Abstract
Current reform efforts in the United States arecalling for substantial changes in the natureand role of proof in secondary schoolmathematics – changes designed to provideall students with rich opportunities andexperiences with proof throughout theentire secondary school mathematicscurriculum. This study examined 17 experiencedsecondary school mathematics teachers'conceptions of proof from their perspectives asteachers of school mathematics. The resultssuggest that implementing ``proof for all'' maybe difficult for teachers; teachers viewedproof as appropriate for the mathematicseducation of a minority of students. Theresults further suggest that teachers tended toview proof in a pedagogically limited way,namely, as a topic of study rather than as atool for communicating and studyingmathematics. Implications for mathematicsteacher education are discussed in light ofthese findings.
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REFERENCES
Alibert, D. (1988). Towards new customs in the classroom. For the Learning of mathematics, 8(2), 31–35.
Alibert, D. & Thomas, M. (1991). Research on mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking (215–230). The Netherlands: Kluwer Academic Publishers.
Balacheff, N. (1991). The benefits and limits of social interaction: The case of mathematical proof. In A. Bishop, S. Mellin-Olsen & J. Van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (175–192). The Netherlands: Kluwer Academic Publishers.
Ball, D. & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. Phillips (Ed.), Constructivism in education. Chicago: University of Chicago Press.
Bell, A. (1976). A study of pupils' proof - explanations in mathematical situations. Educational Studies in Mathematics, 7, 23–40.
Borko, H. & Putnam, R. (1996). Learning to teach. In R. Calfee & D. Berliner (Eds.), Handbook of educational psychology (673–725). New York: Macmillan.
Chazan, D. (1990). Quasi-empirical views of mathematics and mathematics teaching. Interchange, 21(1), 14–23. TEACHERS' CONCEPTIONS OF PROOF 87
Chazan, D. (1993). High school geometry students' justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359–387.
Chazan, D. & Yerushalmy, M. (1998). Charting a course for secondary geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (67–90). Mahwah, NJ: Erlbaum.
Coe, R. & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 20(1), 41–53.
Davis, P. (1986). The nature of proof. In M. Carss (Ed.), Proceedings of the Fifth International Congress on Mathematical Education (352–358). Adelaide, South Australia: Unesco.
de Villiers, M. (1999). Rethinking proof with the Geometer's Sketchpad. Emeryville, CA: Key Curriculum Press.
Edwards, L. (1997). Exploring the territory before proof: Students' generalizations in a computer microworld for transformation geometry. International Journal of Computers for Mathematical Learning, 2, 187–215.
Epstein, D. & Levy, S. (1995). Experimentation and proof in mathematics. Notices of the American Mathematical Society, 42(6), 670–674.
Fennema, E. & Franke, M. (1992). Teachers' knowledge and its impact. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (147–164). NY: Macmillan.
Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9–24.
Goetting, M. (1995). The college students' understanding of mathematical proof (Doctoral dissertation, University of Maryland, 1995). Dissertations Abstracts International, 56, 3016A.
Grossman, P. (1990). The making of a teacher: Teacher knowledge and teacher education. New York: Teachers College Press.
Hanna, G. (1983). Rigorous proof in mathematics education. Toronto, Ontario: OISE Press.
Hanna, G. (1989). More than formal proof. For the Learning of Mathematics, 9(1), 20–23.
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.
Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15(3), 42–49.
Harel, G. & Sowder, L. (1998). Students' proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput & E. Dubinsky (Eds.), Research in collegiate mathematics education III (234–283). Washington, DC: Mathematical Association of America.
Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.
Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24, 389–399.
Horgan, J. (1993). The death of proof. Scientific American, 269(4), 93–103.
Hoyles, C. (1997). The curricular shaping of students' approaches to proof. For the Learning of Mathematics, 17(1), 7–16.
Jones, K. (1997). Student-teachers' conceptions of mathematical proof. Mathematics Education Review, 9, 21–32.
Knuth, E. (In press). Secondary school mathematics teachers' conceptions of proof. Journal for Research in Mathematics Education.
Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.
Maher, C. & Martino, A. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214. 88 ERIC J. KNUTH
Manin, Y. (1977). A course in mathematical logic. New York: Springer-Verlag.
Martin, W. G. & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41–51.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (1991). Professional standards for teaching mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.
Peled, I. & Zaslavsky, O. (1998). Counter-examples that (only) prove and counterexamples that (also) explain. Focus on Learning Problems in Mathematics, 19, 49–61.
Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics, 21, 589–598.
Richards, J. (1991). Mathematical discussions. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (13–51). The Netherlands: Kluwer.
Ross, K. (1998). Doing and proving: The place of algorithms and proof in school mathematics. American Mathematical Monthly, 3, 252–255.
Schoenfeld, A. (1994). What do we know about mathematics curricula? Journal of Mathematical Behavior, 13(1), 55–80.
Senk, S. (1985). How well do students write geometry proofs? Mathematics Teacher, 78(6), 448–456.
Simon, M. & Blume, G. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3–31.
Sowder, L. & Harel, G. (1998). Types of students' justifications. Mathematics Teacher, 91(8), 670–675.
Spradley, J. (1979). The ethnographic interview. New York: Holt, Rinehart and Winston.
Steiner, M. (1978). Mathematical explanations. Philosophical Studies, 34, 135–151.
Thompson, A. (1992). Teachers' beliefs and conceptions: A synthesis of the research. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (127–146). NY: Macmillan.
Thurston, W. (1995). On proof and progress in mathematics. For the Learning of Mathematics, 15(1), 29–37.
Wheeler, D. (1990). Aspects of mathematical proof. Interchange, 21(1), 1–5.
Wu, H. (1996). The role of Euclidean geometry in high school. Journal of Mathematical Behavior, 15, 221–237. Teacher Education Building 225 N. Mills Street University of Wisconsin Madison, WI 537606 E-mail: knuth@education.wisc.edu
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Knuth, E.J. Teachers' Conceptions of Proof in the Context of Secondary School Mathematics. Journal of Mathematics Teacher Education 5, 61–88 (2002). https://doi.org/10.1023/A:1013838713648
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DOI: https://doi.org/10.1023/A:1013838713648