Preservice and novice teachers’ knowledge on preformal proofs: Triangle postulate as an example
Keywords:
mathematical knowledge for teaching, preservice teachers, preformal proofs, teachers' knowledge for proof and proving, triangle postulateAbstract
By considering the example of proving the triangle postulate, this study aimed to explore Hong Kong preservice and novice teachers’ knowledge competencies and their beliefs about preformal and formal proofs. The findings revealed that such teachers are not proficient in using preformal proofs and do not realize that preformal proofs are a useful tool for connecting abstract geometrical concepts with concrete meanings. We conclude by providing strategies for teachers to use preformal proofs effectively in their teaching of geometric propositions.
References
Arici, S., & Aslan-Tutak, F. (2015). The effect of origami-based instruction on spatial visualization, geometry achievement, and geometric reasoning. International Journa of Science and Mathematics Education, 13(1), 179–200.
Balacheff, N. (2010). Bridging knowing and proving in mathematics: A didactical perspective. In G. Hanna, H. N. Jahnke & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 115–135). New York: Springer.
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.
Barlow, A. T., & Reddish, J. M. (2006). Mathematical myths: Teacher candidate’ beliefs and the implication for teacher educators. Teacher Education, 59(5), 389–407.
Battista, M. T., & Clements, D. H. (1995). Geometry and proof. Mathematics Teacher, 88(1), 48–54.
Bayat, H. (2016). Lakatos and Hersh on mathematical proof. Philosophical Investigations, 9(17), 75–93.
Biza, I., Nardi, E., & Zachariades, T. (2009a). Do images disprove but do not prove? Teachers’ beliefs about visualization. In F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proceedings of the 19th Study of the International Commission on Mathematical Instruction: Proof and Proving in Mathematics Education (Vol. 1, pp. 59–64). Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.
Biza, I., Nardi, E., & Zachariades, T. (2009b). Teacher beliefs and the didactical contract on visualization. For the Learning of Mathematics, 29(3), 31–36.
Blum, W., & Kirsch, A. (1991). Pre-formal proving: Examples and reflections. Educational Studies in Mathematics, 22(2), 183–203.
Bostic, J. D. (2016). Fostering justification: A case study of preservice teachers, proof-related tasks, and manipulatives. Journal of Mathematics Education at Teachers College, 7(1), 35–42.
Bruner, J. (1966). Toward a Theory of Instruction. Cambridge: Harvard University Press.
Buchholtz, N., Leung, F. K. S., Ding, L., Kaiser, G., Park, K., & Schwarz, B. (2013). Future mathematics teachers’ professional knowledge of elementary mathematics from an advanced standpoint. ZDM–The International Journal on Mathematics Education, 45(1), 107–120.
Cabassut, R., Conner, A. M., İşçimen, F. A., Furinghetti, F., Jahnke, H. N., & Morselli, F. (2012). Conceptions of Proof – In Research and Teaching. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education: The 19th ICMI study (Vol. 15, pp. 169–190). New York: Springer.
Clausen-May, T., Jones, K., McLean, A., & Rowlands, S. (2000). Perspectives on the design of the geometry curriculum. Proceedings of the British Society for Research into Learning Mathematics 20(1&2), 34–41.
Coad, L. (2006). Paper folding in the middle school classroom and beyond. The Australian Mathematics Teacher, 62(1), 6–13.
Corleis, A., Schwarz, B., Kaiser, G., & Leung, I. K. C. (2008). Content and pedagogical content knowledge in argumentation and proof of future teachers: A comparative case study in Germany and Hong Kong. ZDM–The International Journal on Mathematics Education, 40(1), 813–832.
Crowley, M. L. (1987). The van Hiele model of the development of geometric thought. In M. M. Lindquist & A. P. Shulte (Eds.), 1987 Yearbook of the National Council of Teachers of Mathematics (pp. 1–16). Reston, VA: National Council of Teachers of Mathematics.
Diezmann, C. M., Watters, J. J., & English, L. D. (2002). Teacher behaviours that influence young children’s reasoning. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 27th Annual Conference of the International Group for the Psychology of Mathematics Education 2 (pp. 289–296), Norwick, UK.
Dreyfus, T. (1999). Why Johnny can’t prove. Educational Studies in Mathematics, 38(1), 85–109.
Dreyfus, T., Nardi, E., & Leikin, R. (2012). Forms of proof and proving. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education–the 19th international commission for mathematics instruction study (pp. 111–120). New York: Springer.
Fischer, H. (2014). Angle preserving linear transformations [PDF document]. Retrieved from Lecture Notes Online Web site: http://www.cs.bsu.edu/homepages/fischer/math445/angles.pdf
Fitzpatrick, R. (2007). Euclid’s Elements of Geometry. http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
Fujita, T., Jones, K., & Yamamoto, S. (2004). Geometrical intuition and the learning and teaching of geometry. Paper presented at the Topic Group on Research and Development in the Teaching and Learning of Geometry, 10th International Congress on Mathematical Education (ICME-10), Copenhagen, Denmark.
Hanna, G., & Barbeau, E. (2010). Proofs as Bearers of Mathematical Knowledge. In G. Hanna, H. N. Jahnke & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 85–100). New York: Springer.
Hanna, G., Jahnke, H. N., & Pulte, H. (Eds.). (2010). Explanation and proof in mathematics: Philosophical and educational perspectives. New York: Springer.
Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Researcch in collegiate mathematics education III (pp. 234–283). Providence, RI: American Mathematical Society.
Harel, G., & Sowder, L. (2007). Toward a comprehensive perspective on proof. In F. K. Lester (Ed.), Handbook of research on teaching and learning mathematics (2nd edition, pp. 805–842). Greenwich, CT: Information Age Publishing.
Hilbert, D. (1927/1967). The foundations of mathematics. In J. Van Heijenoort (Ed.), From Frege to Gödel (p. 475). Cambridge: Harvard University Press.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406.
Johnson, D. A. (1999). Paper folding for the mathematics class. Washington: National Council of Teachers of Mathematics.
Knuth, E. J. (2002) Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.
Kolb, D. A. (1984). Experiential learning: Experience as the source of learning and development. Englewood Cliff, NJ: Prentice Hall.
Küchemann, D. & Hoyles, C. (2001a). Identifying differences in students’ evaluation of mathematical reasons. British Society for Research into Learning Mathematics, Proceedings, 21(1), 37–42.
Küchemann, D. & Hoyles, C. (2001b). Investigating factors that influence students’ mathematical reasoning. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 257–264). Utrecht: PME.
Kwong, K. T. (2013). An analysis of geometry in the junior secondary school (grade 7-9) mathematics textbooks from England, Hong Kong and Chinese Taipei (Unpublished doctoral dissertation). The University of Hong Kong, Hong Kong.
Lakatos, I. (1978). What does a mathematical proof prove? In J. Worrall & G. Currie (Eds.), Mathematics, science and epistemology (pp. 61–69). Cambridge: Cambridge University Press.
Larvor, B. (2010). Authoritarian versus authoritative teaching: Polya and Lakatos. In G. Hanna, H. N. Jahnke & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 71–83). New York: Springer.
Leron, U., & Zaslavsky, O. (2013). Generic proving: Reflections on scope and method. For the Learning of Mathematics, 33(3), 24–30.
Leung, I. K. C., Ding, L., Leung, A. Y. L., & Wong, N. Y. (2014). Prospective teachers’ competency in teaching how to compare geometric figures: The concept of congruent triangles as an example. Journal of the Korean Society of Mathematics Education Series D: Research in Mathematical Education, 18(3), 171–185.
Leung, I. K. C., Ding, L., Leung, A. Y. L., & Wong, N. Y. (2016). The correlation between prospective teachers’ knowledge of algebraic inverse operations and teaching competency – using the square root as an example. International Journal for Mathematics Teaching and Learning, 17(3).
Leung, K. S. (2005). Exploring mathematics book. 1A. Hong Kong: Oxford University Press.
Levav-Waynberg, A., & Leikin, R. (2012). The role of multiple solution tasks in developing knowledge and creativity in geometry. The Journal of Mathematical Behavior, 31, 73–90.
Lin, F. L., Yan, K. L., Lo, J. J., Tsamir, P., Tirosh, D., & Stylianides, G. (2012). Teachers’ professional learning of teaching proof and proving. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education: The 19th ICMI study (Vol. 15, pp. 327–346). New York: Springer.
Malerstein, A. J., & Ahern, M. M. (1979). Piaget’s stages of cognitive development and adult character structure. American Journal of Psychotherapy, 33(1), 107–119.
Mayring, P. (2000). Qualitative Inhaltsanalyse. Forum Qualitative Sozialforschung. On-line Journal, 1(2). http://qualitative-research.net/index.php/fqs/article/view/1089/2383
Miller, R. L. (2012). On proofs without words. Whitman College.
Nardi, E. (2014). Reflections on visualization in mathematics and in mathematics education. In M. N. Fried & T. Dreyfus (Eds.), Mathematics & mathematics education: Searching for common ground (pp. 193–220). Dordrecht: Springer.
Natsheh, I., & Karsenty, R. (2014). Exploring the potential role of visual reasoning tasks among inexperienced solvers. ZDM–The International Journal on Mathematics Education, 46(1), 109–122.
Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM project. In A. Gagatsis & S. Papastavridis (Eds.), Proceedings of the 3rd Mediterranean Conference on Mathematical Education (pp.115–124). Athens: Hellenic Mathematical Society and Cyprus Mathematical Society.
Niss, M., & Højgaard, T. (2011). Competencies and mathematical learning. Ideas and Inspiration for the development of mathematics teaching and learning in Denmark (English ed.). Roskilde, Denmark: IMFUMFA.
OECD. (2014). PISA 2012 results: What students know and can do – Student performance in mathematics, reading, and science (Vol 1). Paris: PISA OECD Publishing.
Ojose, B. (2008). Applying Piaget’s theory of cognitive development to mathematics instruction. The Mathematics Educator, 18(1), 26–30.
Poon, K. K., & Leung, C. K. (2016). A study of geometric understanding via logical reasoning in Hong Kong. International Journal for Mathematics Teaching and Learning, 17(3).
Reid, D. A. (1993). Pre-formal, formal, and formulaic proving. In Proceedings of the 1993 Annual Meeting of the Canadian Mathematics Education Study Group (pp. 151–156). Toronto: Canadian Mathematics Education Study Group.
Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8, 255–271.
Rowland, T., & Turner, F. (2007). Developing and using the ‘knowledge quartet’: A framework for the observation of mathematics teaching. The Mathematics Educator, 10(1), 107–123.
Schwarz, B., Leung, I. K. C., Buchholtz, N., Kaiser, G., Stillman, G., Brown, J., & Vale, C. (2008). Future teachers’ professional knowledge on argumentation and proof: A case study from universities in three countries. ZDM–The International Journal on Mathematics Education, 40(1), 791–811.
Stewart, P. W., & Hadley, K. (2014). Investigating the relationship between visual imagery, metacognition and mathematics pedagogical content knowledge. Journal of the International Society for Teacher Education, 18(1), 26–35.
Tatto, M. T., Peck, R., Schwille, J., Bankov, K., Senk, S. L., Rodriguez, M., Ingvarson, L., Reckase, M., & Rowley, G. (2012). Policy, practice, and readiness to teach primary and secondary mathematics in 17 countries. Amsterdam: IEA.
Usiskin, Z., Peressini, A. L., Marchisotto, E., & Stanley, D. (2003). Mathematics for high school teachers: An advanced perspective. Upper Saddle River, NJ: Prentice Hall.
Van Asch, A. G. (1993). To prove, why and how? International Journal of Mathematical Education in Science and Technology, 24(2), 301–313.
Vermunt, J. D. (2007). The power of teaching-learning environments to influence student learning. British Journal of Education Psychology Monograph Series II, 4, 73–90.
Weisstein, E. W., (2003). CRC concise encyclopedia of mathematics (2nd ed.). London: Chapman & Hall.
Whiteley, W. (2009). Refutations: the role of counter-examples in developing proof. In F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proceedings of the 19th Study of the International Commission on Mathematical Instruction: Proof and Proving in Mathematics Education (Vol. 2, pp. 257–262). Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.
