Course Identification

The course will comprise 9 3-hour lectures.

It will attend to three aspects of teaching and learning geometry:

Epistemic: What it means “to know” geometry at different grade levels

Cognitive: How people learn geometry, obstacles and their origins

Didactic: Approaches to teaching geometry

Main topics:

• Acquisition of geometrical concepts
• Definitions, and their role in creating geometric worlds
• Justification and proof
• Theories of learning geometry
• Connections to other mathematical topics
• Problem solving in geometry
• Roles of technology in teaching and learning geometry

Upon successful completion of the course, students should be able to:

1. Describe visual learning processes of basic geometrical concepts
2. Describe approaches to teaching basic concepts, and discuss advantages and disadvantages of different approaches
3. Base their teaching theoretically
4. Demonstrate proficiency in basic research works in the learning and teaching of geometry, and replicate basic research works.
5. Demonstrate insight regarding connections between research on geometry education and its practice (teaching)
6. Utilize dynamic geometry software as a resource for teaching and problem solving
7. Plan teaching sequences that balance aspects of justifying and proving in geometry
8. Make connections in their teaching between topics within geometry and with other mathematical domains.
9. Lead students through diverse geometrical "worlds" based on definitions of basic concepts

The following is a tentative list.

Mason, M. (2002). The van Hiele levels of geometric understanding. Professional Handbook for Teachers, GEOMETRY: EXPLORATIONS AND APPLICATIONS. McDougal Littell.

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241.

De Villiers, M. (2009). To teach definitions in geometry or teach to define. Colección Digital Eudoxus, 1(2).

Hershkowitz, R. (1987). The acquisition of concepts and misconceptions in basic geometry - or when "A little learning is a dangerous thing".  In J. D. Novak (Ed): Misconceptions and Educational Strategies in Science and Mathematics.   Cornell University, Vol. III, pp. 238-251.

Hanna, G. (1998). Proof as explanation in geometry. Focus on Learning Problems in Mathematics, 20, 4–13.

Buchbinder, O., & Zaslavsky, O. (2018). Strengths and inconsistencies in students’ understanding of the roles of examples in proving. The Journal of Mathematical Behavior. https://doi.org/10.1016/j.jmathb.2018.06.010

Fischbein, E. & Kedem, I. (1982). Proof and certitude in the development of mathematical thinking. Proceedings of the 6th International Conference PME (pp. 128-131). Antwerp.

Haj Yahya, A., Hershkowitz, R., & Dreyfus, T. (2014). Investigating students'  geometrical proofs through the lens of students' definitions. In Oesterle, S., Liljedahl, P., Nicol, C., & Allan, D. (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36, Vol. 3. pp. 217-224. Vancouver, Canada.

Krause, E. F. (1973). Taxicab geometry. The Mathematics Teacher, 66(8), 695–706.

Koichu, B., & Leron, U. (2015). Proving as problem solving: The role of cognitive decoupling. The Journal of Mathematical Behavior, 40, 233–244. https://doi.org/10.1016/j.jmathb.2015.10.005

Arcavi, A., & Hadas, N. (2000). Computer mediated Learning: an example of approach. International Journal of Computers for Mathematical Learning 5: 25–45, Kluwer Academic Publishers. Netherlands.

Leung, A., Baccaglini-Frank, A., & Mariotti, M. A. (2013). Discernment of invariants in dynamic geometry environments. Educational Studies in Mathematics, 84(3), 439–460. https://doi.org/10.1007/s10649-013-9492-4

Palatnik, A., & Dreyfus, T. (2018). Students’ reasons for introducing auxiliary lines in proving situations. The Journal of Mathematical Behavior. https://doi.org/10.1016/j.jmathb.2018.10.004

Lachmy, R., & Koichu, B. (2014). The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment. The Journal of Mathematical Behavior, 36, 150–165. https://doi.org/10.1016/j.jmathb.2014.07.002