## 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.

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## References

Allendoefer, C. B. (1969). The dilemma in geometry.

*Mathematics Teacher, 62*,165–169.Arbel, B. (1990).

*Problem-solving strategies*. Tel-Aviv, Israel: Open University. (In Hebrew)Bennett, A. B. (1989). Fraction patterns—visual and numerical.

*Mathematics Teacher, 82*, 254–259.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.Chazan, D. (2000).

*Beyond formulas in mathematics teaching: Dynamics of the high school algebra classroom*. New York, NY: Teachers College.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.Cooper, M. (1992). Three-dimensional symmetry.

*Educational Studies in Mathematics, 23*, 179–202.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.Dreyfus, T., & Eisenberg, T. (1990). Symmetry in mathematics learning.

*International Reviews on Mathematical Education, 2*, 53–59.Eccles, F. M.. (1972). Transformations in high school geometry.

*Mathematics Teacher, 65*, 165–169.Ellis-Davies, A. (1986). Symmetry in the mathematics curriculum.

*Mathematics in School, 15*(3), 27–30.Evered, L. J. (1992). Folded fashions: Symmetry in clothing design.

*Arithmetic Teacher, 40*, 204–206.Fischbein, E. (1987).

*Intuition in science and mathematics: An educational approach*. New York, NY: Kluwer.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.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.Graf, K.-D., & Hodgson, B. (1990). Popularizing geometrical concepts: The case of the kaleidoscope.

*For the Learning of Mathematics, 10*(3), 42–50.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.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.Jaworski, B. (1998). Mathematics teacher research: Process, practice, and the development of teaching,

*Journal of Mathematics Teacher Education, 1*, 3–31.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.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.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)Leikin, R. (2000).

*Teachers’ inclinations to solve mathematical problems: Tlxe case of symmetry*. Manuscript in preparation.Leikin, R., Berman, A., & Zaslavsky, O. (1995). The role of symmetry in mathematical problem solving: An interdisciplinary approach.

*Symmetry: Culture & Science, 6*, 332–335.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.Leikin, R., & Zaslavsky, O. (1997). Facilitating students’ interactions in mathematics in cooperative learning settings.

*Journal for Research in Mathematics Education, 28*, 331–354.Leinhardt, G. (1989). Math lessons: A contrast of novice and expert competence.

*Journal for Research in Mathematics Education, 20*, 52–75.Ma, L. (1999).

*Knoioing and teaching elementary mathematics; Teacher’s understanding of fundamental mathematics in China and the United States*. Mahwah, NJ: Lawrence Erlbaum.Newman, R. S., & Goldin, L. (1990). Children’s reluctance to seek help with schoolwork.

*Journal of Educational Psychology*, 82, 9–100.Polya, G. (1973).

*How to solve it: A new aspect of mathematical method*. Princeton, NJ: Princeton University Press.Polya, G. (1981).

*Mathematical Discovery*. New York, NY: Wiley.Schifter, D. (Ed.). (1996).

*What’s happening in math class? Envisioning new practices through teacher narratives*. New York, NY: Teacher College Press.Schifter, D. (1998). Learning mathematics for teaching: From a teachers’ seminar to the classroom.

*Journal of Mathematics Teacher Education, 1*, 55–87.Schön, D. A. (1983).

*The reflective practitioner: How professionals think in action*. New York, NY: Basic Books.Schoenfeld, A. H. (1985).

*Mathematical problem solving*. New York, NY: Academic Press.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.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.Schultz, K. A., & Austin, J. D. (1983). Directional effects in transformation tasks.

*Journal for Research in Mathematical Education, 14*, 95–101.Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers.

*Journal of Mathematics Teacher Education, 1*,157–189.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.Thompson, P. W. (1985). A Piagetian approach to transformation geometry via microworlds.

*Mathematics Teacher, 78*, 465–471.Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. O. Tall (Ed.),

*Advanced Mathematical Thinking*. Dordrecht: Kluwer.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.Webb, N. M. (1991). Task-related verbal interactions and mathematics learning in small groups.

*Journal for Research in Mathematics Education, 22*, 390–408.Weyl, H. (1952).

*Symmetry*. Princeton, NJ: Princeton University Press.Yaglom, I. M. (1962).

*Geometric transformations: Vol. 1. Displacements and symmetry*. New York, NY: Random House.Zaslavsky, O. (1994). Tracing students’ misconceptions back to their teacher: A case of symmetry.

*Pythagoras, 33*, 10–17.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.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.

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## Additional information

The study was conducted while the first author was at the Department of Education in Technology and Science, Technion—israel Institute of Technology.

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Leikin, R., Berman, A. & Zaslavsky, O. Learning through teaching: The case of symmetry.
*Math Ed Res J* **12, **18–36 (2000). https://doi.org/10.1007/BF03217072

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### Keywords

- Symmetry Axis
- Mathematics Teacher
- Student Teacher
- Berman
- Mathematical Knowledge