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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