On the Appropriateness and Necessity of Proofs and Proving in Middle School Mathematics Classrooms
Keywords:
mathematics education, proofs, middle grades, reasoning and proof, informal proofs, operative proofsAbstract
Proofs are central to mathematics as a discipline, yet outside of high school geometry classes proofs and proving are often absent from school mathematics classes. The absence of proofs is detrimental to later student success in mathematics. Research indicates that proofs and proving, especially informal proofs, are appropriate for middle grades students and improve student understanding and confidence in their knowledge. This article discusses the research and calls for the inclusion of proofs and proving as a primary component of middle grades mathematics instruction.
References
Balecheff, N. (2017). Benefits and limits of social interaction: The case of mathematical proof. In A. J. Bishop, S. Mellin-Olsen, J. van Domrolen (Eds.), Mathematical knowledge: Its growth through teaching (pp.175-192). Kluwer Academic Publishers.
Bass, H. (2009). How do you know that you know? Making believe in mathematics. Distinguished University Lecture, University of Michigan. https://deepblue.lib.umich.edu/bitstream/handle/2027.42/64280/Bass-2009.pdf?sequence=1&isAllowed=y
Kleiner, I. (1991). Rigor and proof in mathematics: A historical perspective. Mathematics Magazine, 64(5), 291-314.
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
National Council of Teachers of Mathematics (2021, June). Executive summary: Principles and standards for school mathematics. https://www.nctm.org/uploadedFiles/Standards_and_Positions/PSSM_ExecutiveSummary.pdf
Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics, 21, 589-598
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(3), 5-41
Stylianides, A. J. (2007a). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65, 1-20.
Stylianides, A. J. (2007b). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 28(3), 289-321
Tall, D. (1998). The cognitive development of proof: Is mathematical proof for all or for some? Conference of the University of Chicago School Mathematics Project. https://www.researchgate.net/publication/247812326_The_Cognitive_Development_of_Proof_Is_Mathematical_Proof_For_All_or_For_Some
