Pre-Service Teachers’ 3D Visualization Strategies

Author(s)
Christos Markopoulos^{1},
Marilyn Chaseling^{1},
Κoralia Petta^{1},
Warren Lake^{2},
William Boyd^{2}

Affiliation(s)

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

^{2}
School of Environment, Science & Engineering, Southern Cross University, Lismore, Australia.

ABSTRACT

This study examines primary and early childhood pre-service teachers’ strategies on a written task that promotes 3D geometric thinking and visualization processes. Visualization and conceptualisation of 3D objects are complex cognitive processes, and both require the development of students’ abilities to decode and encode spatial information. The analysis of 289 pre-service teachers’ written responses resulted in identifying students’ difficulties in decoding and encoding visual information. The visual information dominated student thinking, and they found it hard to identify relationships between the 2D representation and the 3D mental construction of the solids. Most made incorrect claims regarding relative volumes. Neither spatial visualisation nor formula-driven computation provided adequate engagement with the task. Visualization and conceptualisation of 3D objects are complex cognitive processes, and pre-service teachers need to engage with a variety of learning activities to help them develop their abilities to decode and encode spatial information and, it is hoped, develop their 3D geometric thinking. However, from a learning approach perspective, the results indicate a dominant surface learning approach; this may arise from prior inadequate learning. The best lesson the student may get, therefore, from this task is not the mastery of a mathematical computation, but awareness of the importance of teaching design and aligned teaching methods.

This study examines primary and early childhood pre-service teachers’ strategies on a written task that promotes 3D geometric thinking and visualization processes. Visualization and conceptualisation of 3D objects are complex cognitive processes, and both require the development of students’ abilities to decode and encode spatial information. The analysis of 289 pre-service teachers’ written responses resulted in identifying students’ difficulties in decoding and encoding visual information. The visual information dominated student thinking, and they found it hard to identify relationships between the 2D representation and the 3D mental construction of the solids. Most made incorrect claims regarding relative volumes. Neither spatial visualisation nor formula-driven computation provided adequate engagement with the task. Visualization and conceptualisation of 3D objects are complex cognitive processes, and pre-service teachers need to engage with a variety of learning activities to help them develop their abilities to decode and encode spatial information and, it is hoped, develop their 3D geometric thinking. However, from a learning approach perspective, the results indicate a dominant surface learning approach; this may arise from prior inadequate learning. The best lesson the student may get, therefore, from this task is not the mastery of a mathematical computation, but awareness of the importance of teaching design and aligned teaching methods.

Cite this paper

Markopoulos, C. , Chaseling, M. , Petta, Κ. , Lake, W. and Boyd, W. (2015) Pre-Service Teachers’ 3D Visualization Strategies.*Creative Education*, **6**, 1053-1059. doi: 10.4236/ce.2015.610104.

Markopoulos, C. , Chaseling, M. , Petta, Κ. , Lake, W. and Boyd, W. (2015) Pre-Service Teachers’ 3D Visualization Strategies.

References

[1] AITSL (2011). Accreditation of Initial Teacher Education Programs in Australia. Carlton South: Mceedcya.

[2] 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

[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] Biggs, J. B. (1987). The Study Process Questionnaire (SPQ): Manual. Hawthorn: Australian Council for Educational Research.

[5] Biggs, J., Kember, D., & Leung, D. Y. (2001). The Revised Two-Factor Study Process Questionnaire: R-SPQ-2F. The British Journal Of Educational Psychology, 71, 133-149.

http://dx.doi.org/10.1348/000709901158433

[6] 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, 229-236.

[7] 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

[8] 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

[9] 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, 3-19.

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

[11] Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of Teachers’ Mathematical Knowledge for Teaching on Student Achievement. American Educational Research Journal, 42, 371-406.

http://dx.doi.org/10.3102/00028312042002371

[12] 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.

http://dx.doi.org/10.1142/9789814405430_0008

[13] 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.

[14] Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

[15] 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 13rd International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 258-265). Paris: PME.

[16] Mason, J. (2008). PCK and Beyond. In P. Sullivan, & S. Wilson (Eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching Development (Vol. 1, pp. 301-322). Rotterdam/Taipe: Sense Publishers.

[17] 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

[18] 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.

[19] 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

[20] 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.

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

[1] AITSL (2011). Accreditation of Initial Teacher Education Programs in Australia. Carlton South: Mceedcya.

[2] 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

[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] Biggs, J. B. (1987). The Study Process Questionnaire (SPQ): Manual. Hawthorn: Australian Council for Educational Research.

[5] Biggs, J., Kember, D., & Leung, D. Y. (2001). The Revised Two-Factor Study Process Questionnaire: R-SPQ-2F. The British Journal Of Educational Psychology, 71, 133-149.

http://dx.doi.org/10.1348/000709901158433

[6] 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, 229-236.

[7] 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

[8] 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

[9] 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, 3-19.

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

[11] Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of Teachers’ Mathematical Knowledge for Teaching on Student Achievement. American Educational Research Journal, 42, 371-406.

http://dx.doi.org/10.3102/00028312042002371

[12] 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.

http://dx.doi.org/10.1142/9789814405430_0008

[13] 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.

[14] Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.

[15] 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 13rd International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 258-265). Paris: PME.

[16] Mason, J. (2008). PCK and Beyond. In P. Sullivan, & S. Wilson (Eds.), Knowledge and Beliefs in Mathematics Teaching and Teaching Development (Vol. 1, pp. 301-322). Rotterdam/Taipe: Sense Publishers.

[17] 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

[18] 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.

[19] 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

[20] 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.

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