Abstract
Despite increased appreciation of the role of proof in students’ mathematical experiences across all grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major features of any given argument – foundation, formulation, representation, and social dimension – so that the argument could count as proof at the elementary school level. My examination is situated in an episode from a third-grade class, which presents a student’s argument that could potentially count as proof. In order to examine the extent to which this argument could count as proof (given its four major elements), I develop and use a theoretical framework that is comprised of two principles for conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, at once, honest to mathematics as a discipline and honoring of students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that students’ experiences with proof in school have coherence. The two principles offer the basis for certain judgments about whether the particular argument in the episode could count as proof. Also, they support more broadly ideas for a possible conceptualization of the notion of proof in the elementary grades.
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References
Balacheff, N.: 1988, ‘Aspects of proof in pupils' practice of school mathematics’, in D. Pimm (ed.), Mathematics, Teachers and Children, Hodder and Stoughton, London, pp. 216–235.
Balacheff, N.: 1990, ‘Towards a problématique for research on mathematics teaching’, Journal for Research in Mathematics Education 21(4), 258–272.
Balacheff, N.: 1991, ‘The benefits and limits of social interaction: The case of mathematical proof’, in A.J. Bishop (ed.), Mathematical Knowledge: Its Growth through Teaching, Kluwer Academic Publishers, Dordrecht, pp. 175–192.
Ball, D.L.: 1993, ‘With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics’, The Elementary School Journal 93(4), 373–397.
Ball, D.L. and Bass, H.: 2003, ‘Making mathematics reasonable in school’, in J. Kilpatrick, W.G. Martin and D. Schifter (eds.), A Research Companion to Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA, pp. 27–44.
Ball, D.L., Hoyles, C., Jahnke, H.N. and Movshovitz-Hadar, N.: 2002, ‘The teaching of proof’, in L.I. Tatsien (ed.), Proceedings of the International Congress of Mathematicians, Vol. III, Higher Education Press, Beijing, pp. 907–920.
Ball, D.L. and Wilson, S.M.: 1996, ‘Integrity in teaching: Recognizing the fusion of the moral and intellectual’, American Educational Research Journal 33, 155–192.
Brousseau, G.: 1981, ‘Problèmes de didactique des décimaux’, Recherches en Didactique des Mathématiques 2, 37–128.
Bruner, J.: 1960, The Process of Education, Harvard University Press, Cambridge, MA.
Carpenter, T.P., Franke, M.L. and Levi, L.: 2003, Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School, Heinemann, Portsmouth, NH.
Coe, R. and Ruthven, K.: 1994, ‘Proof practices and constructs of advanced mathematics students’, British Educational Research Journal 24(4), 333–344.
De Millo, R., Lipton, R. and Perlis, A: 1979/1998, ‘Social processes and proofs of theorems and programs’, in T. Tymoczko (ed.), New Directions in the Philosophy of Mathematics, Princeton University Press, Princeton, NJ, pp. 267–285.
Ernest, P.: 1991, The Philosophy of Mathematics Education, The Falmer Press, London.
Ernest, P.: 1998, Social Constructivism as a Philosophy of Mathematics, State University of New York Press, Albany, NY.
Fischbein, E.: 1982, ‘Intuition and proof’, For the Learning of Mathematics 3(2), 9–18, 24.
Galotti, K.M., Komatsu, L.K. and Voelz, S.: 1997, ‘Children's differential performance on deductive and inductive syllogisms’, Developmental Psychology 33(1), 70–78.
Hanna, G.: 1983, Rigorous Proof in Mathematics Education, OISE Press, Toronto.
Hanna, G. and Jahnke, H.N.: 1996, ‘Proof and proving’, in A.J. Bishop, K. Clements, C. Keitel, J. Kilpatrick and C. Laborde (eds.), International Handbook of Mathematics Education, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 877–908.
Healy, L. and Hoyles, C.: 2000, ‘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.
Kitcher, P.: 1984, The Nature of Mathematical Knowledge, Oxford University Press, New York, NY.
Lakatos, I.: 1976, Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge. University Press.
Lampert, M.: 1990, ‘When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching’, American Educational Research Journal 27, 29–63.
Lampert, M.: 1992, ‘Practice and problems in teaching authentic mathematics’, in F.K. Oser, A. Dick and J. Patry (eds.), Effective and Responsible Teaching: The New Synthesis, Jossey-Bass Publishers, San Francisco, CA, pp. 295–314.
Light, P., Blaye, A., Gilly, M. and Girotto, V.: 1989, ‘Pragmatic schemas and logical reasoning in 6- to 8-year-old children’, Cognitive Development 4, 49–64.
Maher, C.A. and Martino, A.M.: 1996, ‘The development of the idea of mathematical proof: A 5-year case study’, Journal for Research in Mathematics Education 27, 194–214.
Mariotti, M.A.: 2000, ‘Introduction to proof: The mediation of a dynamic software environment’, Educational Studies in Mathematics 44, 25–53.
Moore, R.C.: 1994, ‘Making the transition to formal proof’, Educational Studies in Mathematics 27, 249–266.
National Council of Teachers of Mathematics: 2000, Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA.
Polya, G.: 1981, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, Wiley Combined Edition, New York, NY.
Schoenfeld, A.H.: 1994, ‘What do we know about mathematics curricula?’ Journal of Mathematical Behavior 13, 55–80.
Simon, M.A. and Blume, G.W.: 1996, ‘Justification in mathematics classrooms: A study of prospective elementary school students’, Journal of Mathematical Behavior 15(1), 3–31.
Sowder, L. and Harel, G.: 1998, ‘Types of students' justifications’, The Mathematics Teacher 91(8), 670–675.
Tymoczko, T. (ed.): 1986/1998, New Directions in the Philosophy of Mathematics, Princeton University Press, Princeton, NJ.
Usiskin, Z.: 1987, ‘Resolving the continuing dilemmas in school geometry’, in M.M. Lindquist and A.P. Shulte (eds.), Learning and Teaching Geometry, K-12, National Council of Teachers of Mathematics, Reston, VA, pp. 17–31.
Yackel, E. and Cobb, P.: 1996, ‘Sociomathematical norms, argumentation, and autonomy in mathematics’, Journal for Research in Mathematics Education 27, 458–477.
Yackel, E. and Hanna, G.: 2003, ‘Reasoning and proof’, in J. Kilpatrick, W.G. Martin and D. Schifter (eds.), A Research Companion to Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA, pp. 227–236.
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stylianides, a.j. The Notion of Proof in the Context of Elementary School Mathematics. Educ Stud Math 65, 1–20 (2007). https://doi.org/10.1007/s10649-006-9038-0
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DOI: https://doi.org/10.1007/s10649-006-9038-0