (Q5), write a definition of
a parallelogram (Q6) and solve problems using their knowledge and understanding of the
relationships (Q7-10). The third section tests students' geometrical thinking in 3D geometry contexts.
The items are almost identical to those used in Fujita et al. (2022).

Participants

It is expected that G8 students who have studied the required Japanese curriculum content would be able to answer all the questions. From February to March 2022, a survey was conducted with 238 G8, and 208 G9 students aged from 14 to 15 years from three state, nonselective lower secondary schools in two different cities in Japan. The recruitment process was exactly the same as described in Fujita et al. (2022). The schools were recruited through the authors’ contacts and the study aims and survey procedures were explained to teachers, and then they agreed to participate in the survey. The period from February to March was chosen as students in each grade have experienced the prescribed mathematical curriculum (the Japanese school academic year is from April to March). The survey question items were directly distributed to the students by their class teachers with paper forms, and in general, students completed answering them within 30 min.

Analysis procedure

In this study, we attempt to identify which survey items might be particularly related to the core skills
in deductive proofs in geometry. In order to identify a few items from 33 questions, we used the
method developed and explored in Fujita et al. (2022), which utilises 2PLM analysis with the
following procedure, in which “we divide the question items (set C) into two groups, set A from the
extracted items and set B from the others (i.e., C=A+B), and calculate the Pearson correlation
coefficient r between the sum of the scores of sets A and B” (p. 446). Also, this step employs the
following principles (ibids., p. 446):

1. Consistency with theory: The extracted question items are closely related to our theoretical framework.

2. Consistency with past research evidence: The extracted question items are also identified as important in past/other related surveys.

3. Less is more (or Occam's razor): Consider the whole set of question items as C. Suppose from C we extract five items (set A) but four items are also suggested (set A'). The rest of the items are B (and B'). It is better if the cardinal number of set A is less than set B(= C-A), that is, |A|<|B| (and |A'|<|B'|). Then we calculate correlations between (set A, set B) and (set A', set B'), and if the correlation value r' between sets A' and B' (r'A'B') is higher than r between sets A and B (rAB) (i.e., |A'|>|A'| but r'A'B' > rAB) then set A' will be more likely to be
considered as a set of core skill items as we use fewer question items but have more explanatory power.

For the analysis, R 4.3.1 with “ltm” package was used.

# RESULT

## Overall performance from the survey

Overall, the average mean scores are 18.27 (out of 33) for G8 and 18.33 for G9, there was no significant difference between G8 and G9, t(428.79)=- 088542, p=.9295, despite G9 (M=18.33, SD=7.15) attaining higher scores than G8 (M=18.27, SD=6.77). The following Tables 1-4 summarise the detailed results of correct answers (%s, N=238(G8) and 208 (G9)).

<table><caption>Table 1: Survey results for Section 1) Deductive proof</caption><thead><tr><th>Grade</th><th>Q1a</th><th>Q1b</th><th>Q1c</th><th>Q2</th><th>Q3a</th><th>Q3b</th><th>Q3c</th><th>Q4a</th><th>Q4b</th></tr></thead><tbody><tr><td>G8</td><td>53.8</td><td>47.9</td><td>73.9</td><td>62.6</td><td>84.0</td><td>81.9</td><td>55.5</td><td>65.1</td><td>38.7</td></tr><tr><td>G9</td><td>41.3</td><td>33.2</td><td>69.2</td><td>63.9</td><td>82.7</td><td>80.8</td><td>52.9</td><td>70.2</td><td>46.2</td></tr></tbody></table>

<table><caption>Table 2: Survey results for Section 2) Geometrical figures</caption><thead><tr><th>Grade</th><th>Q5</th><th>Q6</th><th>Q7a</th><th>Q7b</th><th>Q8a</th><th>Q8b</th><th>Q9</th><th>Q10</th></tr></thead><tbody><tr><td>G8</td><td>57.1</td><td>81.1</td><td>60.1</td><td>9.2</td><td>73.5</td><td>19.7</td><td>37.8</td><td>60.9</td></tr><tr><td>G9</td><td>42.3</td><td>73.6</td><td>53.4</td><td>9.1</td><td>70.7</td><td>15.4</td><td>24.0</td><td>46.6</td></tr></tbody></table>

<table><caption>Table 3: Survey results for Section 3) 3D Geometry (Q1K-Q5K)</caption><thead><tr><th>Grade</th><th>Q1K</th><th>Q2K</th><th>Q3aK</th><th>Q3bK</th><th>Q3cK</th><th>Q4K</th><th>Q5K</th></tr></thead><tbody><tr><td>G8</td><td>92.4</td><td>80.3</td><td>89.1</td><td>78.2</td><td>96.6</td><td>31.1</td><td>32.8</td></tr><tr><td>G9</td><td>93.8</td><td>86.1</td><td>88.0</td><td>81.3</td><td>91.8</td><td>45.2</td><td>26.0</td></tr></tbody></table>

<table><caption>Table 4: Survey results for Section 3) 3D Geometry (Q6aK-Q11K)</caption><thead><tr><th>Grade</th><th>Q6aK</th><th>Q6bK</th><th>Q6cK</th><th>Q7aK</th><th>Q7bK</th><th>Q8K</th><th>Q9K</th><th>Q10K</th><th>Q11K</th></tr></thead><tbody><tr><td>G8</td><td>55.0</td><td>56.3</td><td>52.1</td><td>42.0</td><td>13.9</td><td>63.9</td><td>46.6</td><td>27.3</td><td>6.7</td></tr><tr><td>G9</td><td>65.4</td><td>58.2</td><td>52.9</td><td>51.9</td><td>22.1</td><td>79.3</td><td>64.4</td><td>34.1</td><td>17.3</td></tr></tbody></table>

The key findings from the survey are summarised as follows. First, both G8 and G9 students had a relatively good understanding of proof writing, but struggled with the generality of proofs and proofs with logical circularity: In the first section 1. Deductive Proof, while over 60% of G8/9 students were able to write a simple proof (Q2), less than 50% of them were unsure whether they could accept or reject the use of some examples to prove the sum of the interior angles of a triangle (Q1). Similarly, only 38.7% of the G8 students and 46.2% of the G9 students were able to give their reasons for not accepting a proof with a logical circularity (Q4b). This result is similar to that reported by Kunimune et al. (2010).

Second, both G8 and G9 students had a rather limited understanding of geometric figures in the context of hierarchical relationships between quadrilaterals: By the time the students participated in the survey, they had studied the hierarchical relationships between quadrilaterals, but less than 60% of the students could correctly identify images of parallelograms (Q5). Similarly, less than a quarter of G8 and G9 students were able to state their reasons for not having to prove the lengths of the diagonals of a square again once we have proved the rectangles, because the square is a special type
considered as a set of core skill items as we use fewer question items but have more explanatory power.

For the analysis, R 4.3.1 with “ltm” package was used.

# RESULT

## Overall performance from the survey

Overall, the average mean scores are 18.27 (out of 33) for G8 and 18.33 for G9, there was no significant difference between G8 and G9, t(428.79)=- 088542, p=.9295, despite G9 (M=18.33, SD=7.15) attaining higher scores than G8 (M=18.27, SD=6.77). The following Tables 1-4 summarise the detailed results of correct answers (%s, N=238(G8) and 208 (G9)).

<table><caption>Table 1: Survey results for Section 1) Deductive proof</caption><thead><tr><th>Grade</th><th>Q1a</th><th>Q1b</th><th>Q1c</th><th>Q2</th><th>Q3a</th><th>Q3b</th><th>Q3c</th><th>Q4a</th><th>Q4b</th></tr></thead><tbody><tr><td>G8</td><td>53.8</td><td>47.9</td><td>73.9</td><td>62.6</td><td>84.0</td><td>81.9</td><td>55.5</td><td>65.1</td><td>38.7</td></tr><tr><td>G9</td><td>41.3</td><td>33.2</td><td>69.2</td><td>63.9</td><td>82.7</td><td>80.8</td><td>52.9</td><td>70.2</td><td>46.2</td></tr></tbody></table>

<table><caption>Table 2: Survey results for Section 2) Geometrical figures</caption><thead><tr><th>Grade</th><th>Q5</th><th>Q6</th><th>Q7a</th><th>Q7b</th><th>Q8a</th><th>Q8b</th><th>Q9</th><th>Q10</th></tr></thead><tbody><tr><td>G8</td><td>57.1</td><td>81.1</td><td>60.1</td><td>9.2</td><td>73.5</td><td>19.7</td><td>37.8</td><td>60.9</td></tr><tr><td>G9</td><td>42.3</td><td>73.6</td><td>53.4</td><td>9.1</td><td>70.7</td><td>15.4</td><td>24.0</td><td>46.6</td></tr></tbody></table>

<table><caption>Table 3: Survey results for Section 3) 3D Geometry (Q1K-Q5K)</caption><thead><tr><th>Grade</th><th>Q1K</th><th>Q2K</th><th>Q3aK</th><th>Q3bK</th><th>Q3cK</th><th>Q4K</th><th>Q5K</th></tr></thead><tbody><tr><td>G8</td><td>92.4</td><td>80.3</td><td>89.1</td><td>78.2</td><td>96.6</td><td>31.1</td><td>32.8</td></tr><tr><td>G9</td><td>93.8</td><td>86.1</td><td>88.0</td><td>81.3</td><td>91.8</td><td>45.2</td><td>26.0</td></tr></tbody></table>

<table><caption>Table 4: Survey results for Section 3) 3D Geometry (Q6aK-Q11K)</caption><thead><tr><th>Grade</th><th>Q6aK</th><th>Q6bK</th><th>Q6cK</th><th>Q7aK</th><th>Q7bK</th><th>Q8K</th><th>Q9K</th><th>Q10K</th><th>Q11K</th></tr></thead><tbody><tr><td>G8</td><td>55.0</td><td>56.3</td><td>52.1</td><td>42.0</td><td>13.9</td><td>63.9</td><td>46.6</td><td>27.3</td><td>6.7</td></tr><tr><td>G9</td><td>65.4</td><td>58.2</td><td>52.9</td><td>51.9</td><td>22.1</td><td>79.3</td><td>64.4</td><td>34.1</td><td>17.3</td></tr></tbody></table>

The key findings from the survey are summarised as follows. First, both G8 and G9 students had a relatively good understanding of proof writing, but struggled with the generality of proofs and proofs with logical circularity: In the first section 1. Deductive Proof, while over 60% of G8/9 students were able to write a simple proof (Q2), less than 50% of them were unsure whether they could accept or reject the use of some examples to prove the sum of the interior angles of a triangle (Q1). Similarly, only 38.7% of the G8 students and 46.2% of the G9 students were able to give their reasons for not accepting a proof with a logical circularity (Q4b). This result is similar to that reported by Kunimune et al. (2010).

Second, both G8 and G9 students had a rather limited understanding of geometric figures in the context of hierarchical relationships between quadrilaterals: By the time the students participated in the survey, they had studied the hierarchical relationships between quadrilaterals, but less than 60% of the students could correctly identify images of parallelograms (Q5). Similarly, less than a quarter of G8 and G9 students were able to state their reasons for not having to prove the lengths of the diagonals of a square again once we have proved the rectangles, because the square is a special type
of rectangle (Q8b, 19.7% of G8 and 15.4% of G9 gave correct answers). Only 10% or fewer students were able to give a correct answer and reason for a parallelogram whose four vertices are on the circumference of a circle, e.g. a rectangle (Q7b). Again, this result is quite similar to that reported by Koseki (1987) or Fujita (2012).

Third, while both G8 and G9 students demonstrated their good understanding of simple 3D geometry problems, they struggled to solve problems using both spatial reasoning skills and domain-specific knowledge: The results are very similar to our previous study (Fujita et al., 2022). Both G8 and G9 students struggled to identify the size of angles formed in a cube, implying they have limited spatial reasoning skills.

## Identifying items which represent “core” skills for deductive proving in geometry

Whereas all the 33 items are important to grasp the current Japanese students’ understanding of deductive proving in geometry, our interest is which items can be extracted from the 33, and what core skills might be related to these extracted items (NB: Core skills are those that may be particularly related to students' successful problem solving).

Our analysis with 2PLM analysis suggests each item’s relative difficulty and discrimination values. We took these values with the three principles described in the methodology, and the following 7 items were suggested, summarised in the table 5 and figure 1 below.

<table><caption>Table 5: Suggested items which represent the “core” reasoning skills</caption><thead><tr><td>Suggested item</td><td>Reasons based on the principles</td></tr></thead><tbody><tr><td>From section 1<br/>Q1a, Q3b and Q4b</td><td><ul><li>These 7 items have relatively higher discrimination values from 2PLM analysis.</li></ul></td></tr><tr><td>From section 2<br/>Q8b</td><td><ul><li>rAB = 0.81 (p &lt; .001), that is, the sum of the seven items (set A) correlates very high with the sum of the other twenty six items (set B), see figure 1.</li></ul></td></tr><tr><td>From section 3<br/>Q2K, Q8K and Q11K</td><td><ul><li>Section 1. Deductive proof: Q1a is related to the generality of proofs (e.g., Kunimune et al., 2010; Marco, et al., 2022), and Q3b and Q4b are related to the elemental and holistic aspects of the structure of deductive proofs (e.g., Moore, 1994; Miyazaki et al., 2017).</li><li>Section 2. Geometrical figure: Q8b is related to the economical nature of hierarchical classification of geometrical shapes (e.g., de Villiers, 1994; Fujita, 2012).</li><li>Section 3. 3D geometry: The three items were also suggested as the items related to core skills in Fujita et al. (2022).</li></ul></td></tr><tr><td>Figure 1: Jittered scatter graph for the sum of 7 items (x-axis) and 26 items (y-axis)</td><td></td></tr></tbody></table>

## DISCUSSION AND CONCLUDING REMARK

The results presented in the previous section indicate a few remarks. From Section 1) Deductive proof, we extracted 3 items. For Q1a, if a student accepts an explanation for this question but can
of rectangle (Q8b, 19.7% of G8 and 15.4% of G9 gave correct answers). Only 10% or fewer students were able to give a correct answer and reason for a parallelogram whose four vertices are on the circumference of a circle, e.g. a rectangle (Q7b). Again, this result is quite similar to that reported by Koseki (1987) or Fujita (2012).

Third, while both G8 and G9 students demonstrated their good understanding of simple 3D geometry problems, they struggled to solve problems using both spatial reasoning skills and domain-specific knowledge: The results are very similar to our previous study (Fujita et al., 2022). Both G8 and G9 students struggled to identify the size of angles formed in a cube, implying they have limited spatial reasoning skills.

## Identifying items which represent “core” skills for deductive proving in geometry

Whereas all the 33 items are important to grasp the current Japanese students' understanding of deductive proving in geometry, our interest is which items can be extracted from the 33, and what core skills might be related to these extracted items (NB: Core skills are those that may be particularly related to students' successful problem solving).

Our analysis with 2PLM analysis suggests each item's relative difficulty and discrimination values. We took these values with the three principles described in the methodology, and the following 7 items were suggested, summarised in the table 5 and figure 1 below.

<table><caption>Table 5: Suggested items which represent the “core” reasoning skills</caption><thead><tr><th>Suggested item</th><th>Reasons based on the principles</th></tr></thead><tbody><tr><td>From section 1<br>Q1a, Q3b and Q4b</td><td><ul><li>These 7 items have relatively higher discrimination values from 2PLM analysis.</li></ul></td></tr><tr><td>From section 2<br>Q8b</td><td><ul><li>rAB = 0.81 (p &lt; .001), that is, the sum of the seven items (set A) correlates very high with the sum of the other twenty six items (set B), see figure 1.</li></ul></td></tr><tr><td>From section 3<br>Q2K, Q8K and Q11K</td><td><ul><li>Section 1. Deductive proof: Q1a is related to the generality of proofs (e.g., Kunimune et al., 2010; Marco, et al., 2022), and Q3b and Q4b are related to the elemental and holistic aspects of the structure of deductive proofs (e.g., Moore, 1994; Miyazaki et al., 2017).</li><li>Section 2. Geometrical figure: Q8b is related to the economical nature of hierarchical classification of geometrical shapes (e.g., de Villiers, 1994; Fujita, 2012).</li><li>Section 3. 3D geometry: The three items were also suggested as the items related to core skills in Fujita et al. (2022).</li></ul></td></tr><tr><td>Figure 1: Jittered scatter graph for the sum of 7 items (x-axis) and 26 items (y-axis)</td><td></td></tr></tbody></table>

## DISCUSSION AND CONCLUDING REMARK

The results presented in the previous section indicate a few remarks. From Section 1) Deductive proof, we extracted 3 items. For Q1a, if a student accepts an explanation for this question but can
write a proof, then s/he might not fully understand the generality of proof, which has been reported
as one of the challenges in the learning of proofs (e.g., Koseki, 1987; Hoyles & Healy 2007;
Kunimune et al., 2010; Marco, et al., 2022). Q3b asks why AB=CD (a property of parallelogram) can
be said in a proof. Also, Q4b asks to state a reason why a proof with logical circularity cannot be
accepted. These questions are related to the structure of deductive proofs (Moore, 1994) and identify
the assumptions of a proof and the logical circularity are both indicated as essential for understanding
of the structure of proofs (Miyazaki, et al., 2017). From Section 2). From Section 2) Geometrical
figures, Q8b is indicated to be extracted. This question asks if ‘the lengths of the two diagonals of a
square are equal’ can be true without any proofs or measurement if we have proved for rectangle. If
a student can answer this question correctly, then this student is likely to understand the economical
nature of hierarchical classification of geometrical shapes (e.g., de Villiers, 1994; Fujita, 2012).
Also, the students who answered this question correctly are likely to understand what diagrams drawn
for a proof might be representing in problem contexts, and have a good understanding of geometrical
figures in a sense of Fishbein (1993). From Section 3) 3D geometry, Q2S asks students to draw a
cylinder from a front view, and a drawing skill is suggested as important by Sinclair et al. (2018).
Q8K and Q11K are related how the students use their spatial skills such as spatial visualisation and
mental rotations as well as property-based reasoning, which are again important skills in 3D geometry
(e.g., Bruce & Hawes, 2015; Lowrie et al., 2021). Also, the suggested items are the same as the items
which were also suggested in our previous study with the different data set (Fujita et al., 2022).

Our findings suggest that our students still have challenges in their understanding of deductive
reasoning in geometry, and a more effective teaching sequence might be needed to improve our
students' understanding of proof in geometry. We can identify an effective teaching sequence by
taking the core skills and activities derived from the 7 items. First, the high correlation coefficient
value (0.81, *p* < .001) indicates that if students could answer these 7 items, then they are likely to
answer the other 26 questions correctly in the survey. This implies that the skills to be used for
answering these chosen 7 items might be particularly important for success in not only constructing
and reading simple deductive proofs in geometry, but also answering problem solving which require
deductive reasoning and proving in wider geometry problem solving contexts.

We can use these 7 items for considering the teaching sequence. When designing a teaching sequence, one way is to locate activities and questions related to these 7 questions explicitly as such activities might be related to the core skills in deductive reasoning in geometry, i.e., the generality of proofs, the structure of proofs, the economical nature of hierarchical relationships, and spatial reasoning. For example, when introducing what a geometry proof is, not only check the fundamental structure of deductive proofs such as definition, assumptions and conclusions (Q3a-c in our survey items), but also provide an opportunity for students to critically reflect why such an argument might be necessary or valid (Q1a or Q4b). Also, the roles of geometrical figures should be checked carefully at some point, and the economical nature of the hierarchical relationships between geometrical shapes should be explicitly taught (Q8b). Even where proof tasks might be heavily related to 2D geometry, checking if students can visualise geometrical figures, mentally manipulate them by using 3D geometry tasks (e.g., Q2K, Q11K etc.) can be done before students learn how to write and read deductive proofs in geometry. As our tentative conclusion, a teaching sequence might be considered as follows: 1) first, learning opportunities for the generality of proofs and structure of proofs are included (e.g., Q1 and Q3, Deductive proof) with some spatial reasoning skills training (e.g., Q2K, Q8K, Q11K, 3D
write a proof, then s/he might not fully understand the generality of proof, which has been reported
as one of the challenges in the learning of proofs (e.g., Koseki, 1987; Hoyles & Healy 2007;
Kunimune et al., 2010; Marco, et al., 2022). Q3b asks why AB=CD (a property of parallelogram) can
be said in a proof. Also, Q4b asks to state a reason why a proof with logical circularity cannot be
accepted. These questions are related to the structure of deductive proofs (Moore, 1994) and identify
the assumptions of a proof and the logical circularity are both indicated as essential for understanding
of the structure of proofs (Miyazaki, et al., 2017). From Section 2). From Section 2) Geometrical
figures, Q8b is indicated to be extracted. This question asks if ‘the lengths of the two diagonals of a
square are equal’ can be true without any proofs or measurement if we have proved for rectangle. If
a student can answer this question correctly, then this student is likely to understand the economical
nature of hierarchical classification of geometrical shapes (e.g., de Villiers, 1994; Fujita, 2012).
Also, the students who answered this question correctly are likely to understand what diagrams drawn
for a proof might be representing in problem contexts, and have a good understanding of geometrical
figures in a sense of Fishbein (1993). From Section 3) 3D geometry, Q2S asks students to draw a
cylinder from a front view, and a drawing skill is suggested as important by Sinclair et al. (2018).
Q8K and Q11K are related how the students use their spatial skills such as spatial visualisation and
mental rotations as well as property-based reasoning, which are again important skills in 3D geometry
(e.g., Bruce & Hawes, 2015; Lowrie et al., 2021). Also, the suggested items are the same as the items
which were also suggested in our previous study with the different data set (Fujita et al., 2022).

Our findings suggest that our students still have challenges in their understanding of deductive reasoning in geometry, and a more effective teaching sequence might be needed to improve our students’ understanding of proof in geometry. We can identify an effective teaching sequence by taking the core skills and activities derived from the 7 items. First, the high correlation coefficient value (0.81, *p* < .001) indicates that if students could answer these 7 items, then they are likely to answer the other 26 questions correctly in the survey. This implies that the skills to be used for answering these chosen 7 items might be particularly important for success in not only constructing and reading simple deductive proofs in geometry, but also answering problem solving which require deductive reasoning and proving in wider geometry problem solving contexts.

We can use these 7 items for considering the teaching sequence. When designing a teaching sequence, one way is to locate activities and questions related to these 7 questions explicitly as such activities might be related to the core skills in deductive reasoning in geometry, i.e., the generality of proofs, the structure of proofs, the economical nature of hierarchical relationships, and spatial reasoning. For example, when introducing what a geometry proof is, not only check the fundamental structure of deductive proofs such as definition, assumptions and conclusions (Q3a-c in our survey items), but also provide an opportunity for students to critically reflect why such an argument might be necessary or valid (Q1a or Q4b). Also, the roles of geometrical figures should be checked carefully at some point, and the economical nature of the hierarchical relationships between geometrical shapes should be explicitly taught (Q8b). Even where proof tasks might be heavily related to 2D geometry, checking if students can visualise geometrical figures, mentally manipulate them by using 3D geometry tasks (e.g., Q2K, Q11K etc.) can be done before students learn how to write and read deductive proofs in geometry. As our tentative conclusion, a teaching sequence might be considered as follows: 1) first, learning opportunities for the generality of proofs and structure of proofs are included (e.g., Q1 and Q3, Deductive proof) with some spatial reasoning skills training (e.g., Q2K, Q8K, Q11K, 3D
geometry); 2) students then learn how to write and read proofs, but logical circularity (e.g., Q4, Deductive proof) and the economical nature of hierarchical classification of shapes should be explicitly addressed (e.g., Q8, Geometrical figures); 3) their understanding of deductive proofs and reasoning can be strengthened through more proof tasks (e.g., Q2), and problem solving in 2/3D geometry contexts (e.g., Q7, Q7K etc.). Also, these 7 items can be used to quickly check the extent to which students have understood the basic elements of deductive proofs. For example, in the case of Japanese students, we could use these 7 items with G9 students to quickly check their understanding, who have already completed their study of geometric proofs in G8, but will begin to learn more proofs in similar triangles.

## REFERENCES

Bruce, C. D., & Hawes, Z. (2015). The role of 2D and 3D mental rotation in mathematics for young children: what is it? Why does it matter? And what can we do about it?. *ZDM*, *47*(3), 331-343.

De Villiers, M. (1994). The role and function of a hierarchical classification of quadrilaterals. *For the learning of mathematics*, 14(1), 11-18.

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

Fujita, T., & Jones, K. (2003). The place of experimental tasks in geometry teaching: learning from the textbook designs of the early 20th century. *Research in Mathematics Education*, 5(1), 47-62.

Fujita T. (2012). Learners’ level of understanding of the inclusion relations of quadrilaterals and prototype phenomenon. *Journal of Mathematical Behavior*, 31(1), 60-72.

Fujita, T., Kondo, Y., Kumakura, H., Kunimune, S., & Jones, K. (2020). Spatial reasoning skills about 2D representations of 3D geometrical shapes in grades 4 to 9. *Mathematics Education Research Journal*, 32, 235-255.

Fujita, T., Kondo, Y., Kumakura, H., Miyawaki, S., Kunimune, S., & Shojima, K. (2022). Identifying Japanese students' core spatial reasoning skills by solving 3D geometry problems: An exploration. *Asian Journal for Mathematics Education*, 1(4), 437-454.

Hoyles, C., & Healy, L. (2007). Curriculum change and geometrical reasoning. In P. Boero (Ed.), *Theorems in school* (pp. 81–115). Rotterdam: Sense.

Lowrie, T., Harris, D., Logan, T., & Hegarty, M. (2021). The impact of a spatial intervention program on students’ spatial reasoning and mathematics performance. *The Journal of Experimental Education*, 89(2), 259-277.

Koseki. K. (Ed.). (1987). *The teaching of geometrical proof*, Meiji Tosho Publishers, Tokyo (in Japanese)

Kunimune, S., Fujita, T., & Jones, K. (2010). Strengthening students’ understanding of proof in geometry in lower secondary school. In *Proceedings of the 6th Congress of the European Society for Research in Mathematics Education* (pp. 756-765).

Marco, N., Palatnik, A., & Schwarz, B. B. (2022). When is less more? Investigating gap-filling in proofs without words activities. *Educational Studies in Mathematics*, 111(2), 271-297.

Miyazaki, M., Fujita, T., & Jones, K. (2017). Students’ understanding of the structure of deductive proof. *Educational Studies in Mathematics*, 94, 223-239.

Sinclair, N., Bussi, M. G. B., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2016). Recent research on geometry education: An ICME-13 survey team report. *ZDM*, 48(5), 691-719.

Sinclair, N., Moss, J., Hawes, Z., & Stephenson, C. (2018). Learning through and from drawing in early years geometry. In *Visualizing mathematics* (pp. 229-252). Springer, Cham.
geometry); 2) students then learn how to write and read proofs, but logical circularity (e.g., Q4, Deductive proof) and the economical nature of hierarchical classification of shapes should be explicitly addressed (e.g., Q8, Geometrical figures); 3) their understanding of deductive proofs and reasoning can be strengthened through more proof tasks (e.g., Q2), and problem solving in 2/3D geometry contexts (e.g., Q7, Q7K etc.). Also, these 7 items can be used to quickly check the extent to which students have understood the basic elements of deductive proofs. For example, in the case of Japanese students, we could use these 7 items with G9 students to quickly check their understanding, who have already completed their study of geometric proofs in G8, but will begin to learn more proofs in similar triangles.

## REFERENCES

Bruce, C. D., & Hawes, Z. (2015). The role of 2D and 3D mental rotation in mathematics for young children: what is it? Why does it matter? And what can we do about it?. *ZDM*, *47*(3), 331-343.

De Villiers, M. (1994). The role and function of a hierarchical classification of quadrilaterals. *For the learning of mathematics, 14*(1), 11-18.

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

Fujita, T., & Jones, K. (2003). The place of experimental tasks in geometry teaching: learning from the textbook designs of the early 20th century. *Research in Mathematics Education, 5*(1), 47-62.

Fujita T. (2012). Learners’ level of understanding of the inclusion relations of quadrilaterals and prototype phenomenon. *Journal of Mathematical Behavior, 31*(1), 60-72.

Fujita, T., Kondo, Y., Kumakura, H., Kunimune, S., & Jones, K. (2020). Spatial reasoning skills about 2D representations of 3D geometrical shapes in grades 4 to 9. *Mathematics Education Research Journal, 32*, 235-255.

Fujita, T., Kondo, Y., Kumakura, H., Miyawaki, S., Kunimune, S., & Shojima, K. (2022). Identifying Japanese students' core spatial reasoning skills by solving 3D geometry problems: An exploration. *Asian Journal for Mathematics Education, 1*(4), 437-454.

Hoyles, C., & Healy, L. (2007). Curriculum change and geometrical reasoning. In P. Boero (Ed.), *Theorems in school* (pp. 81–115). Rotterdam: Sense.

Lowrie, T., Harris, D., Logan, T., & Hegarty, M. (2021). The impact of a spatial intervention program on students’ spatial reasoning and mathematics performance. *The Journal of Experimental Education, 89*(2), 259-277.

Koseki. K. (Ed.). (1987). *The teaching of geometrical proof*, Meiji Tosho Publishers, Tokyo (in Japanese)

Kunimune, S., Fujita, T., & Jones, K. (2010). Strengthening students’ understanding of proof in geometry in lower secondary school. In *Proceedings of the 6th Congress of the European Society for Research in Mathematics Education* (pp. 756-765).

Marco, N., Palatnik, A., & Schwarz, B. B. (2022). When is less more? Investigating gap-filling in proofs without words activities. *Educational Studies in Mathematics*, 111(2), 271-297.

Miyazaki, M., Fujita, T., & Jones, K. (2017). Students’ understanding of the structure of deductive proof. *Educational Studies in Mathematics*, 94, 223-239.

Sinclair, N., Bussi, M. G. B., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2016). Recent research on geometry education: An ICME-13 survey team report. *ZDM*, *48*(5), 691-719.

Sinclair, N., Moss, J., Hawes, Z., & Stephenson, C. (2018). Learning through and from drawing in early years geometry. In *Visualizing mathematics* (pp. 229-252). Springer, Cham.
# APPENDIX

<table><thead><tr><th>Section 1 Deductive proof</th><th>Section 2 Geometrical figures</th><th>Section 3D geometry</th></tr></thead><tbody><tr><td>Q1 Which explanation do you agree with as a proof (you can accept as many as you like)?<br>A: 'I have measured each of the angles and they are 50, 53 and 77. 50+53+77=180. So the sum is 180 degrees.'</td><td>Q5 Choose parallelograms from the figure below.</td><td>Q2K Draw and complete the front view of the cylinder in the given box.</td></tr><tr><td>B: 'I drew a triangle and cut off each angle and put them together. They formed a straight line. So the sum is 180 degrees.'<br>C: Demonstration using the properties of a parallel line (an acceptable proof).<br>Q2 Prove that AD=CB when ∠A=∠C, and AE=CE.</td><td>Q6 What is a parallelogram? Describe it in words.<br>Q7b Is it possible to draw a parallelogram whose four vertices are on the circumference of a circle? State your reason as well.<br>Q8b We know that 'the lengths of the two diagonals of a rectangle are equal'. From this, is it also true that 'the lengths of the diagonals of a square are equal' without having to draw a diagram, measure the lengths, etc.? State your reason.</td><td>Q3-2K Next, we stack 5 cubes and make a different solid. When we look at this new solid from the top view, then it is like fig. 4. Choose the correct solid from a)-e).</td></tr><tr><td>Q3 Reading a proof of the diagonals of a parallelogram intersect at their middle points', and point out why we can say a) AB//DC, b) AB=CD, and c) ∠ABO=∠CDO?</td><td>Q9 Name the shape described by the following statement - 'A parallelogram with one vertex as a right angle.'</td><td>Q5K Sketch possible two solids whose plan and elevation.</td></tr><tr><td>Q4 Do you accept the following argument which demonstrates that in an isosceles triangle ABC, the base angles are equal? 'Draw an angle bisector AD from ∠A. In ∠ABD and ∠ACD, AB=AC, ∠BAD=∠CAD and ∠B=∠C. Therefore, ∠ABD=∠ACD and hence ∠B=∠C'. If you do not accept, then write down your reason (Q4b).</td><td>Q10 Tick the ones you think are correct.<br>(a) A rhombus is different from a parallelogram.<br>(b) You can say that a rhombus is a parallelogram.<br>(c) It is acceptable to say that a parallelogram is a rhombus.</td><td>Q8K In a cube, find the size of the angle ABC in each cube.<br>Q11K What is the size of the angle DEB in a cube? Write your reasons why.</td></tr></tbody></table>
APPENDIX

<table><thead><tr><th>Section 1 Deductive proof</th><th>Section 2 Geometrical figures</th><th>Section 3D geometry</th></tr></thead><tbody><tr><td>Q1 Which explanation do you agree with as a proof (you can accept as many as you like)?<br>A: 'I have measured each of the angles and they are 50, 53 and 77. 50+53+77=180. So the sum is 180 degrees.'</td><td>Q5 Choose parallelograms from the figure below.</td><td>Q2K Draw and complete the front view of the cylinder in the given box.</td></tr><tr><td>B: 'I drew a triangle and cut off each angle and put them together. They formed a straight line. So the sum is 180 degrees.'<br>C: Demonstration using the properties of a parallel line (an acceptable proof).<br>Q2 Prove that AD=CB when ∠A=∠C, and AE=CE.</td><td>Q6 What is a parallelogram? Describe it in words.<br>Q7b Is it possible to draw a parallelogram whose four vertices are on the circumference of a circle? State your reason as well.<br>Q8b We know that 'the lengths of the two diagonals of a rectangle are equal'. From this, is it also true that 'the lengths of the diagonals of a square are equal' without having to draw a diagram, measure the lengths, etc.? State your reason.</td><td>Q3-2K Next, we stack 5 cubes and make a different solid. When we look at this new solid from the top view, then it is like fig. 4. Choose the correct solid from a)-e).<br>Q5K Sketch possible two solids whose plan and elevation.</td></tr><tr><td>Q3 Reading a proof of the diagonals of a parallelogram intersect at their middle points', and point out why we can say a) AB//DC, b) AB=CD, and c) ∠ABO=∠CDO?<br>Q4 Do you accept the following argument which demonstrates that in an isosceles triangle ABC, the base angles are equal? 'Draw an angle bisector AD from ∠A. In ∠ABD and ∠ACD, AB=AC, ∠BAD=∠CAD and ∠B=∠C. Therefore, ∠ABD=∠ACD and hence ∠B=∠C'. If you do not accept, then write down your reason (Q4b).</td><td>Q9 Name the shape described by the following statement – 'A parallelogram with one vertex as a right angle.'</td><td>Q8K In a cube, find the size of the angle ABC in each cube.</td></tr><tr><td></td><td>Q10 Tick the ones you think are correct.<br>(a) A rhombus is different from a parallelogram.<br>(b) You can say that a rhombus is a parallelogram.<br>(c) It is acceptable to say that a parallelogram is a rhombus.</td><td>Q11K What is the size of the angle DEB in a cube? Write your reasons why.</td></tr></tbody></table>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>
# THE ROLE OF VISUAL MEDIATORS IN GEOMETRIC LEARNING PROCESSES IN UNIVERSITY EDUCATION

Alessandro Gambini¹, Giada Viola², Federica Ferretti²

¹ Sapienza University of Rome, ² University of Ferrara

> *Our research aims to investigate how the presence of visual mediators can affect the resolution process of university students. Specifically, our study is focused on analyzing students' problem-solving processes in reference to tasks that require them to determine the straight lines intersecting a given parabola. This study allowed us to identify two different categories of strategies implemented by the students, which are framed by Sfard's theory of Commognition. This categorization allowed us to investigate the impact of the presence of visual mediators in the text of the tasks, and their role in the resolution processes enacted by students.*

## RATIONALE

Geometrical spatial thinking is the ability to visualize, represent and manipulate objects in our minds; as shown by literature (e.g. Presmeg, 2006), spatial visualization skills are those cognitive abilities that an individual can acquire to process images in his or her mind and, thus, also provide solutions to various problems. These are fundamental skills for understanding and solving geometry problems and due to their crucial contribution in the process of advancing mathematical knowledge, they are currently further studied in mathematics education (Presmeg, 2020).

As envisaged in the *ICMI Study 26 Discussion Document*, this contribution aims to explore the following key question: "In what ways are visualization and geometry linked, and how does visualization support geometric thinking?". This delicate and crucial issue plays a fundamental role in the teaching and learning processes of geometry and is linked to the management of semiotic transformations.

The management of different representations of a mathematical object, and the difficulties in managing different semiotic representations that may hinder student learning have been the subject of several studies in the literature (e.g., Duval, 2006). As per Duval (2006), in order to address the challenges in comprehension that numerous students face during mathematical tasks, it is necessary to take a cognitive perspective and delve into the fundamental traits that underlie the various mathematical processes. The access to mathematical objects themselves, as well as their manipulation, are strictly linked to the possibility of representing them; the impossibility of direct access to mathematical objects also gave rise to Duval's (1993) well-known paradox.

Sfard's "Theory of Commognition" (Sfard, 2008) highlights the significance of visualization in the context of acquiring these mathematical concepts. This theory is focused on the mathematical discourse; the combination of the terms "cognition" and "communication" highlights the fact that cognition and communication are closely related to each other and represent different manifestations, one intrapersonal and the other interpersonal, of the same phenomenon. In this theory, four main properties characterize mathematical discourse: the visual mediators, the routines, word use and
# THE ROLE OF VISUAL MEDIATORS IN GEOMETRIC LEARNING PROCESSES IN UNIVERSITY EDUCATION

Alessandro Gambini¹, Giada Viola², Federica Ferretti²

¹ Sapienza University of Rome, ² University of Ferrara

> *Our research aims to investigate how the presence of visual mediators can affect the resolution process of university students. Specifically, our study is focused on analyzing students' problem-solving processes in reference to tasks that require them to determine the straight lines intersecting a given parabola. This study allowed us to identify two different categories of strategies implemented by the students, which are framed by Sfard's theory of Commognition. This categorization allowed us to investigate the impact of the presence of visual mediators in the text of the tasks, and their role in the resolution processes enacted by students.*

## RATIONALE

Geometrical spatial thinking is the ability to visualize, represent and manipulate objects in our minds; as shown by literature (e.g. Presmeg, 2006), spatial visualization skills are those cognitive abilities that an individual can acquire to process images in his or her mind and, thus, also provide solutions to various problems. These are fundamental skills for understanding and solving geometry problems and due to their crucial contribution in the process of advancing mathematical knowledge, they are currently further studied in mathematics education (Presmeg, 2020).

As envisaged in the *ICMI Study 26 Discussion Document*, this contribution aims to explore the following key question: "In what ways are visualization and geometry linked, and how does visualization support geometric thinking?". This delicate and crucial issue plays a fundamental role in the teaching and learning processes of geometry and is linked to the management of semiotic transformations.

The management of different representations of a mathematical object, and the difficulties in managing different semiotic representations that may hinder student learning have been the subject of several studies in the literature (e.g., Duval, 2006). As per Duval (2006), in order to address the challenges in comprehension that numerous students face during mathematical tasks, it is necessary to take a cognitive perspective and delve into the fundamental traits that underlie the various mathematical processes. The access to mathematical objects themselves, as well as their manipulation, are strictly linked to the possibility of representing them; the impossibility of direct access to mathematical objects also gave rise to Duval's (1993) well-known paradox.

Sfard's "Theory of Commognition" (Sfard, 2008) highlights the significance of visualization in the context of acquiring these mathematical concepts. This theory is focused on the mathematical discourse; the combination of the terms "cognition" and "communication" highlights the fact that cognition and communication are closely related to each other and represent different manifestations, one intrapersonal and the other interpersonal, of the same phenomenon. In this theory, four main properties characterize mathematical discourse: the visual mediators, the routines, word use and
narratives. In this research study, we focus attention on the role of routines and visual mediators in
resolution processes enacted by university students.

The routines – which in a very broad sense can be seen as the repetition of a previously performed
action – enact the mathematical concept, since the invariant that identifies the concept provides the
schema used to repeat the action done before. Lavie, Steiner and Sfard (2019) propose a refined and
complete definition of routines, which we will adopt throughout our study, and conceptualize the
learning of mathematics as a process of routinization of learners’ actions. Sfard (2008) identified
three kinds of routines: deeds, explorations, and rituals. The first type occurs when there is a physical
change in the mathematical object; the second type occurs in the production of the narratives; the last
type regards the discursive sequence aimed at maintaining a relationship with people. As we will see
later, rituals will play a key role in our categorization.

The visual mediators are means by which objects of discourse are identified within mathematical discourses, and their communication is coordinated. While colloquial discourses are usually mediated by images of independently existing material things, mathematical discourses often involve symbolic artifacts created specifically for this particular form of communication, including graphical representations. Visual mediators are visual elements that play a role during the communication process; they can be gestures, graphs, drawings, but also algebraic representations or material objects used during the teaching of mathematics. The visual mediators help during the communicative process, whether they are actually present or only imagined; in fact, they have been defined as “providers of the images with which discursants identify the object of their talk and coordinate their communication” (Sfard, 2008, p.147). Sfard shows several examples where visual objects (described as realizations of signifiers) are used during communication. The signifiers are the object of discourse, while the realizations of signifiers allow us to perceive the signifiers. There are different types of realizations of signifiers in mathematical discourse, with one of the most important being the visual type as this category allows us to gain a lot of information about the manipulation and representation of mathematical objects (Sfard, 2008). In the field of mathematics education research, there are many studies about the role of visual mediators, especially in their use in group communication (Ryve et al., 2013) and in the students’ resolution processes (Gambini et al., 2023). In Dell’Agnello et al.’s study (2022) it is shown how the use of visual mediators could help students to recognize the presence of polynomial roots.

As pointed out by *ICMI Study 26 - Discussion Document*, visualization plays an important role in the students' learning process, and it is very important to work with students with different inputs. Our study fits in this line of thought; in fact, we presented the same task to students in two different versions with two different inputs, to analyze how this difference could affect their solvings processes. Our research aims to investigate the role of visual mediators in university students' resolution processes for a task that requires them to determine the straight lines intersecting a given parabola. The research questions that guided our study are: in an analytical geometry task concerning the parabola and straight lines intersecting it, are there differences (both in terms of performance and the solution procedures chosen) if the same task is administered with the aid of different visual mediators? Does the use of different visual mediators in solving processes affect performance?
narratives. In this research study, we focus attention on the role of routines and visual mediators in
resolution processes enacted by university students.

The routines – which in a very broad sense can be seen as the repetition of a previously performed
action – enact the mathematical concept, since the invariant that identifies the concept provides the
schema used to repeat the action done before. Lavie, Steiner and Sfard (2019) propose a refined and
complete definition of routines, which we will adopt throughout our study, and conceptualize the
learning of mathematics as a process of routinization of learners’ actions. Sfard (2008) identified
three kinds of routines: deeds, explorations, and rituals. The first type occurs when there is a physical
change in the mathematical object; the second type occurs in the production of the narratives; the last
type regards the discursive sequence aimed at maintaining a relationship with people. As we will see
later, rituals will play a key role in our categorization.

The visual mediators are means by which objects of discourse are identified within mathematical discourses, and their communication is coordinated. While colloquial discourses are usually mediated by images of independently existing material things, mathematical discourses often involve symbolic artifacts created specifically for this particular form of communication, including graphical representations. Visual mediators are visual elements that play a role during the communication process; they can be gestures, graphs, drawings, but also algebraic representations or material objects used during the teaching of mathematics. The visual mediators help during the communicative process, whether they are actually present or only imagined; in fact, they have been defined as “providers of the images with which discursants identify the object of their talk and coordinate their communication” (Sfard, 2008, p.147). Sfard shows several examples where visual objects (described as realizations of signifiers) are used during communication. The signifiers are the object of discourse, while the realizations of signifiers allow us to perceive the signifiers. There are different types of realizations of signifiers in mathematical discourse, with one of the most important being the visual type as this category allows us to gain a lot of information about the manipulation and representation of mathematical objects (Sfard, 2008). In the field of mathematics education research, there are many studies about the role of visual mediators, especially in their use in group communication (Ryve et al., 2013) and in the students’ resolution processes (Gambini et al., 2023). In Dell’Agnello et al.’s study (2022) it is shown how the use of visual mediators could help students to recognize the presence of polynomial roots.

As pointed out by *ICMI Study 26 - Discussion Document*, visualization plays an important role in the students' learning process, and it is very important to work with students with different inputs. Our study fits in this line of thought; in fact, we presented the same task to students in two different versions with two different inputs, to analyze how this difference could affect their solvings processes. Our research aims to investigate the role of visual mediators in university students' resolution processes for a task that requires them to determine the straight lines intersecting a given parabola. The research questions that guided our study are: in an analytical geometry task concerning the parabola and straight lines intersecting it, are there differences (both in terms of performance and the solution procedures chosen) if the same task is administered with the aid of different visual mediators? Does the use of different visual mediators in solving processes affect performance?
## METHODOLOGY

In our study, our attention was focused on the role of visual mediators and routines in the sense of theory of commognition (Sfard, 2008). To investigate the role of visual mediators, a task composed of three multiple-choice questions was designed and administered in two different versions (Type 1 and Type 2, Figure 1). In the first version (Type 1), we provided the algebraic representation of a parabola, while in the second version (Type 2) we showed the graphical representation of the same parabola. In both iterations, the questions remain identical and require the identification of intersections between straight lines passing through specified points and the parabola. Each multiple-choice question offers four answer choices: 0, 1, 2, 3. At the conclusion of each question, students are prompted to provide a rationale for their selected response.

<figure><img src="image_24.png" /><figcaption>Figure 1: Tasks of Type 1 and Type 2</figcaption></figure>

The subjects in this research are 48 freshers of the Faculty of Engineering; the study was conducted at the end of the Calculus course in the first semester. To carry out a comparison between resolution processes and choices in relation to the proposed stimulus, half the students were given Type 1 and the other half Type 2; the students were randomly divided. The answers to the closed questions and corresponding explanations were collected and analysed with the aim to investigate the role of the presence of different visual mediators in the task text and their use in the solving processes.

## CATEGORISATION OF THE ANSWERS

To answer the research questions, we consider students' protocols by performing a qualitative content analysis (Mayring, 2015). Specifically, we sorted the resolution processes according to a concept-driven categorization in the sense of Kuckartz (2019), identifying two main concept-driven categories: Guided by Graph (GG) and Guided by Routines (GR).

The GG category consists of those protocols in which a graphical element, in terms of a visual mediator, is explicitly used and wherein use of the graph is relevant to the resolution process. This category includes both Type 1 solving protocols, in which one or more graphic elements were used even though they were not present in the text, and Type 2 procedures, in which the graphic element
## METHODOLOGY

In our study, our attention was focused on the role of visual mediators and routines in the sense of theory of commognition (Sfard, 2008). To investigate the role of visual mediators, a task composed of three multiple-choice questions was designed and administered in two different versions (Type 1 and Type 2, Figure 1). In the first version (Type 1), we provided the algebraic representation of a parabola, while in the second version (Type 2) we showed the graphical representation of the same parabola. In both iterations, the questions remain identical and require the identification of intersections between straight lines passing through specified points and the parabola. Each multiple-choice question offers four answer choices: 0, 1, 2, 3. At the conclusion of each question, students are prompted to provide a rationale for their selected response.

<figure><img src="image_29.png" /><figcaption>Figure 1: Tasks of Type 1 and Type 2</figcaption></figure>

The subjects in this research are 48 freshers of the Faculty of Engineering; the study was conducted at the end of the Calculus course in the first semester. To carry out a comparison between resolution processes and choices in relation to the proposed stimulus, half the students were given Type 1 and the other half Type 2; the students were randomly divided. The answers to the closed questions and corresponding explanations were collected and analysed with the aim to investigate the role of the presence of different visual mediators in the task text and their use in the solving processes.

## CATEGORISATION OF THE ANSWERS

To answer the research questions, we consider students' protocols by performing a qualitative content analysis (Mayring, 2015). Specifically, we sorted the resolution processes according to a concept-driven categorization in the sense of Kuckartz (2019), identifying two main concept-driven categories: Guided by Graph (GG) and Guided by Routines (GR).

The GG category consists of those protocols in which a graphical element, in terms of a visual mediator, is explicitly used and wherein use of the graph is relevant to the resolution process. This category includes both Type 1 solving protocols, in which one or more graphic elements were used even though they were not present in the text, and Type 2 procedures, in which the graphic element
is also explicitly present in the text. An example of a protocol that belongs in this category is shown
in Figure 2.

<figure><img src="image_25.png" /><figcaption>Figure 2: Example of a GG protocol of Type 2</figcaption></figure>

In the protocol (Fig. 2), the student used graphical representation not only to solve the task but also to explain his answers. In fact, for each case, the student showed the intersection of the line with the parabola and the passage to the assigned point. Specifically, in the figure, the student wrote: “3 lines passing through A(0; 0)”; “2 lines passing through B(0; 1)” and “1 line passing through C(0; 2)”.

In Figure 3 we can see an example from a student who solved Type 1 task and whose resolution strategy is included in the GG category.

<figure><img src="image_26.png" /><figcaption>Figure 3: Example of GG protocol of Type 1</figcaption></figure>

This example is in reference to a Type 1 task, where there was no figure in the text, but only the algebraic representation of the parabola. The student drew the parabola and represented on this the given points and straight lines passing through them. He supported his answers with this drawing and an explanation regarding the position of the point. In particular, he stated: “As we can see in the figure, each point stays on the y-axis”.

The protocols that show an attempt to find the straight lines with the algebraic method and without the aid of a graphic element belong to the GR category. The techniques predominantly used in these protocols allow them to find only tangent lines and, more relevantly, only lines that are functions. These protocols reveal use of the routine for applying the standard solving formula, which consists of setting up the equation of the parabola and the bundle of straight lines passing through a point, and
is also explicitly present in the text. An example of a protocol that belongs in this category is shown
in Figure 2.

<figure><img src="image_30.png" /><figcaption>Figure 2: Example of a GG protocol of Type 2</figcaption></figure>

In the protocol (Fig. 2), the student used graphical representation not only to solve the task but also to explain his answers. In fact, for each case, the student showed the intersection of the line with the parabola and the passage to the assigned point. Specifically, in the figure, the student wrote: “3 lines passing through A(0; 0)”; “2 lines passing through B(0; 1)” and “1 line passing through C(0; 2)”.

In Figure 3 we can see an example from a student who solved Type 1 task and whose resolution strategy is included in the GG category.

<figure><img src="image_31.png" /><figcaption>Figure 3: Example of GG protocol of Type 1</figcaption></figure>

This example is in reference to a Type 1 task, where there was no figure in the text, but only the algebraic representation of the parabola. The student drew the parabola and represented on this the given points and straight lines passing through them. He supported his answers with this drawing and an explanation regarding the position of the point. In particular, he stated: “As we can see in the figure, each point stays on the y-axis”.

The protocols that show an attempt to find the straight lines with the algebraic method and without the aid of a graphic element belong to the GR category. The techniques predominantly used in these protocols allow them to find only tangent lines and, more relevantly, only lines that are functions. These protocols reveal use of the routine for applying the standard solving formula, which consists of setting up the equation of the parabola and the bundle of straight lines passing through a point, and
then finding the angular coefficient, m, by setting the discriminant equal to zero of the second-degree equation dependent on m. This procedure is commonly used and is widespread in Italian high schools; we can consider it both as a ritual technique and, in a broader sense, a Sfard routine. We have named the protocols in which this solving procedure is used ‘Guided by Rituals – GR’. Also in this case, both the protocols referring to Type 1 and those referring to Type 2 can belong to this category. In Figure 4 we can see an example of a protocol belonging to the GR category.

<figure><img src="image_27.png" /><figcaption>Figure 4: Example of a GR’s protocol of Type 1</figcaption></figure>

In the example (Fig. 4), the student tried to find the angular coefficient m of the tangent lines of the curve; in this way, he failed to find the vertical line. In addition, he used the technique to find the tangent lines of the parabola even when the point is internal (point C). As a result of adopting this procedure, the student concluded that there are two straight lines that intercept the parabola through C. This is an example of using routines to solve the task and also to justify one’s answers.

<figure><img src="image_28.png" /><figcaption>Figure 6: Example of GR protocol of Type 2</figcaption></figure>

Figure 6 shows an example of a Type 2 protocol, in which a strategy belonging to the GR category was implemented. In this protocol it is possible to observe that the student tried to find the equations of the straight lines through the ritual procedure, but this technique was not completed, and the student was not confident of the given answer. Despite the presence of the graph in the text, the student made no reference to, or use of, this and he enacted a ritual procedure (probably because he or she was used to applying it in similar situations in high school).

## ANALYSIS

Through the analysis it was possible to investigate, on one hand, the impact of the presence of different visual mediators in the text of the task and, on the other hand, the influence of the use of graphical representation in the resolution processes.

The presence of the graph in the text of the task affected the percentage of correct answers. We consider "correct" answers as those given correctly to all three questions of the task. The overall
then finding the angular coefficient, m, by setting the discriminant equal to zero of the second-degree equation dependent on m. This procedure is commonly used and is widespread in Italian high schools; we can consider it both as a ritual technique and, in a broader sense, a Sfard routine. We have named the protocols in which this solving procedure is used ‘Guided by Rituals – GR’. Also in this case, both the protocols referring to Type 1 and those referring to Type 2 can belong to this category. In Figure 4 we can see an example of a protocol belonging to the GR category.

<figure><img src="image_32.png" /><figcaption>Figure 4: Example of a GR’s protocol of Type 1</figcaption></figure>

In the example (Fig. 4), the student tried to find the angular coefficient m of the tangent lines of the curve; in this way, he failed to find the vertical line. In addition, he used the technique to find the tangent lines of the parabola even when the point is internal (point C). As a result of adopting this procedure, the student concluded that there are two straight lines that intercept the parabola through C. This is an example of using routines to solve the task and also to justify one’s answers.

<figure><img src="image_33.png" /><figcaption>Figure 6: Example of GR protocol of Type 2</figcaption></figure>

Figure 6 shows an example of a Type 2 protocol, in which a strategy belonging to the GR category was implemented. In this protocol it is possible to observe that the student tried to find the equations of the straight lines through the ritual procedure, but this technique was not completed, and the student was not confident of the given answer. Despite the presence of the graph in the text, the student made no reference to, or use of, this and he enacted a ritual procedure (probably because he or she was used to applying it in similar situations in high school).

## ANALYSIS

Through the analysis it was possible to investigate, on one hand, the impact of the presence of different visual mediators in the text of the task and, on the other hand, the influence of the use of graphical representation in the resolution processes.

The presence of the graph in the text of the task affected the percentage of correct answers. We consider "correct" answers as those given correctly to all three questions of the task. The overall
correct answer percentages are very low: in total, 35 % of students answered correctly but there is a relevant difference between the correct answers in reference to the two Types. In fact, only 25% of students responding to the Type 1 task answered correctly, while almost half (46%) of the students who tackled the Type 2 task (in which the graphic element is present in the text) provided the correct answer. Moreover, the use of graphical representation also played a decisive role in the resolution processes. Comparing the GG and GR categories allows us to highlight how graphical representation can facilitate the resolution process and the production of a justification.

Students who used a GG strategy were those who explicitly used graphical representation to solve the task or to justify their answers. Among all students who answered correctly, 88% used graphical representation in their solving processes. Specifically, all students who performed the Type 1 task (without a parabola graph) and used graphical representation all answered correctly (Tab. 1). In GR's category protocols, students used routines to solve the task; in particular, they used the technique to find the angular coefficient of the tangent lines, and this procedure prevents finding the vertical line because it is not a function. Conversely, none of those students in the GR category answered correctly either Type 1 or Type 2.

<table><caption>Table 1: Percentage of correct answers in relation to the GG category</caption><thead><tr><th></th><th>% Correct Answers</th><th>% GG</th><th>% CA of GG</th></tr></thead><tbody><tr><th>Type 1</th><td>25%</td><td>54%</td><td>100%</td></tr><tr><th>Type 2</th><td>45.8%</td><td>58%</td><td>78.5%</td></tr></tbody></table>

This is also evident from the different approach to solving the task and interpreting the questions. In fact, in this task it was not required to find the equations of the lines, but to indicate how many lines intersect the parabola at a single point passing through a given point. Those who used the graphical representation were able to determine the number of the straight lines more easily and were able to draw the straight lines qualitatively. Students who implemented a GR category strategy, on the other hand, had difficulty visualizing the situation, and the algebraic approach did not allow them to find all straight lines. The technique implemented in the protocols of the GR category is a widespread procedure in Italian teaching practices and was most likely used in a ritual manner at high school by the students involved when they encountered tasks with intersections between a parabola and straight line.

## CONCLUDING REMARKS

Spatial visualization skills are fundamental for understanding and solving problems in geometry; in this study we investigated these abilities framing within two elements of the cognitive theory of Sfard: visual mediators and routines. We investigated and analyzed solving analysis tasks interpreting them with these theoretical elements and investigating their influence in the solving procedures implemented by university students. The inclusion of graphical representation in the task had a discernible impact on students' performance, resulting in a notably higher percentage of correct responses compared to the same task conducted without visual mediators. Specifically, it was evident that students who engaged in the Type 2 task achieved a higher proportion of accurate answers compared to those who undertook the Type 1 task. These results are in line with what is stated in the
correct answer percentages are very low: in total, 35 % of students answered correctly but there is a relevant difference between the correct answers in reference to the two Types. In fact, only 25% of students responding to the Type 1 task answered correctly, while almost half (46%) of the students who tackled the Type 2 task (in which the graphic element is present in the text) provided the correct answer. Moreover, the use of graphical representation also played a decisive role in the resolution processes. Comparing the GG and GR categories allows us to highlight how graphical representation can facilitate the resolution process and the production of a justification.

Students who used a GG strategy were those who explicitly used graphical representation to solve the task or to justify their answers. Among all students who answered correctly, 88% used graphical representation in their solving processes. Specifically, all students who performed the Type 1 task (without a parabola graph) and used graphical representation all answered correctly (Tab. 1). In GR's category protocols, students used routines to solve the task; in particular, they used the technique to find the angular coefficient of the tangent lines, and this procedure prevents finding the vertical line because it is not a function. Conversely, none of those students in the GR category answered correctly either Type 1 or Type 2.

<table><caption>Table 1: Percentage of correct answers in relation to the GG category</caption><thead><tr><th></th><th>% Correct Answers</th><th>% GG</th><th>% CA of GG</th></tr></thead><tbody><tr><th>Type 1</th><td>25%</td><td>54%</td><td>100%</td></tr><tr><th>Type 2</th><td>45.8%</td><td>58%</td><td>78.5%</td></tr></tbody></table>

This is also evident from the different approach to solving the task and interpreting the questions. In fact, in this task it was not required to find the equations of the lines, but to indicate how many lines intersect the parabola at a single point passing through a given point. Those who used the graphical representation were able to determine the number of the straight lines more easily and were able to draw the straight lines qualitatively. Students who implemented a GR category strategy, on the other hand, had difficulty visualizing the situation, and the algebraic approach did not allow them to find all straight lines. The technique implemented in the protocols of the GR category is a widespread procedure in Italian teaching practices and was most likely used in a ritual manner at high school by the students involved when they encountered tasks with intersections between a parabola and straight line.

## CONCLUDING REMARKS

Spatial visualization skills are fundamental for understanding and solving problems in geometry; in this study we investigated these abilities framing within two elements of the cognitive theory of Sfard: visual mediators and routines. We investigated and analyzed solving analysis tasks interpreting them with these theoretical elements and investigating their influence in the solving procedures implemented by university students. The inclusion of graphical representation in the task had a discernible impact on students' performance, resulting in a notably higher percentage of correct responses compared to the same task conducted without visual mediators. Specifically, it was evident that students who engaged in the Type 2 task achieved a higher proportion of accurate answers compared to those who undertook the Type 1 task. These results are in line with what is stated in the
ICMI Study 26 document where it is emphasized how much “it is important to consider how students
can be offered a much wider variety of experiences (and inputs)” (p. 4).

Regarding the second research question, it became evident that the choice of visual mediators during
students' performance significantly influenced their ability to provide correct answers. This
phenomenon likely arises from the fact that the use of graphical representation facilitated students in
visually discerning the number of straight lines intersecting with the parabola. Conversely, when
employing the algebraic approach, students attempted to determine the angular coefficient, a method
that did not enable them to identify all the requisite straight lines. This study underscores the efficacy
of incorporating and utilizing graphical representation in successfully tackling the task at hand.
According to Duval’s (2006) research, this study showed the importance of management of different
representations, particularly the graphical representation of a mathematical object. The correct use of
graphical representation to solve the task showed students’ ability to handle the representations of a
parabola. Moreover, those who used the GG strategy correctly in the Type 1 task revealed an ability
to switch between algebraic and graphical representations. This grasp and management of different
representations failed when routines took over.

When tackling a new mathematical topic, the student is involved in mathematical activity pushed by
social needs such as acceptance, expectations and consensus of the teacher or schoolmates. The
mathematical routine has no link with the objective of the task situation and instead of being outcome-
oriented (as it should be in nature), it becomes process-oriented. Students who adopted a process that
belongs to the GR category (probably because they preferred to use a familiar practice that meets the
expectations of the teacher) instead focus attention on the request made by the task.

The use of the GG or GR categories strategies affected not only the percentage of correct answers,
but also the approach to the problem and the aim of this approach; in one case, it is geared toward
solving the task, while in the case of routines it is focused on the procedure employed. Further studies,
combined with interviews, will provide further insight into the causes and features of the choice and
implementation of respective resolution strategies.

References

Dell'Agnello, R., Gambini, A., Maffia, A., & Viola, G. (2022). University students' discourse about irreducible polynomials. In: Fernández, Ceneida, et al. (Eds.). *Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education*. Alicante, Spain, 18-23 of July 2022. Alicante: PME, 2022. Vol. 2. pp. 219-226.

Duval R. (1993). Registres de représentations sémiotiques et fonctionnement cognitif de la pensée. *Annales de Didactique et de Science Cognitives*. ULP, IREM Strasbourg. 5, 37-65.

Duval R. (2006), The cognitive analysis of problems of comprehension in the learning of mathematics. *Educational Studies in Mathematics*, 61(1), 103-131.

Gambini, A., Viola, G. & Ferretti, F. (2023). The influence of visual mediators in university students' resolution processes. In: *Proceedings of the 46th Conference of the International Group for the Psychology of Mathematics Education*, in press.

Kuckartz, U. (2019). Qualitative text analysis: A systematic approach. In *Compendium for early career researchers in mathematics education* (pp. 181-197). Springer.

Lavie, I., Steiner, A., & Sfard, A. (2019). Routines we live by: From ritual to exploration. *Educational Studies in Mathematics*, 101(2), 153-176.