Abstract
This paper indicates that prospective teachers’ familiarity with theoretical models of students’ ways of thinking may contribute to their mathematical subject matter knowledge. This study introduces the intuitive rules theory to address the intuitive, same sides-same angles solutions that prospective teachers of secondary school mathematics come up with, and the proficiency they acquired during the course “Psychological aspects of mathematics education”. The paper illustrates how drawing participants’ attention to their own erroneous applications of same sides-same angles ideas to hexagons, challenged and developed their mathematical knowledge.
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Australian Education Council: 1991, A National Statement on Mathematics for Australian Schools: Curriculum Corporation, Melbourne.
Ball, D.L.: 1991, ‘Research on teaching mathematics: Making subject-matter knowledge part of the equation’, in J. Brophy (ed.), Advances in Research on Teaching, vol. II, JAI Press, London, pp. 1–49.
Ball, D.L.: 1999, ‘Concluding remarks’, Knowing and Learning Mathematics for Teaching: Proceedings of a workshop at the National Academy of Science with the support of the National Science Foundation, National Academy Press, Washington, pp. 127–128.
Barnett, C.: 1999, ‘Case materials’, Knowing and Learning Mathematics for Teaching: Proceedings of a workshop at the National Academy of Science with the support of the National Science Foundation, National Academy Press, Washington, pp. 94–97.
Brown, S.I. and Walter, M.I. (eds.): 1983, The Art of Problem Posing, The Franklin Institute Press, Philadelphia, Pennsylvania, pp. 31–62.
Carpenter, T.P., Franke, M.L. and Levi, L.: 2003, Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School, New Hampshire, USA.
Clement, J.: 1993, ‘Using bridging analogies and anchoring intuitions to deal with students’ preconceptions in physics’, Journal of Research in Science Teaching 30, 1241–1257.
Cooney, T.J.: 1994, ‘Teacher education as an exercise in adaptation’, in D.B. Aichele and A.F. Coxford (eds.), Professional Development for Teachers of Mathematics, NCTM, Reston, VA, pp. 9–22.
Cooney, T.J.: 1999, ‘Conceptualizing teachers’ ways of knowing’, Educational Studies in Mathematics 38, 163–187.
Cooney, T.J. and Wiegel, H.G.: 2003, ‘Examining the mathematics in mathematics teacher education’, in A. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick and F.K.S. Leung (eds.), Second International Handbook of Mathematics Education, Kluwer, Dordrecht, The Netherlands, pp. 795–828.
Dembo, Y., Levin, I. and Siegler, R.S.: 1997, ‘A comparison of the geometric reasoning of students attending Israeli ultra-orthodox and mainstream schools’, Developmental Psychology 33, 92–103.
Even, R.: 1999, ‘Integrating academic and practical knowledge in a teacher leaders’ development program’, Educational Studies in Mathematics 38, 235–252.
Even, R. and Tirosh, D.: 1995, ‘Subject-matter knowledge and knowledge about students as sources of teacher presentations of the subject-matter’, Educational Studies in Mathematics 29, 1–20.
Feibel, W.: 1993, “‘What-if-not?” A technique for involving and motivating students in psychology courses’, in S.I. Brown and M.I. Walter (eds.), Problem Posing: Reflection and Applications, Erlbaum, Hillsdale, New Jersey, pp. 52–57.
Findell, B. and Ball, D.L.: 1999, ‘Video as a delivery mechanism’, Knowing and Learning Mathematics for Teaching: Proceedings of a workshop at the National Academy of Science with the support of the National Science Foundation, National Academy Press, Washington, pp. 98–101.
Fischbein, E.: 1987, Intuition in Science and Mathematics: An Educational Approach, Reidel, Dordrecht, The Netherlands.
Fischbein, E.: 1993a, ‘The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity’, in R. Biehler, R.W. Scholz, R. Straesser and B. Winkelmann (eds.), Didactics of Mathematics as a Scientific Discipline, Kluwer, Dordrecht, The Netherlands, pp. 231–245.
Fischbein, E.: 1993b, ‘The theory of figural concepts’, Educational Studies in Mathematics 24(2), 139–162.
Graeber, A.O.: 1999, ‘Forms of knowing mathematics: What preservice teachers should learn’, Educational Studies in Mathematics 38, 189–208.
Gray, E. and Tall, D.: 1994, ‘Duality, ambiguity and flexibility: A proceptual view of simple arithmetic’, The Journal for Research in Mathematics Education 26, 115–141.
Jaworski, B. and Gellert, U.: 2003, ‘Educating new mathematics teachers: Integrating theory and practice, and the roles of practicing teachers’, in A. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick and F.K.S. Leung (eds.), Second International Handbook of Mathematics Education, Kluwer, Dordrecht, The Netherlands, pp. 829–875.
Lavy, I. and Bershadsky, I.: 2003, ‘Problom posing via “What if not?” strategy in solid geometry–A case study’, Journal of Mathematical Behavior 22, 369–387.
Lee, S.Y.: 1999, ‘Student curriculum materials: Japanese teachers’ manual’, Knowing and Learning Mathematics for Teaching: Proceedings of a workshop at the National Academy of Science with the support of the National Science Foundation, National Academy Press, Washington, pp. 78–85.
Lerman, S.: 2002, ‘Situating research on mathematics teachers’ beliefs and on change’, in G.C. Leder and E. Pehkonen (eds.), Beliefs: A Hidden Variable in Mathematics Education?, Kluwer, Dordrecht, The Netherlands, pp. 233–246.
Loucks-Horsley, S., Hewson, P.W., Love, N. and Stiles, K.E.: 1998, Designing Professional Development for Teachers of Science and Mathematics, Corwin, Thousand Oaks, CA.
Mason, J. and Spence, M.: 1999, ‘Beyond mere knowledge of mathematics: The importance of knowing-to-act in the moment’, Educational Studies in Mathematics 38, 51–66.
Menon, R.: 1998, ‘Preservice teachers’ understanding of perimeter and area’, School Science and Mathematics 98(7), 361–368.
National Council of Teachers of Mathematics [NCTM]: 2000, Professional Standards for Teaching Mathematics, NCTM, Reston, VA.
Noddings, N.: 1992, ‘Professionalization and mathematics teaching’, in D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, pp. 197–208.
Ramirez, M.: 1999, ‘Student curriculum materials: Investigation in number, data, and space’, Knowing and Learning Mathematics for Teaching: Proceedings of a workshop at the National Academy of Science with the support of the National Science Foundation, National Academy Press, Washington, pp. 86–89.
Reinke, K.S.: 1997, ‘Area and perimeter: Prospective teachers’ confusion’, School Science and Mathematics 97(2), 75–77.
Schoenfeld, A.H.: 1985, Mathematical Problem Solving, Academic Press, London.
Schoenfeld, A.H.: 1987, ‘A brief and biased history of problem solving’, in F.R. Curcio (ed.), Teaching and Learning: A Problem-Solving Focus, National Council of Teachers of Mathematics. Reston, Virginia, pp. 27–46.
Schoenfeld, A.H.: 1992, ‘Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics’, in D. Grouws (ed.), Handbook for Research on Mathematics, Macmillan, New York, pp. 334–370.
Shulman, L.S.: 1986, ‘Those who understand: Knowledge growth with teaching’, Educational Researcher 15(2), 4–14.
Skemp, R.R.: 1971, The Psychology of Learning Mathematics, Penguin, Harmondsworth, Middlesex.
Skemp, R.R.: 1979, Intelligence, Learning, and Action: A Foundation for Theory and Practice in Education, J. Wiley, Chichester.
Stavy, R. and Tirosh, D.: 2000, How Students’ (Mis-)Understand Science and Mathematics: Intuitive Rules, Teachers College Press, New York.
Swan, M.: 1983, Teaching Decimal Place Value — A Comparative Study of Conflict and Positive Only Approaches, Shell Centre for Mathematical Education, University of Nottingham, Nottingham, UK.
Tall. D.: 2005, ‘A theory of mathematical growth through embodiment, symbolism and proof’. Plenary Lecture for the International Colloquium on Mathematical Learning from Early Childhood to Adulthood, 5–7 July, Belgium.
1Tall, D. and Vinner, S.: 1981, ‘Concept image and concept definition in mathematics, with special reference to limits and continuity’, Educational Studies in Mathematics 12, 151–169.
Tirosh, D.: 2000, ‘Enhancing prospective teachers’ knowledge of childrenɳs conceptions: The case of division of fractions’, Journal for Research in Mathematics Education 31, 5–25.
Tirosh, D. and Stavy, R.: 1999, ‘Intuitive (USE THE CORRECT TYPE OF APOSTROPHE) rules: A way to explain and predict students’ reasoningɳ, Educational Studies in Mathematics 38, 51–66.
Tirosh, D., Stavy, R. and Cohen, S.: 1998, ‘Cognitive conflict and intuitive rules’, International Journal of Science Education 20, 1257–1269.
Tirosh, D., Stavy, R. and Tsamir, P.: 2001, ‘Using the intuitive rules theory as a basis for educating teachers’, in F.L. Lin and T. Cooney (eds.), Making Sense of Mathematics Education, Kluwer, Dordrecht, The Netherlands.
Tsamir, P.: 2002, ‘The intuitive rule “same A-same B“: The case of triangles and quadrilaterals’, Focus on Learning Problems in Mathematics 24(4), 54–70.
Tsamir, P.: 2003a, ‘Different representations in instruction of vertical angles’, Focus on Learning Problems in Mathematics 25(1), 1–13.
Tsamir, P.: 2003b, ‘From “easy” to “difficult” or vice versa: The case of infinite sets’, Focus on Learning Problems in Mathematics 25(2), 1–16.
Tsamir, P.: 2003c, ‘Basing instruction on theory and research: What is the impact of an extreme case?’, Focus on Learning Problems in Mathematics 25(4), 4–21.
Tsamir, P. and Mandel, N.: 2000, ‘The intuitive rule same A — same B: The case of area and perimeter’, in T. Nakahara and M. Koyama (eds.), Proceedings of the 24th Annual Meeting for the Psychology of Mathematics Education, vol. IV, Hiroshima, Japan, pp. 225–232.
Tsamir, P., Tirosh, D., Stavy, R. and Ronen, I.: 2001, ‘Intuitive rules: A theory and its implications to mathematics and science teacher education’, in H. Behrendt, H. Dahncke, R. Duit, W. Gruber, M. Komorek, A. Kross and P. Reiska, (eds.) Research in Science Education – Past, Present and Future, Kluwer, Dordrecht, The Netherlands, pp. 167–176.
Vinner, S.: 1997, ‘The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning’, Educational Studies in Mathematics 34, 97–129.
Vosniadou, S. (ed.): 1994, ‘Conceptual change’, Special issue of Learning and Instruction, Vol. 4.
Vosniadou, S.: 2002, ‘Exploring the relationships between conceptual change and intentional learning’, in G.M. Sinatra and P.R. Pintrich (eds.), Intentional Conceptual Change. Lawrence Erlbaum, Mahwah, NJ.
Zazkis, R.: 1999, ‘Intuitive rules in number theory: Example of “The more of A, the more of B’ rule implementation”, Educational Studies in Mathematics 40(2), 197–209.
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Tsamir, P. When Intuition Beats Logic: Prospective Teachers’ Awareness of their Same Sides – Same Angles Solutions. Educ Stud Math 65, 255–279 (2007). https://doi.org/10.1007/s10649-006-9053-1
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DOI: https://doi.org/10.1007/s10649-006-9053-1