# Learning through teaching: The case of symmetry

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## Abstract

A study was carried out within the framework of an undergraduate course in teaching skills and strategies. An experimental part of the course was designed to provide an opportunity for the students to learn a mathematical topic through teaching it to eighth grade pupils in a*Learning Through Teaching* (LTT) environment. Symmetry was chosen as the focal mathematical topic for the experiment. This paper focuses on the development of the students’ understanding of line symmetry. The findings show that the implemented LTT environment served as a vehicle for the student teachers to learn mathematics, hi spite of the difficulties they encountered during the study, the students expressed positive dispositions towards symmetry and its role in mathematics.

## Keywords

Symmetry Axis Mathematics Teacher Student Teacher Berman Mathematical Knowledge## Preview

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## References

- Allendoefer, C. B. (1969). The dilemma in geometry.
*Mathematics Teacher, 62*,165–169.Google Scholar - Arbel, B. (1990).
*Problem-solving strategies*. Tel-Aviv, Israel: Open University. (In Hebrew)Google Scholar - Bennett, A. B. (1989). Fraction patterns—visual and numerical.
*Mathematics Teacher, 82*, 254–259.Google Scholar - Brown, C. A., & Borko, H. (1992). Becoming a mathematics teacher. In D. A. Grouws (Ed.),
*Handbook of Research on Mathematics Teaching and Learning*(pp. 209–239). New York, NY: Macmillan.Google Scholar - Chazan, D. (2000).
*Beyond formulas in mathematics teaching: Dynamics of the high school algebra classroom*. New York, NY: Teachers College.Google Scholar - Clements, D. H., & Battista, M. T. (1992). Geometry and special reasoning. In G. A. Grouws (Ed.),
*Handbook of Research on Mathematics Teaching and Learning*(pp. 420–464). New York, NY: Macmillan.Google Scholar - Cooper, M. (1992). Three-dimensional symmetry.
*Educational Studies in Mathematics, 23*, 179–202.CrossRefGoogle Scholar - Davidson N., & Kroll, D. L (1991). An overview of research on cooperative learning related to mathematics.
*Journal for Research in Mathematics Education*, 22, 362–365.CrossRefGoogle Scholar - Dreyfus, T., & Eisenberg, T. (1990). Symmetry in mathematics learning.
*International Reviews on Mathematical Education, 2*, 53–59.Google Scholar - Eccles, F. M.. (1972). Transformations in high school geometry.
*Mathematics Teacher, 65*, 165–169.Google Scholar - Ellis-Davies, A. (1986). Symmetry in the mathematics curriculum.
*Mathematics in School, 15*(3), 27–30.Google Scholar - Evered, L. J. (1992). Folded fashions: Symmetry in clothing design.
*Arithmetic Teacher, 40*, 204–206.Google Scholar - Fischbein, E. (1987).
*Intuition in science and mathematics: An educational approach*. New York, NY: Kluwer.Google Scholar - Geddes, D., & Fortunato, I. (1992). Geometry: Research and classroom activities. In D. T. Owens (Ed.),
*Research ideas for the classroom: Middle grades mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Good, T. L., Mulryan, C., & McCaslin, M. (1992). Grouping for instruction in mathematics: A call for programmatic research on small-group processes. In D. A. Grouws (Ed.),
*Handbook of Research on Mathematics Teaching and Learning*, (pp. 165–196). New York, NY: Macmillan.Google Scholar - Graf, K.-D., & Hodgson, B. (1990). Popularizing geometrical concepts: The case of the kaleidoscope.
*For the Learning of Mathematics, 10*(3), 42–50.Google Scholar - Grenier, D. (1985). Middle school pupils’ conception about reflections according to a task of construction. In L. Streefland (Ed.),
*Proceedings of the 9th annual conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 183–188). Utrecht, The Netherlands: Program Committee.Google Scholar - Hershkowitz, R., Ben-Chaim, D., Hoyles, C., Lappan, G., Mitchelmore, M. C., & Vinner, S. (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, MA: Cambridge University Press.Google Scholar - Jaworski, B. (1998). Mathematics teacher research: Process, practice, and the development of teaching,
*Journal of Mathematics Teacher Education, 1*, 3–31.CrossRefGoogle Scholar - Krainer, K. (1999). Promoting reflection and networking as an intervention strategy in professional development programs for mathematics teachers and mathematics teacher educators. In O. Zaslavsky (Ed.),
*Proceedings of the 23rd annual conference of the International Group for the Psychology of Mathematics Education*. Haifa, Israel: Program Committee.Google Scholar - Lampert, M. (1993). Teachers’ thinking about students’ thinking about geometry: The effects of new teaching tools. In J. L. Schwartz, M. Yerushalmy, & B. Wilson (Eds.),
*The Geometric Supposer: What is it a case of?*(pp.143–1177). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Leikin, R. (1997).
*Symmetry as a way of thought—A tool for professional development of mathematics teachers*. Unpublished doctoral dissertation, Technion—Israel Institute of Technology, Haifa, Israel. (In Hebrew)Google Scholar - Leikin, R. (2000).
*Teachers’ inclinations to solve mathematical problems: Tlxe case of symmetry*. Manuscript in preparation.Google Scholar - Leikin, R., Berman, A., & Zaslavsky, O. (1995). The role of symmetry in mathematical problem solving: An interdisciplinary approach.
*Symmetry: Culture & Science, 6*, 332–335.Google Scholar - Leikin, R., Berman, A., & Zaslavsky, O. (1998, April).
*Difficulties associated with the relationship between definitions and examples: The case of symmetry*. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA.Google Scholar - Leikin, R., & Zaslavsky, O. (1997). Facilitating students’ interactions in mathematics in cooperative learning settings.
*Journal for Research in Mathematics Education, 28*, 331–354.CrossRefGoogle Scholar - Leinhardt, G. (1989). Math lessons: A contrast of novice and expert competence.
*Journal for Research in Mathematics Education, 20*, 52–75.CrossRefGoogle Scholar - Ma, L. (1999).
*Knoioing and teaching elementary mathematics; Teacher’s understanding of fundamental mathematics in China and the United States*. Mahwah, NJ: Lawrence Erlbaum.Google Scholar - Newman, R. S., & Goldin, L. (1990). Children’s reluctance to seek help with schoolwork.
*Journal of Educational Psychology*, 82, 9–100.Google Scholar - Polya, G. (1973).
*How to solve it: A new aspect of mathematical method*. Princeton, NJ: Princeton University Press.Google Scholar - Polya, G. (1981).
*Mathematical Discovery*. New York, NY: Wiley.Google Scholar - Schifter, D. (Ed.). (1996).
*What’s happening in math class? Envisioning new practices through teacher narratives*. New York, NY: Teacher College Press.Google Scholar - Schifter, D. (1998). Learning mathematics for teaching: From a teachers’ seminar to the classroom.
*Journal of Mathematics Teacher Education, 1*, 55–87.CrossRefGoogle Scholar - Schön, D. A. (1983).
*The reflective practitioner: How professionals think in action*. New York, NY: Basic Books.Google Scholar - Schoenfeld, A. H. (1985).
*Mathematical problem solving*. New York, NY: Academic Press.Google Scholar - Shulman, L. S. (1986). Paradigms and research programs in the study of teaching: A contemporary perspective. In M. C. Wittrock (Ed.),
*Handbook of Research in Teaching*(3rd ed., pp. 3–36). New York, NY: Macmillan.Google Scholar - Schultz, K. A. (1978). Variables influencing the difficulty of rigid transformations during the transition between the concrete and formal operational stages of cognitive development. In R. A. Lesh & D. B. Mierkiewicz (Eds.),
*Recent research concerning the development of special and geometric concepts*, (pp. 177–193). Columbus, OH: ERIC/SMEAC.Google Scholar - Schultz, K. A., & Austin, J. D. (1983). Directional effects in transformation tasks.
*Journal for Research in Mathematical Education, 14*, 95–101.CrossRefGoogle Scholar - Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers.
*Journal of Mathematics Teacher Education, 1*,157–189.CrossRefGoogle Scholar - Thomas, D. (1978). Students’ understanding of selected transformation geometry concepts. In R. A. Lesh & D. B. Mierkiewicz (Eds.),
*Recent research concerning the development of special and geometric concepts*(pp. 177–193). Columbus, OH: ERIC/SMEAC.Google Scholar - Thompson, P. W. (1985). A Piagetian approach to transformation geometry via microworlds.
*Mathematics Teacher, 78*, 465–471.Google Scholar - Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. O. Tall (Ed.),
*Advanced Mathematical Thinking*. Dordrecht: Kluwer.Google Scholar - Webb, N. M. (1985). Student interaction and learning in small groups: A research summary. In R. Slavin, S. Sharan, S. Kagan, R. Hertz-Lazarowitz, C. Webb, & R. Schmuck (Eds.),
*Learning to cooperate, cooperating to learn*(pp. 147–172). New York, NY: Plenum Press.Google Scholar - Webb, N. M. (1991). Task-related verbal interactions and mathematics learning in small groups.
*Journal for Research in Mathematics Education, 22*, 390–408.CrossRefGoogle Scholar - Weyl, H. (1952).
*Symmetry*. Princeton, NJ: Princeton University Press.Google Scholar - Yaglom, I. M. (1962).
*Geometric transformations: Vol. 1. Displacements and symmetry*. New York, NY: Random House.Google Scholar - Zaslavsky, O. (1994). Tracing students’ misconceptions back to their teacher: A case of symmetry.
*Pythagoras, 33*, 10–17.Google Scholar - Zaslavsky, O., & Leikin, R. (1999). Interweaving the training of mathematics teacher-educators and the professional development of mathematics teachers. In O. Zaslavsky (Ed.),
*Proceedings of the 23rd annual conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 143–158). Haifa, Israel: Program Committee.Google Scholar - Zaslavsky, O., & Peled, I. (1996). Inhibiting factors in generating examples by mathematics teachers and students: The case of binary operation.
*Journal for Research in Mathematics Education, 27*, 67–78.CrossRefGoogle Scholar