Instructions and Recipes in Mathematical Proofs

In mathematics education research, proofs are often conceptualized as sequences of mathematical assertions. We argue that this ignores proofs that contain instructions to perform mathematical actions, often in the form of imperatives, which are common both in mathematical practice and in undergraduate mathematics textbooks. We consider in detail a specific type of proof which we call a recipe proof, which is comprised of sequence of instructions that direct the reader to produce mathematical objects with desirable properties. We present a model of what it means to understand a recipe proof and use this model in conjunction with process-object theories from mathematics education research, to explain why recipe proofs are inherently difficult for students to understand.