The Development of Primary School Students’ 3D Geometrical Thinking within a Dynamic Transformation Context

Author(s)
Christos Markopoulos^{1},
Despina Potari^{2},
William Boyd^{3},
Κoralia Petta^{1},
Marilyn Chaseling^{1}

Affiliation(s)

^{1}
School of Education, Southern Cross University, Lismore, Australia.

^{2}
Faculty of Mathematics, National and Kapodistrian University of Athens, Athens, Greece.

^{3}
School of Environment, Science & Engineering, Southern Cross University, Lismore, Australia.

ABSTRACT

This study examines children’s thinking about geometrical solids through an investigation of dynamic transformations employed by young children making mental transformations of an orthogonal parallelepiped. The focus of the study is on the investigation of the role that a dynamic environment could play in the development of children’s geometrical thinking concerning geometrical solids and their properties. Twenty 6th grade children, who had previously worked with dynamic transformations of physical models of geometrical solids in their classroom, were interviewed. Analysis of the data resulted in a categorization of children’s thinking, and indicated a development from a perceptual to a geometrical consideration of the solid. Although not all the children reached an advanced level of thinking, the context of dynamic transformations promoted the development of most children’s geometrical thinking. There is also an indication that children’s experience with dynamic transformations of physical models in a mathematics classroom environment can act to allow children to transfer experience to the context of mental transformations.

This study examines children’s thinking about geometrical solids through an investigation of dynamic transformations employed by young children making mental transformations of an orthogonal parallelepiped. The focus of the study is on the investigation of the role that a dynamic environment could play in the development of children’s geometrical thinking concerning geometrical solids and their properties. Twenty 6th grade children, who had previously worked with dynamic transformations of physical models of geometrical solids in their classroom, were interviewed. Analysis of the data resulted in a categorization of children’s thinking, and indicated a development from a perceptual to a geometrical consideration of the solid. Although not all the children reached an advanced level of thinking, the context of dynamic transformations promoted the development of most children’s geometrical thinking. There is also an indication that children’s experience with dynamic transformations of physical models in a mathematics classroom environment can act to allow children to transfer experience to the context of mental transformations.

Cite this paper

Markopoulos, C. , Potari, D. , Boyd, W. , Petta, Κ. and Chaseling, M. (2015) The Development of Primary School Students’ 3D Geometrical Thinking within a Dynamic Transformation Context.*Creative Education*, **6**, 1508-1522. doi: 10.4236/ce.2015.614151.

Markopoulos, C. , Potari, D. , Boyd, W. , Petta, Κ. and Chaseling, M. (2015) The Development of Primary School Students’ 3D Geometrical Thinking within a Dynamic Transformation Context.

References

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http://dx.doi.org/10.1080/10986060903016484

[2] Arnold, L., & Lawson, M. (2003). Spatial Problem-Solving in Year 7 Mathematics: An Examination of the Effects of Use of a Computer-Mediated Software Program. Mathematics Education Research Journal, 15, 187-202.

http://dx.doi.org/10.1007/BF03217378

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[4] Battista, M. T., & Clements, D. H. (1996). Students’ Understanding of Three-Dimensional Rectangular Arrays of Cubes. Journal for Research in Mathematics Education, 27, 258-292.

http://dx.doi.org/10.2307/749365

[5] Bell, A. (1993). Some Experiments in Diagnostic Teaching. Educational Studies in Mathematics, 24, 115-137.

http://dx.doi.org/10.1007/BF01273297

[6] Bishop, A. J. (1985). Visualizing Rectangular Solids Made of Small Cubes: Analyzing and Effecting Students’ Performance. Educational Studies in Mathematics, 16, 389-409.

http://dx.doi.org/10.1007/BF00417194

[7] Bliss, J., Monk, M., & Ogborn, J. (1983). Qualitative Data Analysis for Educational Research. London: Croom Helm.

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http://dx.doi.org/10.2307/749161

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http://dx.doi.org/10.2307/749283

[12] Duval, R. (2006). A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics. Educational Studies in Mathematics, 61, 103-131.

http://dx.doi.org/10.1007/s10649-006-0400-z

[13] Duval, R. (2011). Why Figures Cannot Help Students to See and Understand in Geometry? Analysis of the Role and the Cognitive Functioning of Visualization. In Symposium Mathematics Education Research at the University of Cyprus and Tel Aviv University (pp. 22-23). Cyprus: University of Cyprus.

[14] Evans, J. (1999). Building Bridges: Reflections on the Problem of Transfer of Learning in Mathematics. Educational Studies in Mathematics, 39, 23-44.

http://dx.doi.org/10.1023/A:1003755611058

[15] Fischbein, E. (1993). The Theory of Figural Concepts. Educational Studies in Mathematics, 24, 139-162.

http://dx.doi.org/10.1007/BF01273689

[16] Freudenthal, H. (1991). Revisiting Mathematics Education. Dordrecht: Kluwer Academic.

[17] Gutierrez, A. (1996). Visualization in 3-Dimensional Geometry: In Search of a Framework. In L. Puig, & A. Gutierrez (Eds.), Proceedings of the 20th International Conference for the Psychology of Mathematics Education (Vol. 1, pp. 3-19). Valencia: Universidad de Valencia.

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[19] Hershkowitz, R. (1989). Visualization in Geometry—Two Sides of the Coin. Focus on Learning Problems in Mathematics, 11, 61-76.

[20] Hunting, R. P. (1997). Clinical Interview Methods in Mathematics Education Research and Practice. Journal for Research in Mathematics Education, 16, 145-165.

http://dx.doi.org/10.1016/s0732-3123(97)90023-7

[21] Jones, K. (2000). Providing a Foundation for Deductive Reasoning: Students’ Interpretations When Using Dynamic Geometry Software and Their Evolving Mathematical Explanations. Educational Studies in Mathematics, 44, 55-85.

http://dx.doi.org/10.1023/A:1012789201736

[22] Lakoff, G., & Núñez, R. E. (2000). Where Mathematics Comes from. How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.

[23] Lawrie C., Pegg, J., & Gutierrez, A. (2000). Coding the Nature of Thinking Displayed in Responses on Nets of Solids. In T. Nakahara, & M. Koyama (Eds.), Proceedings of the 24th International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 215-222). Hiroshima: ERIC.

[24] Lawrie, C., Pegg, J., & Gutierrez, A. (2002). Unpacking Student Meaning of Cross-Sections: A Frame for Curriculum Development. In A. D. Cockburn, & E. Nardi (Eds.), Proceedings of the 26th International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 289-296). Norwich: SIME.

[25] Lean, G., & Clements, M. A. (1981). Spatial Ability, Visual Imagery, and Mathematical Performance. Educational Studies in Mathematics, 12, 267-299.

http://dx.doi.org/10.1007/BF00311060

[26] Lowrie, T. (2012). Visual and Spatial Reasoning: The Changing Form of Mathematics Representation and Communication. In B. Kaur, & T. T. Lam (Eds.), Reasoning, Communication and Connections in Mathematics, Yearbook 2012, Association of Mathematics Teachers (pp. 149-168). Singapore: Word Scientific Publishing.

[27] Ma, H. L., Wu, D. B., Chen, J. W., & Hsieh, K. J. (2009). Mitchelmore’s Development stages of the Right Rectangular Prisms of Elementary School Students in Taiwan. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 57-64). Thessaloniki: PME.

[28] Mariotti, M. A. (1989). Mental Images: Some Problems Related to the Development of Solids. In G. Vergnaud, J. Rogalski, & M. Artique (Eds.), Proceedings of the 13th International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 258-265). Paris: PME.

[29] Mariotti, M. A. (2000). Introduction to Proof: The Mediation of a Dynamic Software Environment. Educational Studies in Mathematics, 44, 25-53.

http://dx.doi.org/10.1023/A:1012733122556

[30] Markopoulos, C. (2003). Children’s Thinking of Geometrical Solids in a Computer-Based Environment. In T. Triandafillidis, & K. Hatzikiriakou (Eds.), Proceedings of the 6th International Conference on Technology in Mathematics Education (pp. 152-157). Volos: SIME.

[31] Markopoulos, C., & Potari, D. (2000). Dynamic Transformations of Solids in the Mathematics Education. In T. Nakahara, & M. Koyama (Eds.), Proceedings of the 24th International Conference for the Psychology of Mathematics Education, (Vol. 3, pp. 215-222). Hiroshima: ERIC.

[32] Markopoulos, C., & Potari, D. (1999). Forming Relationships in Three Dimensional Geometry through Dynamic Environments. In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 273-280). Haifa: PME.

[33] Mitchelmore, M. C. (1980). Prediction of Developmental Stages in the Representation of Regular Space Figures. Journal for Research in Mathematics Education, 11, 83-93.

http://dx.doi.org/10.2307/748901

[34] Modenov, P. S., & Parkhomenko, A. S. (1965). Geometric Transformations, Euclidean and Affine Transformations (Vol. 1). New York: Academic Press.

[35] Otte, M. (1997). Mathematics, Semiotics, and the Growth of Social Knowledge. For the Learning of Mathematics, 17, 47-54.

[36] Pandiscio, E., & Orton, R. E. (1998). Geometry and Metacognition: An Analysis of Piaget’s and van Hiele’s Perspectives. Focus on Learning Problems in Mathematics, 20, 78-87.

[37] Pegg, J. (1997). Broadening the Descriptors of van Hiele’s Levels 2 and 3. In Proceedings of the 20th Annual Meeting of the Mathematics Education Research Group of Australasia (pp. 391-396). Hamilton: MERGA.

[38] Pegg, J., & Baker, P. (1999). An Exploration of the Interface between van Hiele’s Levels 1 and 2: Initial Findings. Environments. In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education (Vol. 4, pp. 25-32). Haifa: PME.

[39] Piaget, J. (1972). Psychology and Epistemology. Towards a Theory of Knowledge. Westminster: Penguin Books.

[40] Piaget, J., & Garcia, R. (1989). Psychogenesis and the History of Science. New York: Columbia University Press.

[41] Pittalis, M., & Christou, C. (2010). Types of Reasoning in 3D Geometry Thinking and Their Relations with Spatial Ability. Educational Studies in Mathematics, 75, 191-212.

http://dx.doi.org/10.1007/s10649-010-9251-8

[42] Potari, D., & Spiliotopoulou, V. (1992). Children’s Representations of the Development of Solids. For the Learning of Mathematics, 12, 38-46.

[43] Presmeg, N. (2006). Research on Visualization in Learning and Teaching Mathematics. In A. Gutiérrez, & P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future (pp. 205-236). Rotterdam: Sense.

[44] Sack, J. J. (2013). Development of a Top-View Numeric Coding Teaching-Learning Trajectory within an Elementary Grades 3-D Visualization Design Research Project. The Journal of Mathematical Behavior, 32, 183-196.

http://dx.doi.org/10.1016/j.jmathb.2013.02.006

[45] Sdrolias, K. A., & Triandafillidis, T. A. (2008). The Transition to Secondary School Geometry: Can There Be a “Chain of School Mathematics”? Educational Studies in Mathematics, 67, 159-169.

http://dx.doi.org/10.1007/s10649-007-9093-1

[46] Stylianou, D. A., Leikin, R., & Silver, E. A. (1999). Exploring Students’ Solution Strategies in Solving a Spatial Visualization Problem Involving Nets. In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education (Vol. 4, pp. 241-248). Haifa: PME.

[47] Van Hiele, P. M. (1986). Structure and Insight. A Theory of Mathematics Education. London: Academic Press.

[48] Waldegg, G. (2004). Problem Solving, Collaborative Learning and History of Mathematics: Experiences in Training in-Service Teachers. Mediterranean Journal for Research in Mathematics Education, 3, 63-72.

[49] Wheatley, G. H. (1990). Spatial Sense and Mathematics Learning. Arithmetic Teacher, 37, 10-11.

[1] Ambrose, R., & Kenehan, G. (2009). Children’s Evolving Understanding of Polyhedra in the Classroom. Mathematical Thinking and Learning, 11, 158-176.

http://dx.doi.org/10.1080/10986060903016484

[2] Arnold, L., & Lawson, M. (2003). Spatial Problem-Solving in Year 7 Mathematics: An Examination of the Effects of Use of a Computer-Mediated Software Program. Mathematics Education Research Journal, 15, 187-202.

http://dx.doi.org/10.1007/BF03217378

[3] Battista, M. T. (2007). The Development of Geometric and Spatial Thinking. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 843-908). Charlotte, NC: Information Age.

[4] Battista, M. T., & Clements, D. H. (1996). Students’ Understanding of Three-Dimensional Rectangular Arrays of Cubes. Journal for Research in Mathematics Education, 27, 258-292.

http://dx.doi.org/10.2307/749365

[5] Bell, A. (1993). Some Experiments in Diagnostic Teaching. Educational Studies in Mathematics, 24, 115-137.

http://dx.doi.org/10.1007/BF01273297

[6] Bishop, A. J. (1985). Visualizing Rectangular Solids Made of Small Cubes: Analyzing and Effecting Students’ Performance. Educational Studies in Mathematics, 16, 389-409.

http://dx.doi.org/10.1007/BF00417194

[7] Bliss, J., Monk, M., & Ogborn, J. (1983). Qualitative Data Analysis for Educational Research. London: Croom Helm.

[8] Brown, D. L., & Wheatley, G. H. (1997). Components of Imagery and Mathematical Understanding. Focus on Learning Problems in Mathematics, 19, 45-70.

[9] Cobb, P., Yackel, E., & Wood, T. (1992). A Constructivist Alternative to the Representational View of Mind in Mathematics Education. Journal for Research in Mathematics Education, 23, 2-33.

http://dx.doi.org/10.2307/749161

[10] Cohen, N. (2003). Curved Solid Nets. In N. Paterman, B. J. Doughery, & J. Zillox (Eds.), Proceedings of the 27th International Conference of Psychology in Mathematics Education, Vol. 2, (pp. 229-236). Hawaii: University of Hawaii.

[11] Cooper, M., & Sweller, J. (1989). Secondary School Students’ Representations of Solids. Journal for Research in Mathematics Education, 20, 202-212.

http://dx.doi.org/10.2307/749283

[12] Duval, R. (2006). A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics. Educational Studies in Mathematics, 61, 103-131.

http://dx.doi.org/10.1007/s10649-006-0400-z

[13] Duval, R. (2011). Why Figures Cannot Help Students to See and Understand in Geometry? Analysis of the Role and the Cognitive Functioning of Visualization. In Symposium Mathematics Education Research at the University of Cyprus and Tel Aviv University (pp. 22-23). Cyprus: University of Cyprus.

[14] Evans, J. (1999). Building Bridges: Reflections on the Problem of Transfer of Learning in Mathematics. Educational Studies in Mathematics, 39, 23-44.

http://dx.doi.org/10.1023/A:1003755611058

[15] Fischbein, E. (1993). The Theory of Figural Concepts. Educational Studies in Mathematics, 24, 139-162.

http://dx.doi.org/10.1007/BF01273689

[16] Freudenthal, H. (1991). Revisiting Mathematics Education. Dordrecht: Kluwer Academic.

[17] Gutierrez, A. (1996). Visualization in 3-Dimensional Geometry: In Search of a Framework. In L. Puig, & A. Gutierrez (Eds.), Proceedings of the 20th International Conference for the Psychology of Mathematics Education (Vol. 1, pp. 3-19). Valencia: Universidad de Valencia.

[18] Hershkowitz, R. (1990). Psychological Aspects of Learning Geometry. In P. Nesher, & J. Kilpatrick (Eds.), Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education (pp. 70-95). Cambridge: Cambridge University Press.

[19] Hershkowitz, R. (1989). Visualization in Geometry—Two Sides of the Coin. Focus on Learning Problems in Mathematics, 11, 61-76.

[20] Hunting, R. P. (1997). Clinical Interview Methods in Mathematics Education Research and Practice. Journal for Research in Mathematics Education, 16, 145-165.

http://dx.doi.org/10.1016/s0732-3123(97)90023-7

[21] Jones, K. (2000). Providing a Foundation for Deductive Reasoning: Students’ Interpretations When Using Dynamic Geometry Software and Their Evolving Mathematical Explanations. Educational Studies in Mathematics, 44, 55-85.

http://dx.doi.org/10.1023/A:1012789201736

[22] Lakoff, G., & Núñez, R. E. (2000). Where Mathematics Comes from. How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.

[23] Lawrie C., Pegg, J., & Gutierrez, A. (2000). Coding the Nature of Thinking Displayed in Responses on Nets of Solids. In T. Nakahara, & M. Koyama (Eds.), Proceedings of the 24th International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 215-222). Hiroshima: ERIC.

[24] Lawrie, C., Pegg, J., & Gutierrez, A. (2002). Unpacking Student Meaning of Cross-Sections: A Frame for Curriculum Development. In A. D. Cockburn, & E. Nardi (Eds.), Proceedings of the 26th International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 289-296). Norwich: SIME.

[25] Lean, G., & Clements, M. A. (1981). Spatial Ability, Visual Imagery, and Mathematical Performance. Educational Studies in Mathematics, 12, 267-299.

http://dx.doi.org/10.1007/BF00311060

[26] Lowrie, T. (2012). Visual and Spatial Reasoning: The Changing Form of Mathematics Representation and Communication. In B. Kaur, & T. T. Lam (Eds.), Reasoning, Communication and Connections in Mathematics, Yearbook 2012, Association of Mathematics Teachers (pp. 149-168). Singapore: Word Scientific Publishing.

[27] Ma, H. L., Wu, D. B., Chen, J. W., & Hsieh, K. J. (2009). Mitchelmore’s Development stages of the Right Rectangular Prisms of Elementary School Students in Taiwan. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 57-64). Thessaloniki: PME.

[28] Mariotti, M. A. (1989). Mental Images: Some Problems Related to the Development of Solids. In G. Vergnaud, J. Rogalski, & M. Artique (Eds.), Proceedings of the 13th International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 258-265). Paris: PME.

[29] Mariotti, M. A. (2000). Introduction to Proof: The Mediation of a Dynamic Software Environment. Educational Studies in Mathematics, 44, 25-53.

http://dx.doi.org/10.1023/A:1012733122556

[30] Markopoulos, C. (2003). Children’s Thinking of Geometrical Solids in a Computer-Based Environment. In T. Triandafillidis, & K. Hatzikiriakou (Eds.), Proceedings of the 6th International Conference on Technology in Mathematics Education (pp. 152-157). Volos: SIME.

[31] Markopoulos, C., & Potari, D. (2000). Dynamic Transformations of Solids in the Mathematics Education. In T. Nakahara, & M. Koyama (Eds.), Proceedings of the 24th International Conference for the Psychology of Mathematics Education, (Vol. 3, pp. 215-222). Hiroshima: ERIC.

[32] Markopoulos, C., & Potari, D. (1999). Forming Relationships in Three Dimensional Geometry through Dynamic Environments. In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 273-280). Haifa: PME.

[33] Mitchelmore, M. C. (1980). Prediction of Developmental Stages in the Representation of Regular Space Figures. Journal for Research in Mathematics Education, 11, 83-93.

http://dx.doi.org/10.2307/748901

[34] Modenov, P. S., & Parkhomenko, A. S. (1965). Geometric Transformations, Euclidean and Affine Transformations (Vol. 1). New York: Academic Press.

[35] Otte, M. (1997). Mathematics, Semiotics, and the Growth of Social Knowledge. For the Learning of Mathematics, 17, 47-54.

[36] Pandiscio, E., & Orton, R. E. (1998). Geometry and Metacognition: An Analysis of Piaget’s and van Hiele’s Perspectives. Focus on Learning Problems in Mathematics, 20, 78-87.

[37] Pegg, J. (1997). Broadening the Descriptors of van Hiele’s Levels 2 and 3. In Proceedings of the 20th Annual Meeting of the Mathematics Education Research Group of Australasia (pp. 391-396). Hamilton: MERGA.

[38] Pegg, J., & Baker, P. (1999). An Exploration of the Interface between van Hiele’s Levels 1 and 2: Initial Findings. Environments. In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education (Vol. 4, pp. 25-32). Haifa: PME.

[39] Piaget, J. (1972). Psychology and Epistemology. Towards a Theory of Knowledge. Westminster: Penguin Books.

[40] Piaget, J., & Garcia, R. (1989). Psychogenesis and the History of Science. New York: Columbia University Press.

[41] Pittalis, M., & Christou, C. (2010). Types of Reasoning in 3D Geometry Thinking and Their Relations with Spatial Ability. Educational Studies in Mathematics, 75, 191-212.

http://dx.doi.org/10.1007/s10649-010-9251-8

[42] Potari, D., & Spiliotopoulou, V. (1992). Children’s Representations of the Development of Solids. For the Learning of Mathematics, 12, 38-46.

[43] Presmeg, N. (2006). Research on Visualization in Learning and Teaching Mathematics. In A. Gutiérrez, & P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future (pp. 205-236). Rotterdam: Sense.

[44] Sack, J. J. (2013). Development of a Top-View Numeric Coding Teaching-Learning Trajectory within an Elementary Grades 3-D Visualization Design Research Project. The Journal of Mathematical Behavior, 32, 183-196.

http://dx.doi.org/10.1016/j.jmathb.2013.02.006

[45] Sdrolias, K. A., & Triandafillidis, T. A. (2008). The Transition to Secondary School Geometry: Can There Be a “Chain of School Mathematics”? Educational Studies in Mathematics, 67, 159-169.

http://dx.doi.org/10.1007/s10649-007-9093-1

[46] Stylianou, D. A., Leikin, R., & Silver, E. A. (1999). Exploring Students’ Solution Strategies in Solving a Spatial Visualization Problem Involving Nets. In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education (Vol. 4, pp. 241-248). Haifa: PME.

[47] Van Hiele, P. M. (1986). Structure and Insight. A Theory of Mathematics Education. London: Academic Press.

[48] Waldegg, G. (2004). Problem Solving, Collaborative Learning and History of Mathematics: Experiences in Training in-Service Teachers. Mediterranean Journal for Research in Mathematics Education, 3, 63-72.

[49] Wheatley, G. H. (1990). Spatial Sense and Mathematics Learning. Arithmetic Teacher, 37, 10-11.