What are explanatory proofs in mathematics and how can they contribute to teaching and learning?

IF 1 Q3 EDUCATION & EDUCATIONAL RESEARCH

Journal of Mathematical Behavior Pub Date : 2024-09-26 DOI:10.1016/j.jmathb.2024.101191

Marc Lange

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引用次数: 0

Abstract

This paper will examine several simple examples (drawn from the mathematics literature) where there are multiple proofs of the same theorem, but only some of these proofs are widely regarded by mathematicians as explanatory. These examples will motivate an account of explanatory proofs in mathematics. Along the way, the paper will discuss why deus ex machina proofs are not explanatory, what a mathematical coincidence is, and how a theorem's proper setting reflects the naturalness of various mathematical kinds. The paper will also investigate how context influences which features of a theorem are salient and consequently which proofs are explanatory. The paper will discuss several ways in which explanatory proofs can contribute to teaching and learning, including how shifts in context (and hence in a proof’s explanatory power) can be exploited in a classroom setting, leading students to dig more deeply into why some theorem holds. More generally, the paper will examine how “Why?” questions operate in mathematical thinking, teaching, and learning.

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什么是数学解释性证明，它们如何促进教学？

本文将研究几个简单的例子（摘自数学文献），在这些例子中，同一定理有多个证明，但只有其中一些证明被数学家广泛视为解释性证明。这些例子将促使我们对数学中的解释性证明进行阐述。同时，本文还将讨论为什么神来之笔的证明不是解释性的，什么是数学巧合，以及定理的适当设置如何反映各种数学种类的自然性。本文还将研究背景如何影响定理的哪些特征是突出的，从而影响哪些证明是解释性的。论文将讨论解释性证明如何促进教学，包括如何在课堂上利用情境的变化（以及证明的解释力），引导学生更深入地探究某些定理成立的原因。更广泛地说，本文将探讨 "为什么？"问题如何在数学思考、教学和学习中发挥作用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Mathematical Behavior EDUCATION & EDUCATIONAL RESEARCH-

CiteScore

2.70

自引率

17.60%

发文量

期刊介绍： The Journal of Mathematical Behavior solicits original research on the learning and teaching of mathematics. We are interested especially in basic research, research that aims to clarify, in detail and depth, how mathematical ideas develop in learners. Over three decades, our experience confirms a founding premise of this journal: that mathematical thinking, hence mathematics learning as a social enterprise, is special. It is special because mathematics is special, both logically and psychologically. Logically, through the way that mathematical ideas and methods have been built, refined and organized for centuries across a range of cultures; and psychologically, through the variety of ways people today, in many walks of life, make sense of mathematics, develop it, make it their own.