* How can the concept of mathematical proof in geometry be characterized at different educational levels? What types of geometry proofs should students learn during the different school levels? Are there paths or trajectories during the process of learning to prove geometric statements?

* What types of teaching sequences are helpful to promote the learning of geometric proofs?

* How might teacher and/or student uses of digital technologies provide a set of affordances to think of different types of arguments (visual, empirical, formal)?

* How do undergraduate students value the correctness of geometric proofs? Is this in line with their teachers' values?

* How do dynamic geometry environments affect conceptions of what is meant by geometric proof or what a geometric proof is?

* What affective components (e.g., anxiety, confidence, and efficacy) influence the learning of geometric proof? Are there gender differences in approaches to geometric proofs?

* In which ways can students be engaged in formulating conjectures and doing geometric proofs?

## Sub-topic A2: The role of external visual inputs in geometric reasoning

Visualization is a broad construct that is important in mathematics. Simply put, it involves “the mind seeing things.” Use of imagery is known to support geometric thinking and is instrumental in working in novel or complex situations. Along with many others, Mason’s (2004) work on the “structure of attention” intersects with these notions of visualization. However, challenges prevail in mathematics education where canonical/prototypical and static inputs tend to dominate students’ encoding of visual inputs. Consequently, it is important to consider how students can be offered a much wider variety of experiences (and inputs) as early as possible. Dynamic visualizations, where there is movement, such as in visualizations of compressions and stretches are one example of expanding these cases. Building on the work of Duval (1998), what is seen is quickly classified in the minds’ eye as “triangle”, for example; this is the iconic vision, and these canonical/prototypical constructs can be deconstructed. For instance, a square would need to be seen not only as a shape of square in an iconic way, but also as four lines intersecting orthogonally at equal distance and ultimately four points. New technologies offer interesting new affordances worthy of exploration. A burgeoning area of research related to visualization focuses on objects-to-think-with or implicit tools for use in a broad range of contexts. There is much to explore at this time in the arena of external visual inputs.

The IPC invites contributions that explore these key questions:

* In what ways are visualization and geometry linked, and how does visualization support geometric thinking?

* What affordances and challenges are engendered through technology uses for visualization?

* How might learners break away from canonical visual depictions of geometric figures? Why is this important?

* Are visual and analytic geometric processing distinct? How do they overlap and when should they overlap?
* How can the concept of mathematical proof in geometry be characterized at different educational levels? What types of geometry proofs should students learn during the different school levels? Are there paths or trajectories during the process of learning to prove geometric statements?

* What types of teaching sequences are helpful to promote the learning of geometric proofs?

* How might teacher and/or student uses of digital technologies provide a set of affordances to think of different types of arguments (visual, empirical, formal)?

* How do undergraduate students value the correctness of geometric proofs? Is this in line with their teachers' values?

* How do dynamic geometry environments affect conceptions of what is meant by geometric proof or what a geometric proof is?

* What affective components (e.g., anxiety, confidence, and efficacy) influence the learning of geometric proof? Are there gender differences in approaches to geometric proofs?

* In which ways can students be engaged in formulating conjectures and doing geometric proofs?

## Sub-topic A2: The role of external visual inputs in geometric reasoning

Visualization is a broad construct that is important in mathematics. Simply put, it involves “the mind seeing things.” Use of imagery is known to support geometric thinking and is instrumental in working in novel or complex situations. Along with many others, Mason’s (2004) work on the “structure of attention” intersects with these notions of visualization. However, challenges prevail in mathematics education where canonical/prototypical and static inputs tend to dominate students’ encoding of visual inputs. Consequently, it is important to consider how students can be offered a much wider variety of experiences (and inputs) as early as possible. Dynamic visualizations, where there is movement, such as in visualizations of compressions and stretches are one example of expanding these cases. Building on the work of Duval (1998), what is seen is quickly classified in the minds’ eye as “triangle”, for example; this is the iconic vision, and these canonical/prototypical constructs can be deconstructed. For instance, a square would need to be seen not only as a shape of square in an iconic way, but also as four lines intersecting orthogonally at equal distance and ultimately four points. New technologies offer interesting new affordances worthy of exploration. A burgeoning area of research related to visualization focuses on objects-to-think-with or implicit tools for use in a broad range of contexts. There is much to explore at this time in the arena of external visual inputs.

The IPC invites contributions that explore these key questions:

* In what ways are visualization and geometry linked, and how does visualization support geometric thinking?

* What affordances and challenges are engendered through technology uses for visualization?

* How might learners break away from canonical visual depictions of geometric figures? Why is this important?

* Are visual and analytic geometric processing distinct? How do they overlap and when should they overlap?
* How do objects-to-think-with, implicit structures, and/or forms of visualization operate as a tool for geometric thinking and supporting analytic reasoning?

## Sub-topic A3: Spatial reasoning interventions and transfer to geometry

Spatial reasoning is integral to activities in daily life, such as packing a car for a trip or reading maps. At a broad level, spatial reasoning describes the ability to mentally represent, analyze, and transform objects and their relations, and is comprised of distinct, yet related, skills with strong ties to mathematics achievement. Spatial reasoning involves the location and movement of objects and ourselves, either mentally or physically, in space. It is not a single ability or process but actually refers to a considerable number of concepts, tools, and processes (National Research Council, 2006). The role of spatial reasoning within mathematics problem solving and performance have been noted for decades (Clements & Battista, 1992). In fact, a recent meta-analysis (Hawes et al., 2022) confirms that improving spatial reasoning skills causally supports improvements in mathematics performance.

Spatial reasoning is malleable, with improvements observed even in short periods of explicit instruction or exposure to spatial tools and playful experiences. Spatial reasoning is especially relevant to geometry education, with numerous (recent) studies demonstrating transfer to geometric understanding (Lowrie et al., 2019). What are less known are the mechanisms that support the transfer from spatial development to geometric understanding. This topic seeks advances in the field that describe interventions and learning programs that spatialize the geometry curriculum as well as spatial learning activities that support geometric understanding. Topics include the relationship between spatial reasoning and geometry in 2D and 3D transformations, composing and decomposing shape, perspective taking and activities involving symmetry and rotations of 2D shapes and 3D objects.

The IPC invites proposals that address some of the following questions:

* What are the mechanisms that support transfer from spatial reasoning to geometric understanding?

* What constructs or types of spatial reasoning should be included in school (Pre-K to 12) mathematics?

* Given the strong relationship between spatial reasoning and geometric understanding, what are promising ways of doing this?

* How is spatial reasoning positioned in school curricula and how does this vary from nation to nation? Is this serving students well? What is the impact?

## TOPIC B: CURRICULAR AND METHODOLOGICAL APPROACHES

A core pillar of the 26th ICMI Study relates to geometry curricula and methodological approaches to teaching geometry, from the early years through to higher education. There are novel developments that are changing the landscape of geometry education with intensity of research activity in the early years and in higher education. When developments in curriculum for school geometry are considered, increased attention to spatial reasoning is apparent, alongside and/or within the geometry curriculum. In many nations, curriculum constitutes policy, which carries weight and encourages greater classroom time devoted to geometry learning. Recent research on professional learning focused on geometry teaching and learning has demonstrated that longer and more intensive professional training programs, which are situated in classrooms directly involving students, are proving to have lasting
* How do objects-to-think-with, implicit structures, and/or forms of visualization operate as a tool for geometric thinking and supporting analytic reasoning?

## Sub-topic A3: Spatial reasoning interventions and transfer to geometry

Spatial reasoning is integral to activities in daily life, such as packing a car for a trip or reading maps. At a broad level, spatial reasoning describes the ability to mentally represent, analyze, and transform objects and their relations, and is comprised of distinct, yet related, skills with strong ties to mathematics achievement. Spatial reasoning involves the location and movement of objects and ourselves, either mentally or physically, in space. It is not a single ability or process but actually refers to a considerable number of concepts, tools, and processes (National Research Council, 2006). The role of spatial reasoning within mathematics problem solving and performance have been noted for decades (Clements & Battista, 1992). In fact, a recent meta-analysis (Hawes et al., 2022) confirms that improving spatial reasoning skills causally supports improvements in mathematics performance.

Spatial reasoning is malleable, with improvements observed even in short periods of explicit instruction or exposure to spatial tools and playful experiences. Spatial reasoning is especially relevant to geometry education, with numerous (recent) studies demonstrating transfer to geometric understanding (Lowrie et al., 2019). What are less known are the mechanisms that support the transfer from spatial development to geometric understanding. This topic seeks advances in the field that describe interventions and learning programs that spatialize the geometry curriculum as well as spatial learning activities that support geometric understanding. Topics include the relationship between spatial reasoning and geometry in 2D and 3D transformations, composing and decomposing shape, perspective taking and activities involving symmetry and rotations of 2D shapes and 3D objects.

The IPC invites proposals that address some of the following questions:

* What are the mechanisms that support transfer from spatial reasoning to geometric understanding?

* What constructs or types of spatial reasoning should be included in school (Pre-K to 12) mathematics?

* Given the strong relationship between spatial reasoning and geometric understanding, what are promising ways of doing this?

* How is spatial reasoning positioned in school curricula and how does this vary from nation to nation? Is this serving students well? What is the impact?

## TOPIC B: CURRICULAR AND METHODOLOGICAL APPROACHES

A core pillar of the 26th ICMI Study relates to geometry curricula and methodological approaches to teaching geometry, from the early years through to higher education. There are novel developments that are changing the landscape of geometry education with intensity of research activity in the early years and in higher education. When developments in curriculum for school geometry are considered, increased attention to spatial reasoning is apparent, alongside and/or within the geometry curriculum. In many nations, curriculum constitutes policy, which carries weight and encourages greater classroom time devoted to geometry learning. Recent research on professional learning focused on geometry teaching and learning has demonstrated that longer and more intensive professional training programs, which are situated in classrooms directly involving students, are proving to have lasting
impact (Clements & Sarama, 2011). This includes attending to individual differences of learners and cultural and political contexts of learning.

**Sub-topic B1: Geometric thinking across age levels (from preschool to higher education)**

Despite decades of robust research in geometry education, there continues to be gaps in students’ geometry experiences and learning across age levels from preschool to higher education. The gaps include limited opportunities to engage with real-world and mathematical objects that foster geometric modeling and reasoning, uneven quality of curriculum and teaching that students experience, limited connections made between spatial, geometric, and other mathematical concepts, and what students are asked to “do” (e.g., doing, describing, analyzing, proving, etc.). There is an urgent need to develop and/or refine ways of analyzing these gaps. In the last 30 years, there has been a flurry of research activity in the early years and in higher education that focuses on improving spatial sense and geometric reasoning. The literature on building spatial reasoning for young learners and for those pursuing careers in STEM education is developing at an accelerated pace.

Some of the important issues on which researchers and teachers alike could contribute are:

* What is the current status of geometry curricula across age levels and how do such curricula support student’s conceptual understanding and development?

* What curricular and classroom-based innovations could help to understand, analyze, and fill the gaps among early, primary, secondary, and higher education? How can recent research efforts in the early years and in post-secondary education be built on?

* How do informal and formal geometry curricula seek to include the use of language, gestures, drawing, modelling, or representations to support student thinking? How can the effectiveness of these aspects of conveying meaning be analyzed?

* What are effective ways to investigate children’s developing understanding of 2D and 3D shapes in early years? How are early years spatial-related experiences linked to later formal learning of geometry?

* What efforts have been made to ensure that formal geometry curricula build on earlier understandings?

* How are post-secondary programs in STEM areas supporting the geometric thinking required for professions such as engineering, medicine, chemistry (and beyond)?

**Sub-topic B2: Individual differences, including students with learning difficulties or mathematical giftedness**

The ability to study geometry successfully can be a consequence of learning opportunities provided to students or their innate capabilities, that may impede or advance the development of their geometry knowledge, skills, and interest in learning geometry.

Difficulties in processing geometric concepts can depend on a range of factors. For example, dyslexia or difficulties related to reading comprehension may affect students' capabilities to formulate, describe, and characterize geometrical objects and their properties and read geometrical text, such as definitions and theorems. Students with learning difficulties, for example, may find visual-spatial supports that do not require language and number to provide new pathways into complex geometry. Dyscalculia can influence solving calculation problems in geometry whereas dysgraphia can be an obstacle for drawing geometric figures. Visual impairments also impact students' geometrical
impact (Clements & Sarama, 2011). This includes attending to individual differences of learners and cultural and political contexts of learning.

## Sub-topic B1: Geometric thinking across age levels (from preschool to higher education)

Despite decades of robust research in geometry education, there continues to be gaps in students’ geometry experiences and learning across age levels from preschool to higher education. The gaps include limited opportunities to engage with real-world and mathematical objects that foster geometric modeling and reasoning, uneven quality of curriculum and teaching that students experience, limited connections made between spatial, geometric, and other mathematical concepts, and what students are asked to “do” (e.g., doing, describing, analyzing, proving, etc.). There is an urgent need to develop and/or refine ways of analyzing these gaps. In the last 30 years, there has been a flurry of research activity in the early years and in higher education that focuses on improving spatial sense and geometric reasoning. The literature on building spatial reasoning for young learners and for those pursuing careers in STEM education is developing at an accelerated pace.

Some of the important issues on which researchers and teachers alike could contribute are:

* What is the current status of geometry curricula across age levels and how do such curricula support student’s conceptual understanding and development?

* What curricular and classroom-based innovations could help to understand, analyze, and fill the gaps among early, primary, secondary, and higher education? How can recent research efforts in the early years and in post-secondary education be built on?

* How do informal and formal geometry curricula seek to include the use of language, gestures, drawing, modelling, or representations to support student thinking? How can the effectiveness of these aspects of conveying meaning be analyzed?

* What are effective ways to investigate children’s developing understanding of 2D and 3D shapes in early years? How are early years spatial-related experiences linked to later formal learning of geometry?

* What efforts have been made to ensure that formal geometry curricula build on earlier understandings?

* How are post-secondary programs in STEM areas supporting the geometric thinking required for professions such as engineering, medicine, chemistry (and beyond)?

## Sub-topic B2: Individual differences, including students with learning difficulties or mathematical giftedness

The ability to study geometry successfully can be a consequence of learning opportunities provided to students or their innate capabilities, that may impede or advance the development of their geometry knowledge, skills, and interest in learning geometry.

Difficulties in processing geometric concepts can depend on a range of factors. For example, dyslexia or difficulties related to reading comprehension may affect students’ capabilities to formulate, describe, and characterize geometrical objects and their properties and read geometrical text, such as definitions and theorems. Students with learning difficulties, for example, may find visual-spatial supports that do not require language and number to provide new pathways into complex geometry. Dyscalculia can influence solving calculation problems in geometry whereas dysgraphia can be an obstacle for drawing geometric figures. Visual impairments also impact students’ geometrical
learning and processing. Students with learning disabilities need special programs and approaches to teaching geometry that allow them to study fluently and experience success in geometry lessons.

Moreover, students with high mathematical abilities who enjoy solving challenging geometry problems also need special attention. As Krutetskii (1976) demonstrated, mathematically gifted students are able to memorize mathematical objects, schemes, principles, and relationships and often reason both logically and spatially. Such mathematical processing at these high levels across different branches of mathematics, including geometry, provide opportunities for creativity-directed activities such as conjecturing, proving, or refuting geometry statements using different theorems and with different auxiliary contractions. *Mathematical cast of mind* (Krutetskii, 1976) reflects students' special capability to see mathematical objects within the real world as well as to be creative in mathematics. Mathematically gifted students can be exceptionally creative, and their creativity needs to be encouraged and promoted.

The IPC invites contributions that explore teaching and learning geometry as related to a continuum of mathematical capabilities. Submissions are encouraged of studies that include, but are not limited to, the following questions linked to (a) various student learning disabilities and (b) students with high mathematical abilities:

* How do student capabilities affect studying geometry?

* Are there types of geometry tasks that map to different capabilities?

* What teaching approaches are most effective for students with different characteristics?

* How can mathematics teachers be supported to work with students of varying capabilities?

* What type of assessment tasks are effective when teaching students with special educational needs?

* How can technological tools be used to support and challenge the teaching and learning of geometry by students with special learning needs, including gifted students?

* How can creativity-directed activities or creativity-promoting activities support learning geometry for students with learning disabilities or mathematically gifted students?

**Sub-topic B3: Influence on the curriculum of different cultural and political traditions and contexts around the world**

Curriculum is regarded as the knowledge, skills, values, and attitudes presented to the learners in order to change their behaviors to become active members of society. A school mathematics curriculum should reflect the political and cultural contexts of the students for which it is planned. Political context refers to the way power is achieved or used in a society. Cultural context refers to the shared values, attitudes, beliefs, and practices of a community. Geometry should not be regarded as a value-free curricular content area. In fact, some argue that Mathematics, including geometry, should equip students to deal with political and cultural issues; for example, closing the learning gap between informal and formal geometric knowledge.

The IPC invites proposals that address some of the following questions:

* What is the influence on the curriculum of different cultural traditions and contexts around the world regarding geometric concepts?
learning and processing. Students with learning disabilities need special programs and approaches to teaching geometry that allow them to study fluently and experience success in geometry lessons.

Moreover, students with high mathematical abilities who enjoy solving challenging geometry problems also need special attention. As Krutetskii (1976) demonstrated, mathematically gifted students are able to memorize mathematical objects, schemes, principles, and relationships and often reason both logically and spatially. Such mathematical processing at these high levels across different branches of mathematics, including geometry, provide opportunities for creativity-directed activities such as conjecturing, proving, or refuting geometry statements using different theorems and with different auxiliary contractions. *Mathematical cast of mind* (Krutetskii, 1976) reflects students' special capability to see mathematical objects within the real world as well as to be creative in mathematics. Mathematically gifted students can be exceptionally creative, and their creativity needs to be encouraged and promoted.

The IPC invites contributions that explore teaching and learning geometry as related to a continuum of mathematical capabilities. Submissions are encouraged of studies that include, but are not limited to, the following questions linked to (a) various student learning disabilities and (b) students with high mathematical abilities:

* How do student capabilities affect studying geometry?

* Are there types of geometry tasks that map to different capabilities?

* What teaching approaches are most effective for students with different characteristics?

* How can mathematics teachers be supported to work with students of varying capabilities?

* What type of assessment tasks are effective when teaching students with special educational needs?

* How can technological tools be used to support and challenge the teaching and learning of geometry by students with special learning needs, including gifted students?

* How can creativity-directed activities or creativity-promoting activities support learning geometry for students with learning disabilities or mathematically gifted students?

**Sub-topic B3: Influence on the curriculum of different cultural and political traditions and contexts around the world**

Curriculum is regarded as the knowledge, skills, values, and attitudes presented to the learners in order to change their behaviors to become active members of society. A school mathematics curriculum should reflect the political and cultural contexts of the students for which it is planned. Political context refers to the way power is achieved or used in a society. Cultural context refers to the shared values, attitudes, beliefs, and practices of a community. Geometry should not be regarded as a value-free curricular content area. In fact, some argue that Mathematics, including geometry, should equip students to deal with political and cultural issues; for example, closing the learning gap between informal and formal geometric knowledge.

The IPC invites proposals that address some of the following questions:

* What is the influence on the curriculum of different cultural traditions and contexts around the world regarding geometric concepts?
* What is the influence on the curriculum of different political traditions and contexts around the world regarding geometric concepts?

* How can political and cultural contexts of mathematics curricula help to recognize and respect the history, tradition, and mathematical thinking developed by members of distinct cultural groups through the development of geometric thinking?

* What is the significance of the influence on the curriculum of different political and cultural contexts for researchers, policy makers, and school teachers?

**Sub-topic B4: Professional training and development in geometry education (pre-service and in-service)**

There are a wide range of professional learning models related to geometry education both for pre-service and in-service educators. Early years teachers often receive mathematics professional training that focuses on playful approaches to exploring patterns (repeating patterns), informal volume and capacity activities (emptying and filling containers), size comparisons (ordering objects from smallest to largest), and in the area of geometry, naming and recognizing 2D shapes. Recently, programs that focus on mathematics for young children have intensified the attention on geometry and spatial reasoning (Moss, Bruce & Bobis, 2015).

In terms of educator challenges, some primary teachers have limited knowledge about geometry and do not necessarily gain further understanding in the pre-service years due to a typical focus on general mathematics that affects the amount of attention on geometry in the classroom. For example, there is evidence that teachers concentrate on the precision of geometrical drawings rather than on the validity of the geometrical constructions. Moreover, the integration of the spatial reasoning components in geometry learning are yet to be fully realized in elementary curricula. By contrast, some secondary teachers might find it challenging to present pedagogical practices that are engaging, especially if they lack an appreciation of the nature of geometry, the foundations of geometry and how to teach these principles and baseline concepts. Educators also face challenges when presented with professional training programs. These include a lack of time to engage in professional learning, a lack of resources to implement new strategies (to release teachers and provide support materials), fear of mathematics and a lack of geometry knowledge, and/or a devaluing of geometry as an important strand/area of mathematics.

Despite these challenges, some professional learning opportunities are showing promise in supporting geometry educators. For both pre-service and in-serving teaching, the opportunity to explore models of active teaching and use and design rich geometry tasks can help support the development of effective pedagogical practices. In-service training in geometry can be deeply supported by more intensive forms of professional learning such as lesson study and collaborative action research. Working directly and consistently with students and teachers in the classroom has also been shown to be effective in building effective practices.

Methodologically, many research-based professional learning programs are guided by a design research approach. This involves the collaborative and continual refinement of materials, tools and meaningful tasks that address learning needs which have been identified in the analysis of student thinking. Using this type of approach, classroom trials ultimately engender refinements that are increasingly scalable.
* What is the influence on the curriculum of different political traditions and contexts around the world regarding geometric concepts?

* How can political and cultural contexts of mathematics curricula help to recognize and respect the history, tradition, and mathematical thinking developed by members of distinct cultural groups through the development of geometric thinking?

* What is the significance of the influence on the curriculum of different political and cultural contexts for researchers, policy makers, and school teachers?

**Sub-topic B4: Professional training and development in geometry education (pre-service and in-service)**

There are a wide range of professional learning models related to geometry education both for pre-service and in-service educators. Early years teachers often receive mathematics professional training that focuses on playful approaches to exploring patterns (repeating patterns), informal volume and capacity activities (emptying and filling containers), size comparisons (ordering objects from smallest to largest), and in the area of geometry, naming and recognizing 2D shapes. Recently, programs that focus on mathematics for young children have intensified the attention on geometry and spatial reasoning (Moss, Bruce & Bobis, 2015).

In terms of educator challenges, some primary teachers have limited knowledge about geometry and do not necessarily gain further understanding in the pre-service years due to a typical focus on general mathematics that affects the amount of attention on geometry in the classroom. For example, there is evidence that teachers concentrate on the precision of geometrical drawings rather than on the validity of the geometrical constructions. Moreover, the integration of the spatial reasoning components in geometry learning are yet to be fully realized in elementary curricula. By contrast, some secondary teachers might find it challenging to present pedagogical practices that are engaging, especially if they lack an appreciation of the nature of geometry, the foundations of geometry and how to teach these principles and baseline concepts. Educators also face challenges when presented with professional training programs. These include a lack of time to engage in professional learning, a lack of resources to implement new strategies (to release teachers and provide support materials), fear of mathematics and a lack of geometry knowledge, and/or a devaluing of geometry as an important strand/area of mathematics.

Despite these challenges, some professional learning opportunities are showing promise in supporting geometry educators. For both pre-service and in-serving teaching, the opportunity to explore models of active teaching and use and design rich geometry tasks can help support the development of effective pedagogical practices. In-service training in geometry can be deeply supported by more intensive forms of professional learning such as lesson study and collaborative action research. Working directly and consistently with students and teachers in the classroom has also been shown to be effective in building effective practices.

Methodologically, many research-based professional learning programs are guided by a design research approach. This involves the collaborative and continual refinement of materials, tools and meaningful tasks that address learning needs which have been identified in the analysis of student thinking. Using this type of approach, classroom trials ultimately engender refinements that are increasingly scalable.
The IPC invites proposals that address some of the following questions:

* What models or approaches to professional learning with a focus on geometry are proving to be effective in building geometric reasoning, and helping teachers and students to develop understanding of fundamental ideas in geometry?

* What are some effective approaches to designing meaningful tasks (including tasks that involve technology use) that support and reveal geometric reasoning and/or proof building from intuitive ideas, and theoretical understanding of geometric principles? How can these tasks be leveraged to support pre-service and in-service teacher training? How can teachers themselves be supported in task design that promotes geometry reasoning?

* How can pre-service and in-service teachers be supported in understanding the importance of explicit and implicit use of instruments in the learning of geometry? How does this translate to classroom practice?

* To what extent do pre-service and in-service teachers value geometry and how can their epistemology of the subject be reinforced?

## TOPIC C: RESOURCES FOR TEACHING AND LEARNING GEOMETRY

To teach and promote geometry understanding, teachers need to access resources and promote their use with students. These resources may be technological or non-technological in nature. Depending on the teaching context, these resources may operate as constraints or affordances, and be imposed, overabundant or limited. This topic explores questions about the impacts of various resources in supporting the teaching and learning of geometry. The inherent limitations of resources due to external factors are also considered.

### Sub-topic C1: Digital technologies in learning and teaching geometry

Seven lenses through which to examine technological tools in teaching and learning of geometry, the development of spatial capabilities, and advancement of geometrical reasoning and creativity are as follows: (1) the types of technology: software vs. hardware; (2) the goals of uses of technology: teaching and learning vs. evaluation and analysis; (3) the level of openness of the tools: open vs. half-baked vs. closed; (4) the educational functions: exploration vs. verification vs. proving vs. demonstration; (5) the context: pure mathematics vs. real/practical situation vs. Virtual Reality; (6) the nature of space and shape —2D vs. 3D vs. ... n-dimensional— and how it is represented; and (7) the types of geometry: Euclidean, Projective, Affine, Non-Euclidean geometries.

The tools under consideration could be, for example, dynamic geometry software, 3D spaces, virtual reality, intelligent tutors using AI, learning analytics, videos, programming tools and more. The hardware used in the studies can include, among others, computers, tablets, interactive whiteboards, 3D printers and robots. Information and communication technologies have the power of enhancing visual representations of shapes and space, readily conveying geometrical relations, and connecting multiple representations of concepts which are not possible by the tool of paper-and-pencil; mediating individual (teacher and student) and classroom discourse; broadening the scope of information provided by teachers and textbooks; and promoting students' multimodal, action-based, independent, and self-regulated learning.
The IPC invites proposals that address some of the following questions:

* What models or approaches to professional learning with a focus on geometry are proving to be effective in building geometric reasoning, and helping teachers and students to develop understanding of fundamental ideas in geometry?

* What are some effective approaches to designing meaningful tasks (including tasks that involve technology use) that support and reveal geometric reasoning and/or proof building from intuitive ideas, and theoretical understanding of geometric principles? How can these tasks be leveraged to support pre-service and in-service teacher training? How can teachers themselves be supported in task design that promotes geometry reasoning?

* How can pre-service and in-service teachers be supported in understanding the importance of explicit and implicit use of instruments in the learning of geometry? How does this translate to classroom practice?

* To what extent do pre-service and in-service teachers value geometry and how can their epistemology of the subject be reinforced?

## TOPIC C: RESOURCES FOR TEACHING AND LEARNING GEOMETRY

To teach and promote geometry understanding, teachers need to access resources and promote their use with students. These resources may be technological or non-technological in nature. Depending on the teaching context, these resources may operate as constraints or affordances, and be imposed, overabundant or limited. This topic explores questions about the impacts of various resources in supporting the teaching and learning of geometry. The inherent limitations of resources due to external factors are also considered.

### Sub-topic C1: Digital technologies in learning and teaching geometry

Seven lenses through which to examine technological tools in teaching and learning of geometry, the development of spatial capabilities, and advancement of geometrical reasoning and creativity are as follows: (1) the types of technology: software vs. hardware; (2) the goals of uses of technology: teaching and learning vs. evaluation and analysis; (3) the level of openness of the tools: open vs. half-baked vs. closed; (4) the educational functions: exploration vs. verification vs. proving vs. demonstration; (5) the context: pure mathematics vs. real/practical situation vs. Virtual Reality; (6) the nature of space and shape —2D vs. 3D vs. ... n-dimensional— and how it is represented; and (7) the types of geometry: Euclidean, Projective, Affine, Non-Euclidean geometries.

The tools under consideration could be, for example, dynamic geometry software, 3D spaces, virtual reality, intelligent tutors using AI, learning analytics, videos, programming tools and more. The hardware used in the studies can include, among others, computers, tablets, interactive whiteboards, 3D printers and robots. Information and communication technologies have the power of enhancing visual representations of shapes and space, readily conveying geometrical relations, and connecting multiple representations of concepts which are not possible by the tool of paper-and-pencil; mediating individual (teacher and student) and classroom discourse; broadening the scope of information provided by teachers and textbooks; and promoting students' multimodal, action-based, independent, and self-regulated learning.
Contributions can be about completed or in-progress research or innovation projects. Research can be empirical, experimental, or theoretical. The IPC invites proposals that address some of the following questions:

* What is the impact of using different types of technological tools in relation to one or more lenses proposed above?

* How does the impact of technological tools on geometry education depend on the types of tools, their openness, or their educational functions?

* How do the uses of different kinds of technologies by students, teachers, or teacher educators impact on their practices and thinking in geometry?

* What are some of the opportunities and challenges that are presented by the use of technology in geometry education?

## Sub-topic C2: Manipulatives and visual tools in teaching and learning geometry

Visual tools and concrete manipulatives are often seen as important resources in promoting geometrical thinking. A review of the state of the art of research on the contribution of manipulatives to learning and teaching geometry is needed. Are manipulations and visualization sufficient to allow effective learning processes of geometry, including deductive reasoning, proving, and constructing in geometry? How is the use of manipulatives linked to modelling, abstracting, using language to describe geometrical objects, and proving?

Manipulatives and visual tools can be examined from different theoretical perspectives and regarding different goals of the use of manipulatives and visualization: creating experience, identifying critical properties, formulating conjectures, and proving or refuting the conjectures.

The questions posed can be about different types of visualization and objects such as material tools, photographs, video, and drawings. The contributions may address different uses of visualization and objects, depending on the ages of students and the specific difficulties they might have (e.g., blindness, developmental coordination disorder, etc.).

The IPC invites proposals that address some of the following questions:

* What are the links between the use of manipulatives, visual tools, and modelling activity?

* What is the difference between using the manipulatives or dynamic diagrams constructed by educational designers and constructing manipulatives or working in an open dynamic environment (e.g., GeoGebra)?

* What is the role of visual tools and manipulatives on geometrical proof processes?

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

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

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

## Sub-topic C3: Learning geometry with resource constraints

The scope and focus of this sub-topic address issues relating to the teaching and learning of geometry in resource-constrained classrooms and under-resourced school communities. Resources, in this context, include physical resources (e.g., digital tools and non-digital manipulatives), non-tangible
Contributions can be about completed or in-progress research or innovation projects. Research can be empirical, experimental, or theoretical. The IPC invites proposals that address some of the following questions:

* What is the impact of using different types of technological tools in relation to one or more lenses proposed above?

* How does the impact of technological tools on geometry education depend on the types of tools, their openness, or their educational functions?

* How do the uses of different kinds of technologies by students, teachers, or teacher educators impact on their practices and thinking in geometry?

* What are some of the opportunities and challenges that are presented by the use of technology in geometry education?

## Sub-topic C2: Manipulatives and visual tools in teaching and learning geometry

Visual tools and concrete manipulatives are often seen as important resources in promoting geometrical thinking. A review of the state of the art of research on the contribution of manipulatives to learning and teaching geometry is needed. Are manipulations and visualization sufficient to allow effective learning processes of geometry, including deductive reasoning, proving, and constructing in geometry? How is the use of manipulatives linked to modelling, abstracting, using language to describe geometrical objects, and proving?

Manipulatives and visual tools can be examined from different theoretical perspectives and regarding different goals of the use of manipulatives and visualization: creating experience, identifying critical properties, formulating conjectures, and proving or refuting the conjectures.

The questions posed can be about different types of visualization and objects such as material tools, photographs, video, and drawings. The contributions may address different uses of visualization and objects, depending on the ages of students and the specific difficulties they might have (e.g., blindness, developmental coordination disorder, etc.).

The IPC invites proposals that address some of the following questions:

* What are the links between the use of manipulatives, visual tools, and modelling activity?

* What is the difference between using the manipulatives or dynamic diagrams constructed by educational designers and constructing manipulatives or working in an open dynamic environment (e.g., GeoGebra)?

* What is the role of visual tools and manipulatives on geometrical proof processes?

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

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

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

## Sub-topic C3: Learning geometry with resource constraints

The scope and focus of this sub-topic address issues relating to the teaching and learning of geometry in resource-constrained classrooms and under-resourced school communities. Resources, in this context, include physical resources (e.g., digital tools and non-digital manipulatives), non-tangible
resources (e.g., the language of instruction), and human resources (e.g., specific languages used by visual or hearing-impaired students). Constraints refer to limitations in the use of resources. Three sources of resource constraints can be identified: (1) the resource itself: any physical resource has limitations in the scope of its use, and it is necessary for teachers to be aware of them, so they do not pose tasks to their pupils where the resources are an obstacle more than a help; (2) the users: some students may have difficulties in handling digital or manipulative resources due to some disability, and some teachers may not have adequate training in using resources, with consequences for their pupils' learning; (3) the socio-economic conditions: many schools do not have access to resources for teaching mathematics, so students cannot benefit from their use to help them develop their geometric conceptions.

The IPC aims at composing an overview of the current state of the difficulties encountered by students in resource-constrained classrooms in the learning of geometry by (1) documenting measures and attempts to address the constraints; (2) addressing theoretical issues relating to the teaching and learning of geometry in resource-constrained classrooms; and (3) suggesting ways forward in meeting the needs of students in learning geometry in resource-constrained classroom.

We invite proposals that address some of the following questions:

* What theoretical lenses on geometry learning in resources-constrained classrooms and communities are emerging, and how do these inform the understanding of students' learning of geometry in such classrooms?

* What practices have proved to be effective in resource-constrained geometry classrooms and under-resourced communities?

* How should instructional activities be designed in resource-constrained geometry lessons, in particular for students with disabilities?

* How can the teaching and learning of geometry be improved in resource-constrained contexts where use of software (e.g., dynamic geometry) and manipulatives is limited?

* How are local languages (e.g., indigenous verified mathematics) used as a resource for supporting mathematical access in teaching and learning geometry in multilingual countries?

## TOPIC D: MULTIDISCIPLINARY PERSPECTIVES

The multidisciplinary perspectives can be seen as a polysemic interaction between different knowledge fields such as ethnomathematics. One of the perspectives can be related to how teaching and learning of geometry can contribute to the development of other disciplines regarding professional, social, and cultural activities and practices. Another perspective addresses the possible contributions of other sciences such as neuroscience and psychology disciplines, and professional needs, in the process of teaching and learning geometry. The third perspective relates to the roles that geometry may play within different disciplines and knowledge fields. The main goal is to further advance research and understanding in relation to geometric knowledge and its polysemic multidisciplinary connection with other knowledge fields and sociocultural contexts.

### Sub-topic D1: Connections between geometry and professional contexts

Geometry has inherent importance. There are also practical implications in professional contexts where practitioners are using geometry consistently. Efforts have been made to examine the role of
resources (e.g., the language of instruction), and human resources (e.g., specific languages used by visual or hearing-impaired students). Constraints refer to limitations in the use of resources. Three sources of resource constraints can be identified: (1) the resource itself: any physical resource has limitations in the scope of its use, and it is necessary for teachers to be aware of them, so they do not pose tasks to their pupils where the resources are an obstacle more than a help; (2) the users: some students may have difficulties in handling digital or manipulative resources due to some disability, and some teachers may not have adequate training in using resources, with consequences for their pupils' learning; (3) the socio-economic conditions: many schools do not have access to resources for teaching mathematics, so students cannot benefit from their use to help them develop their geometric conceptions.

The IPC aims at composing an overview of the current state of the difficulties encountered by students in resource-constrained classrooms in the learning of geometry by (1) documenting measures and attempts to address the constraints; (2) addressing theoretical issues relating to the teaching and learning of geometry in resource-constrained classrooms; and (3) suggesting ways forward in meeting the needs of students in learning geometry in resource-constrained classroom.

We invite proposals that address some of the following questions:

* What theoretical lenses on geometry learning in resources-constrained classrooms and communities are emerging, and how do these inform the understanding of students' learning of geometry in such classrooms?

* What practices have proved to be effective in resource-constrained geometry classrooms and under-resourced communities?

* How should instructional activities be designed in resource-constrained geometry lessons, in particular for students with disabilities?

* How can the teaching and learning of geometry be improved in resource-constrained contexts where use of software (e.g., dynamic geometry) and manipulatives is limited?

* How are local languages (e.g., indigenous verified mathematics) used as a resource for supporting mathematical access in teaching and learning geometry in multilingual countries?

## TOPIC D: MULTIDISCIPLINARY PERSPECTIVES

The multidisciplinary perspectives can be seen as a polysemic interaction between different knowledge fields such as ethnomathematics. One of the perspectives can be related to how teaching and learning of geometry can contribute to the development of other disciplines regarding professional, social, and cultural activities and practices. Another perspective addresses the possible contributions of other sciences such as neuroscience and psychology disciplines, and professional needs, in the process of teaching and learning geometry. The third perspective relates to the roles that geometry may play within different disciplines and knowledge fields. The main goal is to further advance research and understanding in relation to geometric knowledge and its polysemic multidisciplinary connection with other knowledge fields and sociocultural contexts.

### Sub-topic D1: Connections between geometry and professional contexts

Geometry has inherent importance. There are also practical implications in professional contexts where practitioners are using geometry consistently. Efforts have been made to examine the role of
mathematics in Science, Technology, Engineering and Mathematics (STEM) professions (Van der Wal, Bakker & Drijvers, 2017). With respect to geometry, it is the case that compulsory schooling, initial professional training and workplace training differ in the nature and the emphasis on geometric learning. Some of these connections are obvious while others not. Interest in the way school geometry is taught in schools and colleges and the mathematics needed in a variety of work-related activities has been evident in the literature for some time (see Roth, 2014, for a description of bridging between formal mathematics and enacted practical geometric experiences). Carpenters for example, rely on Pythagoras' theorem (especially the 3-4-5 triangle) in the form of a set square for much of their daily activities. Construction professionals such as engineers, architects and builders use trigonometry to calculate angles and distances routinely in their drawings and interpretation of plans. Creative professionals including fashion designers use symmetry and tessellations to create patterns while town planners utilize Thales' theorem to level ground accurately over long distances. In a recent example, technological developments allow sonographers to measure the head circumference of an unborn baby in various static and dynamic images of 2D-3D space.

This topic seeks research links between school mathematics and applications for on-the-job training and applications in the real world. Such links raise several questions to be researched:

* Where might school geometry be needed in emerging professions and changing workplaces?

* How can school-based geometric tasks be designed to be meaningful for out-of-school professional applications?

* How can professional activities inform the design and development of innovative geometric experiences in the classroom?

* How does geometry needed in initial professional and on-the-job training differ from geometry taught in schools?

**Sub-topic D2: Ethnomathematics and indigenous ways of understanding geometry**

Cultural artifacts are objects created by members of distinct cultural groups that convey cultural meanings and information about their creators and users. Geometric ideas, procedures, techniques, and practices are related to the development of cultural artifacts that are socio-culturally situated as well as distributed among these members from generation to generation. This approach also includes embodied cognition in which cognitive activities use symbols as external resources that assist these members to develop their mental representations and manipulation of objects. Geometric representations exist in all cultural groups and their meaning may be different from one specific culture to another. It is important to emphasize that ethnomathematics is also a program that studies geometric concepts and theories developed locally as well as it is related to dynamic pedagogical actions that respond to the environmental, social, cultural, political, and economic needs of the students, which enables them to develop their imagination and creativity. Thus, there is a necessity to understand indigenous ways of understanding geometry through ethnomathematics.

The IPC invites proposals that address some of the following questions:

* What role does ethnomathematics play in the understanding of diverse indigenous ways of doing geometry? What are the sociocultural influences of the use of this knowledge in
mathematics in Science, Technology, Engineering and Mathematics (STEM) professions (Van der Wal, Bakker & Drijvers, 2017). With respect to geometry, it is the case that compulsory schooling, initial professional training and workplace training differ in the nature and the emphasis on geometric learning. Some of these connections are obvious while others not. Interest in the way school geometry is taught in schools and colleges and the mathematics needed in a variety of work-related activities has been evident in the literature for some time (see Roth, 2014, for a description of bridging between formal mathematics and enacted practical geometric experiences). Carpenters for example, rely on Pythagoras' theorem (especially the 3-4-5 triangle) in the form of a set square for much of their daily activities. Construction professionals such as engineers, architects and builders use trigonometry to calculate angles and distances routinely in their drawings and interpretation of plans. Creative professionals including fashion designers use symmetry and tessellations to create patterns while town planners utilize Thales' theorem to level ground accurately over long distances. In a recent example, technological developments allow sonographers to measure the head circumference of an unborn baby in various static and dynamic images of 2D-3D space.

This topic seeks research links between school mathematics and applications for on-the-job training and applications in the real world. Such links raise several questions to be researched:

* Where might school geometry be needed in emerging professions and changing workplaces?

* How can school-based geometric tasks be designed to be meaningful for out-of-school professional applications?

* How can professional activities inform the design and development of innovative geometric experiences in the classroom?

* How does geometry needed in initial professional and on-the-job training differ from geometry taught in schools?

**Sub-topic D2: Ethnomathematics and indigenous ways of understanding geometry**

Cultural artifacts are objects created by members of distinct cultural groups that convey cultural meanings and information about their creators and users. Geometric ideas, procedures, techniques, and practices are related to the development of cultural artifacts that are socio-culturally situated as well as distributed among these members from generation to generation. This approach also includes embodied cognition in which cognitive activities use symbols as external resources that assist these members to develop their mental representations and manipulation of objects. Geometric representations exist in all cultural groups and their meaning may be different from one specific culture to another. It is important to emphasize that ethnomathematics is also a program that studies geometric concepts and theories developed locally as well as it is related to dynamic pedagogical actions that respond to the environmental, social, cultural, political, and economic needs of the students, which enables them to develop their imagination and creativity. Thus, there is a necessity to understand indigenous ways of understanding geometry through ethnomathematics.

The IPC invites proposals that address some of the following questions:

* What role does ethnomathematics play in the understanding of diverse indigenous ways of doing geometry? What are the sociocultural influences of the use of this knowledge in
distinct cultural groups? What investigations have been done in this area to date? What has been learnt from these investigations?

* What cultural traits contribute to the construction of local geometric knowledge? Does this lead to the development of embodied cognition and situated learning regarding geometry? How might mainstream/traditional mathematics learn from these constructions of local geometric knowledge?

* In what ways does ethnomathematics contribute to and improve current formal geometry content and teaching in schools?

* How do members of distinct cultural groups extend their cognitive processes beyond the brain, the body, and their immediate environments in order to develop diverse ways of understanding geometry? Does this mean that cognition is not only embodied and situated, but also distributed as evidenced in their ways of creating, processing, accumulating, and diffusing information conjointly, including geometric knowledge?

**Sub-topic D3: Contributions of psychology and neuroscience to research in mathematics education focusing on geometry**

Cognitive scientists posit that student mathematical processing in general and geometrical processing in particular are determined by general cognitive traits and socio-emotional characteristics. Students' cognitive characteristics such as memory, attention, pattern recognition, speed, and types of information processing are proven to affect student mathematics learning and processing. In this sense, cognition research may inform mathematics education research on the role of general cognition and the role of affective factors in studying geometry.

In the field of Neuroscience, results from research using fMRI, EEG, and eye-tracking studies (as examples) suggest that solving mathematical problems, including geometry problems, are connected to student characteristics. The research also suggests that students with varied levels of mathematical performance exhibit different patterns of brain activity (Ansari, 2016; De Smedt & Grabner, 2016; Grabner, 2022).

Teacher efficacy, attitudes, beliefs, and knowledge in geometry is another related area of robust research activity. For example, in parallel work in Psychology and Mathematics Education, research on student self-efficacy in geometry is aiming to explicitly link confidence, motivation, persistence, anxiety, and comfort with the teaching and learning of geometry.

Beyond discrete disciplinary research, it is in the intersection of mind, brain, and education research where novel contributions are increasingly expanding conceptions of geometry and spatial reasoning learning. For example, the work on mental transformations of 2D and 3D figures has been jointly propelled by research activity in clinical and classroom settings and shows significant promise as a predictor of mathematics achievement but is particularly related to geometric reasoning (see Mix et al., 2016, Uttal et al., 2013). This includes research on transformations, mental mapping, coding, perspective taking, and first-person insertion technologies, for example, that may be subsequently linked to improving the teaching and learning of geometry.

The IPC invites proposals that address some of the following questions:
distinct cultural groups? What investigations have been done in this area to date? What has been learnt from these investigations?

* What cultural traits contribute to the construction of local geometric knowledge? Does this lead to the development of embodied cognition and situated learning regarding geometry? How might mainstream/traditional mathematics learn from these constructions of local geometric knowledge?

* In what ways does ethnomathematics contribute to and improve current formal geometry content and teaching in schools?

* How do members of distinct cultural groups extend their cognitive processes beyond the brain, the body, and their immediate environments in order to develop diverse ways of understanding geometry? Does this mean that cognition is not only embodied and situated, but also distributed as evidenced in their ways of creating, processing, accumulating, and diffusing information conjointly, including geometric knowledge?

**Sub-topic D3: Contributions of psychology and neuroscience to research in mathematics education focusing on geometry**

Cognitive scientists posit that student mathematical processing in general and geometrical processing in particular are determined by general cognitive traits and socio-emotional characteristics. Students' cognitive characteristics such as memory, attention, pattern recognition, speed, and types of information processing are proven to affect student mathematics learning and processing. In this sense, cognition research may inform mathematics education research on the role of general cognition and the role of affective factors in studying geometry.

In the field of Neuroscience, results from research using fMRI, EEG, and eye-tracking studies (as examples) suggest that solving mathematical problems, including geometry problems, are connected to student characteristics. The research also suggests that students with varied levels of mathematical performance exhibit different patterns of brain activity (Ansari, 2016; De Smedt & Grabner, 2016; Grabner, 2022).

Teacher efficacy, attitudes, beliefs, and knowledge in geometry is another related area of robust research activity. For example, in parallel work in Psychology and Mathematics Education, research on student self-efficacy in geometry is aiming to explicitly link confidence, motivation, persistence, anxiety, and comfort with the teaching and learning of geometry.

Beyond discrete disciplinary research, it is in the intersection of mind, brain, and education research where novel contributions are increasingly expanding conceptions of geometry and spatial reasoning learning. For example, the work on mental transformations of 2D and 3D figures has been jointly propelled by research activity in clinical and classroom settings and shows significant promise as a predictor of mathematics achievement but is particularly related to geometric reasoning (see Mix et al., 2016, Uttal et al., 2013). This includes research on transformations, mental mapping, coding, perspective taking, and first-person insertion technologies, for example, that may be subsequently linked to improving the teaching and learning of geometry.

The IPC invites proposals that address some of the following questions:
* What insights from the recent research on spatial reasoning (such as mental rotations) can be integrated into research on geometry thinking? What are the implications for teaching and the resultant learning?

* What affective constructs impact on geometry teaching and learning?

* How can students' cognitive characteristics be taken into account and what are the implications for geometry education?

* How can neuro-cognitive research inform geometry education about individual differences in geometric processing and in learning geometry?

* Is there room for transdisciplinary research and what should it focus on as it relates to geometry teaching and learning?

## REFERENCES

Ansari, D. (2016). The neural roots of mathematical expertise. *Proceedings of the National Academy of Sciences, 113*(18), 4887-4889. DOI [10.1073/pnas.1604758113](https://doi.org/10.1073/pnas.1604758113)

Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom. Artifacts and signs after a Vygotskian perspective. In L. English (Ed.), *Handbook of international research in mathematics education* (pp. 746-805). Routledge.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), *Handbook of research on mathematics teaching and learning* (pp. 420-464). Macmillan.

Clements, D. H., & Sarama, J. (2011). Early childhood teacher education: The case of geometry. Journal of Mathematics Teacher Education, 14(2), 133-148. DOI [10.1007/s10857-011-9173-0](https://doi.org/10.1007/s10857-011-9173-0)

De Smedt, B., & Grabner, R. H. (2016). Potential applications of cognitive neuroscience to mathematics education. *ZDM Mathematics Education, 48*(3), 249-253. DOI [10.1007/s11858-016-0784-x](https://doi.org/10.1007/s11858-016-0784-x)

Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), *Perspectives on the teaching of geometry for the 21st Century* (pp. 37-52). Kluwer.

Fischbein, E. (1993). The theory of figural concepts. *Educational Studies in Mathematics, 24*(2), 139-162. DOI [10.1007/BF01273689](https://doi.org/10.1007/BF01273689)

Gal, H., & Linchevski, L. (2010). To see or not to see: analyzing difficulties in geometry from the perspective of visual perception. *Educational Studies in Mathematics, 74*(2), 163-183. DOI [10.1007/s10649-010-9232-y](https://doi.org/10.1007/s10649-010-9232-y)

Grabner, R. H. (2022). Section III. Cognitive neuroscience of mathematics. In M. Danesi (Ed.), *Handbook of cognitive mathematics* (pp. 251-256). Springer. DOI [10.1007/978-3-031-03945-4](https://doi.org/10.1007/978-3-031-03945-4)

Hawes, Z. C. K., Gilligan-Lee, K. A., & Mix, K. S. (2022). Effects of spatial training on mathematics performance: A meta-analysis. *Developmental Psychology, 58*(1), 112-137. DOI [10.1037/dev0001281](https://doi.org/10.1037/dev0001281)

Krutetskii, V. A. (1976). *The psychology of mathematical abilities in schoolchildren*. The University of Chicago Press.

Lowrie, T., Harris, D., Logan, T., & Hegarty, M. (2019). The impact of a spatial intervention program on students' spatial reasoning and mathematics performance, *The Journal of Experimental Education, 89*(2), 259-277. DOI [10.1080/00220973.2019.1684869](https://doi.org/10.1080/00220973.2019.1684869)

Mammana, C., & Villani, V. (Eds.). (1998). *Perspectives on the teaching of geometry for the 21st Century* (New ICMI Study Series, Vol. 5). Kluwer.

Mariotti, M. A., & Fischbein, E. (1997). Defining in classroom activities. *Educational Studies in Mathematics, 34*(3), 219-248. DOI [10.1023/A:1002985109323](https://doi.org/10.1023/A:1002985109323)
* What insights from the recent research on spatial reasoning (such as mental rotations) can be integrated into research on geometry thinking? What are the implications for teaching and the resultant learning?

* What affective constructs impact on geometry teaching and learning?

* How can students' cognitive characteristics be taken into account and what are the implications for geometry education?

* How can neuro-cognitive research inform geometry education about individual differences in geometric processing and in learning geometry?

* Is there room for transdisciplinary research and what should it focus on as it relates to geometry teaching and learning?

## REFERENCES

Ansari, D. (2016). The neural roots of mathematical expertise. *Proceedings of the National Academy of Sciences, 113*(18), 4887-4889. DOI [10.1073/pnas.1604758113](https://doi.org/10.1073/pnas.1604758113)

Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom. Artifacts and signs after a Vygotskian perspective. In L. English (Ed.), *Handbook of international research in mathematics education* (pp. 746-805). Routledge.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), *Handbook of research on mathematics teaching and learning* (pp. 420-464). Macmillan.

Clements, D. H., & Sarama, J. (2011). Early childhood teacher education: The case of geometry. Journal of Mathematics Teacher Education, 14(2), 133-148. DOI [10.1007/s10857-011-9173-0](https://doi.org/10.1007/s10857-011-9173-0)

De Smedt, B., & Grabner, R. H. (2016). Potential applications of cognitive neuroscience to mathematics education. *ZDM Mathematics Education, 48*(3), 249-253. DOI [10.1007/s11858-016-0784-x](https://doi.org/10.1007/s11858-016-0784-x)

Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), *Perspectives on the teaching of geometry for the 21st Century* (pp. 37-52). Kluwer.

Fischbein, E. (1993). The theory of figural concepts. *Educational Studies in Mathematics, 24*(2), 139-162. DOI [10.1007/BF01273689](https://doi.org/10.1007/BF01273689)

Gal, H., & Linchevski, L. (2010). To see or not to see: analyzing difficulties in geometry from the perspective of visual perception. *Educational Studies in Mathematics, 74*(2), 163-183. DOI [10.1007/s10649-010-9232-y](https://doi.org/10.1007/s10649-010-9232-y)

Grabner, R. H. (2022). Section III. Cognitive neuroscience of mathematics. In M. Danesi (Ed.), *Handbook of cognitive mathematics* (pp. 251-256). Springer. DOI [10.1007/978-3-031-03945-4](https://doi.org/10.1007/978-3-031-03945-4)

Hawes, Z. C. K., Gilligan-Lee, K. A., & Mix, K. S. (2022). Effects of spatial training on mathematics performance: A meta-analysis. *Developmental Psychology, 58*(1), 112-137. DOI [10.1037/dev0001281](https://doi.org/10.1037/dev0001281)

Krutetskii, V. A. (1976). *The psychology of mathematical abilities in schoolchildren*. The University of Chicago Press.

Lowrie, T., Harris, D., Logan, T., & Hegarty, M. (2019). The impact of a spatial intervention program on students' spatial reasoning and mathematics performance, *The Journal of Experimental Education, 89*(2), 259-277. DOI [10.1080/00220973.2019.1684869](https://doi.org/10.1080/00220973.2019.1684869)

Mammana, C., & Villani, V. (Eds.). (1998). *Perspectives on the teaching of geometry for the 21st Century* (New ICMI Study Series, Vol. 5). Kluwer.

Mariotti, M. A., & Fischbein, E. (1997). Defining in classroom activities. *Educational Studies in Mathematics, 34*(3), 219-248. DOI [10.1023/A:1002985109323](https://doi.org/10.1023/A:1002985109323)
Mason, J. (2004). Doing ≠ construing and doing + discussing ≠ learning: The importance of the structure of attention. Text of the *ICME 10 Regular Lecture*. Copenhagen, 4-11 July.

Mix, K. S., Levine, S. C., Cheng, Y. L., Young, C., Hambrick, D. Z., Ping, R., & Konstantopoulos, S. (2016). Separate but correlated: the latent structure of space and mathematics across development. *Journal of Experimental Psychology: General*, 145(9), 1206-1227. DOI [10.1037/xge0000182](https://doi.org/10.1037/xge0000182)

Moss, J., Bruce, C., & Bobis, J. (2015). Young children's access to powerful mathematics ideas: a review of current challenges and new developments in the early years. In L. English & D. Kirshner (Eds.), *Handbook of international research in mathematics education*. Routledge. DOI [10.4324/9780203448946](https://doi.org/10.4324/9780203448946)

National Research Council (2006). *Learning to think spatially: GIS as a support system in the K-12 curriculum*. National Academic Press.

Ng, O.-L., & Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment. *ZDM Mathematics Education*, 47(3), 421-434. DOI [10.1007/s11858-014-0660-5](https://doi.org/10.1007/s11858-014-0660-5)

Owens, K. (2014). Diversifying our perspectives on mathematics about space and geometry: an ecocultural approach. *International Journal of Science and Mathematics Education*, 12(4), 941-974. DOI [10.1007/s10763-013-9441-9](https://doi.org/10.1007/s10763-013-9441-9)

Roth, W.-M. (2014). Rules of bending, bending the rules: the geometry of electrical conduit bending in college and workplace. *Educational Studies in Mathematics*, 86(2), 177-192. DOI [10.1007/s10649-011-9376-4](https://doi.org/10.1007/s10649-011-9376-4)

Sinclair, N., Bartolini Bussi, M. G., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2016). Recent research on geometry education: an ICME-13 survey team report. *ZDM Mathematics Education*, 48(5), 691-719. DOI [10.1007/s11858-016-0796-6](https://doi.org/10.1007/s11858-016-0796-6)

Sinclair, N., & Bruce, C. (2015). New opportunities in geometry education at the primary school. *ZDM Mathematics Education* 47(3), 319-329. DOI [10.1007/s11858-015-0693-4](https://doi.org/10.1007/s11858-015-0693-4)

Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. S. (2013). The malleability of spatial skills: a meta-analysis of training studies. *Psychological Bulletin*, 139(2), 352-402. DOI [10.1037/a0028446](https://doi.org/10.1037/a0028446)

Van der Wal, N., Bakker, A., & Drijvers, P. (2017). Which techno-mathematical literacies are essential for future engineers? *International Journal of Science and Mathematics Education*, 15(Supp 1), S87-S104. DOI [10.1007/s10763-017-9810-x](https://doi.org/10.1007/s10763-017-9810-x)

Van Hiele, P. M. (1986). *Structure and insight: a theory of mathematics education*. Academic Press.

## THE 26TH ICMI STUDY PRODUCTS

The ICMI Studies are a major activity of ICMI. Their global aims are to contribute to a better understanding of the challenges faced by mathematics education in our multidisciplinary and culturally diverse world and to collaborate in advancing to their resolution. The first ICMI Study was launched in 1980. More detailed information can be found in the ICMI web page (https://www.mathunion.org/icmi/activities/icmi-studies-activities).

The main products of an ICMI Study are a conference and a volume of the ICMI Studies Series. The conference addresses a topic or issue of particular significance in contemporary mathematics education. It is aimed to gather together leading scholars and practitioners specialized in the field of inquiry of the Study, for them to engage in a productive interaction and collaboration to advance in the knowledge of the topics of the Study.

Based on the activity during and results from the conference, the final and most relevant product of an ICMI Study is a volume offering a coherent state-of-the-art view of the topic addressed by the Study. This volume will be part of the New ICMI Study Series.
Mason, J. (2004). Doing ≠ construing and doing + discussing ≠ learning: The importance of the structure of attention. Text of the *ICME 10 Regular Lecture*. Copenhagen, 4-11 July.

Mix, K. S., Levine, S. C., Cheng, Y. L., Young, C., Hambrick, D. Z., Ping, R., & Konstantopoulos, S. (2016). Separate but correlated: the latent structure of space and mathematics across development. *Journal of Experimental Psychology: General*, 145(9), 1206-1227. DOI [10.1037/xge0000182](https://doi.org/10.1037/xge0000182)

Moss, J., Bruce, C., & Bobis, J. (2015). Young children's access to powerful mathematics ideas: a review of current challenges and new developments in the early years. In L. English & D. Kirshner (Eds.), *Handbook of international research in mathematics education*. Routledge. DOI [10.4324/9780203448946](https://doi.org/10.4324/9780203448946)

National Research Council (2006). *Learning to think spatially: GIS as a support system in the K-12 curriculum*. National Academic Press.

Ng, O.-L., & Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment. *ZDM Mathematics Education*, 47(3), 421-434. DOI [10.1007/s11858-014-0660-5](https://doi.org/10.1007/s11858-014-0660-5)

Owens, K. (2014). Diversifying our perspectives on mathematics about space and geometry: an ecocultural approach. *International Journal of Science and Mathematics Education*, 12(4), 941-974. DOI [10.1007/s10763-013-9441-9](https://doi.org/10.1007/s10763-013-9441-9)

Roth, W.-M. (2014). Rules of bending, bending the rules: the geometry of electrical conduit bending in college and workplace. *Educational Studies in Mathematics*, 86(2), 177-192. DOI [10.1007/s10649-011-9376-4](https://doi.org/10.1007/s10649-011-9376-4)

Sinclair, N., Bartolini Bussi, M. G., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2016). Recent research on geometry education: an ICME-13 survey team report. *ZDM Mathematics Education*, 48(5), 691-719. DOI [10.1007/s11858-016-0796-6](https://doi.org/10.1007/s11858-016-0796-6)

Sinclair, N., & Bruce, C. (2015). New opportunities in geometry education at the primary school. *ZDM Mathematics Education* 47(3), 319-329. DOI [10.1007/s11858-015-0693-4](https://doi.org/10.1007/s11858-015-0693-4)

Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. S. (2013). The malleability of spatial skills: a meta-analysis of training studies. *Psychological Bulletin*, 139(2), 352-402. DOI [10.1037/a0028446](https://doi.org/10.1037/a0028446)

Van der Wal, N., Bakker, A., & Drijvers, P. (2017). Which techno-mathematical literacies are essential for future engineers? *International Journal of Science and Mathematics Education*, 15(Supp 1), S87-S104. DOI [10.1007/s10763-017-9810-x](https://doi.org/10.1007/s10763-017-9810-x)

Van Hiele, P. M. (1986). *Structure and insight: a theory of mathematics education*. Academic Press.

## THE 26TH ICMI STUDY PRODUCTS

The ICMI Studies are a major activity of ICMI. Their global aims are to contribute to a better understanding of the challenges faced by mathematics education in our multidisciplinary and culturally diverse world and to collaborate in advancing to their resolution. The first ICMI Study was launched in 1980. More detailed information can be found in the ICMI web page (https://www.mathunion.org/icmi/activities/icmi-studies-activities).

The main products of an ICMI Study are a conference and a volume of the ICMI Studies Series. The conference addresses a topic or issue of particular significance in contemporary mathematics education. It is aimed to gather together leading scholars and practitioners specialized in the field of inquiry of the Study, for them to engage in a productive interaction and collaboration to advance in the knowledge of the topics of the Study.

Based on the activity during and results from the conference, the final and most relevant product of an ICMI Study is a volume offering a coherent state-of-the-art view of the topic addressed by the Study. This volume will be part of the New ICMI Study Series.
## THE ICMI STUDY CONFERENCE

The 26th ICMI Study focuses on *Advances in Geometry Education* and is planned to provide a platform for teachers, teacher educators, researchers, policy makers, and other stakeholders around the world to share research, practices, projects, and reports that advance the understanding of geometry. As in every ICMI Study, the 26th ICMI Study is built around an International Study Conference that is directed towards the preparation of a volume published in the New ICMI Studies Series. In the conference, substantial time will be allocated for collective work and discussion on significant problems within the four topics and the related sub-topics. Outcomes from the conference activity provide the foundations for the Study volume.

The Study Conference will be organized around working groups on the sub-topics presented above. The working groups will meet in parallel during the conference. It is the work of these groups that is captured as chapters in the ICMI Study volume.

Authors should nominate the topic (one of four) and sub-topic in which they would like their paper considered, by examining the content and questions outlined in the topic sections presented above. Contributions are encouraged that are analytical and innovative rather than solely descriptive in nature. It may be the case that interconnections between sub-topics emerge and warrant attention; consequently, papers may be re-allocated by the IPC if beneficial.

## Participation

As is the usual practice for ICMI studies, participation in the Study Conference is by invitation only. Proposed papers will be reviewed, and a selection made according to the quality of the contribution, the potential to contribute to the advancement of the Study, with explicit links to the sub-topics proposed in this Discussion Document, and the need to ensure diversity among the perspectives and representation. The number of invited participants is limited to approximately 100 delegates. As part of the invitation process, potential participants will receive a registration code, which will be required to complete registration details. Unfortunately, an invitation to participate in the conference does not imply financial support from the organizers, and participants should be prepared to finance their own attendance at the conference.

## THE ICMI STUDY VOLUME

The first product of the 26th ICMI Study is an electronic volume of the Conference proceedings, to be made available prior to the conference on the conference website and later the ICMI website. The proceedings will contain all the accepted papers as reviewed papers, so they can be cited as a refereed (peer-reviewed) publication with an ISBN number. They are published online only.

The second and main product is an edited volume published by Springer as part of the New ICMI Studies Series. The editing process and content of the volume will be the subject of discussion among the International Program Committee (IPC). It is expected that the structure of the volume will follow the organization and topics set out in this Discussion Document, although some changes might be introduced as a consequence of the discussions raised during the conference. The chapters in the volume will collectively and consensually integrate the outcomes of the discussions of the parallel working groups at the conference, informed by the papers presented. It must be appreciated that there is no guarantee that any of the papers accepted in the Study conference proceedings will appear in
## THE ICMI STUDY CONFERENCE

The 26th ICMI Study focuses on *Advances in Geometry Education* and is planned to provide a platform for teachers, teacher educators, researchers, policy makers, and other stakeholders around the world to share research, practices, projects, and reports that advance the understanding of geometry. As in every ICMI Study, the 26th ICMI Study is built around an International Study Conference that is directed towards the preparation of a volume published in the New ICMI Studies Series. In the conference, substantial time will be allocated for collective work and discussion on significant problems within the four topics and the related sub-topics. Outcomes from the conference activity provide the foundations for the Study volume.

The Study Conference will be organized around working groups on the sub-topics presented above. The working groups will meet in parallel during the conference. It is the work of these groups that is captured as chapters in the ICMI Study volume.

Authors should nominate the topic (one of four) and sub-topic in which they would like their paper considered, by examining the content and questions outlined in the topic sections presented above. Contributions are encouraged that are analytical and innovative rather than solely descriptive in nature. It may be the case that interconnections between sub-topics emerge and warrant attention; consequently, papers may be re-allocated by the IPC if beneficial.

## Participation

As is the usual practice for ICMI studies, participation in the Study Conference is by invitation only. Proposed papers will be reviewed, and a selection made according to the quality of the contribution, the potential to contribute to the advancement of the Study, with explicit links to the sub-topics proposed in this Discussion Document, and the need to ensure diversity among the perspectives and representation. The number of invited participants is limited to approximately 100 delegates. As part of the invitation process, potential participants will receive a registration code, which will be required to complete registration details. Unfortunately, an invitation to participate in the conference does not imply financial support from the organizers, and participants should be prepared to finance their own attendance at the conference.

## THE ICMI STUDY VOLUME

The first product of the 26th ICMI Study is an electronic volume of the Conference proceedings, to be made available prior to the conference on the conference website and later the ICMI website. The proceedings will contain all the accepted papers as reviewed papers, so they can be cited as a refereed (peer-reviewed) publication with an ISBN number. They are published online only.

The second and main product is an edited volume published by Springer as part of the New ICMI Studies Series. The editing process and content of the volume will be the subject of discussion among the International Program Committee (IPC). It is expected that the structure of the volume will follow the organization and topics set out in this Discussion Document, although some changes might be introduced as a consequence of the discussions raised during the conference. The chapters in the volume will collectively and consensually integrate the outcomes of the discussions of the parallel working groups at the conference, informed by the papers presented. It must be appreciated that there is no guarantee that any of the papers accepted in the Study conference proceedings will appear in
the volume. Furthermore, chapters in the volume may be an amalgamation of several presented papers.

A progress report on the Study will be presented at the ICME15 (Sydney, 7-14 July, 2024).

**MEMBERS OF THE INTERNATIONAL PROGRAM COMMITTEE**

*Co-chairs*

Angel Gutiérrez (Spain, angel.gutierrez@uv.es)

Thomas Lowrie (Australia, thomas.lowrie@canberra.edu.au)

*IPC members*

Cathy Bruce (Canada, cathybruce@trentu.ca)

Fabien Emprin (France, fabien.emprin@univ-reims.fr)

Keith Jones (United Kingdom, jones.keith2013@gmail.com)

Roza Leikin (Israel, rozal@edu.haifa.ac.il)

Lisnet Mwadzaangati (Malawi, lmwadzaangati@unima.ac.mw)

Oi-Lam Ng (Hong Kong SAR, oilamn@cuhk.edu.hk)

Yukari Okamoto (United States, yukariokamoto@ucsb.edu)

Milton Rosa (Brazil, milton.rosa@ufop.edu.br)

Manuel Santos-Trigo (Mexico, msantos@cinvestav.mx)

*Ex-officio members*

Frederick K.S. Leung (ICMI President, Hong Kong SAR, icmi.president@mathunion.org)

Jean-Luc Dorier (ICMI General-Secretary, Switzerland, icmi.secretary.general@mathunion.org)
the volume. Furthermore, chapters in the volume may be an amalgamation of several presented papers.

A progress report on the Study will be presented at the ICME15 (Sydney, 7-14 July, 2024).

**MEMBERS OF THE INTERNATIONAL PROGRAM COMMITTEE**

*Co-chairs*

Angel Gutiérrez (Spain, angel.gutierrez@uv.es)

Thomas Lowrie (Australia, thomas.lowrie@canberra.edu.au)

*IPC members*

Cathy Bruce (Canada, cathybruce@trentu.ca)

Fabien Emprin (France, fabien.emprin@univ-reims.fr)

Keith Jones (United Kingdom, jones.keith2013@gmail.com)

Roza Leikin (Israel, rozal@edu.haifa.ac.il)

Lisnet Mwadzaangati (Malawi, lmwadzaangati@unima.ac.mw)

Oi-Lam Ng (Hong Kong SAR, oilamn@cuhk.edu.hk)

Yukari Okamoto (United States, yukariokamoto@ucsb.edu)

Milton Rosa (Brazil, milton.rosa@ufop.edu.br)

Manuel Santos-Trigo (Mexico, msantos@cinvestav.mx)

*Ex-officio members*

Frederick K.S. Leung (ICMI President, Hong Kong SAR, icmi.president@mathunion.org)

Jean-Luc Dorier (ICMI General-Secretary, Switzerland, icmi.secretary.general@mathunion.org)
Vignobles de Champagne

30 octobre 2009

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Brumes sur Ambonnay

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Vignobles de Champagne

20 septembre 2009

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Vignobles de Champagne

30 septembre 2009

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Vignoble de Châteauguay

30 septembre 2009

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Vignoble de Châteauguay

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