The reflection task had low success rates for Low Spatial students and those who used embodiment. Mid Spatial students were equally successful whether using mental imagery or analytical strategies. High Spatial students, although still successful, were better off with a mental imagery strategy.

Students had greater success on the paper folding and cutting task when they used an analytical strategy, despite analytical strategies on both tasks involving matching components. High Spatial students had a significant advantage on this task which was closely aligned to the spatial measure. It is possible that High Spatial students had efficient spatial strategies for both tasks. A slight advantage was evident for Low Spatial students on this geometry task compared with the pure spatial measure.

## DISCUSSION

In this study we examined the intersection of visualization, in the form of strategies, and spatial visualization with geometry task performance amongst primary students. As expected, students with strong spatial skills performed better on the GeoT (Xie et al., 2020). Broadly, SV was more strongly associated with GeoT than with the use of visualization strategies. Previous work has considered these terms synonymous (Gutiérrez, 1996; Gutiérrez et al., 2018), but these results suggest that while visualization may be supported by SV skills, there is a degree of dissociation between the concepts. Overall, mental imagery strategies were the most frequently reported, followed by analytical and embodied. Strong spatial skills were associated with greater use of imagery and less embodiment.

Discrepancy scores revealed much greater (positive) scores for High Spatial students compared with Low and Mid, suggesting a tendency towards mental imagery over analytical strategies. Correlations revealed no significant relationship between SV and discrepancy. We interpret this to mean that although high SV scores and more mental imagery strategies were associated with better geometry performance, they were not inextricably linked for all students. That is, High Spatial students who tended to favor either mental imagery or analytic strategies were more successful on the GeoT than those who used the two strategy types equally. While for Low spatial students, the more mental imagery strategies they used, the greater their success on the GeoT. There was no relationship between analytic strategies and GeoT or SV. Literature would suggest that proficient students would resort to analytic strategies (Lowrie, 2020) but in this study, often the most effective and efficient strategies were mental imagery except for a small number of tasks and subset of High Spatial students.

### Item-based comparisons

Item analysis revealed some important considerations often missed when focusing on global scores. Firstly, mathematical language and conventions are important. As predicted by psychologists, geometry conventions can inhibit students from demonstrating their transformations skills without sufficient content knowledge (Newcombe, 2018). This was evident in the net tasks. Students of all spatial levels were successful in performing the required transformation when the task did not include the interpretation of mathematical language. As a first step, students need opportunities to develop and demonstrate their spatial skills that are not predicated on their content knowledge. Subsequently, students need support to link their conceptual, spatial knowledge with their geometric understanding.

The largest proportion of strategies were visualization, and most often mental imagery. This is not surprising given the nature of pre-proof geometry tasks. Visualization requires the integration of multiple representations, either from the page or screen to a mental transformation or from one type of physical representation to another. Spatial skills can support the transfer and integration of this
The reflection task had low success rates for Low Spatial students and those who used embodiment. Mid Spatial students were equally successful whether using mental imagery or analytical strategies. High Spatial students, although still successful, were better off with a mental imagery strategy.

Students had greater success on the paper folding and cutting task when they used an analytical strategy, despite analytical strategies on both tasks involving matching components. High Spatial students had a significant advantage on this task which was closely aligned to the spatial measure. It is possible that High Spatial students had efficient spatial strategies for both tasks. A slight advantage was evident for Low Spatial students on this geometry task compared with the pure spatial measure.

## DISCUSSION

In this study we examined the intersection of visualization, in the form of strategies, and spatial visualization with geometry task performance amongst primary students. As expected, students with strong spatial skills performed better on the GeoT (Xie et al., 2020). Broadly, SV was more strongly associated with GeoT than with the use of visualization strategies. Previous work has considered these terms synonymous (Gutiérrez, 1996; Gutiérrez et al., 2018), but these results suggest that while visualization may be supported by SV skills, there is a degree of dissociation between the concepts. Overall, mental imagery strategies were the most frequently reported, followed by analytical and embodied. Strong spatial skills were associated with greater use of imagery and less embodiment.

Discrepancy scores revealed much greater (positive) scores for High Spatial students compared with Low and Mid, suggesting a tendency towards mental imagery over analytical strategies. Correlations revealed no significant relationship between SV and discrepancy. We interpret this to mean that although high SV scores and more mental imagery strategies were associated with better geometry performance, they were not inextricably linked for all students. That is, High Spatial students who tended to favor either mental imagery or analytic strategies were more successful on the GeoT than those who used the two strategy types equally. While for Low spatial students, the more mental imagery strategies they used, the greater their success on the GeoT. There was no relationship between analytic strategies and GeoT or SV. Literature would suggest that proficient students would resort to analytic strategies (Lowrie, 2020) but in this study, often the most effective and efficient strategies were mental imagery except for a small number of tasks and subset of High Spatial students.

### Item-based comparisons

Item analysis revealed some important considerations often missed when focusing on global scores. Firstly, mathematical language and conventions are important. As predicted by psychologists, geometry conventions can inhibit students from demonstrating their transformations skills without sufficient content knowledge (Newcombe, 2018). This was evident in the net tasks. Students of all spatial levels were successful in performing the required transformation when the task did not include the interpretation of mathematical language. As a first step, students need opportunities to develop and demonstrate their spatial skills that are not predicated on their content knowledge. Subsequently, students need support to link their conceptual, spatial knowledge with their geometric understanding.

The largest proportion of strategies were visualization, and most often mental imagery. This is not surprising given the nature of pre-proof geometry tasks. Visualization requires the integration of multiple representations, either from the page or screen to a mental transformation or from one type of physical representation to another. Spatial skills can support the transfer and integration of this
information but they are not sufficient (Presmeg, 1986, 2014). If the mental representation of the task
is not functional, translating into a different representation may incorporate erroneous information
and leave students without a useful reproduction (Larkin & Simon, 1987). For each of the analyzed
tasks, spatial skills still underpinned the greatest variance in performance, but their influence was
dissociated from visualization strategies.

Some geometry tasks, particularly in primary school, are closely aligned to spatial measures, like 2D
to 3D conversions. These were the items in this study with the strongest evidence for the influence of
spatial skills. However, in some of these tasks, Low Spatial students were outperforming their Mid
spatial peers, even using spatial strategies. This shows the benefit and opportunities for incorporating
the geometry context in spatial tasks (Resnick & Stieff, 2024).

What can we learn from this?

Psychology offers tools to explore and build student spatial visualization in ways that are abstract but powerful for supporting geometry achievement (Hawes et al., 2023). This study has demonstrated the strength of the relationship, on top of learned visualization strategies. Visualization is more common in primary schools before complex geometric proofs become part of standard practice. Therefore, it is an ideal time to build spatial skills that can support future geometry learning (Woolcott et al., 2020). However, we cannot assume that students are going to intuitively connect spatial skills with geometry content, or effectively use visualization strategies despite the inherent links with geometry without considerable support (Gutiérrez et al., 2018; Lowrie & Logan, 2018; Presmeg, 2014).

Limitations and future directions

There were few tasks that could be solved routinely with computations. For most items, analytic
strategies involved conceptual knowledge of the structure of objects and inferences about what occurs
after prescribed changes, both still required visual processing (Larkin & Simon, 1987). This clearly
demonstrates visualization as “a necessary component for teaching and learning in geometry”
(Gutiérrez, 1996, p. 1-4), and both a product and a process (Arcavi, 2003; Bishop, 1980). However,
visualization is not a direct replacement for spatial skills. Future work could explore how these trends
hold up across geometry tasks that require more computation.

Acknowledgements

This project was funded by the Australian Research Council (DP170101755 and DP230100328).

References

Australian Curriculum and Reporting Authority [ACARA] (2023). *Mathematics: Space*. https://rb.gy/gyker

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. *Educational Studies in Mathematics*, *52*, 215-241.

Battista, M. T. (1990). Spatial visualization and gender differences in high school geometry. *Journal for Research in Mathematics Education, 21*(1), 47-60. https://psycnet.apa.org/doi/10.2307/749456

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.

Bishop, A. J. (1980). Spatial abilities and mathematics education – A review. *Educational Studies in Mathematics*, *11*, 257–269.
information but they are not sufficient (Presmeg, 1986, 2014). If the mental representation of the task
is not functional, translating into a different representation may incorporate erroneous information
and leave students without a useful reproduction (Larkin & Simon, 1987). For each of the analyzed
tasks, spatial skills still underpinned the greatest variance in performance, but their influence was
dissociated from visualization strategies.

Some geometry tasks, particularly in primary school, are closely aligned to spatial measures, like 2D
to 3D conversions. These were the items in this study with the strongest evidence for the influence of
spatial skills. However, in some of these tasks, Low Spatial students were outperforming their Mid
spatial peers, even using spatial strategies. This shows the benefit and opportunities for incorporating
the geometry context in spatial tasks (Resnick & Stieff, 2024).

What can we learn from this?

Psychology offers tools to explore and build student spatial visualization in ways that are abstract but powerful for supporting geometry achievement (Hawes et al., 2023). This study has demonstrated the strength of the relationship, on top of learned visualization strategies. Visualization is more common in primary schools before complex geometric proofs become part of standard practice. Therefore, it is an ideal time to build spatial skills that can support future geometry learning (Woolcott et al., 2020). However, we cannot assume that students are going to intuitively connect spatial skills with geometry content, or effectively use visualization strategies despite the inherent links with geometry without considerable support (Gutiérrez et al., 2018; Lowrie & Logan, 2018; Presmeg, 2014).

Limitations and future directions

There were few tasks that could be solved routinely with computations. For most items, analytic
strategies involved conceptual knowledge of the structure of objects and inferences about what occurs
after prescribed changes, both still required visual processing (Larkin & Simon, 1987). This clearly
demonstrates visualization as “a necessary component for teaching and learning in geometry”
(Gutiérrez, 1996, p. 1-4), and both a product and a process (Arcavi, 2003; Bishop, 1980). However,
visualization is not a direct replacement for spatial skills. Future work could explore how these trends
hold up across geometry tasks that require more computation.

Acknowledgements

This project was funded by the Australian Research Council (DP170101755 and DP230100328).

References

Australian Curriculum and Reporting Authority [ACARA] (2023). *Mathematics: Space*. https://rb.gy/gyker

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. *Educational Studies in Mathematics*, *52*, 215-241.

Battista, M. T. (1990). Spatial visualization and gender differences in high school geometry. *Journal for Research in Mathematics Education, 21*(1), 47-60. https://psycnet.apa.org/doi/10.2307/749456

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.

Bishop, A. J. (1980). Spatial abilities and mathematics education – A review. *Educational Studies in Mathematics*, *11*, 257–269.
Ekstrom, R. B., French, J. W., Harman, H. H., & Dermen, D. (1976). *Kit of factor-referenced cognitive tests*. Educational Testing Service.

Gutiérrez, A. (1996). Visualization in 3-dimensional geometry: In search of a framework. In: L. Puig & A. Gutiérrez (Eds.), *Proceedings of the 20th International Conference for the Psychology of Mathematics Education, 1*, (pp 3–19). Valencia, Spain

Gutiérrez, A., Ramírez, R., Benedicto, C., Beltrán-Meneu, M. J., & Jaime, A. (2018). Visualization abilities and complexity of reasoning in mathematically gifted students’ collaborative solutions to a visualization task: A networked analysis. In K. S. Mix & M. T. Battista (Eds.), *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (309-340). Springer.

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. http://dx.doi.org/10.1037/dev0001281

Hawes, Z., Gilligan-Lee, K. A., & Mix, K. S. (2023). Infusing spatial thinking into middle school mathematics: What, why, and how? In K.M. Robinson, A. Dubé, and D. Kostopoulos (Eds.), *Mathematical cognition and understanding* (13-33). Springer Nature Publishing.

Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. *Cognitive Science, 11*(1), 65–100. https://doi.org/10.1111/j.1551-6708.1987.tb00863.x

Lohman, D. F. (2012). *Cognitive Abilities Test, Form 7, book 11: Paper Folding*. Houghton Mifflin Harcourt.

Lowrie, T. (2020). The utility of diagrams in elementary problem-solving. *Cognitive Development, 55*.

Lowrie, T. & Logan, T. (2018). Part I Commentary 3: Proposing a pedagogical framework for the teaching and learning of spatial skills: A commentary on three chapters. In K. S. Mix & M. T. Battista (Eds.), *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (pp. 171–182). Springer.

Lowrie, T., Logan, T., & Ramful, A. (2016). Cross cultural comparison of grade 6 students’ performance and strategy use on graphic and non-graphic tasks. *Learning and Individual Differences, 52*, 97-108.

Mix, K. S., & Levine, S. C. (2018). Part II Commentary 2: Disparities and opportunities: Plotting a new course for research on spatial visualization and mathematics. In K. S. Mix & M. T. Battista (Eds.), *Visualizing Mathematics: The role of spatial reasoning in mathematical thought* (347-353). Springer.

Newcombe, N. S. (2018). Part II Commentary 3: Linking spatial and mathematical thinking: The search for mechanism. In K. S. Mix & M. T. Battista (Eds.), *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (pp. 355–360). Springer.

Presmeg, N. (1986). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42-46.

Presmeg, N. (2014). Contemplating visualization as an epistemological learning tool in mathematics. ZDM, 46(1), 151–157. https://doi.org/10.1007/s11858-013-0561-z

Resnick, I. & Stieff, M. (2024). Characterizing cognition and context within mathematics education. In M. A. Clements, B. Kaur, T. Lowrie, V. Mesa, & J. Prytz (Eds.) *Fourth international handbook of mathematics education: Controversial issues in mathematics education*. Springer

Suwarsono, S. (1982). *Visual imagery in the mathematical thinking of seventh grade students* [Doctoral Dissertation, Monash University].

Uttal, D. H., & Cohen, C. A. (2012). Chapter four - Spatial thinking and STEM education: When, why, and how? *Psychology of Learning and Motivation, 57*, 147-181.

Woolcott, G., Logan, T., Marshman, M., Ramful, A., Whannell, R., & Lowrie, T. (2020). The re-emergence of spatial reasoning within primary years mathematics education. In J. Way, C. Attard, J. Anderson, J. Bobis, H. McMaster, & K. Cartwright. (Eds.), *Research in mathematics education in Australasia 2016–2019* (pp. 245–268). Springer.

Xie, F., Zhang, L., Chen, X., & Xin, Z. (2020). Is spatial ability related to mathematical ability: A meta-analysis. *Educational Psychology Review, 32*, 113–155. https://doi.org/10.1007/s10648-019-09496-y
Ekstrom, R. B., French, J. W., Harman, H. H., & Dermen, D. (1976). *Kit of factor-referenced cognitive tests*. Educational Testing Service.

Gutiérrez, A. (1996). Visualization in 3-dimensional geometry: In search of a framework. In: L. Puig & A. Gutiérrez (Eds.), *Proceedings of the 20th International Conference for the Psychology of Mathematics Education, 1*, (pp 3–19). Valencia, Spain

Gutiérrez, A., Ramírez, R., Benedicto, C., Beltrán-Meneu, M. J., & Jaime, A. (2018). Visualization abilities and complexity of reasoning in mathematically gifted students’ collaborative solutions to a visualization task: A networked analysis. In K. S. Mix & M. T. Battista (Eds.), *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (309-340). Springer.

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. http://dx.doi.org/10.1037/dev0001281

Hawes, Z., Gilligan-Lee, K. A., & Mix, K. S. (2023). Infusing spatial thinking into middle school mathematics: What, why, and how? In K.M. Robinson, A. Dubé, and D. Kostopoulos (Eds.), *Mathematical cognition and understanding* (13-33). Springer Nature Publishing.

Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. *Cognitive Science, 11*(1), 65–100. https://doi.org/10.1111/j.1551-6708.1987.tb00863.x

Lohman, D. F. (2012). *Cognitive Abilities Test, Form 7, book 11: Paper Folding*. Houghton Mifflin Harcourt.

Lowrie, T. (2020). The utility of diagrams in elementary problem-solving. *Cognitive Development, 55*.

Lowrie, T. & Logan, T. (2018). Part I Commentary 3: Proposing a pedagogical framework for the teaching and learning of spatial skills: A commentary on three chapters. In K. S. Mix & M. T. Battista (Eds.), *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (pp. 171–182). Springer.

Lowrie, T., Logan, T., & Ramful, A. (2016). Cross cultural comparison of grade 6 students’ performance and strategy use on graphic and non-graphic tasks. *Learning and Individual Differences, 52*, 97-108.

Mix, K. S., & Levine, S. C. (2018). Part II Commentary 2: Disparities and opportunities: Plotting a new course for research on spatial visualization and mathematics. In K. S. Mix & M. T. Battista (Eds.), *Visualizing Mathematics: The role of spatial reasoning in mathematical thought* (347-353). Springer.

Newcombe, N. S. (2018). Part II Commentary 3: Linking spatial and mathematical thinking: The search for mechanism. In K. S. Mix & M. T. Battista (Eds.), *Visualizing mathematics: The role of spatial reasoning in mathematical thought* (pp. 355–360). Springer.

Presmeg, N. (1986). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42-46.

Presmeg, N. (2014). Contemplating visualization as an epistemological learning tool in mathematics. ZDM, 46(1), 151–157. https://doi.org/10.1007/s11858-013-0561-z

Resnick, I. & Stieff, M. (2024). Characterizing cognition and context within mathematics education. In M. A. Clements, B. Kaur, T. Lowrie, V. Mesa, & J. Prytz (Eds.) *Fourth international handbook of mathematics education: Controversial issues in mathematics education*. Springer

Suwarsono, S. (1982). *Visual imagery in the mathematical thinking of seventh grade students* [Doctoral Dissertation, Monash University].

Uttal, D. H., & Cohen, C. A. (2012). Chapter four - Spatial thinking and STEM education: When, why, and how? *Psychology of Learning and Motivation, 57*, 147-181.

Woolcott, G., Logan, T., Marshman, M., Ramful, A., Whannell, R., & Lowrie, T. (2020). The re-emergence of spatial reasoning within primary years mathematics education. In J. Way, C. Attard, J. Anderson, J. Bobis, H. McMaster, & K. Cartwright. (Eds.), *Research in mathematics education in Australasia 2016–2019* (pp. 245–268). Springer.

Xie, F., Zhang, L., Chen, X., & Xin, Z. (2020). Is spatial ability related to mathematical ability: A meta-analysis. *Educational Psychology Review, 32*, 113–155. https://doi.org/10.1007/s10648-019-09496-y
# DISCOVERING GEOMETRY IN AFRICAN ETHNO ARTIFACTS

Jasmina Milinković, Maria Vorkapić

University of Belgrade, Faculty of Education

The preschool experience opens an avenue for developing spatial reasoning and introducing geometrical concepts. This paper describes a design study, a curricular experiment – implemented at the University of Belgrade in Serbia with prospective preschool teachers. The research project aimed to document the development of students’ math pedagogy when studying potentials for preschoolers’ exploration of mathematical ideas in the context of the Museum of African Art. Through exploration of ethno artifacts students had the opportunity to learn to reflect on the culturally defined presence of geometrical ideas in them. Here we discuss findings related to students discovering geometry while studying ethno art. The findings are based on a sample of 46 prospective preschool teachers’ interviews. The analysis reveals that the students show sensitivity to the presence of geometrical ideas in ethno artifacts from Africa and recognize opportunities for triggering the development of geometrical reasoning in preschool children.

## INTRODUCTION

Geometry is all around us. This statement is one of the ideas of the Methodological class at the Faculty of Education for prospective preschool teachers in their final year of studying to become teachers for children aged 3 to 7. It directs students to search for the presence of geometrical ideas not only in nature but also in elements of culture (art, architecture, customs, home ethno products, etc.).

Research indicates that student achievement in the field of geometry and measurement is weaker than in other areas of mathematics represented at school (Milinković et al., 2017, Outhred & Mitchelmore, 2000, Tan-Sisman & Aksu, 2016). Understanding geometrical concepts is a complex construct that includes several elements that are not always simultaneously in focus (visualization, measurement, relations, or transformations (Sinclair et al., 2016). The main factor of limited achievement in learning geometry in primary grades is too much focus on procedural knowledge (e.g. learning formulas for calculating perimeter, area, or other quantitative characteristics of geometrical objects) instead of putting emphasis on developing conceptual understanding of basic geometrical concepts and attending to the development of spatial reasoning through observing similarities and differences between objects and their position in space (Tan-Sisman & Aksu, 2015). Some elements of geometrical thinking can be developed in preschool age. An early start of learning geometry in preschool is promising to support better achievements in school geometry.

The major objective of the preschool program in Serbia is to support the holistic development of young children situated in culture (MPNTR, 2018). Early intuitive development of spatial reasoning is indicated in research on early mathematics development. (Piaget, 1976, Clements, 2003, Clements & Battista, 1992, Smith et al., 2011). The preschool curriculum is grounded in the same socio-constructivist position. Like other math domains, the development of initial geometrical concepts
# DISCOVERING GEOMETRY IN AFRICAN ETHNO ARTIFACTS

Jasmina Milinković, Maria Vorkapić

University of Belgrade, Faculty of Education

The preschool experience opens an avenue for developing spatial reasoning and introducing geometrical concepts. This paper describes a design study, a curricular experiment – implemented at the University of Belgrade in Serbia with prospective preschool teachers. The research project aimed to document the development of students’ math pedagogy when studying potentials for preschoolers’ exploration of mathematical ideas in the context of the Museum of African Art. Through exploration of ethno artifacts students had the opportunity to learn to reflect on the culturally defined presence of geometrical ideas in them. Here we discuss findings related to students discovering geometry while studying ethno art. The findings are based on a sample of 46 prospective preschool teachers’ interviews. The analysis reveals that the students show sensitivity to the presence of geometrical ideas in ethno artifacts from Africa and recognize opportunities for triggering the development of geometrical reasoning in preschool children.

## INTRODUCTION

Geometry is all around us. This statement is one of the ideas of the Methodological class at the Faculty of Education for prospective preschool teachers in their final year of studying to become teachers for children aged 3 to 7. It directs students to search for the presence of geometrical ideas not only in nature but also in elements of culture (art, architecture, customs, home ethno products, etc.).

Research indicates that student achievement in the field of geometry and measurement is weaker than in other areas of mathematics represented at school (Milinković et al., 2017, Outhred & Mitchelmore, 2000, Tan-Sisman & Aksu, 2016). Understanding geometrical concepts is a complex construct that includes several elements that are not always simultaneously in focus (visualization, measurement, relations, or transformations (Sinclair et al., 2016). The main factor of limited achievement in learning geometry in primary grades is too much focus on procedural knowledge (e.g. learning formulas for calculating perimeter, area, or other quantitative characteristics of geometrical objects) instead of putting emphasis on developing conceptual understanding of basic geometrical concepts and attending to the development of spatial reasoning through observing similarities and differences between objects and their position in space (Tan-Sisman & Aksu, 2015). Some elements of geometrical thinking can be developed in preschool age. An early start of learning geometry in preschool is promising to support better achievements in school geometry.

The major objective of the preschool program in Serbia is to support the holistic development of young children situated in culture (MPNTR, 2018). Early intuitive development of spatial reasoning is indicated in research on early mathematics development. (Piaget, 1976, Clements, 2003, Clements & Battista, 1992, Smith et al., 2011). The preschool curriculum is grounded in the same socio-constructivist position. Like other math domains, the development of initial geometrical concepts
occurs in a shared socio-cultural environment. Ethno museum with cultural artifacts from Africa is a
such context.

THEORETICAL BACKGROUND

Ethnomathematics is a science studying mathematical practices and transmission of mathematical knowledge of a culturally definable group (Fossa, 2023, D'Ambrosio, 1998). The last years have shown fast growth of ethnomathematics studies in geometry teaching yet the number is still modest; only 37 studies were conducted between 2011 and 2021 (Kyeremaeh et al., 2023). For example, Patricia Elena Ceballos Pimentel (2012) explored the possibility of teaching geometry based on the drawings of the Mapuche, a group of indigenous peoples from central Chile. In response, Fossa remarked that identifying triangles or calculating perimeters and areas in the drawings imposes academic mathematics on the native work. He points out that instead a Mapuche geometry or any other group geometry should be explored in the original cultural context. Our interest brought us to explore geometry within a collection of African artifacts. It provided a window into the significant presence of geometrical ideas in African culture.

The way prospective preschool teachers are educated reflects the way we expect them to work on the
development of mathematics concepts for children at preschool. We operate from a social
constructivist position (von Glaserfeld, 1990). The learning process is modeled by culture and evolves
and changes in interactions of individuals with other people and cultural tools (Milinković &
Bogavac, 2011). The preschool curriculum recommends the project approach as a dominant. A
project is an in-depth investigation by children of a topic that is worthy of their time, attention, and
energy (Katz & Chard, 2000). A project involves three phases selection, investigation, and reflection.
Using cultural tools and products (such as what are symbols, objects, and customs...), as well as
different strategies in building meaning, children learn in daily participation of the community's
practices in which they are involved (family, kindergarten, immediate environment). The project
approach provides opportunities for children to represent this learning through the construction of
personally meaningful products. Teachers are expected to facilitate the process in the project
including all phases of the project, planning, exploration, and culmination of the project, towards
shared outcomes or the creation of artifacts, providing opportunities for children to explore in small
groups, children are involved in investigating authentic objects and resources and reflecting on the
outcomes of the project. Contextualized problems function as a source for the learning process. For
example, the idea of searching for mathematical concepts in art was explored earlier (Grzegorczyk &
Stylianou, 2006). The aim of such activities is for the preschool level to develop predispositions for
searching for mathematics and recognizing mathematical ideas in their surroundings.

METHODOLOGY

During the summer semester of the final academic year, within the Methodology of Developing Basic
Mathematical Concepts course for prospective preschool teachers, the topic of the experimental
project-based program was African culture. Our students were already familiarized with the basics of
the integrative approach and have designed preschool activities exploring geometric objects (line,
point, surface), geometric shapes and solids, spatial relations (position, size, visual comparison of
shapes, shapes of objects, symmetry, patterns). The aim of the program was for students to further
develop skills to identify/create opportunities to recognize and connect mathematical ideas and grow
occurs in a shared socio-cultural environment. Ethno museum with cultural artifacts from Africa is a
such context.

THEORETICAL BACKGROUND

Ethnomathematics is a science studying mathematical practices and transmission of mathematical knowledge of a culturally definable group (Fossa, 2023, D'Ambrosio, 1998). The last years have shown fast growth of ethnomathematics studies in geometry teaching yet the number is still modest; only 37 studies were conducted between 2011 and 2021 (Kyeremaeh et al., 2023). For example, Patricia Elena Ceballos Pimentel (2012) explored the possibility of teaching geometry based on the drawings of the Mapuche, a group of indigenous peoples from central Chile. In response, Fossa remarked that identifying triangles or calculating perimeters and areas in the drawings imposes academic mathematics on the native work. He points out that instead a Mapuche geometry or any other group geometry should be explored in the original cultural context. Our interest brought us to explore geometry within a collection of African artifacts. It provided a window into the significant presence of geometrical ideas in African culture.

The way prospective preschool teachers are educated reflects the way we expect them to work on the
development of mathematics concepts for children at preschool. We operate from a social
constructivist position (von Glaserfeld, 1990). The learning process is modeled by culture and evolves
and changes in interactions of individuals with other people and cultural tools (Milinković &
Bogavac, 2011). The preschool curriculum recommends the project approach as a dominant. A
project is an in-depth investigation by children of a topic that is worthy of their time, attention, and
energy (Katz & Chard, 2000). A project involves three phases selection, investigation, and reflection.
Using cultural tools and products (such as what are symbols, objects, and customs...), as well as
different strategies in building meaning, children learn in daily participation of the community's
practices in which they are involved (family, kindergarten, immediate environment). The project
approach provides opportunities for children to represent this learning through the construction of
personally meaningful products. Teachers are expected to facilitate the process in the project
including all phases of the project, planning, exploration, and culmination of the project, towards
shared outcomes or the creation of artifacts, providing opportunities for children to explore in small
groups, children are involved in investigating authentic objects and resources and reflecting on the
outcomes of the project. Contextualized problems function as a source for the learning process. For
example, the idea of searching for mathematical concepts in art was explored earlier (Grzegorczyk &
Stylianou, 2006). The aim of such activities is for the preschool level to develop predispositions for
searching for mathematics and recognizing mathematical ideas in their surroundings.

METHODOLOGY

During the summer semester of the final academic year, within the Methodology of Developing Basic
Mathematical Concepts course for prospective preschool teachers, the topic of the experimental
project-based program was African culture. Our students were already familiarized with the basics of
the integrative approach and have designed preschool activities exploring geometric objects (line,
point, surface), geometric shapes and solids, spatial relations (position, size, visual comparison of
shapes, shapes of objects, symmetry, patterns). The aim of the program was for students to further
develop skills to identify/create opportunities to recognize and connect mathematical ideas and grow
awareness to different cultural contexts. The experimental program started with a guided tour of the Museum of African Art. The student's task was to come up with ideas for initiating research and playful activities with children based on what they observed and learned in the museum. The students created didactic tools to be used in game-like activities with young children. These didactical tools were displayed at the final exhibition.

A sample of 46 students was involved in the study.

The part of the experimental program reported here involved a visit to the Museum of African Art, situated in Belgrade. *The Museum of African Art – the Veda and Dr. Zdravko Pečar Collection* (https://www.mau.rs/en/) is exclusively dedicated to presenting and interpreting the art and culture of Africa and is the only specialized institution of its kind in Southeast Europe. The Museum exhibits a collection of cultural artifacts from different parts of Africa. The collection includes masks, sculptures in wood, bronze sculptures, bronze weights, ceramics, musical instruments, fabrics, jewelry, and items of everyday use. "Geometric representations exist in all cultural groups and their meaning may be different from one specific culture to another", as it was noted in the discussion document for this conference, and African cultural artifacts are no exception from this observation. The Museum guided tour included information about how the objects were originally used, stories related to the history of the African continent, and particular artifacts and different cultures living in Africa. For example, the tour guide remarked that it is visible that African masks incorporate "geometrics", the symbolism of what each mask represents, the fact that a person under a mask "loses personal identity", when they were used, who used them (tribal chiefs, sorcerers, or ritual dancers), the symbolism of animals on masks, and the mask's connecting the heavenly and the earthly.

<figure><img src="image_209.png" /><figcaption>Figure 1: Ntomo mask, Bambara mask, and Chi-wara mask, from left to right</figcaption></figure>

The guided tour finished with a workshop on the traditional African game Mankala. The visit was followed by a wrap-up lesson at the Faculty. The final part of the project was designing a didactical tool activity ending with the exhibition titled *Discovering Mathematics on African Pathways*. Inspired by a visit to the Museum of African Art, the students came up with ideas for initiating research and playful activities with children. The didactical tools that were exhibited at this exhibition were used in activities with children. These tools, based not only on the museum artifacts but also on students' investigation of the topic, demonstrated the creativity and scope of the students' ideas.

Based on the goal of the research, the following tasks were formulated:

1. Determine the scope of geometrical ideas recognized in African cultural artifacts.

2. Analyze how much of and in what way geometrical concepts were present in students' didactical activities and products.
awareness to different cultural contexts. The experimental program started with a guided tour of the Museum of African Art. The student's task was to come up with ideas for initiating research and playful activities with children based on what they observed and learned in the museum. The students created didactic tools to be used in game-like activities with young children. These didactical tools were displayed at the final exhibition.

A sample of 46 students was involved in the study.

The part of the experimental program reported here involved a visit to the Museum of African Art, situated in Belgrade. *The Museum of African Art – the Veda and Dr. Zdravko Pečar Collection* (https://www.mau.rs/en/) is exclusively dedicated to presenting and interpreting the art and culture of Africa and is the only specialized institution of its kind in Southeast Europe. The Museum exhibits a collection of cultural artifacts from different parts of Africa. The collection includes masks, sculptures in wood, bronze sculptures, bronze weights, ceramics, musical instruments, fabrics, jewelry, and items of everyday use. "Geometric representations exist in all cultural groups and their meaning may be different from one specific culture to another", as it was noted in the discussion document for this conference, and African cultural artifacts are no exception from this observation. The Museum guided tour included information about how the objects were originally used, stories related to the history of the African continent, and particular artifacts and different cultures living in Africa. For example, the tour guide remarked that it is visible that African masks incorporate "geometrics", the symbolism of what each mask represents, the fact that a person under a mask "loses personal identity", when they were used, who used them (tribal chiefs, sorcerers, or ritual dancers), the symbolism of animals on masks, and the mask's connecting the heavenly and the earthly.

<figure><img src="image_221.png" /><figcaption>Figure 1: Ntomo mask, Bambara mask, and Chi-wara mask, from left to right</figcaption></figure>

The guided tour finished with a workshop on the traditional African game Mankala. The visit was followed by a wrap-up lesson at the Faculty. The final part of the project was designing a didactical tool activity ending with the exhibition titled *Discovering Mathematics on African Pathways*. Inspired by a visit to the Museum of African Art, the students came up with ideas for initiating research and playful activities with children. The didactical tools that were exhibited at this exhibition were used in activities with children. These tools, based not only on the museum artifacts but also on students' investigation of the topic, demonstrated the creativity and scope of the students' ideas.

Based on the goal of the research, the following tasks were formulated:

1. Determine the scope of geometrical ideas recognized in African cultural artifacts.

2. Analyze how much of and in what way geometrical concepts were present in students' didactical activities and products.
The museum visit provided context for the development of project activities inspired by the visit to the Museum and the implementation of the activity during in-preschool practice month. Major sources for researchers were 1) interviews, 2) students' practice diaries, and 3D didactical game models. The interviewer used an interview protocol consisting of nine focus questions. Respondents were asked to reflect on 1) their memories of the museum visit and 2) their project (plan of activities and results of the activities). All individual interviews were recorded and transcribed for qualitative analysis. In addition, students' practice diaries were used as evidence supporting claims made in interviews. The obtained data were processed by the method of statistical and descriptive qualitative analysis.

## RESULTS

To begin, we will look at what students find remarkable at the exhibition and to what extent it is reflected in their choice of activity. The results show that Masks and the Mancala game earned the most attention but masks and fabrics were the two most explored objects in activities (Table 1).

<table><caption>Table 1: Artifacts observed and used</caption><thead><tr><td>Artifacts</td><td>Remembered</td><td>Used inactivity</td></tr></thead><tbody><tr><td>Masks</td><td>27</td><td>13</td></tr><tr><td>Fabrics</td><td>11</td><td>12</td></tr><tr><td>Mancala</td><td>24</td><td>5</td></tr><tr><td>Jewelry</td><td>5</td><td>3</td></tr><tr><td>Wooden Sculptures (humans, animals)</td><td>10</td><td>2</td></tr><tr><td>Bronze sculpture and weights</td><td>0</td><td>1</td></tr><tr><td>Ceramics</td><td>1</td><td>1</td></tr><tr><td>Musical instruments</td><td>2</td><td>1</td></tr><tr><td>Items of everyday use</td><td>3</td><td>0</td></tr></tbody></table>

An example of the way artifacts can be used in the project is a worksheet with tasks involving photographs of artifacts from the museum (Figure 2). Children were asked to recognize a pattern and count the number of appearances of the pattern. Others involved children in creating similar artwork such as a string of beads.

<figure><img src="image_210.png" /><figcaption>Figure 2: A pattern task from a worksheet</figcaption></figure>
The museum visit provided context for the development of project activities inspired by the visit to the Museum and the implementation of the activity during in-preschool practice month. Major sources for researchers were 1) interviews, 2) students' practice diaries, and 3D didactical game models. The interviewer used an interview protocol consisting of nine focus questions. Respondents were asked to reflect on 1) their memories of the museum visit and 2) their project (plan of activities and results of the activities). All individual interviews were recorded and transcribed for qualitative analysis. In addition, students' practice diaries were used as evidence supporting claims made in interviews. The obtained data were processed by the method of statistical and descriptive qualitative analysis.

## RESULTS

To begin, we will look at what students find remarkable at the exhibition and to what extent it is reflected in their choice of activity. The results show that Masks and the Mancala game earned the most attention but masks and fabrics were the two most explored objects in activities (Table 1).

<table><caption>Table 1: Artifacts observed and used</caption><thead><tr><td>Artifacts</td><td>Remembered</td><td>Used inactivity</td></tr></thead><tbody><tr><td>Masks</td><td>27</td><td>13</td></tr><tr><td>Fabrics</td><td>11</td><td>12</td></tr><tr><td>Mancala</td><td>24</td><td>5</td></tr><tr><td>Jewelry</td><td>5</td><td>3</td></tr><tr><td>Wooden Sculptures (humans, animals)</td><td>10</td><td>2</td></tr><tr><td>Bronze sculpture and weights</td><td>0</td><td>1</td></tr><tr><td>Ceramics</td><td>1</td><td>1</td></tr><tr><td>Musical instruments</td><td>2</td><td>1</td></tr><tr><td>Items of everyday use</td><td>3</td><td>0</td></tr></tbody></table>

An example of the way artifacts can be used in the project is a worksheet with tasks involving photographs of artifacts from the museum (Figure 2). Children were asked to recognize a pattern and count the number of appearances of the pattern. Others involved children in creating similar artwork such as a string of beads.

<figure><img src="image_222.png" /><figcaption>Figure 2: A pattern task from a worksheet</figcaption></figure>
Another student used a bird fabric motif to explore the idea of a pattern. She created a drawing of birds with patterns of colors and discussed with children about these patterns and also the patterns of the birds seen at the museum. So, as the student wrote in her diary of practice the dialog with children included the following questions:

1. What kind of bird is this? 2. Did you notice anything on her body repeating itself? If yes, what? 4. How do colors repeat? In what order? 5. How are these two birds different? 6. What is the tail of another bird? Compare it with the first. 6. What is their difference, and what is their similarity? 7. How are colors repeated? 8. How is this bird different from the previous two birds? 9. What's her tail like? 10. What does the battle consist of? 11. How do they recur?

Children drew birds (Figure 3) with geometrical patterns and discussed the differences between birds. The majority of five-year-old children created simple two-color patterned bird's tails while African birds have had five colors. After finishing their drawing, young children described patterns, which colors were used, what was the order of colors, and how many elements the pattern consisted of. At the end, they have an exhibition of works and plays.

<figure><img src="image_211.png" /><figcaption>Figure 3: The African bird and children's productions</figcaption></figure>

Our first research question was to investigate how much geometry students recognized in the objects displayed at the exhibition. Also, we investigated how many geometry-related ideas they attended to in the projects with young children. Several different ideas were recognized by a considerable number of students during the observation of the exhibition. Notable fewer of these ideas were discussed later, in projects with children (Table 2).

<table><caption>Table 1: Observed and followed in projects' geometrical concepts</caption><thead><tr><th>Geometrical concept</th><th>Observed concept<br>in ethno artifacts</th><th>The concept incorporated<br>into activities with children</th></tr></thead><tbody><tr><td>Patterns (geometric pattern, number pattern)</td><td>46</td><td>35</td></tr><tr><td>Geometric shapes (geometric 2D-figures)</td><td>45</td><td>22</td></tr><tr><td>Symmetry</td><td>43</td><td>6</td></tr><tr><td>Classification</td><td>15</td><td>5</td></tr><tr><td>Line</td><td>24</td><td>4</td></tr><tr><td>Spatial reasoning (relations, position of objects)</td><td>10</td><td>4</td></tr></tbody></table>
Another student used a bird fabric motif to explore the idea of a pattern. She created a drawing of birds with patterns of colors and discussed with children about these patterns and also the patterns of the birds seen at the museum. So, as the student wrote in her diary of practice the dialog with children included the following questions:

1. What kind of bird is this? 2. Did you notice anything on her body repeating itself? If yes, what? 4. How do colors repeat? In what order? 5. How are these two birds different? 6. What is the tail of another bird? Compare it with the first. 6. What is their difference, and what is their similarity? 7. How are colors repeated? 8. How is this bird different from the previous two birds? 9. What's her tail like? 10. What does the battle consist of? 11. How do they recur?

Children drew birds (Figure 3) with geometrical patterns and discussed the differences between birds. The majority of five-year-old children created simple two-color patterned bird's tails while African birds have had five colors. After finishing their drawing, young children described patterns, which colors were used, what was the order of colors, and how many elements the pattern consisted of. At the end, they have an exhibition of works and plays.

<figure><img src="image_223.png" /><figcaption>Figure 3: The African bird and children's productions</figcaption></figure>

Our first research question was to investigate how much geometry students recognized in the objects displayed at the exhibition. Also, we investigated how many geometry-related ideas they attended to in the projects with young children. Several different ideas were recognized by a considerable number of students during the observation of the exhibition. Notable fewer of these ideas were discussed later, in projects with children (Table 2).

<table><caption>Table 1: Observed and followed in projects' geometrical concepts</caption><thead><tr><th>Geometrical concept</th><th>Observed concept<br>in ethno artifacts</th><th>The concept incorporated<br>into activities with children</th></tr></thead><tbody><tr><td>Patterns (geometric pattern, number pattern)</td><td>46</td><td>35</td></tr><tr><td>Geometric shapes (geometric 2D-figures)</td><td>45</td><td>22</td></tr><tr><td>Symmetry</td><td>43</td><td>6</td></tr><tr><td>Classification</td><td>15</td><td>5</td></tr><tr><td>Line</td><td>24</td><td>4</td></tr><tr><td>Spatial reasoning (relations, position of objects)</td><td>10</td><td>4</td></tr></tbody></table>
While some students mentioned only one math concept, some recognized mathematics in many places. For example, student Sonja reflected on many ideas brought about in her project with children.

Sonja: "We a) arranged rows of patterns can be observed on animals (on fabrics). Some animals can observe lines and circles alternate and create a pattern. Also, geometric concepts, such as circles and lines are present. b) On some fabrics the pattern is created by linearly changing a square, a square consisting of four triangles, and an octagon. Symmetry can also be seen in the painted shape below the pattern, c) symmetry can be observed because if we divide each of these wooden spoons into two halves longitudinally, both halves would be identical. An elliptical shape and a circle shape are observed, and d) symmetry in the form of a mask is observable; if we do not ignore the lack of shells on one side, i.e. half of the mask. The pattern is made by arranging the shells in an ordered sequence, linearly and cyclically, e) geometric terms (circle, triangle, lines, rectangle) are observable, patterns are observed on the bodies of animals, f) symmetry and pattern consisting of squares and rectangles are observed, e) symmetry is observed if we cross it, a pattern is observed, a repetition of the field."

For the second research question, we asked students to conceive of the possibility of extending the project and how would he/she proceed with the project. The majority of students responded positively that their project could be prolonged but few of them were thinking about further development of math concepts, and even fewer remembered geometry concepts in a perceived extension of the project (Table 2).

<table><caption>Table 2: Question: In which direction would you further develop the project?</caption><thead><tr><th>Choice of domain for extension of the project</th><th>Topics</th></tr></thead><tbody><tr><td>Geometry</td><td>basic geometry concepts (line, curve, point)<br>symmetry<br>geometric shapes</td></tr><tr><td>Numbers</td><td></td></tr><tr><td>Arts, Science, Language</td><td></td></tr><tr><td>Not developing project</td><td></td></tr></tbody></table>

Ana envisioned moving from Africa to other cultures while Petra thought about ways to further explore artifacts from the Museum of African Art, primarily focusing on patterns.

Ana: "In addition to getting to know Africa, I would introduce children to other cultures.

Petra: "Another student said: I would take the children to the Museum of African Art, to look at the sculptures and masks together. They would still deal with African culture. They would observe them - make sculptures out of dough and clay - draw patterns, make a series and a pattern."

Tara: "In addition, we could continue to deal with Africa, their instruments, masks, clothes, etc. We could visit the museum with the children and see what interests them the most because that will be the easiest way to organize the activity."

Most of the prospective preschool teachers in the class, similarly to Petra remarked that they would go with children to the Museum, to identify what children find interesting to explore further.

The final project activity resulted in a collection of 33 models of didactical tools which were displayed in an exhibition at the Faculty building. For each product, a group of students who designed it explained what they meant to be the purpose of the tool. The analysis showed that students most
While some students mentioned only one math concept, some recognized mathematics in many places. For example, student Sonja reflected on many ideas brought about in her project with children.

Sonja: "We a) arranged rows of patterns can be observed on animals (on fabrics). Some animals can observe lines and circles alternate and create a pattern. Also, geometric concepts, such as circles and lines are present. b) On some fabrics the pattern is created by linearly changing a square, a square consisting of four triangles, and an octagon. Symmetry can also be seen in the painted shape below the pattern, c) symmetry can be observed because if we divide each of these wooden spoons into two halves longitudinally, both halves would be identical. An elliptical shape and a circle shape are observed, and d) symmetry in the form of a mask is observable; if we do not ignore the lack of shells on one side, i.e. half of the mask. The pattern is made by arranging the shells in an ordered sequence, linearly and cyclically, e) geometric terms (circle, triangle, lines, rectangle) are observable, patterns are observed on the bodies of animals, f) symmetry and pattern consisting of squares and rectangles are observed, e) symmetry is observed if we cross it, a pattern is observed, a repetition of the field."

For the second research question, we asked students to conceive of the possibility of extending the project and how would he/she proceed with the project. The majority of students responded positively that their project could be prolonged but few of them were thinking about further development of math concepts, and even fewer remembered geometry concepts in a perceived extension of the project (Table 2).

<table><caption>Table 2: Question: In which direction would you further develop the project?</caption><thead><tr><th>Choice of domain for extension of the project</th><th>Topics</th></tr></thead><tbody><tr><td>Geometry</td><td>basic geometry concepts (line, curve, point)<br>symmetry<br>geometric shapes</td></tr><tr><td>Numbers</td><td></td></tr><tr><td>Arts, Science, Language</td><td></td></tr><tr><td>Not developing project</td><td></td></tr></tbody></table>

Ana envisioned moving from Africa to other cultures while Petra thought about ways to further explore artifacts from the Museum of African Art, primarily focusing on patterns.

Ana: "In addition to getting to know Africa, I would introduce children to other cultures.

Petra: "Another student said: I would take the children to the Museum of African Art, to look at the sculptures and masks together. They would still deal with African culture. They would observe them - make sculptures out of dough and clay - draw patterns, make a series and a pattern."

Tara: "In addition, we could continue to deal with Africa, their instruments, masks, clothes, etc. We could visit the museum with the children and see what interests them the most because that will be the easiest way to organize the activity."

Most of the prospective preschool teachers in the class, similarly to Petra remarked that they would go with children to the Museum, to identify what children find interesting to explore further.

The final project activity resulted in a collection of 33 models of didactical tools which were displayed in an exhibition at the Faculty building. For each product, a group of students who designed it explained what they meant to be the purpose of the tool. The analysis showed that students most
frequently expected that playing with the tool would contribute to children’s formation of concepts of geometrical shapes and symmetry. Their explanations however lack elaboration or complexity. Notable, all references to symmetry were made concerning mirror symmetry only. We show two examples of games students created inspired by Masks (Figure 4). In both, the expectation was that children would identify and name shapes. Similarly simple expectations repeatedly occurred in games involving patterns. All references to geometrical patterns referred only to linear patterns consisting of geometrical shapes (triangle, square, rectangle).

<figure><img src="image_212.png" /><figcaption>Figure 4: Masks as a didactical tool</figcaption></figure>

## DISCUSSION

While most students recognized mathematics in art pieces displayed in the Museum, some did not find a way to include directly any of the observed ideas in their plan of activity. This may be due to a lack of experience in project planning. Over 90 percent of students recognized patterns, geometric shapes, and symmetry in African exhibits. However, only the idea of the pattern was frequently explored in activities. Only about half of the students (47%) had activities involving geometric shapes and numbers. Even less, about 13 percent of students have designed activities involving symmetry.

## CONCLUSIONS

The study explored the presence of geometrical concepts in projects inspired by African ethnic artifacts. Learning about African culture, not only raised their awareness of this culture but also helped them to get various creative ideas on how to incorporate elements of African culture in math activities with preschoolers. Students did recognize geometrical ideas in the artifacts (art pieces, games, legends, etc.). On the other side, the recognition of the potential for the development of geometrical concepts was limited in scope and depth. Inspired by a visit to the Museum of African Art, the students came up with ideas for initiating research and playful activities with children. The didactic tools that were exhibited at this exhibition were practically used in activities with children. These didactical tools based not only on the museum artifacts but also on students' further investigation of the topic, demonstrate the creativity and scope of the students' ideas.

“Geometry is all around us” is one of the basic premises on which preschool teachers should be working with young children. But it requires building up not only sensitivity to the culturally delineated world of geometry but also the geometry content knowledge of prospective teachers. We plan to pursue the experimental program in the future.
frequently expected that playing with the tool would contribute to children’s formation of concepts of geometrical shapes and symmetry. Their explanations however lack elaboration or complexity. Notable, all references to symmetry were made concerning mirror symmetry only. We show two examples of games students created inspired by Masks (Figure 4). In both, the expectation was that children would identify and name shapes. Similarly simple expectations repeatedly occurred in games involving patterns. All references to geometrical patterns referred only to linear patterns consisting of geometrical shapes (triangle, square, rectangle).

<figure><img src="image_224.png" /><figcaption>Figure 4: Masks as a didactical tool</figcaption></figure>

## DISCUSSION

While most students recognized mathematics in art pieces displayed in the Museum, some did not find a way to include directly any of the observed ideas in their plan of activity. This may be due to a lack of experience in project planning. Over 90 percent of students recognized patterns, geometric shapes, and symmetry in African exhibits. However, only the idea of the pattern was frequently explored in activities. Only about half of the students (47%) had activities involving geometric shapes and numbers. Even less, about 13 percent of students have designed activities involving symmetry.

## CONCLUSIONS

The study explored the presence of geometrical concepts in projects inspired by African ethnic artifacts. Learning about African culture, not only raised their awareness of this culture but also helped them to get various creative ideas on how to incorporate elements of African culture in math activities with preschoolers. Students did recognize geometrical ideas in the artifacts (art pieces, games, legends, etc.). On the other side, the recognition of the potential for the development of geometrical concepts was limited in scope and depth. Inspired by a visit to the Museum of African Art, the students came up with ideas for initiating research and playful activities with children. The didactic tools that were exhibited at this exhibition were practically used in activities with children. These didactical tools based not only on the museum artifacts but also on students' further investigation of the topic, demonstrate the creativity and scope of the students' ideas.

"Geometry is all around us" is one of the basic premises on which preschool teachers should be working with young children. But it requires building up not only sensitivity to the culturally delineated world of geometry but also the geometry content knowledge of prospective teachers. We plan to pursue the experimental program in the future.
## References

D'Ambrosio, U. (1989). *A Research Program and a Course in the History of Mathematics: Ethnomathematics*. *Historia Mathematica*, **16**, 285–288.

Clements, D. H. (2003). Teaching and learning geometry. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), *A research companion to Principles and Standards for School Mathematics* (pp. 151-178). National Council of Teachers of Mathematics.

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

Glasersfeld, E. von (1990). "Environment and Education." In L.P. Steffe & T. Wood (Eds.), *Transforming children's mathematics education: international perspectives* (pp. 200-215). Lawrence Erlbaum.

Grzegorczyk, Ivona & Stylianou, Despina A. (2006). Development of abstract mathematical thinking through artistic patterns J. Novotná, H. Moraová, M.Krátká, & N. Stehlíková (Eds.), *Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 216-223). PME.

Katz, L. G., & Chard, C. (2000). *Engaging children's minds: the project approach* (2nd ed.). Ablex Publishing Corporation.

Kyeremaeh, P, Awuah, F.K. & Dorwu, E. (2023). Integration of ethnomathematics in teaching geometry: A systematic review and bibliometric report. *Journal of Urban Mathematics Education, 16*(2), 68-89.

McGraw, S.A. (Ed.) (2003). *Integrated mathematics: choices and challenges*. NCTM.

Milinković, J., Marušić Jablanović, M. & Dabić Boričić, Milana (2017). Postignuće učenika iz matematike; glavni nalazi, trendovi i nastavni program. U Milica Marušić Jablanović, Nikoleta Gutvajn & Ivana Jakšić (Eds.), *TIMSS 2015 u Srbiji: Rezultati međunarodnog istraživanja postignuća učenika 4. razreda osnovne škole iz matematike i prirodnih nauka*, str. 27-50. Institut za pedagoška istraživanja.

Milinković, J. & Bogavac, D. (2011). Montessori method as a basis for integrated learning. *Metodički obzori-Methodological Horizons*, **10**(1), 135 – 143.

MPNTR (2018). *Foundations of preschool education program “Years of Ascend”*. Ministry of Education, Science and Technological Development of Serbia (MPNTR RS).

Outhred, L. & Mitchelmore, M. (1992). Representation of Area: A Pictorial Perspective. In W. Geeslin & K. Graham (Eds.), *Proceedings of the 16th PME Conference* (Vol 2, pp. 194-201). PME.

Piaget, J. & Inhelder, B. (1967). *The child's conception of space*. Norton.

Pimentel, P. E. C. (2012). Elementos geométricos del pueblo originario Mapuche. *Journal of Mathematics and Culture*, **6**(1), 135-147.

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

Smith, J.P., Heuvel-Panhuizen van den, M. & Teppo, A.R. (2011). Learning, teaching, and using measurement: introduction to the issue. *ZDM Mathematics Education*, **43**, 617.

Tan- Sisman, G. and Aksu, M. (2016). The study on sixth grade students’ misconceptions and errors in spatial measurement: length, area, and volume. *International Journal of Science and Mathematics Education*, 12(3), 1293-1319.

Vukobradović, M. &Milinković, J. (2020). Integration of educational contents in the function of initial understanding of spatial relations. *Innovations in teaching - a journal for modern teaching*, 33(2), 97-111.
## References

D'Ambrosio, U. (1989). *A Research Program and a Course in the History of Mathematics: Ethnomathematics*. *Historia Mathematica*, **16**, 285–288.

Clements, D. H. (2003). Teaching and learning geometry. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), *A research companion to Principles and Standards for School Mathematics* (pp. 151-178). National Council of Teachers of Mathematics.

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

Glasersfeld, E. von (1990). "Environment and Education." In L.P. Steffe & T. Wood (Eds.), *Transforming children's mathematics education: international perspectives* (pp. 200-215). Lawrence Erlbaum.

Grzegorczyk, Ivona & Stylianou, Despina A. (2006). Development of abstract mathematical thinking through artistic patterns J. Novotná, H. Moraová, M.Krátká, & N. Stehlíková (Eds.), *Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education* (Vol. 2, pp. 216-223). PME.

Katz, L. G., & Chard, C. (2000). *Engaging children's minds: the project approach* (2nd ed.). Ablex Publishing Corporation.

Kyeremaeh, P, Awuah, F.K. & Dorwu, E. (2023). Integration of ethnomathematics in teaching geometry: A systematic review and bibliometric report. *Journal of Urban Mathematics Education, 16*(2), 68-89.

McGraw, S.A. (Ed.) (2003). *Integrated mathematics: choices and challenges*. NCTM.

Milinković, J., Marušić Jablanović, M. & Dabić Boričić, Milana (2017). Postignuće učenika iz matematike; glavni nalazi, trendovi i nastavni program. U Milica Marušić Jablanović, Nikoleta Gutvajn & Ivana Jakšić (Eds.), *TIMSS 2015 u Srbiji: Rezultati međunarodnog istraživanja postignuća učenika 4. razreda osnovne škole iz matematike i prirodnih nauka*, str. 27-50. Institut za pedagoška istraživanja.

Milinković, J. & Bogavac, D. (2011). Montessori method as a basis for integrated learning. *Metodički obzori-Methodological Horizons*, 10(1), 135 – 143.

MPNTR (2018). *Foundations of preschool education program “Years of Ascend”*. Ministry of Education, Science and Technological Development of Serbia (MPNTR RS).

Outhred, L. & Mitchelmore, M. (1992). Representation of Area: A Pictorial Perspective. In W. Geeslin & K. Graham (Eds.), *Proceedings of the 16th PME Conference* (Vol 2, pp. 194-201). PME.

Piaget, J. & Inhelder, B. (1967). *The child's conception of space*. Norton.

Pimentel, P. E. C. (2012). Elementos geométricos del pueblo originario Mapuche. *Journal of Mathematics and Culture*, 6(1), 135-147.

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

Smith, J.P., Heuvel-Panhuizen van den, M. & Teppo, A.R. (2011). Learning, teaching, and using measurement: introduction to the issue. *ZDM Mathematics Education*, 43, 617.

Tan- Sisman, G. and Aksu, M. (2016). The study on sixth grade students’ misconceptions and errors in spatial measurement: length, area, and volume. *International Journal of Science and Mathematics Education*, 12(93), 1293-1319.

Vukobradović, M. &Milinković, J. (2020). Integration of educational contents in the function of initial understanding of spatial relations. *Innovations in teaching - a journal for modern teaching*, 33(2), 97-111.
# INVESTIGATING GEOMETRIC KNOWLEDGE IN THE ART OF SISAL TAPESTRY IN A LOCAL COMMUNITY THROUGH ETHNOMODELLING

Daniel Clark Orey, Milton Rosa, Osvaldo Rosa Filho

Universidade Federal de Ouro Preto

This research was conducted in a Professional Master's Degree Program in Mathematics Education, at the Universidade Federal de Ouro Preto, in Brazil. The main objective was to develop an ethnomodelling based investigation on the art of sisal tapestry developed by the members of a local community of Cachoeira do Brumado, in the municipality of Mariana, in the state of Minas Gerais, Brazil. This study aimed to translate local geometric practices through the proposition of a pedagogical action from the perspective of mathematics teachers, for the development of geometric contents for Middle School students. The results of this investigation show that participants in this research (teachers and artisans) value and respect local know-how related to the art of tapestry and the making of sisal carpets, as they became aware of the relevance of their own experiences, which made it possible to understand the connection between etic (school) and emic (tapestry) knowledge in the teaching and learning process in mathematics.

## INITIAL CONSIDERATIONS

There is a need to find alternative ways to attract students to learn through the use of pedagogical actions for the teaching and learning process in mathematics through the contextualization of problems and situations involving practices that occur in everyday life, which are permeated with the knowledge and practices developed by members of distinct cultural groups. At all times, these members are comparing, classifying, quantifying, measuring, explaining, generalizing, inferring, modelling, and, in some way, evaluating by using material and intellectual instruments that are specific to their own culture (D'Ambrosio, 2016).

It became important to plan a teaching and learning process in mathematics that encompasses cultural and everyday thinking, while at the same time, to take into account experiences lived by students in their own communities. Given that the experiences experienced by students in their own sociocultural context are often overlooked, this study was interested in how urban and rural communities, located close to the schools they serve, can positively influence the performance of teachers through interventions and pedagogical actions in the classroom.

In this regard, D'Ambrosio (1998) states that geometry education is related to a scientific field of knowledge production linked to the process of understanding geometric knowledge found in the daily life of the members of distinct cultures through the use of procedures, techniques, and practices developed locally. Similarly, Jones and Tzekaki (2016) affirm that geometry education is moving away from focusing on formal spaces and shapes toward a growing role for spatial reasoning and geometric problems encountered in daily life.

We investigated the process of ethnomodelling as the art of translating local geometric practices (emic) related to tapestry that is developed in a rural community in the state of Minas Gerais, Brazil
# INVESTIGATING GEOMETRIC KNOWLEDGE IN THE ART OF SISAL TAPESTRY IN A LOCAL COMMUNITY THROUGH ETHNOMODELLING

Daniel Clark Orey, Milton Rosa, Osvaldo Rosa Filho

Universidade Federal de Ouro Preto

This research was conducted in a Professional Master's Degree Program in Mathematics Education, at the Universidade Federal de Ouro Preto, in Brazil. The main objective was to develop an ethnomodelling based investigation on the art of sisal tapestry developed by the members of a local community of Cachoeira do Brumado, in the municipality of Mariana, in the state of Minas Gerais, Brazil. This study aimed to translate local geometric practices through the proposition of a pedagogical action from the perspective of mathematics teachers, for the development of geometric contents for Middle School students. The results of this investigation show that participants in this research (teachers and artisans) value and respect local know-how related to the art of tapestry and the making of sisal carpets, as they became aware of the relevance of their own experiences, which made it possible to understand the connection between etic (school) and emic (tapestry) knowledge in the teaching and learning process in mathematics.

## INITIAL CONSIDERATIONS

There is a need to find alternative ways to attract students to learn through the use of pedagogical actions for the teaching and learning process in mathematics through the contextualization of problems and situations involving practices that occur in everyday life, which are permeated with the knowledge and practices developed by members of distinct cultural groups. At all times, these members are comparing, classifying, quantifying, measuring, explaining, generalizing, inferring, modelling, and, in some way, evaluating by using material and intellectual instruments that are specific to their own culture (D'Ambrosio, 2016).

It became important to plan a teaching and learning process in mathematics that encompasses cultural and everyday thinking, while at the same time, to take into account experiences lived by students in their own communities. Given that the experiences experienced by students in their own sociocultural context are often overlooked, this study was interested in how urban and rural communities, located close to the schools they serve, can positively influence the performance of teachers through interventions and pedagogical actions in the classroom.

In this regard, D'Ambrosio (1998) states that geometry education is related to a scientific field of knowledge production linked to the process of understanding geometric knowledge found in the daily life of the members of distinct cultures through the use of procedures, techniques, and practices developed locally. Similarly, Jones and Tzekaki (2016) affirm that geometry education is moving away from focusing on formal spaces and shapes toward a growing role for spatial reasoning and geometric problems encountered in daily life.

We investigated the process of ethnomodelling as the art of translating local geometric practices (emic) related to tapestry that is developed in a rural community in the state of Minas Gerais, Brazil
through the development of pedagogical practices that enables students to develop their own ways of
understanding geometric concepts. It is important to highlight that this approach aimed to develop
local (emic), global (etic), and dialogical (glocal/cultural dynamism) ethnomodels, which were
discussed with mathematics teachers in this community. The main objective was related to the
performance of these teachers in developing a pedagogical action using ethnomodelling for the
development of geometric content for their students emphasize the contents related to geometric
knowledge that appear most prominently in the art of tapestry, mainly in relation to the making of
sisal carpets.

In accordance with this context, the main objective of this study was based on ethnomodelling used
to investigate and related to the art of tapestry developed in the local community by aiming to translate
local geometric practices through a pedagogical action that is locally contextualized. It is important
to highlight that the specific objectives of this study are to develop curricular activities based on the
daily practices of a community of sisal carpet artisans by aiming to show how geometric knowledge
can be influenced by actions carried out in daily activities.

Therefore, the problem statement proposed in this study was justified by how geometry education is
full of ideas, procedures, techniques, and general strategies used to develop powerful and meaningful
mathematics in an attractive and relevant way. It is of great value to discover the factors and
characteristics that determine the way teachers act in the use of modelling, ethnomathematics, and
ethnomodelling in the pedagogical action developed by teachers, with the aim of enabling the
development of a teaching and learning process in mathematics that is innovative, attractive, useful,
and efficient for all students and educators.

In this study, researchers proposed a different methodology for the teaching and learning process in
mathematics, in relation to the dialogic approach of ethnomodelling. Thus, this research is based on
the connection of a ethnomathematics perspective to the cultural aspects of mathematical modelling,
which forms the primary assumption of ethnomodelling. Hence, researchers investigated how might
possible contributions that ethnomathematics and modelling through the dialogical approach of
ethnomodelling play in the teaching and learning process in relation to the making of sisal carpets by
members of a local community. This particular pedagogical action is related, in particular, to
academic geometric practices related to the school curriculum.

For example, the recognition of different types of polygons, angles, straight lines, symmetry, acts of
both reflection, rotation and translation in addition to working with the units of measurement, ratio
and proportion, through ethnomodelling, where the geometric contents of 7th grade middle school
that made up this ethnomodelling pedagogical action and related to the recommendations of the
*National Common Curricular Base* (Brasil, 2018), which is a document of a normative nature that
defines the organic and progressive set of essential learning that all students must develop during
their education.

It is noteworthy that the theme proposed in this study enabled the teacher-researcher to understand
the need for, and to seek the data and information necessary for the development of a teaching and
learning process that allowed him to contribute to the development of teaching practices linked to the
proposed innovative pedagogies.
through the development of pedagogical practices that enables students to develop their own ways of
understanding geometric concepts. It is important to highlight that this approach aimed to develop
local (emic), global (etic), and dialogical (glocal/cultural dynamism) ethnomodels, which were
discussed with mathematics teachers in this community. The main objective was related to the
performance of these teachers in developing a pedagogical action using ethnomodelling for the
development of geometric content for their students emphasize the contents related to geometric
knowledge that appear most prominently in the art of tapestry, mainly in relation to the making of
sisal carpets.

In accordance with this context, the main objective of this study was based on ethnomodelling used
to investigate and related to the art of tapestry developed in the local community by aiming to translate
local geometric practices through a pedagogical action that is locally contextualized. It is important
to highlight that the specific objectives of this study are to develop curricular activities based on the
daily practices of a community of sisal carpet artisans by aiming to show how geometric knowledge
can be influenced by actions carried out in daily activities.

Therefore, the problem statement proposed in this study was justified by how geometry education is
full of ideas, procedures, techniques, and general strategies used to develop powerful and meaningful
mathematics in an attractive and relevant way. It is of great value to discover the factors and
characteristics that determine the way teachers act in the use of modelling, ethnomathematics, and
ethnomodelling in the pedagogical action developed by teachers, with the aim of enabling the
development of a teaching and learning process in mathematics that is innovative, attractive, useful,
and efficient for all students and educators.

In this study, researchers proposed a different methodology for the teaching and learning process in
mathematics, in relation to the dialogic approach of ethnomodelling. Thus, this research is based on
the connection of a ethnomathematics perspective to the cultural aspects of mathematical modelling,
which forms the primary assumption of ethnomodelling. Hence, researchers investigated how might
possible contributions that ethnomathematics and modelling through the dialogical approach of
ethnomodelling play in the teaching and learning process in relation to the making of sisal carpets by
members of a local community. This particular pedagogical action is related, in particular, to
academic geometric practices related to the school curriculum.

For example, the recognition of different types of polygons, angles, straight lines, symmetry, acts of
both reflection, rotation and translation in addition to working with the units of measurement, ratio
and proportion, through ethnomodelling, where the geometric contents of 7th grade middle school
that made up this ethnomodelling pedagogical action and related to the recommendations of the
*National Common Curricular Base* (Brasil, 2018), which is a document of a normative nature that
defines the organic and progressive set of essential learning that all students must develop during
their education.

It is noteworthy that the theme proposed in this study enabled the teacher-researcher to understand
the need for, and to seek the data and information necessary for the development of a teaching and
learning process that allowed him to contribute to the development of teaching practices linked to the
proposed innovative pedagogies.
Thus, this study presented in an interactive way, the possibilities for bringing students closer to local and school geometric language, as well as its use in solving everyday problems. In this context, the activities developed in this research were relevant for mathematics teachers, participants in this study, to understand the relationship between the different contents studied in the school mathematics curriculum. These activities were also necessary to verify whether geometric contents have notions and concepts that can be applicable in daily tasks carried out in everyday life, such as, for example, making sisal carpets.

In this regard, Rosa and Orey (2017) state that research conducted in geometry education indicates that the geometric contents taught in schools, as well as the way in which geometry is taught, have not kept up with cultural and social developments. and technology that today’s society craves. Therefore, conducting this study is also justified by the development of a contextualized pedagogical action that had meaning for students, as well it met the new demands of the teaching and learning process in mathematics.

In this context, the mathematical practices used in the daily lives of members of distinct cultural groups are associated with the development of their own techniques, strategies and procedures, which are the techniques developed by sisal carpet artisans, as there are different ways of solving problems that are inherent to the tapestry production process, which can be denoted by different ways of quantifying, classifying and measuring, which are related to the emic (local) knowledge developed by members of this specific cultural group.

According to Mendes (2008), geometric concepts are found in distinct cultures, as well as in art, sculpture, tapestry, and architecture. Similarly, geometric shapes appear in most carpets and ceramics made by nomadic populations that we know and whose patterns have been given new meanings in other societies for millennia. The varied designs related to the art of Persian tapestry, for example, indicate the origin and, often, even the age of the pieces, which highlights the main characteristics of this particular group of carpets. Experiences related to the connection between geometry and activities created through the use of crafts enable students to identify mathematical geometric concepts present in these cultural artifacts, such as embroidery, ceramic, and carpets.

Therefore, in accordance with the legal guidelines that guide Brazilian Basic Education, Rosa and Orey (2017b) emphasize that one of the main objectives of Ethnomodelling is to connect the cultural aspects of Mathematics with the school mathematical concepts to translate problems, situations, and phenomena present in the daily lives of members of distinct cultural groups to other mathematical knowledge systems, such as school and academia.

## THEORETICAL FOUNDATION OF ETHNMODELLING

A search for innovative methodologies is necessary to record historical forms of geometric ideas, procedures, and techniques that occur in diverse sociocultural contexts to take advantage of an emerging *glocalization* process of mathematics and its cultural aspects by exploring and documenting geometric practices rooted in diverse cultural contexts, which recognizes the connection between geometric phenomena and culture.

In this regard, ethnomodelling differs from classical definitions of mathematics, which are often perceived as a form of universal knowledge that is applicable everywhere that is based on the theory of certainty and was imposed through capitalism and colonialism. Because it studies and investigates
Thus, this study presented in an interactive way, the possibilities for bringing students closer to local and school geometric language, as well as its use in solving everyday problems. In this context, the activities developed in this research were relevant for mathematics teachers, participants in this study, to understand the relationship between the different contents studied in the school mathematics curriculum. These activities were also necessary to verify whether geometric contents have notions and concepts that can be applicable in daily tasks carried out in everyday life, such as, for example, making sisal carpets.

In this regard, Rosa and Orey (2017) state that research conducted in geometry education indicates that the geometric contents taught in schools, as well as the way in which geometry is taught, have not kept up with cultural and social developments. and technology that today’s society craves. Therefore, conducting this study is also justified by the development of a contextualized pedagogical action that had meaning for students, as well it met the new demands of the teaching and learning process in mathematics.

In this context, the mathematical practices used in the daily lives of members of distinct cultural groups are associated with the development of their own techniques, strategies and procedures, which are the techniques developed by sisal carpet artisans, as there are different ways of solving problems that are inherent to the tapestry production process, which can be denoted by different ways of quantifying, classifying and measuring, which are related to the emic (local) knowledge developed by members of this specific cultural group.

According to Mendes (2008), geometric concepts are found in distinct cultures, as well as in art, sculpture, tapestry, and architecture. Similarly, geometric shapes appear in most carpets and ceramics made by nomadic populations that we know and whose patterns have been given new meanings in other societies for millennia. The varied designs related to the art of Persian tapestry, for example, indicate the origin and, often, even the age of the pieces, which highlights the main characteristics of this particular group of carpets. Experiences related to the connection between geometry and activities created through the use of crafts enable students to identify mathematical geometric concepts present in these cultural artifacts, such as embroidery, ceramic, and carpets.

Therefore, in accordance with the legal guidelines that guide Brazilian Basic Education, Rosa and Orey (2017b) emphasize that one of the main objectives of Ethnomodelling is to connect the cultural aspects of Mathematics with the school mathematical concepts to translate problems, situations, and phenomena present in the daily lives of members of distinct cultural groups to other mathematical knowledge systems, such as school and academia.

## THEORETICAL FOUNDATION OF ETHNMODELLING

A search for innovative methodologies is necessary to record historical forms of geometric ideas, procedures, and techniques that occur in diverse sociocultural contexts to take advantage of an emerging *glocalization* process of mathematics and its cultural aspects by exploring and documenting geometric practices rooted in diverse cultural contexts, which recognizes the connection between geometric phenomena and culture.

In this regard, ethnomodelling differs from classical definitions of mathematics, which are often perceived as a form of universal knowledge that is applicable everywhere that is based on the theory of certainty and was imposed through capitalism and colonialism. Because it studies and investigates
geometric knowledge developed locally and globally into a body of knowledge that is identified as glocal geometric practices.

In general, many scholars of geometric practices seek to understand traditions, ideologies, cosmologies, and beliefs of a particular culture by advocating the relevance of the complementarity between local (emic) and global (etic) geometric knowledge into a glocal (dialogic) approach through the development of cultural dynamism. In this process, the anthropological terms emic and etic are used as an analogy between observers from the inside (emic, local) and observers from the outside (etic, global) of a given culture (Orey & Rosa, 2021).

The global (etic) approach is related to extrinsic worldviews of the outside observers in relation to, in this case the geometric knowledge developed by the members of distinct cultural groups. It refers to an interpretation of the characteristics of other cultures from the application of analytical categories developed by external observers. This approach forms the external view made by observers, who are looking at a specific culture from the outside perspective, in a cross-cultural, comparative, and prescriptive stance, and which can be equated with objective explanations of sociocultural phenomena (Rosa & Orey, 2019).

Local (emic) approach aims to understand the characteristics of a given culture based on the intrinsic references developed by its members. It is related to their own worldview and cosmovisions regarding to the development of their geometric knowledge. This approach is considered as an internal view of members who are looking at their own culture from the inside perspective, in an intracultural, particular, and descriptive stance, which is identified with the understanding of the subjective experiences that they acquire, develop, accumulate, and diffuse through history (Orey & Rosa, 2021).

These approaches are related to geometric ideas, procedures, and practices that are linked to everyday phenomena, and are organized, interpreted, and evaluated through the elaboration of ethnomodels, which are representations of systems taken from the reality of members of distinct cultures. They enable these members to communicate, diffuse, and transmit their mathematization processes across generations by helping them in attributing meaning to the sociocultural context in which they perform their daily activities. Thus, ethnomodels are small units of information that link the development of geometric practices developed by these members who use their own sociocultural heritage to holistically understand their surroundings (Rosa & Orey 2019).

By applying ethnomodelling, researchers seek to value and respect ethnomathematical knowledge (local, emic), and in this case, the interpretations and contributions to geometric systematization through modelling (etic, global) in a glocal (dialogic) manner. In this context, Rosa & Orey (2021) state that ethnomodelling is an alternative methodological approach that is suited to different sociocultural realities, which proposes the rediscovery of geometric knowledge systems developed, accumulated, adopted, and adapted in other cultural contexts. Thus, this study focused on the glocal (dialogic) approach of ethnomodelling and how the interaction between local (emic) and global (etic) approaches promoted the understanding of cultural dynamism through the elaboration of ethnomodels.

## METHODOLOGICAL PROCEDURES

Again, the study aimed to translate local geometric practices through the proposition of a pedagogical action for the development of geometric contents for students of the 7th year of Middle School, from
geometric knowledge developed locally and globally into a body of knowledge that is identified as glocal geometric practices.

In general, many scholars of geometric practices seek to understand traditions, ideologies, cosmologies, and beliefs of a particular culture by advocating the relevance of the complementarity between local (emic) and global (etic) geometric knowledge into a glocal (dialogic) approach through the development of cultural dynamism. In this process, the anthropological terms emic and etic are used as an analogy between observers from the inside (emic, local) and observers from the outside (etic, global) of a given culture (Orey & Rosa, 2021).

The global (etic) approach is related to extrinsic worldviews of the outside observers in relation to, in this case the geometric knowledge developed by the members of distinct cultural groups. It refers to an interpretation of the characteristics of other cultures from the application of analytical categories developed by external observers. This approach forms the external view made by observers, who are looking at a specific culture from the outside perspective, in a cross-cultural, comparative, and prescriptive stance, and which can be equated with objective explanations of sociocultural phenomena (Rosa & Orey, 2019).

Local (emic) approach aims to understand the characteristics of a given culture based on the intrinsic references developed by its members. It is related to their own worldview and cosmovisions regarding to the development of their geometric knowledge. This approach is considered as an internal view of members who are looking at their own culture from the inside perspective, in an intracultural, particular, and descriptive stance, which is identified with the understanding of the subjective experiences that they acquire, develop, accumulate, and diffuse through history (Orey & Rosa, 2021).

These approaches are related to geometric ideas, procedures, and practices that are linked to everyday phenomena, and are organized, interpreted, and evaluated through the elaboration of ethnomodels, which are representations of systems taken from the reality of members of distinct cultures. They enable these members to communicate, diffuse, and transmit their mathematization processes across generations by helping them in attributing meaning to the sociocultural context in which they perform their daily activities. Thus, ethnomodels are small units of information that link the development of geometric practices developed by these members who use their own sociocultural heritage to holistically understand their surroundings (Rosa & Orey 2019).

By applying ethnomodelling, researchers seek to value and respect ethnomathematical knowledge (local, emic), and in this case, the interpretations and contributions to geometric systematization through modelling (etic, global) in a glocal (dialogic) manner. In this context, Rosa & Orey (2021) state that ethnomodelling is an alternative methodological approach that is suited to different sociocultural realities, which proposes the rediscovery of geometric knowledge systems developed, accumulated, adopted, and adapted in other cultural contexts. Thus, this study focused on the glocal (dialogic) approach of ethnomodelling and how the interaction between local (emic) and global (etic) approaches promoted the understanding of cultural dynamism through the elaboration of ethnomodels.

## METHODOLOGICAL PROCEDURES

Again, the study aimed to translate local geometric practices through the proposition of a pedagogical action for the development of geometric contents for students of the 7th year of Middle School, from
the perspective of three mathematics teachers and two artisans who make sisal carpets. A pedagogical
action based on ethnomodelling was developed that addressed geometric practices related to the
school context, such as the recognition of different types of polygons, angles, lines, symmetry,
reflection, rotation, and translation, in addition to working with the units of measurement related
mainly to length, perimeter, area, volume, capacity, ratio, and proportion. Below is an example of the
activity proposed in this pedagogical action.

After manufacturing the sisal carpet and the completion of the finishing process, the measurements are checked and the area of the carpet is calculated to determine the sales value of the craft. It is known that, currently, each sisal carpet is sold for R$ 65.00 for each square meter (m²).

<figure><img src="image_213.png" /><figcaption>Figure 1: Sisal Carpet made by the local artisan (Source: Authors’ personal file)</figcaption></figure>

Answer the following questions:

a) What is the perimeter of the sisal carpet?

b) What the area of the sisal?

c) What is the final value of this carpet?

d) If every square meter of the carpet needs, on average, 2.5 kilograms of sisal, how many kilograms of this material, on average, did the artisan spend to make this carpet?

e) Regarding geometric shapes, how many parallelograms are there? How many rhombus?
How many triangles? How many rectangles? How many polygons? How many 90°
angles?

f) Regarding transformations in the plane, how many lines of symmetry are there? Are there rotation (tun), reflection (flip), and translation (slide).

This pedagogical action provided a differentiated educational opportunity for participating teachers.
In this study the researcher used the local (emic) geometric knowledge and practices of the
community of artisans of sisal carpets in connection with global (etic) school geometric knowledge
for the elaboration of geometric activities that dialogue with the teaching and learning process in
mathematics.

The problem statement of this study was defined by the research question: *How can ethnomodelling contribute to the proposition of a pedagogical action, to the development of geometric and geometric contents, based on the art of tapestry of a local community, from the perspective of mathematics teachers of 7th year Middle School?* Data collection was conducted through two questionnaires

<footnote n="26">²⁶ The Brazilian real or reais (R$) is the official currency of Brazil that is subdivided into 100 cents. In 01/09/2024, the currency exchange rate is U$ 1.00 = R$ 4.90.</footnote>
the perspective of three mathematics teachers and two artisans who make sisal carpets. A pedagogical
action based on ethnomodelling was developed that addressed geometric practices related to the
school context, such as the recognition of different types of polygons, angles, lines, symmetry,
reflection, rotation, and translation, in addition to working with the units of measurement related
mainly to length, perimeter, area, volume, capacity, ratio, and proportion. Below is an example of the
activity proposed in this pedagogical action.

After manufacturing the sisal carpet and the completion of the finishing process, the measurements are checked and the area of the carpet is calculated to determine the sales value of the craft. It is known that, currently, each sisal carpet is sold for R$ <sup>26</sup>65.00 for each square meter (m<sup>2</sup>).

<figure><img src="image_225.png" /><figcaption>Figure 1: Sisal Carpet made by the local artisan (Source: Authors’ personal file)</figcaption></figure>

Answer the following questions:

a) What is the perimeter of the sisal carpet?

b) What the area of the sisal?

c) What is the final value of this carpet?

d) If every square meter of the carpet needs, on average, 2.5 kilograms of sisal, how many kilograms of this material, on average, did the artisan spend to make this carpet?

e) Regarding geometric shapes, how many parallelograms are there? How many rhombus?
How many triangles? How many rectangles? How many polygons? How many 90°
angles?

f) Regarding transformations in the plane, how many lines of symmetry are there? Are there rotation (tun), reflection (flip), and translation (slide).

This pedagogical action provided a differentiated educational opportunity for participating teachers. In this study the researcher used the local (emic) geometric knowledge and practices of the community of artisans of sisal carpets in connection with global (etic) school geometric knowledge for the elaboration of geometric activities that dialogue with the teaching and learning process in mathematics.

The problem statement of this study was defined by the research question: *How can ethnomodelling contribute to the proposition of a pedagogical action, to the development of geometric and geometric contents, based on the art of tapestry of a local community, from the perspective of mathematics teachers of 7th year Middle School?* Data collection was conducted through two questionnaires

<footnote n="26"><sup>26</sup> The Brazilian real or reais (R$) is the official currency of Brazil that is subdivided into 100 cents. In 01/09/2024, the currency exchange rate is U$ 1.00 = R$ 4.90.</footnote>
(initial and final), three modules of activities, a focus group, notes in the researchers' field diary, and two semi-structured interviews.

This research had a qualitative approach by using the methodological design adapted from the Grounded Theory, which allowed the analysis of data collected through open coding, as well as the interpretation of results through axial coding, which enabled, respectively, the identification of preliminary codes and conceptual categories. In this adaptation, the selective coding, the identification of the central category and the writing of an emergent theory were not used because the researchers' main objective was to answer the research question proposed to this study.

## RESULTS AND DISCUSSION

The main objective of the pedagogical action proposed in this study was to bring together the knowledge and practices originating from the tapestry weaver culture with the geometric knowledge developed in the school environment through the development of a pedagogical action in which the teachers reflected on the curricular activities proposed in this study. action. In this study, ethnomodelling is considered a pedagogical action that aroused the interest of the participating teachers, helping them to structure the way in which they analyzed the problem situations proposed in the activity blocks, as they expanded the geometric knowledge learned in the school system.

According to the interpretation of the results obtained in this study, it is inferred that the contextualization made it possible to raise awareness of the practical application of geometric and geometric concepts in problem situations related to the production of sisal carpets. Continuing with the interpretation of the results of this study, the blocks of activities carried out during this fieldwork contributed to the understanding of the applicability of geometric and geometric contents in the daily lives of tapestry artisans. This approach can minimize the fact that geometry is presented as an obstacle to students' learning because of the way it is taught in classrooms, given that its contents are memorized through the decontextualized application of exercises unrelated to everyday life. from the students.

In this way, the pedagogical action of ethnomodelling enabled the development of cultural dynamism between the students' school culture and the tapestry weaver culture by providing the opportunity for the contextualization of geometric and geometric concepts studied in the geometric curriculum. This demonstrates how through ethnomodelling, teachers and students develop essential tools so that they can reflect on reality, transforming it with the aim of achieving collective good and improving quality of life. of members of their communities (Rosa & Orey 2010).

According to the interpretation of the results obtained in this study, the participating teachers positively evaluated the work carried out while conducting the blocks of activities proposed for this study in relation to the importance of mathematics for carrying out daily tasks related to the activities of making sisal carpets.

For example, these participants evaluated the craft activities related to tapestry as useful and very good, as they witnessed a significant relationship between the problem situations proposed in the blocks and the geometric and geometric concepts present in the production of sisal carpets. However, the participants also highlighted how this craft is characteristic of this district and that it should be valued and recognized.
(initial and final), three modules of activities, a focus group, notes in the researchers' field diary, and two semi-structured interviews.

This research had a qualitative approach by using the methodological design adapted from the Grounded Theory, which allowed the analysis of data collected through open coding, as well as the interpretation of results through axial coding, which enabled, respectively, the identification of preliminary codes and conceptual categories. In this adaptation, the selective coding, the identification of the central category and the writing of an emergent theory were not used because the researchers' main objective was to answer the research question proposed to this study.

## RESULTS AND DISCUSSION

The main objective of the pedagogical action proposed in this study was to bring together the knowledge and practices originating from the tapestry weaver culture with the geometric knowledge developed in the school environment through the development of a pedagogical action in which the teachers reflected on the curricular activities proposed in this study. action. In this study, ethnomodelling is considered a pedagogical action that aroused the interest of the participating teachers, helping them to structure the way in which they analyzed the problem situations proposed in the activity blocks, as they expanded the geometric knowledge learned in the school system.

According to the interpretation of the results obtained in this study, it is inferred that the contextualization made it possible to raise awareness of the practical application of geometric and geometric concepts in problem situations related to the production of sisal carpets. Continuing with the interpretation of the results of this study, the blocks of activities carried out during this fieldwork contributed to the understanding of the applicability of geometric and geometric contents in the daily lives of tapestry artisans. This approach can minimize the fact that geometry is presented as an obstacle to students' learning because of the way it is taught in classrooms, given that its contents are memorized through the decontextualized application of exercises unrelated to everyday life. from the students.

In this way, the pedagogical action of ethnomodelling enabled the development of cultural dynamism between the students' school culture and the tapestry weaver culture by providing the opportunity for the contextualization of geometric and geometric concepts studied in the geometric curriculum. This demonstrates how through ethnomodelling, teachers and students develop essential tools so that they can reflect on reality, transforming it with the aim of achieving collective good and improving quality of life. of members of their communities (Rosa & Orey 2010).

According to the interpretation of the results obtained in this study, the participating teachers positively evaluated the work carried out while conducting the blocks of activities proposed for this study in relation to the importance of mathematics for carrying out daily tasks related to the activities of making sisal carpets.

For example, these participants evaluated the craft activities related to tapestry as useful and very good, as they witnessed a significant relationship between the problem situations proposed in the blocks and the geometric and geometric concepts present in the production of sisal carpets. However, the participants also highlighted how this craft is characteristic of this district and that it should be valued and recognized.
The findings support the need for teachers and students to be inserted in a learning environment that allows the use of geometric knowledge tacitly acquired in the school community itself through the elaboration of a pedagogical action based on complementarity between geometric knowledge (global, ethical) and everyday knowledge and practices related to tapestry (local, emic).

The participating teachers also responded that they did not encounter any difficulties in carrying out the activities proposed for this pedagogical action, as they are suitable for students in the 7th year of Elementary School. For example, participant PF2 stated that these activities were "well explained, clearly and covered various geometric content".

Similarly, participant PM3 stated that "the activities varied between easy and average, requiring more refined reasoning to be able to respond correctly to what was being required" because "they addressed the crafts they know, in addition to clearly working on the predominant contents of the 7th year". Therefore, the connections between these two environments should be encouraged in schools, as they can reduce the gaps between theoretical and practical geometric knowledge.

For these participants, the most attractive geometric and geometric contents in the activities proposed in this pedagogical action, which are related to the sisal weaving, were the operations with natural numbers, the calculation of areas and perimeters of plane figures, the Cartesian plane, geometric figures, and the angles. For example, participant PM3 argued that "students can determine areas and perimeters of the figures that are inside the mat using geometric formulas, counting the number of points that make up each figure".

These participants stated that they previously used artisanal production to prepare mathematics lessons, such as the production of carpets to introduce basic geometric operations and area calculations, however, they highlighted that there is not enough time to carry out this type of pedagogical action. Ethnomodelling values and respects how geometric knowledge and practices specific to the ethnos of the members of this cultural group, which are in accordance with the sociocultural context itself. Consequently, ethnomodelling also provides an understanding of the curricular geometric activities proposed in this pedagogical action in the context of the tapestry.

## FINAL CONSIDERATIONS

The results of this investigation showed that both teachers and artisans, participants in this research, value and respect local know-how related to the art of tapestry, in the making of sisal carpets, as they became aware of the relevance of their own experiences, which made it possible to understand the connection between etic (school) and emic (tapestry) knowledge in the teaching and learning process in mathematics.

This approach was developed through the analysis of a dialogical pedagogical action by the participating teachers, which contributed to the development of the process of mathematization of problem and situations experienced in the daily lives of community members, as well as made it possible to understand the curricular connections that can be conducted within and outside of schools and mathematics itself with the elaboration of ethnomodels.

At the end of the conduction of this investigation, an educational product was elaborated, in the form of a suggestion book, which, in addition to proposing the use of pedagogical activities for teachers, it also seeks to present the possibility of using everyday practices, developed by members of a certain
The findings support the need for teachers and students to be inserted in a learning environment that allows the use of geometric knowledge tacitly acquired in the school community itself through the elaboration of a pedagogical action based on complementarity between geometric knowledge (global, ethical) and everyday knowledge and practices related to tapestry (local, emic).

The participating teachers also responded that they did not encounter any difficulties in carrying out the activities proposed for this pedagogical action, as they are suitable for students in the 7th year of Elementary School. For example, participant PF2 stated that these activities were “well explained, clearly and covered various geometric content”.

Similarly, participant PM3 stated that “the activities varied between easy and average, requiring more refined reasoning to be able to respond correctly to what was being required” because “they addressed the crafts they know, in addition to clearly working on the predominant contents of the 7th year”. Therefore, the connections between these two environments should be encouraged in schools, as they can reduce the gaps between theoretical and practical geometric knowledge.

For these participants, the most attractive geometric and geometric contents in the activities proposed in this pedagogical action, which are related to the sisal weaving, were the operations with natural numbers, the calculation of areas and perimeters of plane figures, the Cartesian plane, geometric figures, and the angles. For example, participant PM3 argued that “students can determine areas and perimeters of the figures that are inside the mat using geometric formulas, counting the number of points that make up each figure”.

These participants stated that they previously used artisanal production to prepare mathematics lessons, such as the production of carpets to introduce basic geometric operations and area calculations, however, they highlighted that there is not enough time to carry out this type of pedagogical action. Ethnomodelling values and respects how geometric knowledge and practices specific to the ethnos of the members of this cultural group, which are in accordance with the sociocultural context itself. Consequently, ethnomodelling also provides an understanding of the curricular geometric activities proposed in this pedagogical action in the context of the tapestry.

## FINAL CONSIDERATIONS

The results of this investigation showed that both teachers and artisans, participants in this research, value and respect local know-how related to the art of tapestry, in the making of sisal carpets, as they became aware of the relevance of their own experiences, which made it possible to understand the connection between etic (school) and emic (tapestry) knowledge in the teaching and learning process in mathematics.

This approach was developed through the analysis of a dialogical pedagogical action by the participating teachers, which contributed to the development of the process of mathematization of problem and situations experienced in the daily lives of community members, as well as made it possible to understand the curricular connections that can be conducted within and outside of schools and mathematics itself with the elaboration of ethnomodels.

At the end of the conduction of this investigation, an educational product was elaborated, in the form of a suggestion book, which, in addition to proposing the use of pedagogical activities for teachers, it also seeks to present the possibility of using everyday practices, developed by members of a certain
cultural group, in this case, members of the culture of art in sisal tapestry, in the teaching and learning
process in mathematics through ethnomodelling.

Consequently, the development of mathematical curricular activities proposed for the pedagogical
action of ethnomodelling is in accordance with the emic (making sisal carpets) and ethical (school
mathematical content) approaches, as it provided the dialogical development of ethnomodels, which
became evident when the participating teachers. This study mathematized the ideas, procedures, and
mathematical practices typical of everyday artisanal production, which had its natural encounter with
activities carried out in the school environment.

References

Brasil. (2018). *Base nacional comum curricular*. Ministério da Educação.

D'Ambrosio, U. (1998). Educação matemática: da teoria à prática. 4ª Edição. Coleção Perspectivas em Educação Matemática. Papirus.

D'Ambrosio, U. (2016). *Educação para uma sociedade em transição*. 3ª Edição. Editora Livraria da Física.

Jones, K., & Tzekaki, M. (2016). Research on the teaching and learning of geometry. In A. Gutiérrez, G. C. Leder and P. Boero (Eds.), *The second handbook of research on the psychology of mathematics education* (pp. 109–149). Sense Publishers.

Mendes, I. A. (2008). Ensino de conceitos geométricos, medidas e simetria: por uma educação (etno)matemática com arte. *Revista Cocar*, 2(4), 35-47.

Orey, D. C., & Rosa, M. (2021). Ethnomodelling as a glocalization process of mathematical practices through cultural dynamism. *The Mathematics Enthusiast*, 18(3), 438-468.

Rios, O. P. (2000). Etnogeometria. In: M. C. S. Domite (Ed.), *Anais do Primeiro Congresso Brasileiro de Etnomatemática-CBEm1* (pp. 367-375). FE-USP.

Rosa, M., & Orey, D. C. (2010). Ethnomodelling: a pedagogical action for uncovering ethnomathematical practices. *Journal of Mathematical Modelling and Application, 1*(3), 58-67.

Rosa, M., & Orey, D. C. (2017). *Etnomodelagem: a arte de traduzir práticas matemáticas locais*. Editora Livraria da Física.

Rosa, M., & Orey, D. C. (2019). Ethnomodelling as the art of translating cultural practices. *For the Learning of Mathematics*, 39(2), 19-24.
cultural group, in this case, members of the culture of art in sisal tapestry, in the teaching and learning
process in mathematics through ethnomodelling.

Consequently, the development of mathematical curricular activities proposed for the pedagogical
action of ethnomodelling is in accordance with the emic (making sisal carpets) and ethical (school
mathematical content) approaches, as it provided the dialogical development of ethnomodels, which
became evident when the participating teachers. This study mathematized the ideas, procedures, and
mathematical practices typical of everyday artisanal production, which had its natural encounter with
activities carried out in the school environment.

References

Brasil. (2018). *Base nacional comum curricular*. Ministério da Educação.

D'Ambrosio, U. (1998). Educação matemática: da teoria à prática. 4ª Edição. Coleção Perspectivas em Educação Matemática. Papirus.

D'Ambrosio, U. (2016). *Educação para uma sociedade em transição*. 3ª Edição. Editora Livraria da Física.

Jones, K., & Tzekaki, M. (2016). Research on the teaching and learning of geometry. In A. Gutiérrez, G. C. Leder and P. Boero (Eds.), *The second handbook of research on the psychology of mathematics education* (pp. 109–149). Sense Publishers.

Mendes, I. A. (2008). Ensino de conceitos geométricos, medidas e simetria: por uma educação (etno)matemática com arte. *Revista Cocar*, 2(4), 35-47.

Orey, D. C., & Rosa, M. (2021). Ethnomodelling as a glocalization process of mathematical practices through cultural dynamism. *The Mathematics Enthusiast*, 18(3), 438-468.

Rios, O. P. (2000). Etnogeometria. In: M. C. S. Domite (Ed.), *Anais do Primeiro Congresso Brasileiro de Etnomatemática-CBEm1* (pp. 367-375). FE-USP.

Rosa, M., & Orey, D. C. (2010). Ethnomodelling: a pedagogical action for uncovering ethnomathematical practices. *Journal of Mathematical Modelling and Application, 1*(3), 58-67.

Rosa, M., & Orey, D. C. (2017). *Etnomodelagem: a arte de traduzir práticas matemáticas locais*. Editora Livraria da Física.

Rosa, M., & Orey, D. C. (2019). Ethnomodelling as the art of translating cultural practices. *For the Learning of Mathematics*, 39(2), 19-24.
# A PEDAGOGICAL ACTION BASED ON AN ETHNOMATHEMATICAL PERSPECTIVE FOR THE DEVELOPMENT OF GEOMETRIC CONTENT FOR VISUALLY IMPAIRED STUDENTS TO IMPROVE THE TEACHING PRACTICE OF A VISUALLY IMPAIRED MATHEMATICS TEACHER

Milton Rosa, Daniel Clark Orey, Giovana Aparecida Pereira da Silva

Universidade Federal de Ouro Preto

This qualitative study was conducted through a study based on the pedagogical action of ethnomathematics for the development of geometric contents for a visually impaired student, enrolled in the 9th grade of middle school, of a public school in Belo Horizonte, Minas Gerais, in the perspective of a visually impaired mathematics teacher, using manipulatives and concrete materials as mediators of this educational process. Its main objective was related to the following research question: "How can an ethnomathematics pedagogical action contribute to the development of geometric content for a visually impaired student to improve the teaching practice of a visually impaired mathematics teacher, based on ethnomathematics". Data were collected by using questionnaires (initial and final), semi-structured interviews, 10 modules of activities adapted for the visually impaired student with the help of his visually impaired mathematics teacher, and the researchers' field diary, which made up the theoretical sample of this investigation. Data analysis and the interpretation and discussion of the results were conducted by using an adaptation of Grounded Theory.

## INITIAL CONSIDERATIONS

Results obtained in the last demographic census conducted in Brazil in 2022, by the *Brazilian Institute of Geography and Statistics* (Brasil 2022a), shows that the Brazilian population is 203.1 million inhabitants, in which 18.6 million inhabitants who are 2 years old or older have some type of disability. In the context of this study, there 6.5 million Brazilians who have severe visual impairment, 6 million who have great difficulty seeing, and 506 thousand Brazilians who are legally visually impaired. According to the 2022 Brazilian School Census (Brasil, 2022b), there are 80.429 students with low vision, 7.308 visually impaired students, and 628 deafblind students who enrolled in Brazilian basic education schools. This context allowed researchers to understand how a visually impaired student enrolled in a 9th year of Middle School with help of his visually impaired mathematics teacher can develop geometric content by using manipulatives and concrete materials in the classroom.

Thus, this research investigated the inclusion of a visually impaired student in the teaching and learning process in mathematics, from the perspective of a visually impaired mathematics teacher, which aimed to understand the geometric contents related to plane geometry and the Pythagorean Theorem by this student through the use of manipulate materials in an ethnomathematical perspective. In this study, inclusion is understood as the adaptation and transformation of society so that people with disabilities (including the visually impaired) have their needs and differences respected and
# A PEDAGOGICAL ACTION BASED ON AN ETHNOMATHEMATICAL PERSPECTIVE FOR THE DEVELOPMENT OF GEOMETRIC CONTENT FOR VISUALLY IMPAIRED STUDENTS TO IMPROVE THE TEACHING PRACTICE OF A VISUALLY IMPAIRED MATHEMATICS TEACHER

Milton Rosa, Daniel Clark Orey, Giovana Aparecida Pereira da Silva

Universidade Federal de Ouro Preto

This qualitative study was conducted through a study based on the pedagogical action of ethnomathematics for the development of geometric contents for a visually impaired student, enrolled in the 9th grade of middle school, of a public school in Belo Horizonte, Minas Gerais, in the perspective of a visually impaired mathematics teacher, using manipulatives and concrete materials as mediators of this educational process. Its main objective was related to the following research question: "How can an ethnomathematics pedagogical action contribute to the development of geometric content for a visually impaired student to improve the teaching practice of a visually impaired mathematics teacher, based on ethnomathematics". Data were collected by using questionnaires (initial and final), semi-structured interviews, 10 modules of activities adapted for the visually impaired student with the help of his visually impaired mathematics teacher, and the researchers' field diary, which made up the theoretical sample of this investigation. Data analysis and the interpretation and discussion of the results were conducted by using an adaptation of Grounded Theory.

## INITIAL CONSIDERATIONS

Results obtained in the last demographic census conducted in Brazil in 2022, by the *Brazilian Institute of Geography and Statistics* (Brasil 2022a), shows that the Brazilian population is 203.1 million inhabitants, in which 18.6 million inhabitants who are 2 years old or older have some type of disability. In the context of this study, there 6.5 million Brazilians who have severe visual impairment, 6 million who have great difficulty seeing, and 506 thousand Brazilians who are legally visually impaired. According to the 2022 Brazilian School Census (Brasil, 2022b), there are 80.429 students with low vision, 7.308 visually impaired students, and 628 deafblind students who enrolled in Brazilian basic education schools. This context allowed researchers to understand how a visually impaired student enrolled in a 9th year of Middle School with help of his visually impaired mathematics teacher can develop geometric content by using manipulatives and concrete materials in the classroom.

Thus, this research investigated the inclusion of a visually impaired student in the teaching and learning process in mathematics, from the perspective of a visually impaired mathematics teacher, which aimed to understand the geometric contents related to plane geometry and the Pythagorean Theorem by this student through the use of manipulate materials in an ethnomathematical perspective. In this study, inclusion is understood as the adaptation and transformation of society so that people with disabilities (including the visually impaired) have their needs and differences respected and
valued, which provides opportunities for equality and equity in their coexistence in society (Brasil, 2015).

Given the complexity of the inclusive process, there is a need for both initial and continuing teacher training to develop teaching practices that are directed towards creating pedagogical actions contextualized for visually impaired students that seek an understanding of classroom geometric content from a ethnomathematical perspective. This includes educational policies that influence and support the development of inclusive practices in schools, and that establish the equal right of members of distinct cultural groups by supporting methodologies, educators, students and leadership for all learners (Unesco, 2015).

Teaching and learning processes of students with disabilities need to combine these two aspects, the professional and the intellectual, and for this it is necessary to develop the ability to re-elaborate knowledge (Pinheiro & Rosa, 2020). As well, other skills need to be worked on, such as elaboration, definition, and reinterpretation of curricula and programs that promote professionalization, appreciation, and teacher identification regarding students from minority groups. The pedagogical actions in this study are related to the use of a diversity of pedagogical resources, such as Geoboards, Multiplanes, and Cuisenaire manipulative materials. These were used to assist in identifying strategies used to resolve problems and situations proposed as pedagogical action developed from an ethnomathematical perspective.

The pedagogical actions outlined in this study identified methods used by the visually impaired teacher in developing his teaching practice. As well, this study investigated strategies and techniques for visually impaired students in the context of geometric content based on the theoretical and methodological basis of an ethnomathematics program. This qualitative research was conducted through a study based on a pedagogical action of ethnomathematics for the development of geometric contents of a visually impaired student, enrolled in the 9th grade in a public middle school in Belo Horizonte, Minas Gerais, Brazil, in accordance with the perspective of a visually impaired mathematics teacher, through the use of manipulatives and concrete materials as mediators of this educational process.

For many visually impaired students, “manipulation of a concrete resource is essential so that, through touch, they can perceive the shape, size, textures, which will determine the characteristics of the manipulative element (Kaleff, 2016). These resources enable students to understand geometric concepts through tactile perception. Visually impaired students need to use manipulative teaching materials that have different textures, sizes, and shapes, as it is through their manipulation that these students can develop geometric concepts. Thus, the main purpose of this perspective is to make the process of teaching and learning mathematics contextualized in the daily lives of visually impaired students, as well as analyzing which pedagogical resources must be used to assist them in the development of geometric content.

## INCLUSION, EQUITY, AND ETHNOMATHEMATICS

Inclusion is a process that helps students overcome barriers that limit their full participation and achievement in society as citizens. This pedagogical action includes the need to adapt schools for all students. For example, in the educational process as described here, inclusion valued and respected distinct identities, differences, and diversity, central to the elaboration of pedagogical materials and
valued, which provides opportunities for equality and equity in their coexistence in society (Brasil, 2015).

Given the complexity of the inclusive process, there is a need for both initial and continuing teacher training to develop teaching practices that are directed towards creating pedagogical actions contextualized for visually impaired students that seek an understanding of classroom geometric content from a ethnomathematical perspective. This includes educational policies that influence and support the development of inclusive practices in schools, and that establish the equal right of members of distinct cultural groups by supporting methodologies, educators, students and leadership for all learners (Unesco, 2015).

Teaching and learning processes of students with disabilities need to combine these two aspects, the professional and the intellectual, and for this it is necessary to develop the ability to re-elaborate knowledge (Pinheiro & Rosa, 2020). As well, other skills need to be worked on, such as elaboration, definition, and reinterpretation of curricula and programs that promote professionalization, appreciation, and teacher identification regarding students from minority groups. The pedagogical actions in this study are related to the use of a diversity of pedagogical resources, such as Geoboards, Multiplanes, and Cuisenaire manipulative materials. These were used to assist in identifying strategies used to resolve problems and situations proposed as pedagogical action developed from an ethnomathematical perspective.

The pedagogical actions outlined in this study identified methods used by the visually impaired teacher in developing his teaching practice. As well, this study investigated strategies and techniques for visually impaired students in the context of geometric content based on the theoretical and methodological basis of an ethnomathematics program. This qualitative research was conducted through a study based on a pedagogical action of ethnomathematics for the development of geometric contents of a visually impaired student, enrolled in the 9th grade in a public middle school in Belo Horizonte, Minas Gerais, Brazil, in accordance with the perspective of a visually impaired mathematics teacher, through the use of manipulatives and concrete materials as mediators of this educational process.

For many visually impaired students, “manipulation of a concrete resource is essential so that, through touch, they can perceive the shape, size, textures, which will determine the characteristics of the manipulative element (Kaleff, 2016). These resources enable students to understand geometric concepts through tactile perception. Visually impaired students need to use manipulative teaching materials that have different textures, sizes, and shapes, as it is through their manipulation that these students can develop geometric concepts. Thus, the main purpose of this perspective is to make the process of teaching and learning mathematics contextualized in the daily lives of visually impaired students, as well as analyzing which pedagogical resources must be used to assist them in the development of geometric content.

## INCLUSION, EQUITY, AND ETHNOMATHEMATICS

Inclusion is a process that helps students overcome barriers that limit their full participation and achievement in society as citizens. This pedagogical action includes the need to adapt schools for all students. For example, in the educational process as described here, inclusion valued and respected distinct identities, differences, and diversity, central to the elaboration of pedagogical materials and
the development of curricular materials in order to address the unique educational needs of the visually impaired student (Silva & Rosa, 2023).

Given this context, it is necessary to highlight how inclusion and equity form comprehensive principles that guide educational policies, plans, and practices. These principles recognize how education is a basic human right and forms a foundation for the development of more inclusive, equitable and cohesive communities (Vitello & Mithaug, 1998). This ensures that all students have access to quality education by recognizing their intrinsic value and diversity by developing respect for their human dignity. In this regard, differences are perceived positively as a stimulus to foster learning and promote educational equality (Unesco, 2019). At the core, the concept of inclusion forms a principle that both supports and welcomes the cultural identity of all students.

Consequently, the main goal of inclusive education is to eliminate social exclusion that results from discriminatory attitudes related to race, social class, ethnicity, religion, gender, abilities, and disabilities. It is important to emphasize here that discussions about equity were only introduced in Brazil through the debate related to the document named: *Education Action Framework 2030*, which discusses the implications of education for social justice. The document places inclusive education as a priority, as it serves as a basic human right that lays a foundation for the pursuit of peace and the promotion of sustainable development (Unesco, 2017).

According to this perspective, the guidance for the development of inclusive and equitable education can serve as an opportunity for Brazilian educators to renew the discourse for public policies in favor of public education, particularly, in terms of its financial support. Therefore, the perspective of a more inclusive and fairer education in Brazil. This includes educational opportunities for poor students and those from minority groups, which are in accordance with the principles of the 2030 Educational Agenda.

Hence, the search for improving the teaching and learning process in mathematics is still one of the main challenges faced by educators around the world in relation to the development of pedagogical actions that includes students, without distinction, in the educational process (Rosa, 2010). However, although there is recognition of the importance of inclusion for the success of students with disabilities in the school environment, there are still arguments that the inclusive process is restricted and that the search for inclusive pedagogical actions is very complex, requiring time and resources that are unavailable for educators can act in this direction (Rocha, 2017).

For this reason, educators have the duty to guarantee access to rich and diverse learning experiences, which contribute to the academic performance and success for all students. This includes the provision of opportunities for visually impaired students and mathematics teaching and learning experiences that promote the development of mathematical and sociocultural skills through the perspective of ethnomathematics (Pinheiro & Rosa, 2020).

Given this, D'Ambrosio (2001) stated that the importance of mathematics is indisputable in a global context. However, there is a need for this field of knowledge to interact and promote dialogue with other fields of knowledge and other forms of mathematics, enabling the development of a pedagogical action that is based on a sociocultural basis for an inclusive mathematical curriculum.

There is a need for pedagogical action in the classroom to focus on the generation, production, organization, transmission, and dissemination of mathematical knowledge developed and
the development of curricular materials in order to address the unique educational needs of the visually impaired student (Silva & Rosa, 2023).

Given this context, it is necessary to highlight how inclusion and equity form comprehensive principles that guide educational policies, plans, and practices. These principles recognize how education is a basic human right and forms a foundation for the development of more inclusive, equitable and cohesive communities (Vitello & Mithaug, 1998). This ensures that all students have access to quality education by recognizing their intrinsic value and diversity by developing respect for their human dignity. In this regard, differences are perceived positively as a stimulus to foster learning and promote educational equality (Unesco, 2019). At the core, the concept of inclusion forms a principle that both supports and welcomes the cultural identity of all students.

Consequently, the main goal of inclusive education is to eliminate social exclusion that results from discriminatory attitudes related to race, social class, ethnicity, religion, gender, abilities, and disabilities. It is important to emphasize here that discussions about equity were only introduced in Brazil through the debate related to the document named: *Education Action Framework 2030*, which discusses the implications of education for social justice. The document places inclusive education as a priority, as it serves as a basic human right that lays a foundation for the pursuit of peace and the promotion of sustainable development (Unesco, 2017).

According to this perspective, the guidance for the development of inclusive and equitable education can serve as an opportunity for Brazilian educators to renew the discourse for public policies in favor of public education, particularly, in terms of its financial support. Therefore, the perspective of a more inclusive and fairer education in Brazil. This includes educational opportunities for poor students and those from minority groups, which are in accordance with the principles of the 2030 Educational Agenda.

Hence, the search for improving the teaching and learning process in mathematics is still one of the main challenges faced by educators around the world in relation to the development of pedagogical actions that includes students, without distinction, in the educational process (Rosa, 2010). However, although there is recognition of the importance of inclusion for the success of students with disabilities in the school environment, there are still arguments that the inclusive process is restricted and that the search for inclusive pedagogical actions is very complex, requiring time and resources that are unavailable for educators can act in this direction (Rocha, 2017).

For this reason, educators have the duty to guarantee access to rich and diverse learning experiences, which contribute to the academic performance and success for all students. This includes the provision of opportunities for visually impaired students and mathematics teaching and learning experiences that promote the development of mathematical and sociocultural skills through the perspective of ethnomathematics (Pinheiro & Rosa, 2020).

Given this, D'Ambrosio (2001) stated that the importance of mathematics is indisputable in a global context. However, there is a need for this field of knowledge to interact and promote dialogue with other fields of knowledge and other forms of mathematics, enabling the development of a pedagogical action that is based on a sociocultural basis for an inclusive mathematical curriculum.

There is a need for pedagogical action in the classroom to focus on the generation, production, organization, transmission, and dissemination of mathematical knowledge developed and
accumulated by members of different cultural groups (Rosa, 2010). In this regard, D'Ambrosio (2005) criticized the traditional mathematical curriculum practiced in schools, which was designed and detailed in an obsolete, uninteresting, and not very useful way, and that consists of a set of repetitive and abstract techniques that become uninteresting and unnecessary.

In this context, ethnomathematics includes inclusive forms of educational opportunities. These opportunities include an ethics that is fundamental to principles of respect, solidarity, cooperation, and a respectful dialogue with alterity. The pedagogical action proposed by ethnomathematics can be considered as a response to the demographic dynamics of society and social inequality, as this program values and respects the knowledge of members of the different groups that make up contemporary society (Rosa & Orey, 2017). In this regard, D'Ambrosio and Borba (2010) stated that ethnomathematics values and supports the production of knowledge developed by members of distinct cultural groups that have been marginalized in the process of globalization experienced by humanity.

From this perspective, Rosa (2010) states that ethnomathematics helps in the identification and interpretation of forms of mathematical knowledge developed by members of different cultural groups, as well as seeking to value mathematical knowledge and practices that have been historically neglected, such as those developed by visually impaired students. The development of the concept of culture for a group of visually impaired students can, according to Pinheiro and Rosa (2020), be based on an anthropological concepts of culture and the theoretical implications of ethnomathematics by envisioning the possibility of developing pedagogical actions towards the promotion of sociocultural inclusion.

## METHODOLOGICAL PROCEDURES

In this qualitative study data were collected by using the following methodological instruments: questionnaires (initial and final), semi-structured interviews, and ten modules of activities adapted for both the visually impaired student and his visually impaired mathematics teacher. As well as the observations registered in the researchers' field diary were used to provide theoretical samples for this investigation. To conduct this investigation, researchers transcribed semi-structured interviews, coded and analyzed the answers given by the participants to the questionnaires and to the ten blocks of activities proposed in these data collection instruments (Silva & Rosa, 2023).

Data analysis was performed by using an adaptation of Grounded Theory in which *preliminary codes* (Table 1) were identified during the conduction of open coding and then they were grouped through their common concepts and characteristics in *conceptual categories* (Table 2) during the conduction of axial coding.

It is important to state here that in selective coding, the identification of a central category, and the writing of an emerging theory were not employed in this study by considering that our objective was to answer the research question.

The analytical phase of data and the interpretative phase of the results obtained in this study enabled the researchers to determine answers to the problem statement developed for this study. Two conceptual categories emerged from the analysis of the data: a) *Process of Teaching and Learning in Mathematics in the School Context* and b) *Ethnomathematics as a Pedagogical Action for an Inclusive Education*. During the interpretation and discussion of the results obtained in this study, the
accumulated by members of different cultural groups (Rosa, 2010). In this regard, D'Ambrosio (2005) criticized the traditional mathematical curriculum practiced in schools, which was designed and detailed in an obsolete, uninteresting, and not very useful way, and that consists of a set of repetitive and abstract techniques that become uninteresting and unnecessary.

In this context, ethnomathematics includes inclusive forms of educational opportunities. These opportunities include an ethics that is fundamental to principles of respect, solidarity, cooperation, and a respectful dialogue with alterity. The pedagogical action proposed by ethnomathematics can be considered as a response to the demographic dynamics of society and social inequality, as this program values and respects the knowledge of members of the different groups that make up contemporary society (Rosa & Orey, 2017). In this regard, D'Ambrosio and Borba (2010) stated that ethnomathematics values and supports the production of knowledge developed by members of distinct cultural groups that have been marginalized in the process of globalization experienced by humanity.

From this perspective, Rosa (2010) states that ethnomathematics helps in the identification and interpretation of forms of mathematical knowledge developed by members of different cultural groups, as well as seeking to value mathematical knowledge and practices that have been historically neglected, such as those developed by visually impaired students. The development of the concept of culture for a group of visually impaired students can, according to Pinheiro and Rosa (2020), be based on an anthropological concepts of culture and the theoretical implications of ethnomathematics by envisioning the possibility of developing pedagogical actions towards the promotion of sociocultural inclusion.

## METHODOLOGICAL PROCEDURES

In this qualitative study data were collected by using the following methodological instruments: questionnaires (initial and final), semi-structured interviews, and ten modules of activities adapted for both the visually impaired student and his visually impaired mathematics teacher. As well as the observations registered in the researchers' field diary were used to provide theoretical samples for this investigation. To conduct this investigation, researchers transcribed semi-structured interviews, coded and analyzed the answers given by the participants to the questionnaires and to the ten blocks of activities proposed in these data collection instruments (Silva & Rosa, 2023).

Data analysis was performed by using an adaptation of Grounded Theory in which *preliminary codes* (Table 1) were identified during the conduction of open coding and then they were grouped through their common concepts and characteristics in *conceptual categories* (Table 2) during the conduction of axial coding.

It is important to state here that in selective coding, the identification of a central category, and the writing of an emerging theory were not employed in this study by considering that our objective was to answer the research question.

The analytical phase of data and the interpretative phase of the results obtained in this study enabled the researchers to determine answers to the problem statement developed for this study. Two conceptual categories emerged from the analysis of the data: a) *Process of Teaching and Learning in Mathematics in the School Context* and b) *Ethnomathematics as a Pedagogical Action for an Inclusive Education*. During the interpretation and discussion of the results obtained in this study, the
description of the categories allowed the quotes given by the participants, which were identified in this interpretative process, to be used to provide a holistic comprehension of its problem statement.

<table><caption>Table 1: Preliminary codes identified in the open coding process (Source: Authors' personal file)</caption><thead><tr><td>Collected Data</td><td>Open Coding (Preliminary codes)</td></tr></thead><tbody><tr><td><strong>Student:</strong> Manipulative materials helped me in learning geometric content (1), as geometry is related to the use of figures that are present in my daily life (2) in the environment that surrounds me (3). I would like to work more with manipulative materials in the classroom (4).<br><strong>Student:</strong> Manipulative materials helped me in understanding geometric contents (1). It certainly helped me a lot (5) because it was much easier for me to measure figurers with them (6).<br><strong>Student:</strong> I have experience with manipulative material (7): a) Geoplane (8) - it was very good because I managed to learn a lot from it (1). I found it much easier with it (4), b) Multiplane (8) - it was very different because if I can't do it in my head (5), I can do it with this material (4), and c) Cuisenaire rods (8) - it was good (4) because I could make several triangles (9) and do several problems with it (7) to solve the Theorem of Pythagoras (9).<br><strong>Teacher:</strong> What is the formula for the area of a square? (9).<br><strong>Student:</strong> The area of the square is like this! (10). The area of the square is side times side (9).<br><strong>Teacher:</strong> This material contributed to his experience of abstract geometry (11) in terms of plane figures (9), so that he had experience to work with manipulative materials (7) to have tactile contact with geometry (12) and get to know the issue of the figures (9), so this helped a lot with his cognitive construction (11).</td><td>(1) Using manipulative material to learn/understand geometric content.<br>(2) Presence of geometry in everyday life.<br>(3) Sociocultural context.<br>(4) Preference to use manipulative materials.<br>(5) Importance of manipulative materials.<br>(6) Manipulative materials as pedagogical tools.<br>(7) Experience with Manipulative materials.<br>(8) Types of Manipulative materials.<br>(9) Geometric content.<br>(10) Concept of the aera of the square.<br>(11) Contributions of manipulative materials.<br>(12) Ethnomathematical technique.</td></tr></tbody></table>

<table><caption>Table 2: Conceptual categories identified in the axial coding process (Source: Authors' personal file)</caption><thead><tr><td>Open Coding (Preliminary Codes)</td><td>Axial Coding (Conceptual Categories)</td></tr></thead><tbody><tr><td>(1) Using manipulative material to learn/understand geometric content<br>(4) Preference to use manipulative materials<br>(5) Importance of manipulative materials<br>(6) Manipulative materials as pedagogical tools<br>(7) Experience with Manipulative materials<br>(8) Types of Manipulative materials<br>(9) Geometric content<br>(10) Concept of the aera of the square<br>(11) Contributions of manipulative materials<br>(2) Presence of geometry in everyday life<br>(3) Sociocultural context<br>(12) Ethnomathematical technique</td><td>Process of Teaching and Learning in Mathematics in the School Context<br>Ethnomathematics as a Pedagogical Action for an Inclusive Education.</td></tr></tbody></table>
description of the categories allowed the quotes given by the participants, which were identified in this interpretative process, to be used to provide a holistic comprehension of its problem statement.

<table><caption>Table 1: Preliminary codes identified in the open coding process (Source: Authors' personal file)</caption><thead><tr><td>Collected Data</td><td>Open Coding (Preliminary codes)</td></tr></thead><tbody><tr><td><strong>Student:</strong> Manipulative materials helped me in learning geometric content (1), as geometry is related to the use of figures that are present in my daily life (2) in the environment that surrounds me (3). I would like to work more with manipulative materials in the classroom (4).<br><strong>Student:</strong> Manipulative materials helped me in understanding geometric contents (1). It certainly helped me a lot (5) because it was much easier for me to measure figurers with them (6).<br><strong>Student:</strong> I have experience with manipulative material (7): a) Geoplane (8) - it was very good because I managed to learn a lot from it (1). I found it much easier with it (4), b) Multiplane (8) - it was very different because if I can't do it in my head (5), I can do it with this material (4), and c) Cuisenaire rods (8) - it was good (4) because I could make several triangles (9) and do several problems with it (7) to solve the Theorem of Pythagoras (9).<br><strong>Teacher:</strong> What is the formula for the area of a square? (9).<br><strong>Student:</strong> The area of the square is like this! (10). The area of the square is side times side (9).<br><strong>Teacher:</strong> This material contributed to his experience of abstract geometry (11) in terms of plane figures (9), so that he had experience to work with manipulative materials (7) to have tactile contact with geometry (12) and get to know the issue of the figures (9), so this helped a lot with his cognitive construction (11).</td><td>(1) Using manipulative material to learn/understand geometric content.<br>(2) Presence of geometry in everyday life.<br>(3) Sociocultural context.<br>(4) Preference to use manipulative materials.<br>(5) Importance of manipulative materials.<br>(6) Manipulative materials as pedagogical tools.<br>(7) Experience with Manipulative materials.<br>(8) Types of Manipulative materials.<br>(9) Geometric content.<br>(10) Concept of the aera of the square.<br>(11) Contributions of manipulative materials.<br>(12) Ethnomathematical technique.</td></tr></tbody></table>

<table><caption>Table 2: Conceptual categories identified in the axial coding process (Source: Authors' personal file)</caption><thead><tr><td>Open Coding (Preliminary Codes)</td><td>Axial Coding (Conceptual Categories)</td></tr></thead><tbody><tr><td>(1) Using manipulative material to learn/understand geometric content<br>(4) Preference to use manipulative materials<br>(5) Importance of manipulative materials<br>(6) Manipulative materials as pedagogical tools<br>(7) Experience with Manipulative materials<br>(8) Types of Manipulative materials<br>(9) Geometric content<br>(10) Concept of the aera of the square<br>(11) Contributions of manipulative materials<br>(2) Presence of geometry in everyday life<br>(3) Sociocultural context<br>(12) Ethnomathematical technique</td><td>Process of Teaching and Learning in Mathematics in the School Context<br>Ethnomathematics as a Pedagogical Action for an Inclusive Education.</td></tr></tbody></table>
## RESULTS AND DISCUSSION

Results obtained in this study provided evidence that the development of differentiated pedagogical actions in the classroom stimulated the use of other senses of the participants, such as hearing, speech, and touch, by enabling to obtain positive results in the development of their cognitive process using a comprehensive communicative process. By using manipulative and concrete materials, the visually impaired mathematics teacher promoted the improvement of his teaching practice by promoting a pedagogical action for a more holistic teaching and learning process in mathematics, which enabled the development of mathematical and geometric content for the visually impaired student.

The adaptation of manipulatives and concrete materials such as the geoplane, the multiplane, and the cuisenaire rods, helped the student in exploration, investigation, and understanding of mathematical and geometric content proposed by the pedagogical actions conducted in the classroom. This was done by encouraging the student to associate aspects of the concrete to the abstract and included concepts of perimeter, area, and angles, as well as the characteristics and properties of geometric figures, and the identification and use of the Pythagorean Theorem.

These results, aimed at the improvement of an inclusive mathematics education, also showed the importance of the sociocultural differences that agreed with assumptions of ethnomathematics. The visually impaired student stated that "manipulative materials helped me in learning geometric content, as geometry is related to the use of figures that are present in my daily life in the environment that surrounds me. I would like to work more with manipulative materials in the classroom".

Thus, Costa (2018) highlights that it is necessary for the teaching dynamics to be organized with the objective of shifting the emphasis from the visual and/or spatial aspect of geometric objects to the tactile through appropriate materials by given that visually impaired students can grasp the information arising from mathematical propositions embodied in manipulative materials through tactile sensations and the communicative process.

In the final questionnaire, the visually impaired student argued that "manipulative materials helped me in understanding geometric contents. It certainly helped me a lot because it was much easier for me to measure figures with them". This pedagogical action enables the establishment of relations between the properties and characteristics of geometric objects through the mediation of manipulative materials. From this perspective, Moreira (2011) highlights that manipulative materials assume an organizing function in mathematical learning because they establish relations between different geometric figures and objects.

In the final questionnaire, the visually impaired student commented that: "I have experience with manipulative material: a) Geoplane - it was very good because I managed to learn a lot from it. I found it much easier with it, b) Multiplane - it was very different because if I can't do it in my head, I can do it with this material, and c) Cuisenaire rods - it was good because I could make several triangles and do several problems with it to solve the Theorem of Pythagoras". In this context, Moreira and Masini (2001) affirm that teachers transfer the emphasis of learning from the visual aspect to the remaining senses, especially, the tactile one. Similarly, through the communicative process, students can associate information from teachers, the use of pedagogical resources and tacit knowledge existing in their cognitive structure.
## RESULTS AND DISCUSSION

Results obtained in this study provided evidence that the development of differentiated pedagogical actions in the classroom stimulated the use of other senses of the participants, such as hearing, speech, and touch, by enabling to obtain positive results in the development of their cognitive process using a comprehensive communicative process. By using manipulative and concrete materials, the visually impaired mathematics teacher promoted the improvement of his teaching practice by promoting a pedagogical action for a more holistic teaching and learning process in mathematics, which enabled the development of mathematical and geometric content for the visually impaired student.

The adaptation of manipulatives and concrete materials such as the geoplane, the multiplane, and the cuisenaire rods, helped the student in exploration, investigation, and understanding of mathematical and geometric content proposed by the pedagogical actions conducted in the classroom. This was done by encouraging the student to associate aspects of the concrete to the abstract and included concepts of perimeter, area, and angles, as well as the characteristics and properties of geometric figures, and the identification and use of the Pythagorean Theorem.

These results, aimed at the improvement of an inclusive mathematics education, also showed the importance of the sociocultural differences that agreed with assumptions of ethnomathematics. The visually impaired student stated that "manipulative materials helped me in learning geometric content, as geometry is related to the use of figures that are present in my daily life in the environment that surrounds me. I would like to work more with manipulative materials in the classroom".

Thus, Costa (2018) highlights that it is necessary for the teaching dynamics to be organized with the objective of shifting the emphasis from the visual and/or spatial aspect of geometric objects to the tactile through appropriate materials by given that visually impaired students can grasp the information arising from mathematical propositions embodied in manipulative materials through tactile sensations and the communicative process.

In the final questionnaire, the visually impaired student argued that "manipulative materials helped me in understanding geometric contents. It certainly helped me a lot because it was much easier for me to measure figures with them". This pedagogical action enables the establishment of relations between the properties and characteristics of geometric objects through the mediation of manipulative materials. From this perspective, Moreira (2011) highlights that manipulative materials assume an organizing function in mathematical learning because they establish relations between different geometric figures and objects.

In the final questionnaire, the visually impaired student commented that: "I have experience with manipulative material: a) Geoplane - it was very good because I managed to learn a lot from it. I found it much easier with it, b) Multiplane - it was very different because if I can't do it in my head, I can do it with this material, and c) Cuisenaire rods - it was good because I could make several triangles and do several problems with it to solve the Theorem of Pythagoras". In this context, Moreira and Masini (2001) affirm that teachers transfer the emphasis of learning from the visual aspect to the remaining senses, especially, the tactile one. Similarly, through the communicative process, students can associate information from teachers, the use of pedagogical resources and tacit knowledge existing in their cognitive structure.
In the final questionnaire, the visually impaired teacher explained about the development of the activity blocks proposed to the visually impaired student in the classroom by stating that: "This material contributed to his experience of abstract geometry in terms of plane figures, so that he could experience to work with manipulative materials to have tactile contact with geometry and get to know the issue of the figures, so this helped a lot with his cognitive construction".

In one of the activity modules, the student identified a right-angled triangle made of a concrete material and, when feeling its sides with his fingers, he found that this geometric figure had the shape of a small chair and a toboggan. Thus, this student remembered and compared the right triangle with the little chair and its hypotenuse with the toboggan by commenting that it was "a 90° angle". In this regard, the chair jargon was used by the teacher to explain the 90° angle and also to relate it to the right angle by referring his explanation to the notion of perpendicularity. Similarly, the jargon of toboggan was used by the teacher to tell the visually impaired student the side of this right-angled triangle with its hypotenuse.

The interpretation of the results obtained in this study showed that the visually impaired teacher asked the visually impaired student: "What is the formula for the area of a square?". Thus, the visually impaired student took the teacher's finger and placed it inside of the geometric figure that represented the square made of concrete material and then sliding it across the interior of the figure. This participant stated that: "The area of the square is like this!" by pointing his finger towards the internal region of this figure. As well as he slid his fingers down the sides of this figure and answered that: "The area of the square is side times side".

The interpretation of these results shows that the student determined the area of squares by using his own techniques. From this perspective, Domingues (2010) comments on the need for teachers to promote specific learning opportunities for visually impaired students through the use of strategies and specific resources in order to improve their learning.

## FINAL CONSIDERATIONS

The results obtained in this study show that the visually impaired student's active touch was triggered by the exploration and manipulation of manipulative or concrete materials that enable the provision of care targeted at the specificity of their disability. Therefore, it is noteworthy that this pedagogical approach is related to respect for the culture of visually impaired students, as well as the writing system itself and the values and rules of behavior in accordance with the specificities and needs of this school population.

This context enabled, according to Vergani (2007), the development of a conception of Ethnomathematics that also deals with the rational, psychic, emotional, social, and cultural integrity of members of distinct cultural groups through a stance that echoes from and to the different educational levels according to their different degrees of depth.

In the pedagogical action proposed for this study, the assumptions of ethnomathematics and inclusive education coincided, with the aim of respecting and valuing the social, cultural, and educational inclusion of the visually impaired student and the visually impaired mathematics teacher. According to the epistemological dimension of ethnomathematics, there is a need to implement a set of pedagogical actions aimed at developing an inclusive mathematical education that values and respects the differences by promoting cultural plurality in classrooms.
In the final questionnaire, the visually impaired teacher explained about the development of the activity blocks proposed to the visually impaired student in the classroom by stating that: "This material contributed to his experience of abstract geometry in terms of plane figures, so that he could experience to work with manipulative materials to have tactile contact with geometry and get to know the issue of the figures, so this helped a lot with his cognitive construction".

In one of the activity modules, the student identified a right-angled triangle made of a concrete material and, when feeling its sides with his fingers, he found that this geometric figure had the shape of a small chair and a toboggan. Thus, this student remembered and compared the right triangle with the little chair and its hypotenuse with the toboggan by commenting that it was "a 90° angle". In this regard, the chair jargon was used by the teacher to explain the 90° angle and also to relate it to the right angle by referring his explanation to the notion of perpendicularity. Similarly, the jargon of toboggan was used by the teacher to tell the visually impaired student the side of this right-angled triangle with its hypotenuse.

The interpretation of the results obtained in this study showed that the visually impaired teacher asked the visually impaired student: "What is the formula for the area of a square?". Thus, the visually impaired student took the teacher's finger and placed it inside of the geometric figure that represented the square made of concrete material and then sliding it across the interior of the figure. This participant stated that: "The area of the square is like this!" by pointing his finger towards the internal region of this figure. As well as he slid his fingers down the sides of this figure and answered that: "The area of the square is side times side".

The interpretation of these results shows that the student determined the area of squares by using his own techniques. From this perspective, Domingues (2010) comments on the need for teachers to promote specific learning opportunities for visually impaired students through the use of strategies and specific resources in order to improve their learning.

## FINAL CONSIDERATIONS

The results obtained in this study show that the visually impaired student's active touch was triggered by the exploration and manipulation of manipulative or concrete materials that enable the provision of care targeted at the specificity of their disability. Therefore, it is noteworthy that this pedagogical approach is related to respect for the culture of visually impaired students, as well as the writing system itself and the values and rules of behavior in accordance with the specificities and needs of this school population.

This context enabled, according to Vergani (2007), the development of a conception of Ethnomathematics that also deals with the rational, psychic, emotional, social, and cultural integrity of members of distinct cultural groups through a stance that echoes from and to the different educational levels according to their different degrees of depth.

In the pedagogical action proposed for this study, the assumptions of ethnomathematics and inclusive education coincided, with the aim of respecting and valuing the social, cultural, and educational inclusion of the visually impaired student and the visually impaired mathematics teacher. According to the epistemological dimension of ethnomathematics, there is a need to implement a set of pedagogical actions aimed at developing an inclusive mathematical education that values and respects the differences by promoting cultural plurality in classrooms.
Therefore, the need for teachers to de-condition themselves from their academic classificatory
systems in understanding the geometric knowledge and techniques developed by visually impaired
students in their relationship with different sociocultural contexts stands out. Thus, in this study, the
ethnomathematical perspective assumed the appreciation and respect for the culture of students from
minority groups, whose geometric knowledge is despised by hegemonic and dominant cultures, given
that it aims to include ideas, procedures, techniques and practices mathematics developed by visually
impaired students in the school community itself.

References

Brasil (2015). *Plano Nacional de Educação: PNE-2014-2024*. Linha de Base. Diretoria de Estudos Educacionais - DIRED. MEC/INEP.

Brasil (2022a). *Demographic census 2022*. IBGE.

Brasil (2022b). *School census 2022*. MEC/INEP.

Costa, L. F. M. (2-018). *Metodologia do ensino da matemática: fragmentos possíveis*. BK Editora.

D'Ambrosio, U. (2001). *Etnomatemática: elo entre as tradições e a modernidade*. Autêntica Editora.

D'Ambrosio, U. (2005). Sociedade, cultura, matemática e seu ensino. *Educação e Pesquisa, 31*(1), 99-120.

D'Ambrosio, U., & Borba, M. C. (2010). Dynamics of change of mathematics education in Brazil and a scenario of current research. *ZDM Mathematics Education, 42*(3-4), 271–279.

Domingues, C. A. (2010). *A educação especial na perspectiva da inclusão escolar: os alunos com deficiência visual: baixa visão e cegueira*. MEC/SEE.

Kaleff, A. M. M. R. (2016). *Vendo com as mãos, olhos e mente: recursos didáticos para laboratório e museu de Educação Matemática Inclusiva do aluno com deficiência visual*. CEAD/UFF.

Moreira, M. A. (2011). *Aprendizagem significativa: a teoria e textos complementares*. Livraria da Física.

Moreira, M. A., & Masini, E. F. S. (2001). *Aprendizagem significativa: a teoria de David Ausubel*. Centauro.

Pinheiro, R. C., & Rosa, M. (2020). Ethnomathematical paths to Financial Education: analysing cultural traits of Deaf students. *Revemop, 2*(1), 1-18.

Rocha, A. B. O. (2017). O papel do professor na educação inclusiva. *Ensaios Pedagógicos, 7*(2), 1-11.

Rosa, M. (2010). *A mixed-methods study to understand the perceptions of high-school leaders about ELL students: the case of mathematics*. College of Education. California State University.

Rosa, M., & Orey, D. C. (2017). *Influências etnomatemáticas em salas de aula: caminhando para a ação pedagógica*. Appris Editora.

Silva, G. A.P., & Rosa, M. (2023). Etnomatemática e educação inclusiva: reflexões sobre a ação pedagógica. *Journal of Mathematics and Culture, 17*(2), 1-28.

Unesco (2015). *Declaração de Incheon educação 2030: rumo a uma educação de qualidade inclusiva e equitativa e à educação ao longo da vida para todos*. UNESCO.

Unesco (2017). *A guide for ensuring inclusion and equity in education*. UNESCO.

Unesco (2019). *Manual para garantir inclusão e equidade na educação: educação 2030*. UNESCO, 2019.

Vergani, T. (2007). *Educação etnomatemática: o que é? Flecha do Tempo*.

Vitello, S. J., & Mithaug, D. E. (1998). *Inclusive schooling: national and international perspectives*. Lawrence Erlbaum.
Therefore, the need for teachers to de-condition themselves from their academic classificatory
systems in understanding the geometric knowledge and techniques developed by visually impaired
students in their relationship with different sociocultural contexts stands out. Thus, in this study, the
ethnomathematical perspective assumed the appreciation and respect for the culture of students from
minority groups, whose geometric knowledge is despised by hegemonic and dominant cultures, given
that it aims to include ideas, procedures, techniques and practices mathematics developed by visually
impaired students in the school community itself.

References

Brasil (2015). *Plano Nacional de Educação: PNE-2014-2024*. Linha de Base. Diretoria de Estudos Educacionais - DIRED. MEC/INEP.

Brasil (2022a). *Demographic census 2022*. IBGE.

Brasil (2022b). *School census 2022*. MEC/INEP.

Costa, L. F. M. (2-018). *Metodologia do ensino da matemática: fragmentos possíveis*. BK Editora.

D'Ambrosio, U. (2001). *Etnomatemática: elo entre as tradições e a modernidade*. Autêntica Editora.

D'Ambrosio, U. (2005). Sociedade, cultura, matemática e seu ensino. *Educação e Pesquisa, 31*(1), 99-120.

D'Ambrosio, U., & Borba, M. C. (2010). Dynamics of change of mathematics education in Brazil and a scenario of current research. *ZDM Mathematics Education, 42*(3-4), 271–279.

Domingues, C. A. (2010). *A educação especial na perspectiva da inclusão escolar: os alunos com deficiência visual: baixa visão e cegueira*. MEC/SEE.

Kaleff, A. M. M. R. (2016). *Vendo com as mãos, olhos e mente: recursos didáticos para laboratório e museu de Educação Matemática Inclusiva do aluno com deficiência visual*. CEAD/UFF.

Moreira, M. A. (2011). *Aprendizagem significativa: a teoria e textos complementares*. Livraria da Física.

Moreira, M. A., & Masini, E. F. S. (2001). *Aprendizagem significativa: a teoria de David Ausubel*. Centauro.

Pinheiro, R. C., & Rosa, M. (2020). Ethnomathematical paths to Financial Education: analysing cultural traits of Deaf students. *Revemop, 2*(1), 1-18.

Rocha, A. B. O. (2017). O papel do professor na educação inclusiva. *Ensaios Pedagógicos, 7*(2), 1-11.

Rosa, M. (2010). *A mixed-methods study to understand the perceptions of high-school leaders about ELL students: the case of mathematics*. College of Education. California State University.

Rosa, M., & Orey, D. C. (2017). *Influências etnomatemáticas em salas de aula: caminhando para a ação pedagógica*. Appris Editora.

Silva, G. A.P., & Rosa, M. (2023). Etnomatemática e educação inclusiva: reflexões sobre a ação pedagógica. *Journal of Mathematics and Culture, 17*(2), 1-28.

Unesco (2015). *Declaração de Incheon educação 2030: rumo a uma educação de qualidade inclusiva e equitativa e à educação ao longo da vida para todos*. UNESCO.

Unesco (2017). *A guide for ensuring inclusion and equity in education*. UNESCO.

Unesco (2019). *Manual para garantir inclusão e equidade na educação: educação 2030*. UNESCO, 2019.

Vergani, T. (2007). *Educação etnomatemática: o que é? Flecha do Tempo*.

Vitello, S. J., & Mithaug, D. E. (1998). *Inclusive schooling: national and international perspectives*. Lawrence Erlbaum.
DISCUSSION DOCUMENT
DISCUSSION DOCUMENT
<figure><img src="image_214.png" /></figure>
<figure><img src="image_226.png" /></figure>
# THE 26TH ICMI STUDY: ADVANCES IN GEOMETRY EDUCATION
DISCUSSION DOCUMENT

## INTRODUCTION AND BACKGROUND

This document announces a new *ICMI Study* being conducted by the International Commission on Mathematical Instruction (ICMI). The Study, the 26th led by ICMI, addresses the advancements and current challenges in the teaching and learning of geometry, with particular attention to how the landscape has changed in the past three decades since the ICMI Study 9 on “Perspectives on the Teaching of Geometry for the 21st Century” (Mammana & Villani, 1998). Fundamentally, the International Program Committee (IPC) considers that geometry is positioned as a powerful pathway to support the understanding of complex ideas within school mathematics and beyond. In fact, we boldly suggest that geometry education is at its most exciting and quickly advancing period of the past 100 years. This ICMI Study examines the nature and role of geometry in mathematics education and related fields with an eye to the increasing depth and range of geometry education research, knowledge, pedagogies, and approaches on an international scale.

## AIMS AND RATIONALE

The primary aims of the Study are first to report the state of the field of mathematics education in the area of geometry education with respect to theory, research, practice, and policy; and, second, to suggest new directions of research that consider contextual, cultural, national, and political dimensions of practice.

Since there are different ways of understanding geometry education, the Study aims to include multiple theoretical perspectives and methodological approaches. Thus, the IPC encourages contributions that use a variety of methodological strategies including, among others, large-scale experimental and descriptive studies, case studies, and iterative or cyclical processes such as design research and action research. The IPC welcomes contributions from mathematics education researchers and from mathematics teachers, to ensure that teachers' voices are given prominence in accounts of their learning and teaching. Moreover, in this new Study, the IPC invites contributions from those working in the wider geometry space (such as cognitive scientists and educational psychologists). This interdisciplinary approach aims to provide innovative and collaborative opportunities not afforded by the current state of play that can leave work siloed within individual disciplines.

This Study aims to produce and share new knowledge about promoting effective pedagogy in the teaching and learning of geometry for all. It takes into account the large body of existing theory and research, socio-cultural diversity, cultural differences in curricular and institutional constraints, as well as progress the field. In particular, the IPC acknowledges the following specific goals for geometry education and anticipates that these will be developed by participants through the course of the ICMI Study:
# THE 26TH ICMI STUDY: ADVANCES IN GEOMETRY EDUCATION
## DISCUSSION DOCUMENT

## INTRODUCTION AND BACKGROUND

This document announces a new *ICMI Study* being conducted by the International Commission on Mathematical Instruction (ICMI). The Study, the 26th led by ICMI, addresses the advancements and current challenges in the teaching and learning of geometry, with particular attention to how the landscape has changed in the past three decades since the ICMI Study 9 on “Perspectives on the Teaching of Geometry for the 21st Century” (Mammana & Villani, 1998). Fundamentally, the International Program Committee (IPC) considers that geometry is positioned as a powerful pathway to support the understanding of complex ideas within school mathematics and beyond. In fact, we boldly suggest that geometry education is at its most exciting and quickly advancing period of the past 100 years. This ICMI Study examines the nature and role of geometry in mathematics education and related fields with an eye to the increasing depth and range of geometry education research, knowledge, pedagogies, and approaches on an international scale.

## AIMS AND RATIONALE

The primary aims of the Study are first to report the state of the field of mathematics education in the area of geometry education with respect to theory, research, practice, and policy; and, second, to suggest new directions of research that consider contextual, cultural, national, and political dimensions of practice.

Since there are different ways of understanding geometry education, the Study aims to include multiple theoretical perspectives and methodological approaches. Thus, the IPC encourages contributions that use a variety of methodological strategies including, among others, large-scale experimental and descriptive studies, case studies, and iterative or cyclical processes such as design research and action research. The IPC welcomes contributions from mathematics education researchers and from mathematics teachers, to ensure that teachers' voices are given prominence in accounts of their learning and teaching. Moreover, in this new Study, the IPC invites contributions from those working in the wider geometry space (such as cognitive scientists and educational psychologists). This interdisciplinary approach aims to provide innovative and collaborative opportunities not afforded by the current state of play that can leave work siloed within individual disciplines.

This Study aims to produce and share new knowledge about promoting effective pedagogy in the teaching and learning of geometry for all. It takes into account the large body of existing theory and research, socio-cultural diversity, cultural differences in curricular and institutional constraints, as well as progress the field. In particular, the IPC acknowledges the following specific goals for geometry education and anticipates that these will be developed by participants through the course of the ICMI Study:
* Bring together an expert reference group to analyze the state-of-the-art of research and practice in geometry education in order to contribute to better understanding of the challenges that geometry education faces in diverse contexts.

* Bring together communities of international scholars, with diverse representation within the ICMI community, and across regions and nationalities, to address research and practices in geometry education, ultimately resulting in the production of an ICMI Study volume.

* Emphasize the role of geometry as a facilitator of advances in logical reasoning, strategic thinking, and understanding of hierarchical relationships between mathematical (geometrical) objects.

* Analyze the influence of affective components of teaching and learning such as confidence, motivation, anxiety, efficacy, and others, on teachers' decisions and methodological choices and on students' behavior and related learning results.

* Facilitate advances in multi- and inter-disciplinary approaches (including cooperation with other bodies and scientific communities) to research and development in geometry education.

* Explore the role of Geometry as an arena or lever for developing children's and adolescents' mathematical creativity and flexible thinking by means of conjecturing, problem posing, and proving theorems.

* Disseminate scholarship in mathematics education — research, practices, methodologies, theories, findings and results, curricula design, etc. — in geometry teaching and learning.

* Identify and anticipate new research and development possibilities, challenges, and questions related to geometry education and research in geometry education.

* Produce sets of recommendations on effective resources for researchers, teachers, teacher educators, policy makers, curriculum developers, analysts, and the broad range of practitioners in mathematics and mathematics education.

* Act as an instigator of new research and innovation to be produced in the future.

* Promote and assist in the discussion of geometry education and related research in action at local and international levels.

## STRUCTURE: TOPICS, SUB-TOPICS, AND QUESTIONS

The activity of the Study is organized based on four focused topics, aimed to provide complementary perspectives and approaches to the teaching and learning of geometry. Contributions to the topics will be organized around sets of related sub-topics, each sub-topic focusing on a specific issue and stating a set of questions aimed to lead discussions. The four topics are:

A. Theoretical perspectives

B. Curricular and methodological approaches

C. Resources for teaching and learning geometry

D. Multidisciplinary perspectives
* Bring together an expert reference group to analyze the state-of-the-art of research and practice in geometry education in order to contribute to better understanding of the challenges that geometry education faces in diverse contexts.

* Bring together communities of international scholars, with diverse representation within the ICMI community, and across regions and nationalities, to address research and practices in geometry education, ultimately resulting in the production of an ICMI Study volume.

* Emphasize the role of geometry as a facilitator of advances in logical reasoning, strategic thinking, and understanding of hierarchical relationships between mathematical (geometrical) objects.

* Analyze the influence of affective components of teaching and learning such as confidence, motivation, anxiety, efficacy, and others, on teachers' decisions and methodological choices and on students' behavior and related learning results.

* Facilitate advances in multi- and inter-disciplinary approaches (including cooperation with other bodies and scientific communities) to research and development in geometry education.

* Explore the role of Geometry as an arena or lever for developing children's and adolescents' mathematical creativity and flexible thinking by means of conjecturing, problem posing, and proving theorems.

* Disseminate scholarship in mathematics education — research, practices, methodologies, theories, findings and results, curricula design, etc. — in geometry teaching and learning.

* Identify and anticipate new research and development possibilities, challenges, and questions related to geometry education and research in geometry education.

* Produce sets of recommendations on effective resources for researchers, teachers, teacher educators, policy makers, curriculum developers, analysts, and the broad range of practitioners in mathematics and mathematics education.

* Act as an instigator of new research and innovation to be produced in the future.

* Promote and assist in the discussion of geometry education and related research in action at local and international levels.

## STRUCTURE: TOPICS, SUB-TOPICS, AND QUESTIONS

The activity of the Study is organized based on four focused topics, aimed to provide complementary perspectives and approaches to the teaching and learning of geometry. Contributions to the topics will be organized around sets of related sub-topics, each sub-topic focusing on a specific issue and stating a set of questions aimed to lead discussions. The four topics are:

A. Theoretical perspectives

B. Curricular and methodological approaches

C. Resources for teaching and learning geometry

D. Multidisciplinary perspectives
# TOPIC A: THEORETICAL PERSPECTIVES

Several theoretical and conceptual perspectives have been used to study geometry education —some have emerged within the field itself while others have been drawn from the education disciplines or applied from psychology or sociology. Theories specifically about geometry education that continue to be evident in research include the Van Hiele model (Van Hiele, 1986), the theory of figural concepts (Fischbein, 1993; Mariotti & Fischbein, 1997), and the theory of figural apprehension (Duval, 1998). Researchers are continuing to develop, refine and apply these theories. More recently, theories emerging in geometry education include embodied (Ng & Sinclair, 2015) and ecocultural perspectives (Owens, 2014), prototype phenomenon (Gal & Linchevski, 2010), and semiotic mediation (Bartolini & Mariotti, 2008).

In the recent ICMI Survey of geometry education, Sinclair et al. (2016) maintained that over the past decade, there has been an increased focus on theories related to visuospatial reasoning, the role of gestures and diagrams, and the use of digital technologies. As Sinclair and Bruce (2015) reinforce, post-Euclidian developments have moved beyond the emphasis of naming shapes and their properties to “a more active meaning-making orientation to geometry (including composing/decomposing, classifying, mapping and orienting, comparing and mentally manipulating two- and three-dimensional figures)” (p. 320).

Rather than revisiting the well-documented theoretical perspectives, this topic aims to dive more deeply into several emerging areas of research activity in geometry teaching and learning. Such emerging areas include new conceptions of the teaching and learning of proofs in geometry contexts, the role of spatial reasoning and the implications for geometry teaching and learning, and, in a related area, the role of visualization in geometry reasoning.

## Sub-topic A1: Teaching and learning of proof in geometrical contexts

In the study of geometry, the concept of proof appears as a key element to support and validate mathematical statements. The IPC conceives of the proving process as a way to convince oneself or another person, through the presentation of different types of arguments in favor of the pertinence and validity of mathematical statements. In Euclidean geometry contexts, it is important to recognize that, depending on the education level of the students, the type of proofs to support conjectures vary. For instance, at elementary education, concrete examples could be sufficient to present and support mathematical relationships, but when students’ progress or advance in their studies, they should refine their ideas and transit from empirical examples to present informal deductive proofs and, finally, formal proofs. In this context, teachers’ educational programs should explicitly include the development of knowledge and experience in problem posing, formulating conjectures, and validating them. These are creative processes as much as they are logical thought processes.

It is also important to acknowledge that the consistent and systematic use of digital technologies opens different routes for teachers and students to represent and explore geometrical properties and also to support the truth of mathematical conjectures or statements.

Some overarching questions for contributions on the teaching and learning of mathematical proof in geometry include:
# TOPIC A: THEORETICAL PERSPECTIVES

Several theoretical and conceptual perspectives have been used to study geometry education —some have emerged within the field itself while others have been drawn from the education disciplines or applied from psychology or sociology. Theories specifically about geometry education that continue to be evident in research include the Van Hiele model (Van Hiele, 1986), the theory of figural concepts (Fischbein, 1993; Mariotti & Fischbein, 1997), and the theory of figural apprehension (Duval, 1998). Researchers are continuing to develop, refine and apply these theories. More recently, theories emerging in geometry education include embodied (Ng & Sinclair, 2015) and ecocultural perspectives (Owens, 2014), prototype phenomenon (Gal & Linchevski, 2010), and semiotic mediation (Bartolini & Mariotti, 2008).

In the recent ICMI Survey of geometry education, Sinclair et al. (2016) maintained that over the past decade, there has been an increased focus on theories related to visuospatial reasoning, the role of gestures and diagrams, and the use of digital technologies. As Sinclair and Bruce (2015) reinforce, post-Euclidian developments have moved beyond the emphasis of naming shapes and their properties to “a more active meaning-making orientation to geometry (including composing/decomposing, classifying, mapping and orienting, comparing and mentally manipulating two- and three-dimensional figures)” (p. 320).

Rather than revisiting the well-documented theoretical perspectives, this topic aims to dive more deeply into several emerging areas of research activity in geometry teaching and learning. Such emerging areas include new conceptions of the teaching and learning of proofs in geometry contexts, the role of spatial reasoning and the implications for geometry teaching and learning, and, in a related area, the role of visualization in geometry reasoning.

## Sub-topic A1: Teaching and learning of proof in geometrical contexts

In the study of geometry, the concept of proof appears as a key element to support and validate mathematical statements. The IPC conceives of the proving process as a way to convince oneself or another person, through the presentation of different types of arguments in favor of the pertinence and validity of mathematical statements. In Euclidean geometry contexts, it is important to recognize that, depending on the education level of the students, the type of proofs to support conjectures vary. For instance, at elementary education, concrete examples could be sufficient to present and support mathematical relationships, but when students' progress or advance in their studies, they should refine their ideas and transit from empirical examples to present informal deductive proofs and, finally, formal proofs. In this context, teachers' educational programs should explicitly include the development of knowledge and experience in problem posing, formulating conjectures, and validating them. These are creative processes as much as they are logical thought processes.

It is also important to acknowledge that the consistent and systematic use of digital technologies opens different routes for teachers and students to represent and explore geometrical properties and also to support the truth of mathematical conjectures or statements.

Some overarching questions for contributions on the teaching and learning of mathematical proof in geometry include: