The most common explanations for transfer are built on the idea that specific spatial skills are utilised within specific mathematics tasks. Although this contrasts Mix (2019), who argues for a general connection, Mix’s (2019) work is based on correlational research. The experimental work described here suggests there may be specific connections, including within geometry (Adams et al., 2022; Gilligan et al., 2020). This experimental work also highlights a separate role for vsWM. Hawes and Ansari (2020) consider vsWM in the broader context of improvements in general cognition; however, this is not supported by the experimental findings, which instead finds a unique role for vsWM separate from improved spatial skills (e.g., Zhang et al., 2021). Spatial skills and vsWM may work together by providing a larger “mental blackboard” to keep more spatial information in mind as well as the skills required to mentally manipulate that spatial information. Given the small number of experimental studies, and the inclusion of broad geometry measures, more experimental research is needed to characterize the kinds of spatial skills that support specific geometry tasks.

**A new model for mechanism – the role of strategic choice**

Independent of any gains due to improved execution of spatial skills and vsWM (as described above), it is possible that spatial learning might encourage students to apply different strategic approaches when solving geometry problems. Changes in strategy are largely absent from correlational and experimental literatures (Lowrie et al., 2021; 2023), and, when included, described as a priming effect that cannot be ruled out (e.g., Gilligan et al., 2020). However, students know multiple ways of solving problems and will balance efficiency and accuracy when selecting an appropriate strategy (Siegler, 2006). These strategic decisions are influenced by the nature of the problem, their familiarity with different strategies, and their beliefs about potential success. This is aligned with the finding that spatial skills versus vsWM are relatively more important when solving mathematics problems as familiarity and task demands change (Mix et al., 2020), and that strategic processing of spatial-mathematics relations is more predictive of mathematics performance than automatic processing (Mix, 2019). Importantly, completing geometry problems can involve a multi-step problem-solving process, and students may use different cognitive and meta-cognitive strategies at any given step.

**Future directions – how to capture mechanism**

We suggest that it is possible to characterize mechanism with experimental research explicitly designed to assess potential mechanisms. Such work would be benefited from process models (Young et al., 2018) to identify the specific tasks involved in geometry (previous work often uses broad measures). Mixed-methods approaches would also be helpful, with qualitative methods well suited to characterize when and how strategies may change. Also missing from the literature is analysis of changes in quality, accuracy, and relevancy of (external and internal) representations, which would inform how spatial learning supports mathematics performance. For example, future research may explore if spatial learning leads to representations becoming more detailed, abstract, or focused on relevant spatial information; or perhaps the quality of representation remains the same, but the ability to manipulate it improves. Another key approach to determining mechanism is the inclusion of key mediators and moderators to parse out how basic cognitive skills may manifest in different mathematics contexts (Resnick & Stieff, 2024).
## Educational implications for incorporating spatial learning in geometry

Although more research is required to determine mechanism, there is growing evidence that spatial learning does, indeed, support geometry learning. Successful spatial learning interventions tend to be embedded within authentic mathematics contexts, and not separate decontextualized training (Lowrie et al., 2023). This likely provides better alignment between the spatial reasoning skill and relevant mathematics tasks, which enables students to see the connections. Interestingly, most of the research described in the current study came from the field of cognitive science, which tended to focus on broad mathematics achievement. However, given that geometry is the part of the curriculum most associated with spatial reasoning (e.g., Battista et al., 2018), the development of effective spatial intervention to support geometry seems particularly important.

## Acknowledgements

This research was supported by the Australian Research Council Grant DE210101881.

## References

Adams, J., Resnick, I., & Lowrie, T. (2022). Supporting senior high-school students' measurement and geometry performance: Does spatial training transfer to mathematics achievement? *Mathematics Education Research Journal*.

Arıcı, S., & Aslan-Tutak, F. (2015). The effect of origami-based instruction on spatial visualization, geometry achievement, and geometric reasoning. *International Journal of Science and Mathematics Education, 13*(1), 179–200.

Battista, M. T., Frazee, L. M., & Winer, M. L. (2018). Analyzing the relation between spatial and geometric reasoning for elementary and middle school students. In K. S. Mix & M. T. Battista (Eds.), *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (pp. 195–228). Springer International Publishing.

Boakes, N. J. (2009). Origami Instruction in the middle school mathematics classroom: its impact on spatial visualization and geometry knowledge of students. *RMLE Online, 32*(7), 1–12.

Burte, H., Gardony, A. L., Hutton, A., & Taylor, H. A. (2017). Think3d!: Improving mathematics learning through embodied spatial training. *Cognitive Research: Principles and Implications, 2*(1).

Cheng, Y.-L., & Mix, K. S. (2014). Spatial training improves children's mathematics ability. *Journal of Cognition and Development, 15*(1), 2–11.

Cheung, C. N., Sung, J. Y., & Lourenco, S. F. (2020). Does training mental rotation transfer to gains in mathematical competence? Assessment of an at-home visuospatial intervention. *Psychological Research, 84*, 2000-2017.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. *Handbook of research on mathematics teaching and learning*.

Cornu, V., Schiltz, C., Pazouki, T., & Martin, R. (2019). Training early visuo-spatial abilities: A controlled classroom-based intervention study. *Applied Developmental Science, 23*(1), 1-21.

Gilligan-Lee, K. A., Hawes, Z. C. K., Williams, A. Y., Farran, E. K., & Mix, K. S. (2023). Hands-On: Investigating the role of physical manipulatives in spatial training. *Child Development, 94*(5), 1205–1221.

Gilligan, K. A., Hodgkiss, A., Thomas, M. S. C., & Farran, E. K. (2018). The developmental relations between spatial cognition and mathematics in primary school children. *Developmental Science*, e12786.

Gilligan, K. A., Thomas, M. S., & Farran, E. K. (2020). First demonstration of effective spatial training for near transfer to spatial performance and far transfer to a range of mathematics skills at 8 years. *Developmental Science, 23*(4), e12909.
## Educational implications for incorporating spatial learning in geometry

Although more research is required to determine mechanism, there is growing evidence that spatial learning does, indeed, support geometry learning. Successful spatial learning interventions tend to be embedded within authentic mathematics contexts, and not separate decontextualized training (Lowrie et al., 2023). This likely provides better alignment between the spatial reasoning skill and relevant mathematics tasks, which enables students to see the connections. Interestingly, most of the research described in the current study came from the field of cognitive science, which tended to focus on broad mathematics achievement. However, given that geometry is the part of the curriculum most associated with spatial reasoning (e.g., Battista et al., 2018), the development of effective spatial intervention to support geometry seems particularly important.

## Acknowledgements

This research was supported by the Australian Research Council Grant DE210101881.

## References

Adams, J., Resnick, I., & Lowrie, T. (2022). Supporting senior high-school students' measurement and geometry performance: Does spatial training transfer to mathematics achievement? *Mathematics Education Research Journal*.

Arıcı, S., & Aslan-Tutak, F. (2015). The effect of origami-based instruction on spatial visualization, geometry achievement, and geometric reasoning. *International Journal of Science and Mathematics Education, 13*(1), 179–200.

Battista, M. T., Frazee, L. M., & Winer, M. L. (2018). Analyzing the relation between spatial and geometric reasoning for elementary and middle school students. In K. S. Mix & M. T. Battista (Eds.), *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (pp. 195–228). Springer International Publishing.

Boakes, N. J. (2009). Origami Instruction in the middle school mathematics classroom: its impact on spatial visualization and geometry knowledge of students. *RMLE Online, 32*(7), 1–12.

Burte, H., Gardony, A. L., Hutton, A., & Taylor, H. A. (2017). Think3d!: Improving mathematics learning through embodied spatial training. *Cognitive Research: Principles and Implications, 2*(1).

Cheng, Y.-L., & Mix, K. S. (2014). Spatial training improves children's mathematics ability. *Journal of Cognition and Development, 15*(1), 2–11.

Cheung, C. N., Sung, J. Y., & Lourenco, S. F. (2020). Does training mental rotation transfer to gains in mathematical competence? Assessment of an at-home visuospatial intervention. *Psychological Research, 84*, 2000-2017.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. *Handbook of research on mathematics teaching and learning*.

Cornu, V., Schiltz, C., Pazouki, T., & Martin, R. (2019). Training early visuo-spatial abilities: A controlled classroom-based intervention study. *Applied Developmental Science, 23*(1), 1-21.

Gilligan-Lee, K. A., Hawes, Z. C. K., Williams, A. Y., Farran, E. K., & Mix, K. S. (2023). Hands-On: Investigating the role of physical manipulatives in spatial training. *Child Development, 94*(5), 1205–1221.

Gilligan, K. A., Hodgkiss, A., Thomas, M. S. C., & Farran, E. K. (2018). The developmental relations between spatial cognition and mathematics in primary school children. *Developmental Science*, e12786.

Gilligan, K. A., Thomas, M. S., & Farran, E. K. (2020). First demonstration of effective spatial training for near transfer to spatial performance and far transfer to a range of mathematics skills at 8 years. *Developmental Science, 23*(4), e12909.
Guest, G., MacQueen, K., & Namey, E. (2012). Applied thematic analysis Thousand Oaks California SAGE Publications.

Hawes, Z., & Ansari, D. (2020). What explains the relationship between spatial and mathematical skills? A review of evidence from brain and behavior. *Psych. Bull. & Review*, *27*(3), 465–482.

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

Hegarty, M., & Waller, D. (2004). A dissociation between mental rotation and perspective-taking spatial abilities. *Intelligence*, *32*(2), 175–191. https://doi.org/10.1016/j.intell.2003.12.001

Kyttälä, M., & Lehto, J. E. (2008). Some factors underlying mathematical performance: The role of visuospatial working memory and non-verbal intelligence. *European Journal of Psychology of Education*, *23*(1), 77–94.

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

Lowrie, T., & Logan, T. (2023). Spatial visualization supports students’ math: mechanisms for spatial transfer. *Journal of Intelligence*, *11*(6).

Lowrie, T., Logan, T., & Hegarty, M. (2019). The influence of spatial visualization training on students’ spatial reasoning and mathematics performance. *Journal of Cognition and Development*, *20*(5), 729–751.

Lowrie, T., Logan, T., & Ramful, A. (2017). Visuospatial training improves elementary students’ mathematics performance. *British Journal of Educational Psychology*, *87*(2), 170–186.

Mix, K. S. (2019). Why are spatial skill and mathematics related? *Child Development Perspectives*, *13*(2), 121–126.

Mix, K. S., Levine, S. C., Cheng, Y.-L., Stockton, J. D., & Bower, C. (2021). Effects of spatial training on mathematics in first and sixth grade children. *Journal of Educational Psychology*, *113*(2), 304–314.

Mulligan, J., Woolcott, G., Mitchelmore, M., Busatto, S., Lai, J., & Davis, B. (2020). Evaluating the impact of a Spatial Reasoning Mathematics Program (SRMP) intervention in the primary school. *Mathematics Education Research Journal*, *32*, 285–305.

Newcombe, N. S., Booth, J. L., & Gunderson, E. A. (2019). Spatial skills, reasoning, and mathematics. In *The Cambridge handbook of cognition and education* (pp. 100–123). Cambridge University Press.

Resnick, I., Newcombe, N. S., & Jordan, N. C. (2019). The relation between spatial reasoning and mathematical achievement in children with mathematical learning difficulties. In *International Handbook of Mathematical Learning Difficulties* (pp. 423–435). Springer.

Resnick, I., & Stieff, M. (2024). The role of theory in research and practice: Characterizing cognition and context within mathematics education. In M. A. Clements, B. Kaur, T. Lowrie, V. Mesa, & J. Prytz. (Eds.), *International handbook of Mathematics Education*. Springer.

Rohrer, J. M. (2018). Thinking clearly about correlations and causation: graphical causal models for observational data. *Advances in Methods and Practices in Psych Science*, *1*(1), 27–42.

Siegler, R. S. (2006). Microgenetic analyses of learning. In D. Kuhn, R. S. Siegler, W. Damon, & R. M. Lerner (Eds.), *Handbook of child psychology: Cognition, perception, and language* (pp. 464–510). John Wiley & Sons, Inc..

Young, C., Levine, S. C., & Mix, K. S. (2018). What processes underlie the relation between spatial skill and mathematics? In K. S. Mix & M. T. Battista (Eds.), *Visualizing Mathematics: The Role of Spatial Reasoning in Mathematical Thought* (pp. 117–148). Springer International.

Zhang D. W., Zaphf, A., & Klingberg, T. (2021). Resting state EEG related to mathematical improvement after spatial training in children. *Frontiers Human Neuroscience*, *15*.
Guest, G., MacQueen, K., & Namey, E. (2012). Applied thematic analysis Thousand Oaks California SAGE Publications.

Hawes, Z., & Ansari, D. (2020). What explains the relationship between spatial and mathematical skills? A review of evidence from brain and behavior. *Psych. Bull. & Review*, *27*(3), 465–482.

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

Hegarty, M., & Waller, D. (2004). A dissociation between mental rotation and perspective-taking spatial abilities. *Intelligence*, *32*(2), 175–191. https://doi.org/10.1016/j.intell.2003.12.001

Kyttälä, M., & Lehto, J. E. (2008). Some factors underlying mathematical performance: The role of visuospatial working memory and non-verbal intelligence. *European Journal of Psychology of Education*, *23*(1), 77–94.

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

Lowrie, T., & Logan, T. (2023). Spatial visualization supports students’ math: mechanisms for spatial transfer. *Journal of Intelligence*, *11*(6).

Lowrie, T., Logan, T., & Hegarty, M. (2019). The influence of spatial visualization training on students’ spatial reasoning and mathematics performance. *Journal of Cognition and Development*, *20*(5), 729–751.

Lowrie, T., Logan, T., & Ramful, A. (2017). Visuospatial training improves elementary students’ mathematics performance. *British Journal of Educational Psychology*, *87*(2), 170–186.

Mix, K. S. (2019). Why are spatial skill and mathematics related? *Child Development Perspectives*, *13*(2), 121–126.

Mix, K. S., Levine, S. C., Cheng, Y.-L., Stockton, J. D., & Bower, C. (2021). Effects of spatial training on mathematics in first and sixth grade children. *Journal of Educational Psychology*, *113*(2), 304–314.

Mulligan, J., Woolcott, G., Mitchelmore, M., Busatto, S., Lai, J., & Davis, B. (2020). Evaluating the impact of a Spatial Reasoning Mathematics Program (SRMP) intervention in the primary school. *Mathematics Education Research Journal*, *32*, 285–305.

Newcombe, N. S., Booth, J. L., & Gunderson, E. A. (2019). Spatial skills, reasoning, and mathematics. In *The Cambridge handbook of cognition and education* (pp. 100–123). Cambridge University Press.

Resnick, I., Newcombe, N. S., & Jordan, N. C. (2019). The relation between spatial reasoning and mathematical achievement in children with mathematical learning difficulties. In *International Handbook of Mathematical Learning Difficulties* (pp. 423–435). Springer.

Resnick, I., & Stieff, M. (2024). The role of theory in research and practice: Characterizing cognition and context within mathematics education. In M. A. Clements, B. Kaur, T. Lowrie, V. Mesa, & J. Prytz. (Eds.), *International handbook of Mathematics Education*. Springer.

Rohrer, J. M. (2018). Thinking clearly about correlations and causation: graphical causal models for observational data. *Advances in Methods and Practices in Psych Science*, *1*(1), 27–42.

Siegler, R. S. (2006). Microgenetic analyses of learning. In D. Kuhn, R. S. Siegler, W. Damon, & R. M. Lerner (Eds.), *Handbook of child psychology: Cognition, perception, and language* (pp. 464–510). John Wiley & Sons, Inc..

Young, C., Levine, S. C., & Mix, K. S. (2018). What processes underlie the relation between spatial skill and mathematics? In K. S. Mix & M. T. Battista (Eds.), *Visualizing Mathematics: The Role of Spatial Reasoning in Mathematical Thought* (pp. 117–148). Springer International.

Zhang D. W., Zaphf, A., & Klingberg, T. (2021). Resting state EEG related to mathematical improvement after spatial training in children. *Frontiers Human Neuroscience*, *15*.
# GUIDED PLAY THAT FOSTERS THE DEVELOPMENT OF CHILDREN'S SPATIAL ABILITY

Satoshi Watanabe

Jissen Women's University

One type of spatial reasoning involves the skill of mental imaging, a capacity initially acquired by children through physical experiences and gradually refined through mental manipulation (Clements et al., 2004). The development of children's mental manipulation skills necessitates the cultivation of concrete manipulation skills in geometry. However, there is a significant research gap concerning the specific classroom instruction and experiences that contribute to the enhancement of children's mental imaging abilities. In this study, we initiated a lesson centered around free play with geometric toys to gauge children's geometric aptitude, and then we implemented guided play sessions enhance their geometric abilities. By incorporating discussion using appropriate materials at the beginning of guided play and allowing children to select and work on three-dimensional objects themselves, children can create three-dimensional objects at a more advanced level.

## INTRODUCTION

Recent studies have highlighted the close relationship between spatial ability and mathematical performance (Casey et al., 1995; Geary et al., 2000; Mix & Cheng, 2012). Researchers have not only observed a correlation between spatial thinking and mathematics but have identified spatial thinking as a predictor of subsequent mathematics performance (Mix & Cheng, 2012). Moreover, early spatial experiences contribute significantly to the later development of spatial and geometric reasoning skills (Okamoto et al., 2014).

It is noteworthy that the spatial ability of young children demonstrates greater improvement than that of adolescents and adults, which aligns with the prediction of the sensitive period hypothesis (Baenninger & Newcombe, 1989). Research on spatial abilities in young children involves educational resources such as blocks, origami, and computer games (e.g., Cakmak et al., 2014). A variety of measures, including verbalization, pictorial imagery, and construction, have been employed to assess children's spatial abilities (e.g., Thom & McGarvey, 2014). As spatial training transfers to other spatial skills that have not been directly trained (Uttal et al., 2013), it is imperative to consider specific training that enhances children's spatial abilities.

Recent studies have shifted their focus to classroom-based learning, with teachers opting for shorter interventions rather than semester-long training sessions (e.g., Sinclair & Bruce, 2014). These studies suggest that young children who engage in diverse activities, particularly those who initially perform poorly on spatial skills tests, may benefit from training (Baenninger & Newcombe, 1989). However, it is important to note that performance ratings in these studies are often based on children's mathematics assessments, which do not fully reflect their individual spatial skills. Tailoring classroom instruction to each child's spatial ability has been proposed as a more effective approach to increasing students' overall spatial ability.
# GUIDED PLAY THAT FOSTERS THE DEVELOPMENT OF CHILDREN'S SPATIAL ABILITY

Satoshi Watanabe

Jissen Women's University

One type of spatial reasoning involves the skill of mental imaging, a capacity initially acquired by children through physical experiences and gradually refined through mental manipulation (Clements et al., 2004). The development of children's mental manipulation skills necessitates the cultivation of concrete manipulation skills in geometry. However, there is a significant research gap concerning the specific classroom instruction and experiences that contribute to the enhancement of children's mental imaging abilities. In this study, we initiated a lesson centered around free play with geometric toys to gauge children's geometric aptitude, and then we implemented guided play sessions to enhance their geometric abilities. By incorporating discussion using appropriate materials at the beginning of guided play and allowing children to select and work on three-dimensional objects themselves, children can create three-dimensional objects at a more advanced level.

## INTRODUCTION

Recent studies have highlighted the close relationship between spatial ability and mathematical performance (Casey et al., 1995; Geary et al., 2000; Mix & Cheng, 2012). Researchers have not only observed a correlation between spatial thinking and mathematics but have identified spatial thinking as a predictor of subsequent mathematics performance (Mix & Cheng, 2012). Moreover, early spatial experiences contribute significantly to the later development of spatial and geometric reasoning skills (Okamoto et al., 2014).

It is noteworthy that the spatial ability of young children demonstrates greater improvement than that of adolescents and adults, which aligns with the prediction of the sensitive period hypothesis (Baenninger & Newcombe, 1989). Research on spatial abilities in young children involves educational resources such as blocks, origami, and computer games (e.g., Cakmak et al., 2014). A variety of measures, including verbalization, pictorial imagery, and construction, have been employed to assess children's spatial abilities (e.g., Thom & McGarvey, 2014). As spatial training transfers to other spatial skills that have not been directly trained (Uttal et al., 2013), it is imperative to consider specific training that enhances children's spatial abilities.

Recent studies have shifted their focus to classroom-based learning, with teachers opting for shorter interventions rather than semester-long training sessions (e.g., Sinclair & Bruce, 2014). These studies suggest that young children who engage in diverse activities, particularly those who initially perform poorly on spatial skills tests, may benefit from training (Baenninger & Newcombe, 1989). However, it is important to note that performance ratings in these studies are often based on children's mathematics assessments, which do not fully reflect their individual spatial skills. Tailoring classroom instruction to each child's spatial ability has been proposed as a more effective approach to increasing students' overall spatial ability.
Development of the mental imaging in young children

One facet of spatial reasoning involves mental imaging, which entails visualizing and mentally transforming objects, often transitioning between two-dimensional (2-D) and three-dimensional (3-D) representations. Children acquire this skill through physical experiences and subsequently progress to mental manipulation (Clements et al., 2004). The memory of physically manipulating figures observed through hand movements and visual perception contributes to reducing working memory demands for spatial tasks. However, the developmental progression of children's physical manipulation of 2-D figures and 3-D solids is crucial, as without such advancement, children may struggle with mental manipulation.

In this study, the author explored the developmental stages of 3-D construction leading to mental imaging, drawing upon Clements' developmental framework for the composition of 2-D plane figures. Clements delineated four stages in children's developmental progression for the composition of 2-D shapes (Clements et al., 2004, p. 168). The first stage, termed "Precomposer" denotes a phase where children are unable to combine shapes to form a larger shape. The second stage, "Piece Assembler," involves children placing shapes contiguously to create pictures. The third stage, "Picture Maker," signifies the ability of children to concatenate shapes to form pictures where multiple shapes serve a unified role. The fourth stage, "Shape Composer," indicates children's capacity to combine shapes to generate new shapes or complete puzzles. In addition to these four stages, Clements et al. (2004) described three detailed developmental progressions in shape composition. "Substitution Composition" is the stage where children intentionally form composite units of shapes, understanding, for instance, that two trapezoidal pattern blocks can construct a hexagon. "Shape Composite Iterator" marks the stage where children construct and manipulate composition units while maintaining the integrity of their pattern shapes. Lastly, "Shape Composer with Superordinate Units" represents the stage wherein children build a unit and iterate it to create new units, showcasing an advanced level of spatial reasoning.

Preliminary observations were conducted to assess the developmental stages of 2-D and 3-D construction in four- and five-year-old children, particularly in geometric toy play with Polydrons. Polydron shapes, including equilateral triangles, isosceles triangles, squares, pentagons, and hexagons, were employed because of their ease of manipulation, even for young children. During free play at a nursery school, four-year-olds demonstrated the developmental composition stage of "Picture Maker," connecting 2-D images to represent people and flowers using Polydrons. Additionally, both four- and five-year-olds exhibited the "Shape Composure" stage, connecting the same shapes on a plane to create larger 2-D flat shapes.

The developmental stages of constructing 3-D solids were further observed in the work of five-year-olds. The initial stage involved constructing cylinders and cones surrounded by squares or isosceles triangles, representing the "Shape Composure" stage for 3-D solids. Subsequently, the five-year-olds created 3-D pyramids by connecting triangles to the sides based on squares, pentagons, and hexagons. The second stage featured the construction of solids with multiple faces at the bottom and sides, including a child producing a large composite solid by combining multiple cubes, reflecting the "Shape Composure Iterator." stage. The third stage saw the creation of complex 3-D structures combining various 3-D objects, representing the "Shape Composer with Superordinate Units" stage. To enhance a child's mental imaging, progressing through the developmental stages of 3-D solid
Development of the mental imaging in young children

One facet of spatial reasoning involves mental imaging, which entails visualizing and mentally transforming objects, often transitioning between two-dimensional (2-D) and three-dimensional (3-D) representations. Children acquire this skill through physical experiences and subsequently progress to mental manipulation (Clements et al., 2004). The memory of physically manipulating figures observed through hand movements and visual perception contributes to reducing working memory demands for spatial tasks. However, the developmental progression of children’s physical manipulation of 2-D figures and 3-D solids is crucial, as without such advancement, children may struggle with mental manipulation.

In this study, the author explored the developmental stages of 3-D construction leading to mental imaging, drawing upon Clements’ developmental framework for the composition of 2-D plane figures. Clements delineated four stages in children’s developmental progression for the composition of 2-D shapes (Clements et al., 2004, p. 168). The first stage, termed “Precomposer” denotes a phase where children are unable to combine shapes to form a larger shape. The second stage, “Piece Assembler,” involves children placing shapes contiguously to create pictures. The third stage, “Picture Maker,” signifies the ability of children to concatenate shapes to form pictures where multiple shapes serve a unified role. The fourth stage, “Shape Composer,” indicates children’s capacity to combine shapes to generate new shapes or complete puzzles. In addition to these four stages, Clements et al. (2004) described three detailed developmental progressions in shape composition. “Substitution Composition” is the stage where children intentionally form composite units of shapes, understanding, for instance, that two trapezoidal pattern blocks can construct a hexagon. “Shape Composite Iterator” marks the stage where children construct and manipulate composition units while maintaining the integrity of their pattern shapes. Lastly, “Shape Composer with Superordinate Units” represents the stage wherein children build a unit and iterate it to create new units, showcasing an advanced level of spatial reasoning.

Preliminary observations were conducted to assess the developmental stages of 2-D and 3-D construction in four- and five-year-old children, particularly in geometric toy play with Polydrons. Polydron shapes, including equilateral triangles, isosceles triangles, squares, pentagons, and hexagons, were employed because of their ease of manipulation, even for young children. During free play at a nursery school, four-year-olds demonstrated the developmental composition stage of “Picture Maker,” connecting 2-D images to represent people and flowers using Polydrons. Additionally, both four- and five-year-olds exhibited the “Shape Composure” stage, connecting the same shapes on a plane to create larger 2-D flat shapes.

The developmental stages of constructing 3-D solids were further observed in the work of five-year-olds. The initial stage involved constructing cylinders and cones surrounded by squares or isosceles triangles, representing the “Shape Composure” stage for 3-D solids. Subsequently, the five-year-olds created 3-D pyramids by connecting triangles to the sides based on squares, pentagons, and hexagons. The second stage featured the construction of solids with multiple faces at the bottom and sides, including a child producing a large composite solid by combining multiple cubes, reflecting the “Shape Composure Iterator.” stage. The third stage saw the creation of complex 3-D structures combining various 3-D objects, representing the “Shape Composer with Superordinate Units” stage. To enhance a child’s mental imaging, progressing through the developmental stages of 3-D solid
construction through rich geometric experiences is crucial (Clements et al., 2004). Encouraging geometric play involves providing free geometric toys; however, not all children may be equally interested. Furthermore, some children may face challenges owing to insufficient finger dexterity. Therefore, it is essential to establish an environment that promotes 3-D object creation and offers support based on each child's developmental stage.

However, current spatial training programs often lack child-driven choices (Vogt et al., 2018) and may be distant from children's play-based lifestyles (Bruce & Hawes, 2015). To address this issue, research that focuses on play-based lifestyles that foster spatial abilities tailored to individual stages of geometric development is needed (Weisberg et al., 2013). Therefore, this study aimed to create an environment that motivates children to play and develop their geometric abilities, mainly focusing on the following research question:

RQ: How can teachers and classes motivate children's geometric play and develop their individual geometric abilities?

## CONCEPTIONAL FRAMEWORK: GUIDED PLAY

Kindergarten educators often employ mathematics training programs to explicitly develop mathematical abilities (Sakakibara, 2004). However, few comprehensive studies have compared the effectiveness of these approaches in terms of learning outcomes for all children, including those with different abilities (Vogt et al., 2018). In preschool classrooms, teachers must create an environment in which children are free to interact with the content and participate in play according to their abilities. (Chien et al., 2010).

Two prominent pedagogical methods frequently contrasted in preschool education are direct instruction and free-play (Brant, 2013). Direct instruction involves the teacher playing an active role in instructing students, who largely assume passive roles. In contrast, free play allows children to choose their own activities and play independently. Children engaged in free play may encounter challenges in achieving learning goals as they might not be guided to focus on appropriate dimensions (Weisberg et al., 2013). Weisberg et al. (2013) emphasized that playful and child-centered approaches, incorporating some level of adult scaffolding, prove more effective than directed instruction in achieving outcomes in young children. Guided play has emerged as a balanced and effective pedagogical approach in preschool education that integrate the learning objectives of adult scaffolding while maintaining a child-directed environment (Weisberg et al., 2013). In guided play, adults initiate the learning process, set learning goals, and are responsible for maintaining focus on the objectives that guide children's discoveries. The training program demonstrates significant benefits for children with very low competency, whereas the play-based approach serves all children, irrespective of their competency levels, ranging from low to high (Vogt et al., 2018).

In early childhood education in Japan, as outlined by the Ministry of Education, Culture, Sports, Science and Technology (2018), interaction between children and teachers plays a crucial role in fostering curiosity and inquisitiveness. Teachers strive to establish an environment that encourages children to take initiative, allowing them to act independently and engage with both peers and educators to deepen their learning experiences. Therefore, guided play aligns seamlessly with the principles of Japanese early childhood education and has emerged as an effective method for nurturing individual geometric abilities. This study specifically investigated how guided play can
construction through rich geometric experiences is crucial (Clements et al., 2004). Encouraging geometric play involves providing free geometric toys; however, not all children may be equally interested. Furthermore, some children may face challenges owing to insufficient finger dexterity. Therefore, it is essential to establish an environment that promotes 3-D object creation and offers support based on each child's developmental stage.

However, current spatial training programs often lack child-driven choices (Vogt et al., 2018) and may be distant from children's play-based lifestyles (Bruce & Hawes, 2015). To address this issue, research that focuses on play-based lifestyles that foster spatial abilities tailored to individual stages of geometric development is needed (Weisberg et al., 2013). Therefore, this study aimed to create an environment that motivates children to play and develop their geometric abilities, mainly focusing on the following research question:

RQ: How can teachers and classes motivate children's geometric play and develop their individual geometric abilities?

## CONCEPTIONAL FRAMEWORK: GUIDED PLAY

Kindergarten educators often employ mathematics training programs to explicitly develop mathematical abilities (Sakakibara, 2004). However, few comprehensive studies have compared the effectiveness of these approaches in terms of learning outcomes for all children, including those with different abilities (Vogt et al., 2018). In preschool classrooms, teachers must create an environment in which children are free to interact with the content and participate in play according to their abilities. (Chien et al., 2010).

Two prominent pedagogical methods frequently contrasted in preschool education are direct instruction and free-play (Brant, 2013). Direct instruction involves the teacher playing an active role in instructing students, who largely assume passive roles. In contrast, free play allows children to choose their own activities and play independently. Children engaged in free play may encounter challenges in achieving learning goals as they might not be guided to focus on appropriate dimensions (Weisberg et al., 2013). Weisberg et al. (2013) emphasized that playful and child-centered approaches, incorporating some level of adult scaffolding, prove more effective than directed instruction in achieving outcomes in young children. Guided play has emerged as a balanced and effective pedagogical approach in preschool education that integrate the learning objectives of adult scaffolding while maintaining a child-directed environment (Weisberg et al., 2013). In guided play, adults initiate the learning process, set learning goals, and are responsible for maintaining focus on the objectives that guide children's discoveries. The training program demonstrates significant benefits for children with very low competency, whereas the play-based approach serves all children, irrespective of their competency levels, ranging from low to high (Vogt et al., 2018).

In early childhood education in Japan, as outlined by the Ministry of Education, Culture, Sports, Science and Technology (2018), interaction between children and teachers plays a crucial role in fostering curiosity and inquisitiveness. Teachers strive to establish an environment that encourages children to take initiative, allowing them to act independently and engage with both peers and educators to deepen their learning experiences. Therefore, guided play aligns seamlessly with the principles of Japanese early childhood education and has emerged as an effective method for nurturing individual geometric abilities. This study specifically investigated how guided play can
support children in learning about geometry. As part of our approach, while guiding children to tackle
more complex 3-D figures, we encouraged them to set their own guided play goals to enhance their
geometric abilities. The rationale behind this approach is rooted in the belief that by allowing children
to establish their own goals, they would be more motivated to undertake the construction of more
intricate 3-D solids. This personalized goal-setting process was intended to empower the children and
engage them in their geometric learning journey.

METHODS

The data for this study were gathered through experimental lessons on shapes conducted at a local nursery in Tokyo, Japan. The objectives of this study were to assess the efficacy of guided play in teaching and learning 3-D shapes and to identify the outcomes of children’s learning experiences through guided play. A design-based research method (Bakker & Van Eerde, 2014) was employed for a detailed retrospective analysis to elucidate the educational environment and teacher guidance that contribute to the development of individual geometric ability. Three experimental classes were held on February 18, 19, and 21, 2020, at a local nursery school in Tokyo. Each lesson lasted 40 minutes and was conducted by the author with teacher support. In all, 18 children (8 girls and 10 boys) aged five and six years participated in the class. This guidance plan was approved by the nursery school. The purpose of the class was communicated in advance to parents and to the children whose parents agreed to participate. Three lessons were video recorded to understand the stages of geometric development of all the children and were used to reflect on each child’s 3D-making history. All children’s works were photographed and recorded by trained research assistants for use in personal development records. Excerpted photos were used for documentation.

In the first class, children played freely with the geometric toy Polydron and created their own geometric works. After the class, the author and class teacher classified and organized the objects created by the children. We then discussed and selected some three-dimensional objects that could challenge the children based on their abilities and created documentation for the next lesson.

Guided play was conducted in the second and third lessons. At the beginning of the first guided play session, we presented documentation of the previous three-dimensional work to the children and discussed the names and characteristics of the three-dimensional objects to encourage them to think about what they wanted to create next and guide their own goals. After the discussion, the author told the children that they could try new three-dimensional objects or imitate the work that their friends had made. During the class, the author and class teacher asked each student what they were making and encouraged them to create it. In addition, if a child was not motivated to work on it, we asked them what they wanted to make and helped them create it.

After the first guided play, the author and class teacher classified and organized the objects that the children created. Depending on the children’s abilities, we discussed the goal of creating a three-dimensional object which was more difficult than that in the previous lesson, selected multiple three-dimensional objects, and created documentation. At the beginning of the second guided play, the author facilitated a discussion of the characteristics of the three-dimensional object that we wanted the children to try and proceeded with the discussion in a way that encouraged them to want to create that object. During the class, the author and teacher asked each child what they had created and encouraged them. The author worked with the class teacher to ask children who were not keen on
support children in learning about geometry. As part of our approach, while guiding children to tackle
more complex 3-D figures, we encouraged them to set their own guided play goals to enhance their
geometric abilities. The rationale behind this approach is rooted in the belief that by allowing children
to establish their own goals, they would be more motivated to undertake the construction of more
intricate 3-D solids. This personalized goal-setting process was intended to empower the children and
engage them in their geometric learning journey.

METHODS

The data for this study were gathered through experimental lessons on shapes conducted at a local nursery in Tokyo, Japan. The objectives of this study were to assess the efficacy of guided play in teaching and learning 3-D shapes and to identify the outcomes of children’s learning experiences through guided play. A design-based research method (Bakker & Van Eerde, 2014) was employed for a detailed retrospective analysis to elucidate the educational environment and teacher guidance that contribute to the development of individual geometric ability. Three experimental classes were held on February 18, 19, and 21, 2020, at a local nursery school in Tokyo. Each lesson lasted 40 minutes and was conducted by the author with teacher support. In all, 18 children (8 girls and 10 boys) aged five and six years participated in the class. This guidance plan was approved by the nursery school. The purpose of the class was communicated in advance to parents and to the children whose parents agreed to participate. Three lessons were video recorded to understand the stages of geometric development of all the children and were used to reflect on each child’s 3D-making history. All children’s works were photographed and recorded by trained research assistants for use in personal development records. Excerpted photos were used for documentation.

In the first class, children played freely with the geometric toy Polydron and created their own geometric works. After the class, the author and class teacher classified and organized the objects created by the children. We then discussed and selected some three-dimensional objects that could challenge the children based on their abilities and created documentation for the next lesson.

Guided play was conducted in the second and third lessons. At the beginning of the first guided play session, we presented documentation of the previous three-dimensional work to the children and discussed the names and characteristics of the three-dimensional objects to encourage them to think about what they wanted to create next and guide their own goals. After the discussion, the author told the children that they could try new three-dimensional objects or imitate the work that their friends had made. During the class, the author and class teacher asked each student what they were making and encouraged them to create it. In addition, if a child was not motivated to work on it, we asked them what they wanted to make and helped them create it.

After the first guided play, the author and class teacher classified and organized the objects that the children created. Depending on the children’s abilities, we discussed the goal of creating a three-dimensional object which was more difficult than that in the previous lesson, selected multiple three-dimensional objects, and created documentation. At the beginning of the second guided play, the author facilitated a discussion of the characteristics of the three-dimensional object that we wanted the children to try and proceeded with the discussion in a way that encouraged them to want to create that object. During the class, the author and teacher asked each child what they had created and encouraged them. The author worked with the class teacher to ask children who were not keen on
playing with geometric toys what they wanted to create and then helped them create it. This collaborative approach aims to ensure a supportive and engaging learning environment for all participants.

## RESULTS

The first lesson involved free play with geometric toys, specifically Polydron, in which the author gathered the children and instructed them to create something independently or collaboratively with their peers. The observations during this lesson revealed five developmental stages in the construction of 2-D and 3-D figures. The first stage saw the connection of the same shapes to create a 2-D surface, exemplified by three children creating a flat surface with hexagons, marking the “Shape Composure” stage for 2-D figures.

The second stage involved creating cones or columns by connecting the same shape to the sides of a base, resulting in four children constructing triangular and hexagonal prisms, signifying the “Shape Composer” stage for 3-D solids. Two of the four children constructed hexagonal prisms, while the remaining two constructed a triangular pyramid and a hexagonal pyramid. The third stage entailed creating solids with multiple faces at the bottom and sides, with five children developing solids, such as a spaceship and a slide, representing the “Shape Composure Iterator” stage. The fourth stage involved creating solids by connecting multiple shapes, with four children making a star and a pillar, reflecting the “Substitution Composure” stage. The fifth stage comprised complex 3-D solids, including spheres created by connecting pentagons and hexagons, illustrating the “Shape Composer with Superordinate Units” stage.

The second lesson introduced guided play, in which the author showed documentation of the shapes created individually by the children and discussed their characteristics. Following the discussion, the children were encouraged to replicate their friends’ creations or design new 3-D objects. Peer influence was leveraged to challenge the children to create more complex 3-D objects. The children were instructed to seek assistance from the teacher if needed. Table 1 presents the 2-D and 3-D objects created during the second lesson.

The following characteristics were observed in the works created by the children during the second lesson: First, three children connected squares to create a flat 2-D surface. In the previous lesson, these three children made a flat 2-D surface by connecting hexagons. These children were still at the “Shape Composer” stage of developmental progression for the composition of 2-D shapes (Clements et al., 2004). Second, six children started to spin tops connected to eight isosceles triangles. In addition, the number of children who formed closes polyhedral (henceforth, “spheres”) by connecting pentagons and hexagons increased from two to five. After completing the second lesson, the author and classroom teacher discussed two tasks: assisting children with difficulty constructing 3-D solids and challenging children who were enthusiastic about spinning tops to create more difficult 3-D objects. For the first task, we showed the documentation of tops at the beginning of the third lesson to guide children who had difficulty making 3-D solids and to inspire them to make tops. They could connect the faces by pushing them together on the floor and easily expanding the flat 2-D faces. However, when making a top, it is necessary to connect the isosceles triangles with the fingers; therefore, we thought that making a top would develop hand dexterity. The second task was to inspire the children to construct more complex 3-D solids. For this purpose, we presented the documentation
playing with geometric toys what they wanted to create and then helped them create it. This collaborative approach aims to ensure a supportive and engaging learning environment for all participants.

## RESULTS

The first lesson involved free play with geometric toys, specifically Polydron, in which the author gathered the children and instructed them to create something independently or collaboratively with their peers. The observations during this lesson revealed five developmental stages in the construction of 2-D and 3-D figures. The first stage saw the connection of the same shapes to create a 2-D surface, exemplified by three children creating a flat surface with hexagons, marking the “Shape Composure” stage for 2-D figures.

The second stage involved creating cones or columns by connecting the same shape to the sides of a base, resulting in four children constructing triangular and hexagonal prisms, signifying the “Shape Composer” stage for 3-D solids. Two of the four children constructed hexagonal prisms, while the remaining two constructed a triangular pyramid and a hexagonal pyramid. The third stage entailed creating solids with multiple faces at the bottom and sides, with five children developing solids, such as a spaceship and a slide, representing the “Shape Composure Iterator” stage. The fourth stage involved creating solids by connecting multiple shapes, with four children making a star and a pillar, reflecting the “Substitution Composure” stage. The fifth stage comprised complex 3-D solids, including spheres created by connecting pentagons and hexagons, illustrating the “Shape Composer with Superordinate Units” stage.

The second lesson introduced guided play, in which the author showed documentation of the shapes created individually by the children and discussed their characteristics. Following the discussion, the children were encouraged to replicate their friends' creations or design new 3-D objects. Peer influence was leveraged to challenge the children to create more complex 3-D objects. The children were instructed to seek assistance from the teacher if needed. Table 1 presents the 2-D and 3-D objects created during the second lesson.

The following characteristics were observed in the works created by the children during the second lesson: First, three children connected squares to create a flat 2-D surface. In the previous lesson, these three children made a flat 2-D surface by connecting hexagons. These children were still at the “Shape Composer” stage of developmental progression for the composition of 2-D shapes (Clements et al., 2004). Second, six children started to spin tops connected to eight isosceles triangles. In addition, the number of children who formed closes polyhedral (henceforth, “spheres”) by connecting pentagons and hexagons increased from two to five. After completing the second lesson, the author and classroom teacher discussed two tasks: assisting children with difficulty constructing 3-D solids and challenging children who were enthusiastic about spinning tops to create more difficult 3-D objects. For the first task, we showed the documentation of tops at the beginning of the third lesson to guide children who had difficulty making 3-D solids and to inspire them to make tops. They could connect the faces by pushing them together on the floor and easily expanding the flat 2-D faces. However, when making a top, it is necessary to connect the isosceles triangles with the fingers; therefore, we thought that making a top would develop hand dexterity. The second task was to inspire the children to construct more complex 3-D solids. For this purpose, we presented the documentation
of the sphere and discussed its characteristics. In the second lesson, several friends became interested in the large sphere that one girl created by connecting pentagons and hexagons. To create a large sphere, three hexagons must be connected at one vertex and pentagons must be connected between them. This requires strength in the children's fingertips because they must connect them face-to-face in the air. Creating a sphere can assist children in developing their fingertips and understanding the characteristics of 3-D solids that can be made by connecting pentagons and hexagons.

<table><caption>Table1. Types and numbers of 2-D planes and 3-D solid figures created by children</caption><thead><tr><th></th><th>First lesson<br>(Free play)</th><th>Second lesson<br>(First guide play)</th><th>Third lesson<br>(Second guided play)</th></tr></thead><tbody><tr><td>2-D Shape<br>Composer</td><td>3<br>Hexagonal tiling (3)</td><td>3<br>Square tiling (3)</td><td>0</td></tr><tr><td>3-D Shape<br>Composer</td><td>4<br>Hexagonal prism (2)<br>Triangular pyramid (1)<br>Hexagonal pyramid (1)</td><td>6<br>Tops (6)</td><td>5<br>Tops (5)</td></tr><tr><td>Shape Composure<br>Iterator</td><td>5<br>Spaceship (4)<br>Slider (1)</td><td>0</td><td>1<br>Castle (1)</td></tr><tr><td>Substitution<br>Composure</td><td>4<br>Star column (4)</td><td>3<br>Star column (3)</td><td>1<br>Star column (1)</td></tr><tr><td>Shape Composer<br>with Superordinate<br>Units</td><td>2<br>Sphere (2)</td><td>6<br>Sphere (5)<br>3-D cat face (1)</td><td>9<br>Sphere (8)<br>Hexagonal composite<br>solid (1)</td></tr></tbody></table>

The third lesson consisted of guided play session. Commencing the lesson, the author presented documentation of the previously constructed tops and engaged the children in a discussion of their characteristics. One boy suggested that alternately arranging triangles of two colors to create a frame would result in a visually appealing effect during spinning. Another boy recommended spinning the top as flat as possible to prolong the spin duration. Following this, the author showcased the documentation of the sphere and flat surfaces, prompting the children to identify the differences. One boy noted that the sphere was round, and the surface was flat.

Concluding the discussion, the author asked the children what they wanted to construct next, guiding them toward the challenge of creating spheres. Eight children formed three groups and collaborated to construct the spheres. Simultaneously, five children chose to construct tops and derive enjoyment from playing with them. Notably, two of the children created a 2-D square tiling during the second lesson. Additionally, a child who had constructed a 2-D square tiling in the preceding lesson evolved
of the sphere and discussed its characteristics. In the second lesson, several friends became interested in the large sphere that one girl created by connecting pentagons and hexagons. To create a large sphere, three hexagons must be connected at one vertex and pentagons must be connected between them. This requires strength in the children's fingertips because they must connect them face-to-face in the air. Creating a sphere can assist children in developing their fingertips and understanding the characteristics of 3-D solids that can be made by connecting pentagons and hexagons.

<table><caption>Table1. Types and numbers of 2-D planes and 3-D solid figures created by children</caption><thead><tr><th></th><th>First lesson<br>(Free play)</th><th>Second lesson<br>(First guide play)</th><th>Third lesson<br>(Second guided play)</th></tr></thead><tbody><tr><td>2-D Shape<br>Composer</td><td>3<br>Hexagonal tiling (3)</td><td>3<br>Square tiling (3)</td><td>0</td></tr><tr><td>3-D Shape<br>Composer</td><td>4<br>Hexagonal prism (2)<br>Triangular pyramid (1)<br>Hexagonal pyramid (1)</td><td>6<br>Tops (6)</td><td>5<br>Tops (5)</td></tr><tr><td>Shape Composure<br>Iterator</td><td>5<br>Spaceship (4)<br>Slider (1)</td><td>0</td><td>1<br>Castle (1)</td></tr><tr><td>Substitution<br>Composure</td><td>4<br>Star column (4)</td><td>3<br>Star column (3)</td><td>1<br>Star column (1)</td></tr><tr><td>Shape Composer<br>with Superordinate<br>Units</td><td>2<br>Sphere (2)</td><td>6<br>Sphere (5)<br>3-D cat face (1)</td><td>9<br>Sphere (8)<br>Hexagonal composite<br>solid (1)</td></tr></tbody></table>

The third lesson consisted of guided play session. Commencing the lesson, the author presented documentation of the previously constructed tops and engaged the children in a discussion of their characteristics. One boy suggested that alternately arranging triangles of two colors to create a frame would result in a visually appealing effect during spinning. Another boy recommended spinning the top as flat as possible to prolong the spin duration. Following this, the author showcased the documentation of the sphere and flat surfaces, prompting the children to identify the differences. One boy noted that the sphere was round, and the surface was flat.

Concluding the discussion, the author asked the children what they wanted to construct next, guiding them toward the challenge of creating spheres. Eight children formed three groups and collaborated to construct the spheres. Simultaneously, five children chose to construct tops and derive enjoyment from playing with them. Notably, two of the children created a 2-D square tiling during the second lesson. Additionally, a child who had constructed a 2-D square tiling in the preceding lesson evolved
in geometric development, progressing from the “2-D Shape Composure” stage to the “Substitution Composure” stage. This advancement involved using composite units of shapes to construct 3-D solids, as exemplified by the creation of a star-shaped cylinder while referencing the documentation. The child ingeniously used five equilateral triangles to form a pentagon, showcasing a more sophisticated level of geometric composition.

## DISCUSSION

The objective of this study was to explore how teachers and classes can enhance the motivation for individual geometric play and foster unique geometric abilities through guided play using documentation. Preliminary discussions using documentation before the two guided play sessions facilitated the children's contemplation of the challenge of creating more intricate 3-D solids.

An analysis of the changes in the 3-D works created by the children, as depicted in Table 1, reveals a noteworthy increase in the number of children engaging in 3-D creations at more advanced developmental stages in each successive round. Furthermore, guided play allows us to offer personalized guidance based on individual geometric abilities. For instance, a girl who had previously created a 2-D flat plane twice in a row expressed uncertainty about what to make the third time, but when prompted by the classroom teacher, she chose to make a top. While connecting the triangles with the teacher's assistance, the girl gained confidence and successfully connected the last five triangles, creating a top independently. This illustrates how environmental settings, including playing with geometric toys and engaging in discussions using documentation, have the potential to motivate children and propel them through various developmental stages of geometry.

In future research, we will examine the broader impact of guided play on children’s everyday play and the ongoing development of their geometric skills.

## References

* Baenninger, M., & Newcombe, N. (1989). The role of experience in spatial test performance: A meta-analysis. *Sex Roles, 20*(5-6), 327-344.

* Bakker, A. & Van Eerde, D. (2014). An introduction to design-based research education with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), *Approaches to qualitative research in mathematics education* (pp. 429-466). Springer.

* Brant, B. (2013). Everyday pedagogical practices in mathematical play situation in German “Kindergarten”. *Educational Studies in Mathematics, 84*(2), 227-248.

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

* Cakmak, S., Isiksal, M., & Koc, Y. (2014). Investigation effect of origami-based instruction on elementary students' spatial skills and perceptions. *Journal of Education Research, 107*(1), 59-68.

* Casey, M., B., Andrew, N., Schindler., Kersh, J. E., Samper, A., & Copley, J. (2008). The development of spatial skills through interventions involving block building activities. *Cognition and Instruction, 26*(3), 269-309.

* Casey, M. B., Nuttall, R. L., & Pezaris, E. (1997). Mediators of gender differences in mathematics college entrance test score across diverse samples. *Developmental Psychology, 33*, 669-680.

* Casey, M. B., Nuttall, R., Pezaris, E., & Benbow, C. P. (1995). The influence of spatial ability on gender differences in math college entrance test scores diverse samples. *Developmental Psychology, 31*, 697-705.
in geometric development, progressing from the “2-D Shape Composure” stage to the “Substitution Composure” stage. This advancement involved using composite units of shapes to construct 3-D solids, as exemplified by the creation of a star-shaped cylinder while referencing the documentation. The child ingeniously used five equilateral triangles to form a pentagon, showcasing a more sophisticated level of geometric composition.

## DISCUSSION

The objective of this study was to explore how teachers and classes can enhance the motivation for individual geometric play and foster unique geometric abilities through guided play using documentation. Preliminary discussions using documentation before the two guided play sessions facilitated the children's contemplation of the challenge of creating more intricate 3-D solids.

An analysis of the changes in the 3-D works created by the children, as depicted in Table 1, reveals a noteworthy increase in the number of children engaging in 3-D creations at more advanced developmental stages in each successive round. Furthermore, guided play allows us to offer personalized guidance based on individual geometric abilities. For instance, a girl who had previously created a 2-D flat plane twice in a row expressed uncertainty about what to make the third time, but when prompted by the classroom teacher, she chose to make a top. While connecting the triangles with the teacher's assistance, the girl gained confidence and successfully connected the last five triangles, creating a top independently. This illustrates how environmental settings, including playing with geometric toys and engaging in discussions using documentation, have the potential to motivate children and propel them through various developmental stages of geometry.

In future research, we will examine the broader impact of guided play on children’s everyday play and the ongoing development of their geometric skills.

## References

* Baenninger, M., & Newcombe, N. (1989). The role of experience in spatial test performance: A meta-analysis. *Sex Roles, 20*(5-6), 327-344.

* Bakker, A. & Van Eerde, D. (2014). An introduction to design-based research education with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), *Approaches to qualitative research in mathematics education* (pp. 429-466). Springer.

* Brant, B. (2013). Everyday pedagogical practices in mathematical play situation in German “Kindergarten”. *Educational Studies in Mathematics, 84*(2), 227-248.

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

* Cakmak, S., Isiksal, M., & Koc, Y. (2014). Investigation effect of origami-based instruction on elementary students' spatial skills and perceptions. *Journal of Education Research, 107*(1), 59-68.

* Casey, M., B., Andrew, N., Schindler, Kersh, J. E., Samper, A., & Copley, J. (2008). The development of spatial skills through interventions involving block building activities. *Cognition and Instruction, 26*(3), 269-309.

* Casey, M. B., Nuttall, R. L., & Pezaris, E. (1997). Mediators of gender differences in mathematics college entrance test score across diverse samples. *Developmental Psychology, 33*, 669-680.

* Casey, M. B., Nuttall, R., Pezaris, E., & Benbow, C. P. (1995). The influence of spatial ability on gender differences in math college entrance test scores diverse samples. *Developmental Psychology, 31*, 697-705.
Chien, N. C., Howes, C., Burchinal, M., Pianta, R. C., Ritchie, S., Bryant, D.M., Clifford, R. M., Early, D. M., & Barbarin, O. A. (2010). Children's classroom engagement and school readiness gains in prekindergarten. *Child Development, 81*(5), 1534-1549.

Clements, D. H., Wilson, C. D., & Sarama, J. (2004). Young children's composition of geometric figures: A learning trajectory. *Mathematical Thinking and Learning, 6*(2), pp. 163-184.

Geary, D. C., Saults, S. J., Liu, F., & Hoard, M. K. (2000). Sex differences in spatial cognition, computational fluency, and arithmetical reasoning. *Journal of Experimental Child Psychology, 77*, 337-353.

Ministry of Education, Culture, Sports, Science and Technology-Japan. (2018). *Explanation of Kindergarten Curriculum Standards*. Froebel-Kan Press.

Mix, K. S., & Cheng, Y. L. (2012). The relation between space and math: Developmental and education implications. In J.B. Benson (Ed.), *Advances in child development and behavior, 42*, 197-243.

Okamoto, Y., Weckbacher, L. M., Hallowell, D., Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (2014). How is spatial reasoning related to mathematical thinking and important is early exposure to spatial activities? In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.), *Proceedings of the Joint Meeting of PME 38 and PME-NA* (Vol. 1, pp. 177-179). PME.

Rogoff, B. (1994). Developing understanding of idea of communities of learners. *Mind, Culture, and Activity, 1*(4), 209-229.

Sakakibara, T. (2004). The role of Japanese preschool teacher's support in young children's mathematical development. *The Japanese Journal of Developmental Psychology, 17*(1), pp.50-61.

Sinclair, N., & Bruce, C. (2014). Spatial reasoning for young learners. In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.), *Proceedings of the Joint Meeting of PME 38 and PME-NA* (Vol. 1, pp. 173-203). Vancouver, Canada.

Thom, J. S. & McGarvey, L. M. (2014). Children's drawings: A bodying-fourth of spatial reasoning. In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.), *Proceedings of the Joint Meeting of PME 38 and PME-NA* (Vol. 1, pp. 191-192). Vancouver, Canada.

Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C. H., & Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. *Psychological Bulletin, 139*(2), pp. 352-402.

Vogt, F., Hauser, B., Stebler, R., Rechsteiner, K., & Urech, C. (2018). Learning though play-pedagogy and learning outcomes in early childhood mathematics. *European Early Childhood Education Research Journal, 26*(4), pp. 589-603.

Weisberg, D. S., Pasek, K. H., & Golinkoff, R. M. (2013). Guided play: Where curricular goals meet a playful pedagogy. *Mind and Education, 7*(2), pp.104-112.
Chien, N. C., Howes, C., Burchinal, M., Pianta, R. C., Ritchie, S., Bryant, D.M., Clifford, R. M., Early, D. M., & Barbarin, O. A. (2010). Children's classroom engagement and school readiness gains in prekindergarten. *Child Development, 81*(5), 1534-1549.

Clements, D. H., Wilson, C. D., & Sarama, J. (2004). Young children's composition of geometric figures: A learning trajectory. *Mathematical Thinking and Learning, 6*(2), pp. 163-184.

Geary, D. C., Saults, S. J., Liu, F., & Hoard, M. K. (2000). Sex differences in spatial cognition, computational fluency, and arithmetical reasoning. *Journal of Experimental Child Psychology, 77*, 337-353.

Ministry of Education, Culture, Sports, Science and Technology-Japan. (2018). *Explanation of Kindergarten Curriculum Standards*. Froebel-Kan Press.

Mix, K. S., & Cheng, Y. L. (2012). The relation between space and math: Developmental and education implications. In J.B. Benson (Ed.), *Advances in child development and behavior, 42*, 197-243.

Okamoto, Y., Weckbacher, L. M., Hallowell, D., Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (2014). How is spatial reasoning related to mathematical thinking and important is early exposure to spatial activities? In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.), *Proceedings of the Joint Meeting of PME 38 and PME-NA* (Vol. 1, pp. 177-179). PME.

Rogoff, B. (1994). Developing understanding of idea of communities of learners. *Mind, Culture, and Activity, 1*(4), 209-229.

Sakakibara, T. (2004). The role of Japanese preschool teacher's support in young children's mathematical development. *The Japanese Journal of Developmental Psychology, 17*(1), pp.50-61.

Sinclair, N., & Bruce, C. (2014). Spatial reasoning for young learners. In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.), *Proceedings of the Joint Meeting of PME 38 and PME-NA* (Vol. 1, pp. 173-203). Vancouver, Canada.

Thom, J. S. & McGarvey, L. M. (2014). Children's drawings: A bodying-fourth of spatial reasoning. In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.), *Proceedings of the Joint Meeting of PME 38 and PME-NA* (Vol. 1, pp. 191-192). Vancouver, Canada.

Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C. H., & Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. *Psychological Bulletin, 139*(2), pp. 352-402.

Vogt, F., Hauser, B., Stebler, R., Rechsteiner, K., & Urech, C. (2018). Learning though play-pedagogy and learning outcomes in early childhood mathematics. *European Early Childhood Education Research Journal, 26*(4), pp. 589-603.

Weisberg, D. S., Pasek, K. H., & Golinkoff, R. M. (2013). Guided play: Where curricular goals meet a playful pedagogy. *Mind and Education, 7*(2), pp.104-112.
TOPIC B

Curricular and methodological
approaches
TOPIC B

Curricular and methodological
approaches
<figure><img src="image_44.png" /></figure>
<figure><img src="image_50.png" /></figure>
# CLASSROOM SIMULATORS: A NEW TRAINING APPROACH TO INVESTIGATE TEACHERS' PROFESSIONAL KNOWLEDGE AND SUPPORT ITS DEVELOPMENT

Maha Abboud¹, Fabien Emprin²

¹ Cergy Paris Université - LDAR, ² URCA – CEREP

This paper offers an empirical study for supporting teachers' educators. It presents a new form of professional learning based on the use of a classroom simulator in training courses. Among the various research questions that this training approach makes it possible to tackle, we choose to focus in this paper on the way in which teachers' beliefs and geometry knowledge can be revealed by means of the simulation.

## INTRODUCTION

This paper offers an empirical study for supporting teachers' educators. It presents a new form of professional learning based on the use of a classroom simulator in training courses. The design of the simulator is guided both by the outcomes of research into the characteristics and needs of geometric learning at lower secondary school level, and by the effective difficulties experienced by these students in solving geometric tasks. The simulated class situation consists of solving a problem within a Dynamic Geometry Environment (DGE) starting by drawing and manipulation, to pass afterwards to conjecturing and analytic visualization and then to proving.

Among the various research questions that this training approach makes it possible to tackle, we choose to focus in this paper on the way in which teachers' beliefs and geometry knowledge can be revealed by means of the simulation. Indeed, Thomas and Palmer (2014) highlight that teachers not only need a special kind of knowledge for technology implementation, but that beliefs play a crucial role since they frame, guide and filter situations, actions and intentions. Training courses that directly address teachers' competencies are therefore regarded as important in supporting the development of knowledge, beliefs and practices (Hegedus et al. 2017). The specific research question we address is: what do trainee teachers' simulated practices reveal about their professional knowledge and beliefs about geometry teaching and particularly about the role of DGEs in solving geometric problems?

The rationale behind our questioning is informed by a range of factors and contextual elements. First, in French curriculum, students are expected to know how to draw figures before engaging in geometrical reasoning. Constructing with instruments is compulsory and is expected to help students passing from tangible to abstract and understanding geometric concepts. DGEs could be the appropriate place to make this passage. Constructing tasks play thus a key role in curricula, as they help students to move from spatial to geometrical thinking from “seeing on a drawing” to proving (Mithalal & Balacheff, 2019). Second, it is nowadays widely shared that DGEs could contribute in developing geometrical learning, particularly in supporting solving geometrical problems. It is acknowledged that they have the potential to “encourage both exploration and proof, because it makes it so easy to pose and test conjectures” (Hanna, 2000, p. 13). Nevertheless, researchers, such as
# CLASSROOM SIMULATORS: A NEW TRAINING APPROACH TO INVESTIGATE TEACHERS' PROFESSIONAL KNOWLEDGE AND SUPPORT ITS DEVELOPMENT

Maha Abboud¹, Fabien Emprin²

¹ Cergy Paris Université - LDAR, ² URCA – CEREP

This paper offers an empirical study for supporting teachers' educators. It presents a new form of professional learning based on the use of a classroom simulator in training courses. Among the various research questions that this training approach makes it possible to tackle, we choose to focus in this paper on the way in which teachers' beliefs and geometry knowledge can be revealed by means of the simulation.

## INTRODUCTION

This paper offers an empirical study for supporting teachers' educators. It presents a new form of professional learning based on the use of a classroom simulator in training courses. The design of the simulator is guided both by the outcomes of research into the characteristics and needs of geometric learning at lower secondary school level, and by the effective difficulties experienced by these students in solving geometric tasks. The simulated class situation consists of solving a problem within a Dynamic Geometry Environment (DGE) starting by drawing and manipulation, to pass afterwards to conjecturing and analytic visualization and then to proving.

Among the various research questions that this training approach makes it possible to tackle, we choose to focus in this paper on the way in which teachers' beliefs and geometry knowledge can be revealed by means of the simulation. Indeed, Thomas and Palmer (2014) highlight that teachers not only need a special kind of knowledge for technology implementation, but that beliefs play a crucial role since they frame, guide and filter situations, actions and intentions. Training courses that directly address teachers' competencies are therefore regarded as important in supporting the development of knowledge, beliefs and practices (Hegedus et al. 2017). The specific research question we address is: what do trainee teachers' simulated practices reveal about their professional knowledge and beliefs about geometry teaching and particularly about the role of DGEs in solving geometric problems?

The rationale behind our questioning is informed by a range of factors and contextual elements. First, in French curriculum, students are expected to know how to draw figures before engaging in geometrical reasoning. Constructing with instruments is compulsory and is expected to help students passing from tangible to abstract and understanding geometric concepts. DGEs could be the appropriate place to make this passage. Constructing tasks play thus a key role in curricula, as they help students to move from spatial to geometrical thinking from “seeing on a drawing” to proving (Mithalal & Balacheff, 2019). Second, it is nowadays widely shared that DGEs could contribute in developing geometrical learning, particularly in supporting solving geometrical problems. It is acknowledged that they have the potential to “encourage both exploration and proof, because it makes it so easy to pose and test conjectures” (Hanna, 2000, p. 13). Nevertheless, researchers, such as
Mariotti (2006) underline that even if dragging within DGEs provides the students with strong perceptual evidence that a certain property is true, this may reinforce a critical point in the teaching of geometry concerning the relationship between empirical evidence and theoretical reasons. Finally, one of the challenges that research has attempted to meet is to design and/or analyze situations where the connection between spatial and geometric reasoning is effective. Fischbein (1993) has emphasized that one of the main tasks of mathematics education (in the domain of geometry) is “to create types of didactical situations which would systematically ask for a strict cooperation between the two aspects [image and concepts], up to their fusion in unitary mental objects” (p. 161). However, many researchers highlighted that although the situations designed and analyzed were very promising, they have been little used in teaching. The problem of the usability of these situations and “their communication to teachers has not been yet solved” (Bloch & Pressiat, 2009, p 66 [personal translation]).

To address these issues in the context of teacher training, we designed and used a computerized classroom simulator (SIC). The advantage of using such simulator is that it enables teachers to experiment classroom practices in an environment that is both safe - all types of experiments are allowed (no impact on real students) - and allows experiments to be repeated. Indeed, simulation is often used in professional training because it is less risky and speeds up the process of acquiring experience (Pastré, 2005). Moreover, the designed training course includes an analysis of the geometric problem, as well as a collective discussion of the concepts involved and the role of DGEs.

## LEARNING GEOMETRY IN THE TRANSITION FROM PRIMARY TO SECONDARY SCHOOL

This section is far from being exhaustive, its aim is rather to highlight some of the research findings on which our work is based, firstly with regard to the learning of geometry in general, and secondly in the specific context of DGEs. A significant amount of research has been devoted to studying students’ learning in the transition from the tangible geometry of elementary school to the reasoning geometry of early secondary school; characteristics and difficulties have been identified.

Within French research, two main conceptual perspectives inform the field of geometry didactics. Houdement and Kuzniak (1998) distinguish three paradigms of geometry: natural geometry (GI), natural axiomatic geometry (GII) and formalist axiomatic geometry (GIII). The passage from GI to GII is identified as difficult to implement in the classroom and even if students entering secondary school have certain knowledge within GI, they don’t necessarily link geometric constructions to the properties and theorems that justify them; their geometry tends to amalgamate GI and GII. Berthelot & Salin (1998) differentiate between spatial and geometric knowledge and Parzysz (1991) shows that spatial objects and physical representations play an important role in the conceptualization of geometrical concepts. Moreover, Sinclair et al. (2019) underline the strong link between drawing, spatial reasoning and the learning of geometry. This leads us to reflect on the place of construction in geometric work. It is not the precise technical mastery of drawing processes with instruments, but rather the mental objects constructed by the student to accomplish this process, and what ensues for a better apprehension of the figure (Duval, 2006) that is at stake in the school-level transition that interests us here.
Mariotti (2006) underline that even if dragging within DGEs provides the students with strong
perceptual evidence that a certain property is true, this may reinforce a critical point in the teaching
of geometry concerning the relationship between empirical evidence and theoretical reasons. Finally,
one of the challenges that research has attempted to meet is to design and/or analyze situations where
the connection between spatial and geometric reasoning is effective. Fischbein (1993) has
emphasized that one of the main tasks of mathematics education (in the domain of geometry) is “to
create types of didactical situations which would systematically ask for a strict cooperation between
the two aspects [image and concepts], up to their fusion in unitary mental objects” (p. 161). However,
many researchers highlighted that although the situations designed and analyzed were very
promising, they have been little used in teaching. The problem of the usability of these situations and
“their communication to teachers has not been yet solved” (Bloch & Pressiat, 2009, p 66 [personal
translation]).

To address these issues in the context of teacher training, we designed and used a computerized
classroom simulator (SIC). The advantage of using such simulator is that it enables teachers to
experiment classroom practices in an environment that is both safe - all types of experiments are
allowed (no impact on real students) - and allows experiments to be repeated. Indeed, simulation is
often used in professional training because it is less risky and speeds up the process of acquiring
experience (Pastré, 2005). Moreover, the designed training course includes an analysis of the
geometric problem, as well as a collective discussion of the concepts involved and the role of DGEs.

LEARNING GEOMETRY IN THE TRANSITION FROM PRIMARY TO SECONDARY SCHOOL

This section is far from being exhaustive, its aim is rather to highlight some of the research findings
on which our work is based, firstly with regard to the learning of geometry in general, and secondly
in the specific context of DGEs. A significant amount of research has been devoted to studying
students' learning in the transition from the tangible geometry of elementary school to the reasoning
geometry of early secondary school; characteristics and difficulties have been identified.

Within French research, two main conceptual perspectives inform the field of geometry didactics. Houdement and Kuzniak (1998) distinguish three paradigms of geometry: natural geometry (GI), natural axiomatic geometry (GII) and formalist axiomatic geometry (GIII). The passage from GI to GII is identified as difficult to implement in the classroom and even if students entering secondary school have certain knowledge within GI, they don't necessarily link geometric constructions to the properties and theorems that justify them; their geometry tends to amalgamate GI and GII. Berthelot & Salin (1998) differentiate between spatial and geometric knowledge and Parzysz (1991) shows that spatial objects and physical representations play an important role in the conceptualization of geometrical concepts. Moreover, Sinclair et al. (2019) underline the strong link between drawing, spatial reasoning and the learning of geometry. This leads us to reflect on the place of construction in geometric work. It is not the precise technical mastery of drawing processes with instruments, but rather the mental objects constructed by the student to accomplish this process, and what ensues for a better apprehension of the figure (Duval, 2006) that is at stake in the school-level transition that interests us here.
The use of DGEs in geometry learning and teaching is nowadays recognized as offering novel ways of carrying out geometrical activities in mathematics education and participating to the progress in students' conceptualization, influencing thereby different aspects of geometry learning. Within an exploratory approach, research studies stress the key role of dragging in forming a mathematical conjecture (Healy, 2000). By making it possible to drag and drop points and to multiply experiments, DGEs encourage access to conjecture and possibly reasoning to validate this conjecture. Furthermore, Hoyles and Jones (1998) claim that dynamic geometry, supported by "what if" and "what if not" questions, has the potential to promote links between empirical and deductive reasoning. Yet, teaching approaches involving a cycle of exploration-conjecturing-proving require thoughtful design; "engaging students in situations which make them aware of the constructive character of mathematical activities, especially those involving conjecture and proof, possess complex challenges" (Durand-Guerrier et al., 2012, p. 364). Even if dragging is a powerful tool for conjecturing, the resistance of the objects drawn brings a conviction that could slow down students' understanding of the need to prove by arguments. Similarly, while the above comments show the importance of manipulating geometric figures in a DGE, constructing these figures using the software's tools is a separate task from the conjecturing/proving process. Thus, the importance of designing appropriate tasks and the role the teacher should play in their implementation seem even more important in order to meet some learning aims.

# SIMULATOR AND TEACHER TRAINING COURSE DESIGN

## Tasks implemented in SIC: transition between spatial and geometric reasoning within DGEs

Designing a Classroom Simulator means first and foremost to find a teaching situation that is conducive to raising the professional and mathematical questions that will become the focus of training. To encourage teachers to question the role of geometric construction within DGEs in the proof process, we used the problem described in Figure 1. This task considers the epistemological aspects identified by Lesnes-Cuisiniez (2021) in his synthesis of research on the double break between physical and theoretical geometry. These include the need to distinguish figure drawing, instrumental mobilization, heuristic arguments, and theoretical validation. We also added a promising idea: using a problem in which perception is challenged to highlight the need to rely on reasoning rather than measurement or perception. For SIC, we have chosen a problem where perception is distorted (segment [AC] seems larger than segment [EG] in Figure 1), which is reinforced by the question posed which is not: "are the lengths the same?" but "Say which of the two segments is larger."

Make the figure shown in the opposite. Say which of the two segments, [EG] or [AC], is larger. Explain why.

<figure><img src="image_45.png" /><figcaption>Figure 1: Problem implemented in SIC.</figcaption></figure>

This problem is available in several resources for lower secondary school teachers in paper-and-pencil environment; we have adapted it to be solved in a DGE. It is considered both in these resources and by the teachers who have tested it as an open problem that is simple to understand and enables
The use of DGEs in geometry learning and teaching is nowadays recognized as offering novel ways of carrying out geometrical activities in mathematics education and participating to the progress in students' conceptualization, influencing thereby different aspects of geometry learning. Within an exploratory approach, research studies stress the key role of dragging in forming a mathematical conjecture (Healy, 2000). By making it possible to drag and drop points and to multiply experiments, DGEs encourage access to conjecture and possibly reasoning to validate this conjecture. Furthermore, Hoyles and Jones (1998) claim that dynamic geometry, supported by "what if" and "what if not" questions, has the potential to promote links between empirical and deductive reasoning. Yet, teaching approaches involving a cycle of exploration-conjecturing-proving require thoughtful design; "engaging students in situations which make them aware of the constructive character of mathematical activities, especially those involving conjecture and proof, possess complex challenges" (Durand-Guerrier et al., 2012, p. 364). Even if dragging is a powerful tool for conjecturing, the resistance of the objects drawn brings a conviction that could slow down students' understanding of the need to prove by arguments. Similarly, while the above comments show the importance of manipulating geometric figures in a DGE, constructing these figures using the software's tools is a separate task from the conjecturing/proving process. Thus, the importance of designing appropriate tasks and the role the teacher should play in their implementation seem even more important in order to meet some learning aims.

## SIMULATOR AND TEACHER TRAINING COURSE DESIGN

### Tasks implemented in SIC: transition between spatial and geometric reasoning within DGEs

Designing a Classroom Simulator means first and foremost to find a teaching situation that is conducive to raising the professional and mathematical questions that will become the focus of training. To encourage teachers to question the role of geometric construction within DGEs in the proof process, we used the problem described in Figure 1. This task considers the epistemological aspects identified by Lesnes-Cuisiniez (2021) in his synthesis of research on the double break between physical and theoretical geometry. These include the need to distinguish figure drawing, instrumental mobilization, heuristic arguments, and theoretical validation. We also added a promising idea: using a problem in which perception is challenged to highlight the need to rely on reasoning rather than measurement or perception. For SIC, we have chosen a problem where perception is distorted (segment [AC] seems larger than segment [EG] in Figure 1), which is reinforced by the question posed which is not: "are the lengths the same?" but "Say which of the two segments is larger."

Make the figure shown in the opposite. Say which of the two segments, [EG] or [AC], is larger. Explain why.

<figure><img src="image_51.png" /><figcaption>Figure 1: Problem implemented in SIC.</figcaption></figure>

This problem is available in several resources for lower secondary school teachers in paper-and-pencil environment; we have adapted it to be solved in a DGE. It is considered both in these resources and by the teachers who have tested it as an open problem that is simple to understand and enables
the learner to engage in an experimental approach. To begin, the student is asked to draw the figure
in a DGE. This requires specific knowledge that is not the same as that required to draw the figure in
paper-and-pencil, notably the order of construction steps, the commands available in the software
(the software commands that the teacher can choose to authorize or not such as "perpendicular
bisector") and the robustness of drawing under dragging (Healy, 2000). Afterwards, the student is
asked to conjecture the answer to the question: "Say which of the two segments is larger." Finally,
the student is asked to explain why, which refers to the process of argumentation and/or proof. From
a didactic point of view, there are several analyzable difficulties. The one we are aiming at here is
that there is little connection between the construction work in the DGE and the conjecture work. In
fact, the conjecture can be made by using the "measurement" command or by moving the points
connected to the circle to superimpose them; the construction command adds nothing to the
conjecture. The proof phase requires to have noticed that the two segments are the diagonals of two
rectangles and that the second one is in fact the radius of the circle. The construction does not allow
identifying the rectangles (since it rather directs towards the idea of an orthogonal projection). In
conclusion, being able to manipulate the figure in the DGE is important and sufficient to conjecture
and prove; constructing it step by step doesn't provide necessarily additional means to complete these
two tasks.

The Design of SIC

Our goals in building SIC (available at http://www.fabien-emprin.ovh) are to get trainees to reflect on their practices, using their professional knowledge to solve a professional problem and to test several hypotheses by having the possibility to try again and again. To achieve these goals, we used part-scale simulators, which enable the user to make choices (what the teacher says, to whom he/she says it: one student or the whole classroom, what he/she does: act directly on a student's screen or projecting the same screen on the white board, etc.) and to see the effect of his/her choices on what students do (actions on the DGE, verbal responses, etc.). According to Patré (2005), it is a way of "impoverishing the situation to make it more accessible to learning" (p. 27). We use a non-random model in which an action always has the same effect each time it is repeated, with no introduction of random phenomena, enabling to carry out analyses between trials by the same user and between users. In this sense, it's an experience-building accelerator.

To design a simulation, combined with a tool for computing these interactions, we needed to define
the possible choices of the teacher and the effects of these choices for students in real classrooms.
Our method was based on recording enough classroom settings. To do this, we set up a learning
situation and a support-resource accessible to the teacher to implement in his/her classroom (which
could have been found in an educational resource (textbook, website)). We carried out an a priori
analysis of the situation, based on didactic knowledge of geometry, highlighting possible choices in
terms of tasks and class management. We then provided the situation to teachers, who were free to
interpret it according to how they plan to carry it out in their classrooms. By observing these
classrooms' settings, we gathered information that enabled us to compare the a priori analysis with
actual implementations, and to identify the training knowledge that can be updated in the simulation.
We observed around ten classroom settings, then built a simulation reproducing the teachers' choices
and the students' reactions, while respecting the average proportion observed.
the learner to engage in an experimental approach. To begin, the student is asked to draw the figure
in a DGE. This requires specific knowledge that is not the same as that required to draw the figure in
paper-and-pencil, notably the order of construction steps, the commands available in the software
(the software commands that the teacher can choose to authorize or not such as “perpendicular
bisector”) and the robustness of drawing under dragging (Healy, 2000). Afterwards, the student is
asked to conjecture the answer to the question: “Say which of the two segments is larger.” Finally,
the student is asked to explain why, which refers to the process of argumentation and/or proof. From
a didactic point of view, there are several analyzable difficulties. The one we are aiming at here is
that there is little connection between the construction work in the DGE and the conjecture work. In
fact, the conjecture can be made by using the “measurement” command or by moving the points
connected to the circle to superimpose them; the construction command adds nothing to the
conjecture. The proof phase requires to have noticed that the two segments are the diagonals of two
rectangles and that the second one is in fact the radius of the circle. The construction does not allow
identifying the rectangles (since it rather directs towards the idea of an orthogonal projection). In
conclusion, being able to manipulate the figure in the DGE is important and sufficient to conjecture
and prove; constructing it step by step doesn’t provide necessarily additional means to complete these
two tasks.

The Design of SIC

Our goals in building SIC (available at http://www.fabien-emprin.ovh) are to get trainees to reflect
on their practices, using their professional knowledge to solve a professional problem and to test
several hypotheses by having the possibility to try again and again. To achieve these goals, we used
part-scale simulators, which enable the user to make choices (what the teacher says, to whom he/she
says it: one student or the whole classroom, what he/she does: act directly on a student's screen or
projecting the same screen on the white board, etc.) and to see the effect of his/her choices on what
students do (actions on the DGE, verbal responses, etc.). According to Patré (2005), it is a way of
“impoverishing the situation to make it more accessible to learning” (p. 27). We use a non-random
model in which an action always has the same effect each time it is repeated, with no introduction of
random phenomena, enabling to carry out analyses between trials by the same user and between users.
In this sense, it's an experience-building accelerator.

To design a simulation, combined with a tool for computing these interactions, we needed to define
the possible choices of the teacher and the effects of these choices for students in real classrooms.
Our method was based on recording enough classroom settings. To do this, we set up a learning
situation and a support-resource accessible to the teacher to implement in his/her classroom (which
could have been found in an educational resource (textbook, website)). We carried out an a priori
analysis of the situation, based on didactic knowledge of geometry, highlighting possible choices in
terms of tasks and class management. We then provided the situation to teachers, who were free to
interpret it according to how they plan to carry it out in their classrooms. By observing these
classrooms' settings, we gathered information that enabled us to compare the a priori analysis with
actual implementations, and to identify the training knowledge that can be updated in the simulation.
We observed around ten classroom settings, then built a simulation reproducing the teachers' choices
and the students' reactions, while respecting the average proportion observed.
The feedback provided by the simulator is twofold: during the simulated session, the user can access the student's work, either by observing it (on the DGE screen, for example) or by questioning it; at the end of the session, the user can access what each student found (his/her construction, his/her answer to the question) and what the student remembers one week after the session. The latter could help the user measure the effect on students' learning.

## A TRAINING COURSE USING SIC

First, we emphasize that we do not consider that SIC has any intrinsic value; playing the simulation doesn't inherently develop professional skills. It's the debriefing that follows the simulation that allows questions and analyses to emerge and brings out the trainees' professional knowledge and thus possibly enables effective training to take place (Pastré, 2005). Secondly, we consider that the trainee teacher's professional knowledge develops by going back and forth between the deployment of existing knowledge and the information derived from the interactions first with SIC, and afterwards with other trainees (with and without the trainer's interventions). The training course centered on the simulator, as described below, aims both to immerse teachers in a (simulated) professional context and to provide reflective feedback on their choices and actions with the simulator. An alternation of moments of experimentation and discussions is therefore supposed to develop the teachers' professional knowledge.

The course, designed for in-service secondary school mathematics teachers, is organized in five phases following the more general training approach proposed by Abboud et al. (2022). This process is presented to trainees at the beginning of the course. The aim of the first phase is to make trainees analyze the mathematical task possibly by carrying it out themselves and anticipating possible students' difficulties. The second phase takes the form of a group discussion on what the trainees were able to anticipate throughout the task analysis: what problems and difficulties the students are likely to encounter, what objective should be assigned to the teaching/learning situation... During the third phase, the trainees use SIC. Thanks to the fact that the time is simulated (independent of real time), they can make several trials, but without having to complete the simulation each time. At the end of each simulation, they obtain summary information on the students' learning, which enables them to adapt their choices and test new hypotheses during the next trial. The fourth phase is again a collective discussion on the performance of the virtual students and the comparison with the real students (their own) and ways of helping them to learn geometry when accomplishing the task. Finally, during the fifth phase, the trainees are asked to design a class session based on the geometric task they had tried out and its implications in terms of promoting students' geometric reasoning.

It is during phases 2 and 4 that the trainer can pick up on elements of the teachers' practices and can intervene by supplying/sharing knowledge for professional development. It is these two phases that interest us in this article. In what follows we propose to analyze some exchanges from phase 2 and others from phase 4. In addition, as SIC makes it possible to keep track, in the form of a chronological table, of the actions undertaken by the trainee teacher during the various trails made during phase 3, we provide examples of these traces in order to gain an overall view of how the training session unfolded.
The feedback provided by the simulator is twofold: during the simulated session, the user can access the student's work, either by observing it (on the DGE screen, for example) or by questioning it; at the end of the session, the user can access what each student found (his/her construction, his/her answer to the question) and what the student remembers one week after the session. The latter could help the user measure the effect on students' learning.

## A TRAINING COURSE USING SIC

First, we emphasize that we do not consider that SIC has any intrinsic value; playing the simulation doesn't inherently develop professional skills. It's the debriefing that follows the simulation that allows questions and analyses to emerge and brings out the trainees' professional knowledge and thus possibly enables effective training to take place (Pastré, 2005). Secondly, we consider that the trainee teacher's professional knowledge develops by going back and forth between the deployment of existing knowledge and the information derived from the interactions first with SIC, and afterwards with other trainees (with and without the trainer's interventions). The training course centered on the simulator, as described below, aims both to immerse teachers in a (simulated) professional context and to provide reflective feedback on their choices and actions with the simulator. An alternation of moments of experimentation and discussions is therefore supposed to develop the teachers' professional knowledge.

The course, designed for in-service secondary school mathematics teachers, is organized in five phases following the more general training approach proposed by Abboud et al. (2022). This process is presented to trainees at the beginning of the course. The aim of the first phase is to make trainees analyze the mathematical task possibly by carrying it out themselves and anticipating possible students' difficulties. The second phase takes the form of a group discussion on what the trainees were able to anticipate throughout the task analysis: what problems and difficulties the students are likely to encounter, what objective should be assigned to the teaching/learning situation... During the third phase, the trainees use SIC. Thanks to the fact that the time is simulated (independent of real time), they can make several trials, but without having to complete the simulation each time. At the end of each simulation, they obtain summary information on the students' learning, which enables them to adapt their choices and test new hypotheses during the next trial. The fourth phase is again a collective discussion on the performance of the virtual students and the comparison with the real students (their own) and ways of helping them to learn geometry when accomplishing the task. Finally, during the fifth phase, the trainees are asked to design a class session based on the geometric task they had tried out and its implications in terms of promoting students' geometric reasoning.

It is during phases 2 and 4 that the trainer can pick up on elements of the teachers' practices and can intervene by supplying/sharing knowledge for professional development. It is these two phases that interest us in this article. In what follows we propose to analyze some exchanges from phase 2 and others from phase 4. In addition, as SIC makes it possible to keep track, in the form of a chronological table, of the actions undertaken by the trainee teacher during the various trials made during phase 3, we provide examples of these traces in order to gain an overall view of how the training session unfolded.
## Analysis of interactions from Phase 2

After clarifying, at the request of the trainees, that the virtual students had already experienced the DGE, the discussion turns to the difference between construction within a DGE and within paper-and-pencil environment; the trainer provides then several clarifications on this point. Given the risk of multiple difficulties, one of the trainees suggests that the drawing should be carried out on paper-and-pencil instead. This first exchange informs us that the teachers are primarily concerned with the students' ability to construct the figure correctly so that they can then engage in conjecture. They have the feeling that using the DGE could generate several difficulties. This leads them to favor to abandon its use for the construction phase. Another exchange takes place with the trainer on the construction process, which is different in paper-and-pencil, particularly about constraints on the order of construction. We can notice here that, although the task was chosen for its potential in the DG environment in terms of two aspects - construction and manipulation - the teachers seemed doubtful about the usefulness of the first and preferred to start with paper-and-pencil to ensure less difficulty in the construction phase and favorable conditions to engage in the manipulation one. In the course of the ensuing discussion, we observe an exchange relating to the conjecture and the usefulness of using a DGE to “see it” in the sense highlighted by Mariotti (2006) where dragging provides the students with strong perceptual evidence that a certain property is true. The trainees emphasize the dynamic nature of the software and the fact that it allows them to visualize an infinite number of figures (thus permitting the generalizing the observed property).

The whole discussion in phase 2 thus seems to show teachers' beliefs that distinguish between the role of the DGE in the drawing process and its role in the conjecturing own. In the first case, it seems more efficient to go back to the usual paper-and-pencil tools to make sure that the students are constructing the figure correctly (by using skills they have already acquired) before engaging in a more complex (for this class level) conjecturing task, where the dynamic aspect of DGEs seems to be unequaled in paper-and-pencil and is therefore very useful.

## Analysis of interactions from Phase 4

Following the same thread, we observe an exchange in phase 4 initiated by a trainee who suggests giving the students the “right” figure from the start, so that they can move on to conjecture. The trainer then informs him that there were three variations of this proposal within SIC to help students make the right figure, without giving them everything straight away. This gives rise to a discussion about how to help the students make the conjecture. In the ensuing discussion, we observe the trainer attempting to provoke a reflection on the fact that giving the students the correct figure without them having constructed it themselves doesn't guarantee that they can engage in the conjecturing process. This attempt fails to achieve its objective, and the trainees prefer to direct the discussion towards ways of finding and/or validating the conjecture. They suggest providing “instrumental” help to students: make the software display the measurements so that equality can be seen. Another possibility of instrumental help was raised in the discussion: asking students to superimpose the points but recognizing it as a special case that doesn't lead to generalizing the conjecture.

Using the simulator seems thus to make it possible to question the trainees' professional knowledge and make them aware of the link that can exist between the construction phase and the conjecture phase and the role that the DGE's functionalities can play in the latter.
## Analysis of interactions from Phase 2

After clarifying, at the request of the trainees, that the virtual students had already experienced the DGE, the discussion turns to the difference between construction within a DGE and within paper-and-pencil environment; the trainer provides then several clarifications on this point. Given the risk of multiple difficulties, one of the trainees suggests that the drawing should be carried out on paper-and-pencil instead. This first exchange informs us that the teachers are primarily concerned with the students' ability to construct the figure correctly so that they can then engage in conjecture. They have the feeling that using the DGE could generate several difficulties. This leads them to favor to abandon its use for the construction phase. Another exchange takes place with the trainer on the construction process, which is different in paper-and-pencil, particularly about constraints on the order of construction. We can notice here that, although the task was chosen for its potential in the DG environment in terms of two aspects - construction and manipulation - the teachers seemed doubtful about the usefulness of the first and preferred to start with paper-and-pencil to ensure less difficulty in the construction phase and favorable conditions to engage in the manipulation one. In the course of the ensuing discussion, we observe an exchange relating to the conjecture and the usefulness of using a DGE to “see it” in the sense highlighted by Mariotti (2006) where dragging provides the students with strong perceptual evidence that a certain property is true. The trainees emphasize the dynamic nature of the software and the fact that it allows them to visualize an infinite number of figures (thus permitting the generalizing the observed property).

The whole discussion in phase 2 thus seems to show teachers' beliefs that distinguish between the role of the DGE in the drawing process and its role in the conjecturing own. In the first case, it seems more efficient to go back to the usual paper-and-pencil tools to make sure that the students are constructing the figure correctly (by using skills they have already acquired) before engaging in a more complex (for this class level) conjecturing task, where the dynamic aspect of DGEs seems to be unequaled in paper-and-pencil and is therefore very useful.

## Analysis of interactions from Phase 4

Following the same thread, we observe an exchange in phase 4 initiated by a trainee who suggests giving the students the “right” figure from the start, so that they can move on to conjecture. The trainer then informs him that there were three variations of this proposal within SIC to help students make the right figure, without giving them everything straight away. This gives rise to a discussion about how to help the students make the conjecture. In the ensuing discussion, we observe the trainer attempting to provoke a reflection on the fact that giving the students the correct figure without them having constructed it themselves doesn't guarantee that they can engage in the conjecturing process. This attempt fails to achieve its objective, and the trainees prefer to direct the discussion towards ways of finding and/or validating the conjecture. They suggest providing “instrumental” help to students: make the software display the measurements so that equality can be seen. Another possibility of instrumental help was raised in the discussion: asking students to superimpose the points but recognizing it as a special case that doesn't lead to generalizing the conjecture.

Using the simulator seems thus to make it possible to question the trainees' professional knowledge and make them aware of the link that can exist between the construction phase and the conjecture phase and the role that the DGE's functionalities can play in the latter.
## Analysis of traces from Phase 3

The analysis of these traces aims first to identify the moments of transition from the construction phase to the conjecture phase in the various trials and second to examine how trainees use SIC as a professional tool for experimentation. We give here the example of the traces of trials of two trainee teachers. *Greg* carried out 3 trials. On the first one, the transition to the conjecture took place at the 38th minute. Noticing that this does not leave enough time within an average duration of a classroom session (55 min) to complete the conjecture phase, he suggests that the task has to be carried out in two sessions. In the second trial, he thus took the time needed for the construction (43 min with session duration of 109 min). He then questioned the need for students to engage in the construction, and rather chose at the third trial to give the figure that had already been constructed switching to the conjecture phase after 12 minutes. *Chris* carried out 4 trials. The first two trials were used to explore the possibilities of SIC, in particular by going to check on all the virtual students during the construction phase and asking some of them whether the figure resists or constructing the figure for them. On trial 3, he accelerated the construction phase by showing the correct figure to the class (at the 33rd min) before asking for the conjecture and giving two minutes later the indication to display the measurements of the two segments. He didn't move on to the proof phase until the 75th minute and he accelerated this phase by immediately asking students to put OB in the radius position and measure it. On trial 4, he first did much the same thing before moving to a pooling of the conjecture at the 62nd minute and then writing down the results at the 73rd minute to close the session.

In broad terms, the analyses show that the trainees who appropriate the simulator as a tool for testing and experimenting, carry out trials that evolve into shorter construction phase and realize that it is the manipulation of the figure and not the drawing that is useful for the proof.

## CONCLUSION

This exploratory study highlights the potential of simulation-based training approach to reveal teachers' professional knowledge of geometry and particularly about supporting students to link geometrical construction and geometrical reasoning.

The aim of using SIC was also to engage teachers in collective reflection on their classroom practices in geometry, with SIC acting as a revealer of these practices. We showed how it has revealed to the trainer some of the trainees' geometric beliefs and practices. We also looked at the knowledge that the trainer considered relevant to contribute in situ to promote professional learning on key elements to geometry teaching enhanced by dynamic geometry.

While geometric construction work, under certain conditions, is conducive to the transition from GI to GII, the technical skill of drawing, whether on paper-and-pencil environment or in a DGE, is a specific skill. However, simulated practices show that teachers remain attached to this activity, with certain confusion between constructing and conjecturing. Feedback from the simulator may lead them to reconsider this role.

This study shows the potential of simulated practices for both the trainer and the researcher. For the moment, the data extracted from the software can only be used by the researcher because of its complex format, but it is possible to imagine transforming it into learning analytics so that the trainer can also use it directly in his/her training sessions.
## Analysis of traces from Phase 3

The analysis of these traces aims first to identify the moments of transition from the construction phase to the conjecture phase in the various trials and second to examine how trainees use SIC as a professional tool for experimentation. We give here the example of the traces of trials of two trainee teachers. *Greg* carried out 3 trials. On the first one, the transition to the conjecture took place at the 38th minute. Noticing that this does not leave enough time within an average duration of a classroom session (55 min) to complete the conjecture phase, he suggests that the task has to be carried out in two sessions. In the second trial, he thus took the time needed for the construction (43 min with session duration of 109 min). He then questioned the need for students to engage in the construction, and rather chose at the third trial to give the figure that had already been constructed switching to the conjecture phase after 12 minutes. *Chris* carried out 4 trials. The first two trials were used to explore the possibilities of SIC, in particular by going to check on all the virtual students during the construction phase and asking some of them whether the figure resists or constructing the figure for them. On trial 3, he accelerated the construction phase by showing the correct figure to the class (at the 33rd min) before asking for the conjecture and giving two minutes later the indication to display the measurements of the two segments. He didn't move on to the proof phase until the 75th minute and he accelerated this phase by immediately asking students to put OB in the radius position and measure it. On trial 4, he first did much the same thing before moving to a pooling of the conjecture at the 62nd minute and then writing down the results at the 73rd minute to close the session.

In broad terms, the analyses show that the trainees who appropriate the simulator as a tool for testing and experimenting, carry out trials that evolve into shorter construction phase and realize that it is the manipulation of the figure and not the drawing that is useful for the proof.

## CONCLUSION

This exploratory study highlights the potential of simulation-based training approach to reveal teachers' professional knowledge of geometry and particularly about supporting students to link geometrical construction and geometrical reasoning.

The aim of using SIC was also to engage teachers in collective reflection on their classroom practices in geometry, with SIC acting as a revealer of these practices. We showed how it has revealed to the trainer some of the trainees' geometric beliefs and practices. We also looked at the knowledge that the trainer considered relevant to contribute in situ to promote professional learning on key elements to geometry teaching enhanced by dynamic geometry.

While geometric construction work, under certain conditions, is conducive to the transition from GI to GII, the technical skill of drawing, whether on paper-and-pencil environment or in a DGE, is a specific skill. However, simulated practices show that teachers remain attached to this activity, with certain confusion between constructing and conjecturing. Feedback from the simulator may lead them to reconsider this role.

This study shows the potential of simulated practices for both the trainer and the researcher. For the moment, the data extracted from the software can only be used by the researcher because of its complex format, but it is possible to imagine transforming it into learning analytics so that the trainer can also use it directly in his/her training sessions.
References

Abboud, M., Robert, A. & Rogalski. J. (2022). Interroger les pratiques de formation des professeurs de mathématiques: orientations de recherche et perspectives (un agenda). *ADSC*, SI 2, 261-285.

Berthelot, R., & Salin, M.-H. (1994). Common spatial representations and their effects upon teaching and learning of space and geometry, *PME 18 Proceedings*, vol. II, 72-80.

Bloch, I., et Pressiat, A. (2009). L'enseignement de la géométrie, de l'école au début du collège: situations et connaissances. In I. Bloch & F. Conne (Eds.) *Nouvelles perspectives en didactique des mathématiques*. XIVème école d'été de didactique des mathématiques, 65-88.

Durand-Guerrier, V., Boero, P., Douek, N., Epp, S. S., & Tanguay, D. (2012). Examining the role of logic in teaching proof. *Proof and proving in mathematics education: The 19th ICMI study*, 369-389.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. *ESM*, 61 (1-2), 103-131.

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

Hanna, G. (2000). Proof, explanation and exploration: An overview. *Educational Studies in Mathematics*, 44, 5-23.

Healy, L. (2000). Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions. In *PME proceedings* (Vol. 1, pp. 1-103).

Hegedus, S., Laborde, C., Brady, C., Dalton, S., Siller, H-St, Tabach, M., et al. (2017). *Uses of technology in upper secondary mathematics education*. Springer.

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

Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry contexts. In C. Mammana & V. Villani (Eds.), *Perspectives on the teaching of geometry for the 21st century: An ICMI study* (pp. 121–128). Kluwer Academic Publishers.

Lesnes-Cuisiniez, E. (2021). *Modélisation didactique de parcours d’apprentissage dans un EIAH* (Doctoral dissertation). Université Paris Cité.

Mariotti, M. A. (2006). Proof and proving in mathematics education. In *Handbook of research on the psychology of mathematics education* (pp. 173-204). Brill.

Mithalal, J., & Balacheff, N. (2019). The instrumental deconstruction as a link between drawing and geometrical figure. *Educational Studies in Mathematics*, 100 (2), 161-176.

Parzysz, B. (1991). Representation of space and students’ conceptions at high school level. *ESM*, 22(6), 575-593.

Patré, P. (2005). Apprendre par la simulation: de l’analyse du travail aux apprentissages professionnels. *Formation*.

Sinclair, N., Moss, J., Hawes, Z., & Stephenson, C. (2019). Learning Through and from Drawing in Early Years Geometry. In K. S. Mix, M. T. Battista (eds.), *Visualizing Mathematics, Research in Mathematics Education* (pp. 229–252). Springer.

Thomas, M. O., & Palmer, J. M. (2014). Teaching with digital technology: Obstacles and opportunities. *The mathematics teacher in the digital era: An international perspective on technology focused professional development*, 71-89.
References

Abboud, M., Robert, A. & Rogalski. J. (2022). Interroger les pratiques de formation des professeurs de mathématiques: orientations de recherche et perspectives (un agenda). *ADSC*, SI 2, 261-285.

Berthelot, R., & Salin, M.-H. (1994). Common spatial representations and their effects upon teaching and learning of space and geometry, *PME 18 Proceedings*, vol. II, 72-80.

Bloch, I., et Pressiat, A. (2009). L'enseignement de la géométrie, de l'école au début du collège: situations et connaissances. In I. Bloch & F. Conne (Eds.) *Nouvelles perspectives en didactique des mathématiques*. XIVème école d'été de didactique des mathématiques, 65-88.

Durand-Guerrier, V., Boero, P., Douek, N., Epp, S. S., & Tanguay, D. (2012). Examining the role of logic in teaching proof. *Proof and proving in mathematics education: The 19th ICMI study*, 369-389.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. *ESM*, 61 (1-2), 103-131.

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

Hanna, G. (2000). Proof, explanation and exploration: An overview. *Educational Studies in Mathematics*, 44, 5-23.

Healy, L. (2000). Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions. In *PME proceedings* (Vol. 1, pp. 1-103).

Hegedus, S., Laborde, C., Brady, C., Dalton, S., Siller, H-St, Tabach, M., et al. (2017). *Uses of technology in upper secondary mathematics education*. Springer.

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

Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry contexts. In C. Mammana & V. Villani (Eds.), *Perspectives on the teaching of geometry for the 21st century: An ICMI study* (pp. 121–128). Kluwer Academic Publishers.

Lesnes-Cuisiniez, E. (2021). *Modélisation didactique de parcours d'apprentissage dans un EIAH* (Doctoral dissertation). Université Paris Cité.

Mariotti, M. A. (2006). Proof and proving in mathematics education. In *Handbook of research on the psychology of mathematics education* (pp. 173-204). Brill.

Mithalal, J., & Balacheff, N. (2019). The instrumental deconstruction as a link between drawing and geometrical figure. *Educational Studies in Mathematics*, 100 (2), 161-176.

Parzysz, B. (1991). Representation of space and students’ conceptions at high school level. *ESM*, 22(6), 575-593.

Patré, P. (2005). Apprendre par la simulation: de l'analyse du travail aux apprentissages professionnels. *Formation*.

Sinclair, N., Moss, J., Hawes, Z., & Stephenson, C. (2019). Learning Through and from Drawing in Early Years Geometry. In K. S. Mix, M. T. Battista (eds.), *Visualizing Mathematics, Research in Mathematics Education* (pp. 229–252). Springer.

Thomas, M. O., & Palmer, J. M. (2014). Teaching with digital technology: Obstacles and opportunities. *The mathematics teacher in the digital era: An international perspective on technology focused professional development*, 71-89.
# EXPERT JUDGMENT FOR CONTENT VALIDATION OF A
QUESTIONNAIRE ON THE LEVEL 5 DEFINITION PROCESS WITHIN
THE VAN HIELE FRAMEWORK

Alberto Arnal-Bailera, Víctor Manero

University of Zaragoza. Mathematics Department

This paper presents the process of designing a questionnaire aimed at measuring the degree of acquisition in the definition process in Geometry among individuals at advanced levels of the Van Hiele model. Our theoretical framework encompasses not only the seminal references of the Van Hiele model but also prior work by the authors in which indicators were designed to characterize the abilities of a high-level Geometry student. In conjunction with the questionnaire, we elucidate the content validation process, conducted through expert judgment. This process entailed semi-structured interviews to refine the task design and closed-ended questions regarding the agreement between the questionnaire items and the indicators describing the definition processes at Van Hiele level 5. As a result, the questionnaire is presented along with its corresponding agreement indices as assessed by Fleiss's Kappa.

## THEORETICAL FRAMEWORK

In this section, we will briefly introduce the fundamental aspects of the Van Hiele model (Van Hiele, 1957) and the prior research that underpins the questionnaire's design. Our primary focus lies on the fifth level of the model, as an in-depth understanding of this level is of particular interest for analyzing geometry learning at the university level. Given the limited existing research on this level, we initially had to design and describe its indicators for each process (Arnal-Bailera & Manero, 2023). Subsequently, our task is to design a questionnaire that facilitates an assessment of the acquisition of this level among undergraduate mathematics students. The specific objective here is to delineate the ongoing validation process of a questionnaire designed for assessing the definition process.

The Van Hiele model (Van Hiele, 1957) stands as one of the most significant theoretical frameworks pertaining to the teaching and learning of geometry across all educational tiers. This model posits the existence of five distinct levels of geometric reasoning. The highest level (fifth-rigor) can be succinctly summarized as follows: individuals at this stage can compare systems grounded in different axioms and can investigate various geometries in the absence of concrete models. What is essential is not merely the ability to work within different geometries or with varying metrics, but rather the capacity to establish relationships among them. It is imperative to comprehend that shifting geometric contexts (by changing the geometry or metric) yields objects of distinct natures, propositions with differing demonstrations, and varying mathematical facts that are transferable between contexts, while others are not.

In previous works (Arnal-Bailera & Manero, 2023), we have developed specific indicators delineating how individuals operating at a Van Hiele level 5 should manifest these abilities when mobilizing skills associated with the processes of definition, proof, and classification. Specifically,
# EXPERT JUDGMENT FOR CONTENT VALIDATION OF A
QUESTIONNAIRE ON THE LEVEL 5 DEFINITION PROCESS WITHIN
THE VAN HIELE FRAMEWORK

Alberto Arnal-Bailera, Víctor Manero

University of Zaragoza. Mathematics Department

This paper presents the process of designing a questionnaire aimed at measuring the degree of acquisition in the definition process in Geometry among individuals at advanced levels of the Van Hiele model. Our theoretical framework encompasses not only the seminal references of the Van Hiele model but also prior work by the authors in which indicators were designed to characterize the abilities of a high-level Geometry student. In conjunction with the questionnaire, we elucidate the content validation process, conducted through expert judgment. This process entailed semi-structured interviews to refine the task design and closed-ended questions regarding the agreement between the questionnaire items and the indicators describing the definition processes at Van Hiele level 5. As a result, the questionnaire is presented along with its corresponding agreement indices as assessed by Fleiss's Kappa.

## THEORETICAL FRAMEWORK

In this section, we will briefly introduce the fundamental aspects of the Van Hiele model (Van Hiele, 1957) and the prior research that underpins the questionnaire's design. Our primary focus lies on the fifth level of the model, as an in-depth understanding of this level is of particular interest for analyzing geometry learning at the university level. Given the limited existing research on this level, we initially had to design and describe its indicators for each process (Arnal-Bailera & Manero, 2023). Subsequently, our task is to design a questionnaire that facilitates an assessment of the acquisition of this level among undergraduate mathematics students. The specific objective here is to delineate the ongoing validation process of a questionnaire designed for assessing the definition process.

The Van Hiele model (Van Hiele, 1957) stands as one of the most significant theoretical frameworks pertaining to the teaching and learning of geometry across all educational tiers. This model posits the existence of five distinct levels of geometric reasoning. The highest level (fifth-rigor) can be succinctly summarized as follows: individuals at this stage can compare systems grounded in different axioms and can investigate various geometries in the absence of concrete models. What is essential is not merely the ability to work within different geometries or with varying metrics, but rather the capacity to establish relationships among them. It is imperative to comprehend that shifting geometric contexts (by changing the geometry or metric) yields objects of distinct natures, propositions with differing demonstrations, and varying mathematical facts that are transferable between contexts, while others are not.

In previous works (Arnal-Bailera & Manero, 2023), we have developed specific indicators delineating how individuals operating at a Van Hiele level 5 should manifest these abilities when mobilizing skills associated with the processes of definition, proof, and classification. Specifically,
when delving into the process of definition at level 5, we have identified five indicators (see Table 1): indicators Def1 and Def2 constitute the initial set of indicators that pertain to the influence of working within diverse geometrical contexts, whereas indicators Def3, Def4, and Def5 are associated with the rationale underlying the formulation of new definitions or the selection between existing ones. In the context of the definition process, the utilization and formulation of definitions are typically investigated as distinct aspects. At this level, the indicators that have emerged are predominantly linked to the formulation of definitions.

<table><caption>Table 1: Indicators for the definition process (Arnal-Bailera & Manero, 2023)</caption><thead><tr><td>Indicator</td><td>Description</td></tr></thead><tbody><tr><td>Def1</td><td>Constructs and uses definitions in different axiomatic systems.</td></tr><tr><td>Def2</td><td>Understands that defining a given mathematical object is not absolute, but is an action relative to the geometric context in which one works, implying for example that the defined object may have different properties in each context.</td></tr><tr><td>Def3</td><td>Defines new objects, for example, because it may be necessary to generalize existing ones or to prove a statement.</td></tr><tr><td>Def4</td><td>Understands that a definition arises out of the necessity to introduce a new mathematical object or to emphasize a property.</td></tr><tr><td>Def5</td><td>Compares equivalent definitions to choose the most interesting one, depending on the work to be done.</td></tr></tbody></table>

## METHODOLOGY

The methodology employed in the design and validation of the questionnaire consisted of two phases. The first phase involved conducting three semi-structured interviews with experts who possessed teaching and/or research experience in the field of Geometry at the university level. During these interviews, the questionnaire was presented to the experts, and their opinions were sought regarding i) the mathematical difficulty of the proposed activities, considering Mathematics undergraduate students as future respondents, and ii) the correspondence between items and indicators in the process of definition. Based on the insights gathered from each of these interviews, certain aspects of the questionnaire were reformulated to ensure a clearer evaluation of certain items or to enhance the comprehensibility of the wording. This revised version was then presented to the following expert.

In the second phase, closed questionnaires regarding the agreement between indicators and items were sent to five Ph.D. holders in Mathematics with research experience. These questionnaires required a dichotomous response (Yes/No) concerning this agreement. Subsequently, using the SPSS software, the Fleiss's Kappa coefficient was analyzed to assess the content validity of the questionnaire (Meyer & Booker, 2001). Future validation processes require the administration and analysis of the questionnaire to undergraduate students.

## RESULTS

We present below some of the results obtained in each of the two phases. Due to space constraints, we summarize the most significant findings derived from the semi-structured interviews with experts, which contributed to the improvement of the initial versions of the questionnaire. After that, we provide an interpretation of a summary of the output obtained using SPSS about the Fleiss Kappa.
when delving into the process of definition at level 5, we have identified five indicators (see Table 1): indicators Def1 and Def2 constitute the initial set of indicators that pertain to the influence of working within diverse geometrical contexts, whereas indicators Def3, Def4, and Def5 are associated with the rationale underlying the formulation of new definitions or the selection between existing ones. In the context of the definition process, the utilization and formulation of definitions are typically investigated as distinct aspects. At this level, the indicators that have emerged are predominantly linked to the formulation of definitions.

<table><caption>Table 1: Indicators for the definition process (Arnal-Bailera & Manero, 2023)</caption><thead><tr><td>Indicator</td><td>Description</td></tr></thead><tbody><tr><td>Def1</td><td>Constructs and uses definitions in different axiomatic systems.</td></tr><tr><td>Def2</td><td>Understands that defining a given mathematical object is not absolute, but is an action relative to the geometric context in which one works, implying for example that the defined object may have different properties in each context.</td></tr><tr><td>Def3</td><td>Defines new objects, for example, because it may be necessary to generalize existing ones or to prove a statement.</td></tr><tr><td>Def4</td><td>Understands that a definition arises out of the necessity to introduce a new mathematical object or to emphasize a property.</td></tr><tr><td>Def5</td><td>Compares equivalent definitions to choose the most interesting one, depending on the work to be done.</td></tr></tbody></table>

## METHODOLOGY

The methodology employed in the design and validation of the questionnaire consisted of two phases. The first phase involved conducting three semi-structured interviews with experts who possessed teaching and/or research experience in the field of Geometry at the university level. During these interviews, the questionnaire was presented to the experts, and their opinions were sought regarding i) the mathematical difficulty of the proposed activities, considering Mathematics undergraduate students as future respondents, and ii) the correspondence between items and indicators in the process of definition. Based on the insights gathered from each of these interviews, certain aspects of the questionnaire were reformulated to ensure a clearer evaluation of certain items or to enhance the comprehensibility of the wording. This revised version was then presented to the following expert.

In the second phase, closed questionnaires regarding the agreement between indicators and items were sent to five Ph.D. holders in Mathematics with research experience. These questionnaires required a dichotomous response (Yes/No) concerning this agreement. Subsequently, using the SPSS software, the Fleiss's Kappa coefficient was analyzed to assess the content validity of the questionnaire (Meyer & Booker, 2001). Future validation processes require the administration and analysis of the questionnaire to undergraduate students.

## RESULTS

We present below some of the results obtained in each of the two phases. Due to space constraints, we summarize the most significant findings derived from the semi-structured interviews with experts, which contributed to the improvement of the initial versions of the questionnaire. After that, we provide an interpretation of a summary of the output obtained using SPSS about the Fleiss Kappa.
## Semi-structured interviews

The first interview (conducted with Expert 1) reviewed the initial version of the questionnaire, revealing that the indicator def4 was not clearly presented. Consequently, the questionnaire would not be able to accurately assess the level of acquisition of this indicator.

**Item 3**

Normally to measure distances in the plane we use the Euclidean metric, which is defined as follows: Given two points, the distance between them is the length of the segment joining them. However, we can define other distances, such as the so-called postman's (or Taxicab) distance, which is defined as follows: the distance between two points is given by the shortest route joining those using only horizontal and vertical lines.

**Item 3.1:** If we define the circumference as the set of points that are equidistant (at the same distance) from another point, what is the shape of the circumferences with the Taxicab metric? Justify your answer.

**Item 3.2:** If we define a circumference as a set of points that are equidistant from another point, what shape do they have with the distance from the postman? Justify your answer. You can use the grid to draw the points that are equidistant from the point indicated.

<figure><img src="image_46.png" /></figure>

**Item 3.3:** If we define a circle as a set of points that are equidistant (at the same distance) from another point. Look at the different drawings below:

All the marked points are, with the Taxicab metric, at distance ___ from point Q.

<figure><img src="image_47.png" /></figure>

All the marked points are, with the Taxicab metric, at distance ___ from point P.

<figure><img src="image_48.png" /></figure>

What is the shape of the circles with the Taxicab metric? Justify your answer.

<figure><img src="image_49.png" /><figcaption>Figure 1. Item 3 as presented to the first expert.</figcaption></figure>

We analyze the moment when Expert 1 was reviewing the third item (see Figure 1): a discussion ensued regarding whether someone who correctly answers item 3.1 demonstrates an understanding that a definition arises from the need to introduce a new mathematical object or emphasize a property. The interview excerpt illustrates the utility of the item, with the expert suggesting a reversal of the ideas presented in this item. It was proposed to start by presenting specific cases and then inquire about the property that the red points have in relation to the blue, with the (imprecise at that moment) idea that asking about the property would prompt the need to define distance.

**Expert 1:** I don't quite see it. This task is similar to the previous one. It's essentially the same question with a different distance. Maybe we can approach it differently and start by asking what the red points in the middle square have in common with the blue ones. Let's see if someone mentions that they are at the same distance, without defining distance. Let's see if the need to define distance arises.

**Interviewer:** Yes, we could use diagrams like the one in the center and ask for a distance so that the red points are equidistant from P.

**Expert 1:** Or, without using the word "distance", what do the red points have in common with the blue ones? Or, define something that all these points have in relation to the blue ones.

Between the interview with the first expert and the second one (Expert 2), the idea explained earlier was implemented. However, the researchers found it more suitable to implement it in item 2, whose initial formulation can be seen in Figure 2, and its subsequent formulation in Figure 3. This decision
## Semi-structured interviews

The first interview (conducted with Expert 1) reviewed the initial version of the questionnaire, revealing that the indicator def4 was not clearly presented. Consequently, the questionnaire would not be able to accurately assess the level of acquisition of this indicator.

**Item 3**

Normally to measure distances in the plane we use the Euclidean metric, which is defined as follows: Given two points, the distance between them is the length of the segment joining them. However, we can define other distances, such as the so-called postman's (or Taxicab) distance, which is defined as follows: the distance between two points is given by the shortest route joining those using only horizontal and vertical lines.

**Item 3.1:** If we define the circumference as the set of points that are equidistant (at the same distance) from another point, what is the shape of the circumferences with the Taxicab metric? Justify your answer.

**Item 3.2:** If we define a circumference as a set of points that are equidistant from another point, what shape do they have with the distance from the postman? Justify your answer. You can use the grid to draw the points that are equidistant from the point indicated.

<figure><img src="image_52.png" /></figure>

**Item 3.3:** If we define a circle as a set of points that are equidistant (at the same distance) from another point. Look at the different drawings below:

All the marked points are, with the Taxicab metric, at distance ___ from point Q.

<figure><img src="image_53.png" /></figure>

All the marked points are, with the Taxicab metric, at distance ___ from point P.

<figure><img src="image_54.png" /></figure>

What is the shape of the circles with the Taxicab metric? Justify your answer.

<figure><img src="image_55.png" /><figcaption>Figure 1. Item 3 as presented to the first expert.</figcaption></figure>

We analyze the moment when Expert 1 was reviewing the third item (see Figure 1): a discussion ensued regarding whether someone who correctly answers item 3.1 demonstrates an understanding that a definition arises from the need to introduce a new mathematical object or emphasize a property. The interview excerpt illustrates the utility of the item, with the expert suggesting a reversal of the ideas presented in this item. It was proposed to start by presenting specific cases and then inquire about the property that the red points have in relation to the blue, with the (imprecise at that moment) idea that asking about the property would prompt the need to define distance.

**Expert 1:** I don't quite see it. This task is similar to the previous one. It's essentially the same question with a different distance. Maybe we can approach it differently and start by asking what the red points in the middle square have in common with the blue ones. Let's see if someone mentions that they are at the same distance, without defining distance. Let's see if the need to define distance arises.

**Interviewer:** Yes, we could use diagrams like the one in the center and ask for a distance so that the red points are equidistant from P.

**Expert 1:** Or, without using the word "distance", what do the red points have in common with the blue ones? Or, define something that all these points have in relation to the blue ones.

Between the interview with the first expert and the second one (Expert 2), the idea explained earlier was implemented. However, the researchers found it more suitable to implement it in item 2, whose initial formulation can be seen in Figure 2, and its subsequent formulation in Figure 3. This decision
was made to maintain the structure of the item 3 super-item (Jaime & Gutiérrez, 1994), which is
convenient if the test would aim to measure the acquisition of lower levels than level 5.

Item 2

Normally, to measure distances in the plane, we use the Euclidean distance, which is defined as follows: given points A=(x₁,y₁) and B=(x₂,y₂), the distance between them is given by the length of the line segment that connects them:

$$
d_E(A, B) := \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
$$

However, we can define other different distances, such as the **radial distance**, which is defined as follows:

$$
d_R(A, B) := \begin{cases} d_E(A, B) & \text{if } A, B \text{ and } O \text{ are aligned} \\ d_E(A, O) + d_E(O, B) & \text{otherwise} \end{cases}
$$

Here, O represents the origin of coordinates, i.e., the point with coordinates O=(0,0). If we define a circle as the set of points equidistant (at the same distance) from another point, **what shape do the circles have with the radial distance**?

<caption>Figure 2. Item 2 as presented to the first expert.</caption>

The modifications to item 2 (see Figure 3) involved adding images with discrete sets of points that have a relationship with another point. Firstly, in Sub-item 2.1, the question asked was about identifying that relationship (i.e., what property these points satisfy). Then, in Sub-item 2.2, respondents were asked to construct the definition of the (continuous) set of points that satisfy that property. The continuous set was presented based on a single image.

On the other hand, Expert 1 pointed out that def5 was not reflected in item 2 because equivalent definitions were not being compared. Since it was also necessary to reinforce this indicator in the questionnaire, it was decided to expand this item with a Sub-item 2.3, including a new formulation of the definition of radial distance for comparison with the original. This new formulation is of a verbal nature and provides a more descriptive explanation of the process by which radial distance between two points is determined: radial distance between two points is the minimum length of the segment or segments that connect them and lie on lines passing through the origin.

In the following, we present an excerpt from the interview with the second expert (Expert 2) when
discussing the revised version of item 2. On one hand, the expert finds it challenging to identify a
specific relationship among the discrete set of points based on the examples presented, as multiple
different relationships appear to exist. Consequently, the expert considers it appropriate to increase
the number of images for both discrete and continuous cases. On the other hand, the expert
acknowledges the presence of indicator def4 in question 1 but expresses doubts about its accurate
evaluation in the form it had in that version. Additionally, the expert agrees that question 2 in the
Sub-item 2.3 assesses indicator def5.

Expert 2: When I first thought about it, I gave an answer that does not say much: all points are at a positive distance... then I thought of something else... that the absolute values of the coordinates of the other points are larger... (...)

Expert 2: This one does not really evaluate anything [item 2.2]; it puts you in a situation to do this one [item 2.3], prepares you for the next one... It's like good marketing; they've created the need to construct a definition... now this one [item 2.3] will really evaluate something... def4 is important... you understand... it's a complicated thing, understanding... (...)

Interviewer: We could give item 2.2 a super-item structure with several images...

Expert 2: Yes, I could have four examples, and I want a definition for all four cases. That way, it generalizes, going from the concrete to the abstract... It's about understanding... you guys
was made to maintain the structure of the item 3 super-item (Jaime & Gutiérrez, 1994), which is
convenient if the test would aim to measure the acquisition of lower levels than level 5.

**Item 2**

Normally, to measure distances in the plane, we use the Euclidean distance, which is defined as follows: given points A=(x₁,y₁) and B=(x₂,y₂), the distance between them is given by the length of the line segment that connects them:

$$
d_{E}(A, B) := \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}.
$$

However, we can define other different distances, such as the **radial distance**, which is defined as follows:

$$
d_R(A, B) := \begin{cases} d_E(A, B) & \text{if } A, B \text{ and } O \text{ are aligned} \\ d_E(A, O) + d_E(O, B) & \text{otherwise} \end{cases}
$$

Here, O represents the origin of coordinates, i.e., the point with coordinates O=(0,0). If we define a circle as the set of points equidistant (at the same distance) from another point, **what shape do the circles have with the radial distance**?

<caption>Figure 2. Item 2 as presented to the first expert.</caption>

The modifications to item 2 (see Figure 3) involved adding images with discrete sets of points that have a relationship with another point. Firstly, in Sub-item 2.1, the question asked was about identifying that relationship (i.e., what property these points satisfy). Then, in Sub-item 2.2, respondents were asked to construct the definition of the (continuous) set of points that satisfy that property. The continuous set was presented based on a single image.

On the other hand, Expert 1 pointed out that def5 was not reflected in item 2 because equivalent definitions were not being compared. Since it was also necessary to reinforce this indicator in the questionnaire, it was decided to expand this item with a Sub-item 2.3, including a new formulation of the definition of radial distance for comparison with the original. This new formulation is of a verbal nature and provides a more descriptive explanation of the process by which radial distance between two points is determined: radial distance between two points is the minimum length of the segment or segments that connect them and lie on lines passing through the origin.

In the following, we present an excerpt from the interview with the second expert (Expert 2) when
discussing the revised version of item 2. On one hand, the expert finds it challenging to identify a
specific relationship among the discrete set of points based on the examples presented, as multiple
different relationships appear to exist. Consequently, the expert considers it appropriate to increase
the number of images for both discrete and continuous cases. On the other hand, the expert
acknowledges the presence of indicator def4 in question 1 but expresses doubts about its accurate
evaluation in the form it had in that version. Additionally, the expert agrees that question 2 in the
Sub-item 2.3 assesses indicator def5.

Expert 2: When I first thought about it, I gave an answer that does not say much: all points are at a positive distance... then I thought of something else... that the absolute values of the coordinates of the other points are larger... (...)

Expert 2: This one does not really evaluate anything [item 2.2]; it puts you in a situation to do this one [item 2.3], prepares you for the next one... It's like good marketing; they've created the need to construct a definition... now this one [item 2.3] will really evaluate something... def4 is important... you understand... it's a complicated thing, understanding... (...)

Interviewer: We could give item 2.2 a super-item structure with several images...

Expert 2: Yes, I could have four examples, and I want a definition for all four cases. That way, it generalizes, going from the concrete to the abstract... It's about understanding... you guys
will know about that. About understanding, I don't know how to answer that. (...)

Interviewer: So, here we could put 4 images, ask for the property, and then here, another 4 images, and ask for the definition. That way, we would convince more of the presence of def4.

**Item 2.1.A:** Could you describe a property that all the blue points (B-G) have in relation to the red point (A) in the given image?

<figure><img src="image_50.png" /></figure>

**Item 2.1.B:** Could you describe a property that all the blue points (B-G) have in relation to the red point (A) in the given images?

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

**Item 2.2:** In the given image, the blue points share a property with respect to the red point (A). Construct a definition for the set of blue points with respect to the red point (A). We will call this set S<sub>A</sub>.

<figure><img src="image_53.png" /></figure>

**Item 2.3:** Normally, to measure distances in the plane, we use the Euclidean distance, which is defined as follows. Given two points A=(x<sub>1</sub>,y<sub>1</sub>) y B=(x<sub>2</sub>,y<sub>2</sub>), the distance between them is given by the length of the line segment that connects them, that is,

$$d_E(A, B) := \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

However, we can define other different distances, such as the **radial distance**, which can be defined as follows:

**Definition 1:**

$$d_R(A, B) := \begin{cases} d_E(A, B) & \text{if } A, B \text{ and } O \text{ are aligned} \\ d_E(A, O) + d_E(O, B) & \text{otherwise} \end{cases}$$

Here, O represents the origin of coordinates, that is, the point with coordinates O=(0,0).

**Question 1.** Could you describe a situation/moment/context that can be modeled using this metric or in which this metric may be necessary/useful?

**PAGE BREAK**

Similarly, the radial distance can be described as follows:

**Definition 2:** The radial distance between two points is the minimum length of the segment or segments that connect them, with these segments contained in lines that pass through the origin. As you can see, these two definitions, one algebraic and the other verbal, are of different natures.

**Question 2:** Compare both definitions (algebraic and verbal) by describing in which contexts you would use one and when you would use the other.

<caption>Figure 3. Item 2 as presented to the second expert.</caption>

It should be noticed that a page break has been included in Sub-item 2.3 since the second part of this Sub-item could produce answers to the previous part, thus the page break indicates that when the second part is reached the previous responses cannot be changed.

Through the questions and answers in the interview with the second expert, the conclusion was reached to reformulate item 2 to facilitate its resolution. In Sub-item 2.1, four discrete sets of points were presented (in addition to the ones shown in Figure 3-Sub-item 2.1), and participants were asked
will know about that. About understanding, I don't know how to answer that. (...)

Interviewer: So, here we could put 4 images, ask for the property, and then here, another 4 images, and ask for the definition. That way, we would convince more of the presence of def4.

**Item 2.1.A:** Could you describe a property that all the blue points (B-G) have in relation to the red point (A) in the given image?

<figure><img src="image_56.png" /></figure>

**Item 2.1.B:** Could you describe a property that all the blue points (B-G) have in relation to the red point (A) in the given images?

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /></figure>

**Item 2.2:** In the given image, the blue points share a property with respect to the red point (A). Construct a definition for the set of blue points with respect to the red point (A). We will call this set S<sub>A</sub>.

<figure><img src="image_59.png" /></figure>

**Item 2.3:** Normally, to measure distances in the plane, we use the Euclidean distance, which is defined as follows. Given two points A=(x<sub>1</sub>,y<sub>1</sub>) y B=(x<sub>2</sub>,y<sub>2</sub>), the distance between them is given by the length of the line segment that connects them, that is,

$$d_E(A, B) := \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

However, we can define other different distances, such as the **radial distance**, which can be defined as follows:

**Definition 1:**

$$d_R(A, B) := \begin{cases} d_E(A, B) & \text{if } A, B \text{ and } O \text{ are aligned} \\ d_E(A, O) + d_E(O, B) & \text{otherwise} \end{cases}$$

Here, O represents the origin of coordinates, that is, the point with coordinates O=(0,0).

**Question 1.** Could you describe a situation/moment/context that can be modeled using this metric or in which this metric may be necessary/useful?

PAGE BREAK

Similarly, the radial distance can be described as follows:

**Definition 2:** The radial distance between two points is the minimum length of the segment or segments that connect them, with these segments contained in lines that pass through the origin. As you can see, these two definitions, one algebraic and the other verbal, are of different natures.

**Question 2:** Compare both definitions (algebraic and verbal) by describing in which contexts you would use one and when you would use the other.

<caption>Figure 3. Item 2 as presented to the second expert.</caption>

It should be noticed that a page break has been included in Sub-item 2.3 since the second part of this Sub-item could produce answers to the previous part, thus the page break indicates that when the second part is reached the previous responses cannot be changed.

Through the questions and answers in the interview with the second expert, the conclusion was reached to reformulate item 2 to facilitate its resolution. In Sub-item 2.1, four discrete sets of points were presented (in addition to the ones shown in Figure 3-Sub-item 2.1), and participants were asked
to infer, for each set, the property that connects them to another point marked in a different color. Then, in Sub-item 2.2, another four sets were presented (in addition to the one shown in Figure 3-Sub-item 2.2), this time continuous, and participants were asked to construct the definition for all of them. The objective was for the first group of sets to prompt reflection on the existence of a common property among all blue points with respect to the red one. Subsequently, a definition would be constructed based on this property, which could be related to the presence of def4. No changes were made in Sub-item 2.3.

This new version of item 2 was presented to the third expert who was interviewed (Expert 3). Despite the changes made, this expert concurred with expert 2 regarding the difficulty in eliciting a relevant property when discrete sets of points were presented in Sub-item 2.1.

**Expert 3:** The first Sub-item is exceedingly challenging to solve; it is conceivable that some students may provide an unexpected property, in the sense that a discrete set of points always possesses a connecting property, even if it takes the form of a 27th-degree equation. This approach would not be beneficial for addressing the following Sub-item.

**Interviewer:** Would you consider a version without the first Sub-item (the one concerning discrete sets) to be better?

**Expert 3:** Yes, as the first Sub-item does not contribute anything, and no indicators from the ones you propose can be identified.

**Item 2.1:** In the given images, the blue points share a property with respect to the red point (A). Construct a definition for the set of blue points with respect to the red point (A). We will call this set $S_A$

<figure><img src="image_54.png" /></figure>

**Item 2.2:** Normally, to measure distances in the plane, we use the Euclidean distance, which is defined as follows. Given two points A=(x₁,y₁) y B=(x₂,y₂), the distance between them is given by the length of the line segment that connects them, that is,

$$d_E(A, B) := \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$

However, we can define other different distances, such as the **radial distance**, which can be defined as follows:

**Definition 1:**

$$d_R(A, B) := \begin{cases} d_E(A, B) & \text{if } A, B \text{ and } O \text{ are aligned} \\ d_E(A, O) + d_E(O, B) & \text{otherwise} \end{cases}$$

Here, O represents the origin of coordinates, that is, the point with coordinates O=(0,0).

**Question 1.** Could you describe a situation/moment/context that can be modeled using this metric or in which this metric may be necessary/useful?

**PAGE BREAK**

Similarly, the radial distance can be described as follows:

**Definition 2:** The radial distance between two points is the minimum length of the segment or segments that connect them, with these segments contained in lines that pass through the origin.

**Question 2:** Compare both definitions (algebraic and verbal) by describing in which contexts you would use one and when you would use the other.

<caption>Figure 4. Final version of Item 2</caption>
to infer, for each set, the property that connects them to another point marked in a different color. Then, in Sub-item 2.2, another four sets were presented (in addition to the one shown in Figure 3-Sub-item 2.2), this time continuous, and participants were asked to construct the definition for all of them. The objective was for the first group of sets to prompt reflection on the existence of a common property among all blue points with respect to the red one. Subsequently, a definition would be constructed based on this property, which could be related to the presence of def4. No changes were made in Sub-item 2.3.

This new version of item 2 was presented to the third expert who was interviewed (Expert 3). Despite the changes made, this expert concurred with expert 2 regarding the difficulty in eliciting a relevant property when discrete sets of points were presented in Sub-item 2.1.

**Expert 3:** The first Sub-item is exceedingly challenging to solve; it is conceivable that some students may provide an unexpected property, in the sense that a discrete set of points always possesses a connecting property, even if it takes the form of a 27th-degree equation. This approach would not be beneficial for addressing the following Sub-item.

**Interviewer:** Would you consider a version without the first Sub-item (the one concerning discrete sets) to be better?

**Expert 3:** Yes, as the first Sub-item does not contribute anything, and no indicators from the ones you propose can be identified.

**Item 2.1:** In the given images, the blue points share a property with respect to the red point (A). Construct a definition for the set of blue points with respect to the red point (A). We will call this set $S_A$

<figure><img src="image_60.png" /></figure>

**Item 2.2:** Normally, to measure distances in the plane, we use the Euclidean distance, which is defined as follows. Given two points A=(x₁,y₁) y B=(x₂,y₂), the distance between them is given by the length of the line segment that connects them, that is,

$$d_E(A, B) := \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$

However, we can define other different distances, such as the **radial distance**, which can be defined as follows:

**Definition 1:**

$$d_R(A, B) := \begin{cases} d_E(A, B) & \text{if } A, B \text{ and } O \text{ are aligned} \\ d_E(A, O) + d_E(O, B) & \text{otherwise} \end{cases}$$

Here, O represents the origin of coordinates, that is, the point with coordinates O=(0,0).

**Question 1.** Could you describe a situation/moment/context that can be modeled using this metric or in which this metric may be necessary/useful?

**PAGE BREAK**

Similarly, the radial distance can be described as follows:

**Definition 2:** The radial distance between two points is the minimum length of the segment or segments that connect them, with these segments contained in lines that pass through the origin.

**Question 2:** Compare both definitions (algebraic and verbal) by describing in which contexts you would use one and when you would use the other.

<caption>Figure 4. Final version of Item 2</caption>
Furthermore, Expert 3 suggested that both definitions were not truly of distinct natures, as they were different only in terms of the representation but similar in terms of the aspects of the concept used to construct them. This suggestion was incorporated by removing the phrase "are of different natures" in the final version of item 2 (see Figure 4), once Sub-item 2.1 had been eliminated, and the following items had been renumbered.

Based on the discussions with the experts during the interviews, we wish to emphasize that when an expert validates an indicator in an item, they are not necessarily thinking about the possibility of finding written evidence. Rather, they infer that the indicator is necessary to address the question posed by the item. We bring this up with the understanding that when students in the Mathematics Bachelor's program respond to this questionnaire, they may not necessarily provide written evidence for all these indicators in every item. Nevertheless, we will consider that an item serves to assess specific indicators because this is the expert's opinion, rather than relying on the presence of specific sentences that can be associated with each indicator.

## Fleiss Kappa Indicator

After the interviews, the questionnaire was dispatched to four experts in the field of Geometry, who provided responses concerning the presence of each indicator across various Sub-items. Consequently, each expert provided 35 dichotomous (Yes/No) responses, which were subsequently analyzed using the SPSS software to calculate Fleiss's Kappa statistic, both for the entire questionnaire and for each individual item (see Table 2). This statistic expresses the level of consensus among experts regarding which process definition indicators are measured by the entire questionnaire or each individual item.

<table><caption>Table 2: Results for the Fleiss Kappa statistic</caption><thead><tr><td>Analysis type</td><td>Kappa and agreement strength (Landis & Koch, 1977)</td><td>p-value</td></tr></thead><tbody><tr><td>Questionnaire</td><td>0,533 – Moderate</td><td>0,000</td></tr><tr><td>Item 1</td><td>0,306 – Fair</td><td>0,018</td></tr><tr><td>Item 2</td><td>0,231 – Fair</td><td>0,073</td></tr><tr><td>Item 3</td><td>0,864 – Almost perfect</td><td>0,000</td></tr></tbody></table>

Notably, there is a statistically significant moderate agreement concerning the entire questionnaire. However, when analyzing the items separately, almost perfect and significant agreement is observed for Item 3, while only acceptable agreement is found for the other two items.

From the descriptive analysis of expert responses, it can be inferred that there is unanimity among experts that indicator def1 is measured in all three items, while def2 is measured in Item 3 and def5 is measured in Items 1 and 2. Additionally, a majority of experts believe that def2 is also measured by items 1 and 2, and def3 is measured by Item 2.

## CONCLUSIONS

In the initial phase of questionnaire design, three interviews were conducted proving cautious optimism regarding the potential completion of item validation. The analysis of the interviews contributed to the refinement of the questionnaire: some items were clarified in their wording, others were removed, and some others were adapted to encompass a greater number of indicators.
Furthermore, Expert 3 suggested that both definitions were not truly of distinct natures, as they were different only in terms of the representation but similar in terms of the aspects of the concept used to construct them. This suggestion was incorporated by removing the phrase "are of different natures" in the final version of item 2 (see Figure 4), once Sub-item 2.1 had been eliminated, and the following items had been renumbered.

Based on the discussions with the experts during the interviews, we wish to emphasize that when an expert validates an indicator in an item, they are not necessarily thinking about the possibility of finding written evidence. Rather, they infer that the indicator is necessary to address the question posed by the item. We bring this up with the understanding that when students in the Mathematics Bachelor's program respond to this questionnaire, they may not necessarily provide written evidence for all these indicators in every item. Nevertheless, we will consider that an item serves to assess specific indicators because this is the expert's opinion, rather than relying on the presence of specific sentences that can be associated with each indicator.

## Fleiss Kappa Indicator

After the interviews, the questionnaire was dispatched to four experts in the field of Geometry, who provided responses concerning the presence of each indicator across various Sub-items. Consequently, each expert provided 35 dichotomous (Yes/No) responses, which were subsequently analyzed using the SPSS software to calculate Fleiss's Kappa statistic, both for the entire questionnaire and for each individual item (see Table 2). This statistic expresses the level of consensus among experts regarding which process definition indicators are measured by the entire questionnaire or each individual item.

<table><caption>Table 2: Results for the Fleiss Kappa statistic</caption><thead><tr><td>Analysis type</td><td>Kappa and agreement strength (Landis & Koch, 1977)</td><td>p-value</td></tr></thead><tbody><tr><td>Questionnaire</td><td>0,533 – Moderate</td><td>0,000</td></tr><tr><td>Item 1</td><td>0,306 – Fair</td><td>0,018</td></tr><tr><td>Item 2</td><td>0,231 – Fair</td><td>0,073</td></tr><tr><td>Item 3</td><td>0,864 – Almost perfect</td><td>0,000</td></tr></tbody></table>

Notably, there is a statistically significant moderate agreement concerning the entire questionnaire. However, when analyzing the items separately, almost perfect and significant agreement is observed for Item 3, while only acceptable agreement is found for the other two items.

From the descriptive analysis of expert responses, it can be inferred that there is unanimity among experts that indicator def1 is measured in all three items, while def2 is measured in Item 3 and def5 is measured in Items 1 and 2. Additionally, a majority of experts believe that def2 is also measured by items 1 and 2, and def3 is measured by Item 2.

## CONCLUSIONS

In the initial phase of questionnaire design, three interviews were conducted proving cautious optimism regarding the potential completion of item validation. The analysis of the interviews contributed to the refinement of the questionnaire: some items were clarified in their wording, others were removed, and some others were adapted to encompass a greater number of indicators.
Secondly, the analysis of the Fleiss Kappa indicator suggests that the work is not yet entirely finished. Item 3 shows nearly perfect agreement in its ability to measure indicators def1 and def2. Nevertheless, experts did not express a high level of consensus about Item 2 despite the careful reflection process undertaken during its design. Specifically, our intention was for Sub-item 2.1 to measure def3 as it introduces a new object (radial distance) through graphic language and requests the construction of its definition. Similarly, in Sub-item 2.2, we expected experts to recognize that def4 is being evaluated, as we understood that the first definition of radial distance would be necessary to describe the objects appearing in Sub-item 2.1 more simply. Another expected response for this Sub-item was consider this distance as modeling real-life situations, such as the operation of a train network passing through a station. Only 2 out of 4 experts agreed with this interpretation, indicating that a more explicit wording of Sub-item 2.2 and others, could enhance expert agreement. For instance, revising Sub-item 2.2 as follows: "What is the need for introducing this new distance? What intra- mathematical or extra-mathematical situations or contexts can it model?" may improve consensus.

In addition to deliberating on the content of the items, there appears to be a need for enhancing certain methodological aspects, such as offering a more thorough elucidation of the meaning of each item to enhance experts' comprehension of the questionnaire they are assessing. This recommendation is rooted in informal feedback received from some experts and is substantiated by existing literature (Meyer & Booker, 2001).

We understand that the process we have presented in this contribution corresponds only to content validation via experts in the area. Subsequently, a second validation process is requisite, including the analysis of the responses of actual students answering the questionnaire and analyzing if in such responses the corresponding indicators appear.

## Acknowledgements

We want to thank the group of experts for their patience during the study. Their valuable contributions and dedication have made this research possible.

The first author is partially supported by Spanish MICINN (grant number PID2019-104964 GB-I00), and the second author is partially supported by Spanish AEI (grant number PID2020-115652 GB-I00). Both authors belong to the research group S60_23R (Gobierno de Aragón, Spain).

## References

* Arnal-Bailera, A., & Manero, V. (2023). A characterization of Van Hiele's level 5 of geometric reasoning using the delphi methodology. *International Journal of Science and Mathematics Education, 1* - 24.

* Gutiérrez, A., & Jaime, A. (1995). Towards the design of a standard test for the assessment of the students' reasoning in geometry. In L. Meira & D. Carraher (Eds.), *Proceedings of the 19th PME conference* (vol. 3, pp. 11–18). PME.

* Gutiérrez, A., Jaime, A., & Fortuny, J. M. (1991). An alternative paradigm to evaluate the acquisition of the van Hiele levels. *Journal for Research in Mathematics Education, 22*(3), 237–251.

* Jaime, A. & Gutiérrez, A. (1994). A model of test design to assess the Van Hiele levels. In J. P. da Ponte & J. F. Matos (Eds.), *Proceedings of the 18th PME conference* (vol. 3, pp. 41–48). PME.

* Meyer, M. A., & Booker, J. M. (2001). *Eliciting and analyzing expert judgment: a practical guide*. Society for Industrial and Applied Mathematics.

* Van Hiele, P. M. (1986). *Structure and insight*. Academic Press.
Secondly, the analysis of the Fleiss Kappa indicator suggests that the work is not yet entirely finished. Item 3 shows nearly perfect agreement in its ability to measure indicators def1 and def2. Nevertheless, experts did not express a high level of consensus about Item 2 despite the careful reflection process undertaken during its design. Specifically, our intention was for Sub-item 2.1 to measure def3 as it introduces a new object (radial distance) through graphic language and requests the construction of its definition. Similarly, in Sub-item 2.2, we expected experts to recognize that def4 is being evaluated, as we understood that the first definition of radial distance would be necessary to describe the objects appearing in Sub-item 2.1 more simply. Another expected response for this Sub-item was consider this distance as modeling real-life situations, such as the operation of a train network passing through a station. Only 2 out of 4 experts agreed with this interpretation, indicating that a more explicit wording of Sub-item 2.2 and others, could enhance expert agreement. For instance, revising Sub-item 2.2 as follows: "What is the need for introducing this new distance? What intra- mathematical or extra-mathematical situations or contexts can it model?" may improve consensus.

In addition to deliberating on the content of the items, there appears to be a need for enhancing certain methodological aspects, such as offering a more thorough elucidation of the meaning of each item to enhance experts' comprehension of the questionnaire they are assessing. This recommendation is rooted in informal feedback received from some experts and is substantiated by existing literature (Meyer & Booker, 2001).

We understand that the process we have presented in this contribution corresponds only to content validation via experts in the area. Subsequently, a second validation process is requisite, including the analysis of the responses of actual students answering the questionnaire and analyzing if in such responses the corresponding indicators appear.

## Acknowledgements

We want to thank the group of experts for their patience during the study. Their valuable contributions and dedication have made this research possible.

The first author is partially supported by Spanish MICINN (grant number PID2019-104964 GB-I00), and the second author is partially supported by Spanish AEI (grant number PID2020-115652 GB-I00). Both authors belong to the research group S60_23R (Gobierno de Aragón, Spain).

## References

* Arnal-Bailera, A., & Manero, V. (2023). A characterization of Van Hiele's level 5 of geometric reasoning using the delphi methodology. *International Journal of Science and Mathematics Education, 1* - 24.

* Gutiérrez, A., & Jaime, A. (1995). Towards the design of a standard test for the assessment of the students' reasoning in geometry. In L. Meira & D. Carraher (Eds.), *Proceedings of the 19th PME conference* (vol. 3, pp. 11–18). PME.

* Gutiérrez, A., Jaime, A., & Fortuny, J. M. (1991). An alternative paradigm to evaluate the acquisition of the van Hiele levels. *Journal for Research in Mathematics Education, 22*(3), 237–251.

* Jaime, A. & Gutiérrez, A. (1994). A model of test design to assess the Van Hiele levels. In J. P. da Ponte & J. F. Matos (Eds.), *Proceedings of the 18th PME conference* (vol. 3, pp. 41–48). PME.

* Meyer, M. A., & Booker, J. M. (2001). *Eliciting and analyzing expert judgment: a practical guide*. Society for Industrial and Applied Mathematics.

* Van Hiele, P. M. (1986). *Structure and insight*. Academic Press.
# TEACHING AND LEARNING GEOMETRY OF PRE-SERVICE PRIMARY EDUCATION TEACHERS BASED ON THE VISUAL-ANALYTICAL METHOD OF DIRECTED OBSERVATION

Nives Baranović

Faculty of Humanities and Social Sciences University of Split, Croatia

This paper presents the results of a quasi-experimental research study conducted with pre-service primary education teachers in Croatia aged 19 to 23. The purpose of the research was to evaluate an alternative approach to learning and teaching geometry to develop visualization skills and geometric thinking and achieve better learning outcomes. The alternative teaching approach was based on van Hiele's theoretical framework, on the visual-analytical method of directed observation, and on the balancing of visual, linguistic, and symbolic expressions. The control group was taught in the traditional way. In order to collect data on the research participants, the three tests were used before and after teaching: the first to measure the level of geometric thinking, the second to gain insight into geometric knowledge and visual skills, and the third to measure special visual-spatial ability. At the beginning of teaching, there were no statistically significant differences among the research participants. After teaching, the progress of the participants in the experimental group was statistically significant compared to the participants of the control group.

## INTRODUCTION

The results of research in education indicate that students at all levels of mathematics education have the most difficulties when learning geometry (e.g., PISA, 2012; TIMMS 2017), especially when establishing connections between geometric concepts (e.g., Baranović, 2019). This problem comes to the fore especially when students transfer from school to higher mathematics education, because the discourses of learning mathematics at school and at the university level are significantly different. At the university level it is difficult for students to resolve cognitive conflicts that they develop during pre-tertiary education which they are often not even aware of (e.g., Thoma & Nardi, 2018).

Mastering the geometry program at the university level requires at least the fourth level of geometric thinking according to van Hiele, but practice and various research in education show that students are not ready for this (e.g., Usiskin; 1982; Crowley 1987; De Villiers, 2009). To learn geometric concepts with understanding and their efficient application in solving problems and proving statements, students should have appropriate visual-spatial skills, since perception alone is not sufficient for mathematical processing of visual representations (Duval 1995; 1998). The causes of weak visualization skills of students of all ages, which are indicated by practice and various research in education, are primarily a dislike of the mathematical community to visualization as well as the difficulties that arise during visual processing (Presmeg, 1986; 2014). In the last few decades, the need for continuous and systematic development and use of visual-spatial abilities in teaching of mathematics throughout the entire educational vertical has been recognized and intensive work is being done to build a theoretical framework that would enable this (Presmeg, 2014).
# TEACHING AND LEARNING GEOMETRY OF PRE-SERVICE PRIMARY EDUCATION TEACHERS BASED ON THE VISUAL-ANALYTICAL METHOD OF DIRECTED OBSERVATION

Nives Baranović

Faculty of Humanities and Social Sciences University of Split, Croatia

This paper presents the results of a quasi-experimental research study conducted with pre-service primary education teachers in Croatia aged 19 to 23. The purpose of the research was to evaluate an alternative approach to learning and teaching geometry to develop visualization skills and geometric thinking and achieve better learning outcomes. The alternative teaching approach was based on van Hiele's theoretical framework, on the visual-analytical method of directed observation, and on the balancing of visual, linguistic, and symbolic expressions. The control group was taught in the traditional way. In order to collect data on the research participants, the three tests were used before and after teaching: the first to measure the level of geometric thinking, the second to gain insight into geometric knowledge and visual skills, and the third to measure special visual-spatial ability. At the beginning of teaching, there were no statistically significant differences among the research participants. After teaching, the progress of the participants in the experimental group was statistically significant compared to the participants of the control group.

## INTRODUCTION

The results of research in education indicate that students at all levels of mathematics education have the most difficulties when learning geometry (e.g., PISA, 2012; TIMMS 2017), especially when establishing connections between geometric concepts (e.g., Baranović, 2019). This problem comes to the fore especially when students transfer from school to higher mathematics education, because the discourses of learning mathematics at school and at the university level are significantly different. At the university level it is difficult for students to resolve cognitive conflicts that they develop during pre-tertiary education which they are often not even aware of (e.g., Thoma & Nardi, 2018).

Mastering the geometry program at the university level requires at least the fourth level of geometric thinking according to van Hiele, but practice and various research in education show that students are not ready for this (e.g., Usiskin; 1982; Crowley 1987; De Villiers, 2009). To learn geometric concepts with understanding and their efficient application in solving problems and proving statements, students should have appropriate visual-spatial skills, since perception alone is not sufficient for mathematical processing of visual representations (Duval 1995; 1998). The causes of weak visualization skills of students of all ages, which are indicated by practice and various research in education, are primarily a dislike of the mathematical community to visualization as well as the difficulties that arise during visual processing (Presmeg, 1986; 2014). In the last few decades, the need for continuous and systematic development and use of visual-spatial abilities in teaching of mathematics throughout the entire educational vertical has been recognized and intensive work is being done to build a theoretical framework that would enable this (Presmeg, 2014).
As a result of the above, geometric concepts are largely mastered procedurally by memorizing appropriate definitions, rules and formulas without conceptual understanding. Hence, students are usually not ready to continue studying geometry at the university level. Therefore, it is possible to ask: How can we teach geometry to future teachers and prepare them to effectively teach geometry in primary education if they are not ready to continue learning according to the university discourse? In order to answer this question, an alternative teaching approach was prepared and implemented.

## THEORETICAL BACKGROUND

For effective learning and teaching of Euclidean geometry, it is necessary to know: the characteristics of the axiomatic construction of the deductive system, possible difficulties in learning and teaching geometric concepts, the appropriate theoretical framework for the development of geometric thinking as well as the characteristics of visualization elements in the context of geometry.

### Learning and teaching geometry as a deductive system

In the process of axiomatic construction of a deductive system, the processes of defining terms and the classifications based on them, the processes of setting statements, testing and proving their truth are important. The results of mathematics education research indicate the multiple benefits of learning formal definitions through the defining process, because in the case of giving ready-made definitions as they are usually learned by heart and without understanding (e.g., de Villiers, 2009). In addition, one should also take into account the duality of the process of acquiring a mathematical (as well as geometrical) concepts, especially the mutuality and completeness of these two processes (Viner, 1983) as well as the duality of the nature of geometric concepts and the balancing between its abstract and concrete properties (Fischbein & Nachlieli, 1998). In particular, in working with concrete visual representations, which are used to gain intuitive insight into abstract geometric concepts, various difficulties arise due to the particularity of the representation, the use of prototypes or the impossibility of "seeing" the representation in different ways (Yerushalmy & Chazan, 1990).

The defining process naturally continues by establishing connections between properties, of the same or different concepts, and ends with assertion (de Villiers, 2003). Making statements is based on basic logical operations, principles and reasoning, which serve as "glue" to connect the two essential parts of the statement (the assumption and the conclusion) and to ensure its exact meaning (Hammack, 2013). The skill of recognizing and parsing the assumption and the conclusion of a statement is a necessary prerequisite in setting up a converse statement, contrapositive and negation of a statement and proving or disproving its truth. Given that proven statements are most often used in mathematics classes, proof is used more for the purpose of persuasion, explanation, discovery, systematization, communication or intellectual challenge instead of determining the truth of the statement. In addition, a proof is a means of transmitting mathematical knowledge because a whole series of mathematical concepts, strategies, methods, established connections, and systematization are woven into the proof (de Villiers, 2003).

Finally, in the process of learning and teaching mathematics, one of the important activities is solving problems. Therefore, every teacher should know the characteristics of different types of tasks, different strategies and solving skills in order to provide his students with a suitable environment in which everyone will develop the necessary knowledge and skills according to their abilities. In this paper, three tasks classifications are considered: according to goal, demandingness and
As a result of the above, geometric concepts are largely mastered procedurally by memorizing appropriate definitions, rules and formulas without conceptual understanding. Hence, students are usually not ready to continue studying geometry at the university level. Therefore, it is possible to ask: How can we teach geometry to future teachers and prepare them to effectively teach geometry in primary education if they are not ready to continue learning according to the university discourse? In order to answer this question, an alternative teaching approach was prepared and implemented.

## THEORETICAL BACKGROUND

For effective learning and teaching of Euclidean geometry, it is necessary to know: the characteristics of the axiomatic construction of the deductive system, possible difficulties in learning and teaching geometric concepts, the appropriate theoretical framework for the development of geometric thinking as well as the characteristics of visualization elements in the context of geometry.

### Learning and teaching geometry as a deductive system

In the process of axiomatic construction of a deductive system, the processes of defining terms and the classifications based on them, the processes of setting statements, testing and proving their truth are important. The results of mathematics education research indicate the multiple benefits of learning formal definitions through the defining process, because in the case of giving ready-made definitions as they are usually learned by heart and without understanding (e.g., de Villiers, 2009). In addition, one should also take into account the duality of the process of acquiring a mathematical (as well as geometrical) concepts, especially the mutuality and completeness of these two processes (Viner, 1983) as well as the duality of the nature of geometric concepts and the balancing between its abstract and concrete properties (Fischbein & Nachlieli, 1998). In particular, in working with concrete visual representations, which are used to gain intuitive insight into abstract geometric concepts, various difficulties arise due to the particularity of the representation, the use of prototypes or the impossibility of "seeing" the representation in different ways (Yerushalmy & Chazan, 1990).

The defining process naturally continues by establishing connections between properties, of the same or different concepts, and ends with assertion (de Villiers, 2003). Making statements is based on basic logical operations, principles and reasoning, which serve as "glue" to connect the two essential parts of the statement (the assumption and the conclusion) and to ensure its exact meaning (Hammack, 2013). The skill of recognizing and parsing the assumption and the conclusion of a statement is a necessary prerequisite in setting up a converse statement, contrapositive and negation of a statement and proving or disproving its truth. Given that proven statements are most often used in mathematics classes, proof is used more for the purpose of persuasion, explanation, discovery, systematization, communication or intellectual challenge instead of determining the truth of the statement. In addition, a proof is a means of transmitting mathematical knowledge because a whole series of mathematical concepts, strategies, methods, established connections, and systematization are woven into the proof (de Villiers, 2003).

Finally, in the process of learning and teaching mathematics, one of the important activities is solving problems. Therefore, every teacher should know the characteristics of different types of tasks, different strategies and solving skills in order to provide his students with a suitable environment in which everyone will develop the necessary knowledge and skills according to their abilities. In this paper, three tasks classifications are considered: according to goal, demandingness and