# Instructions and recipes in mathematical proofs

## Abstract

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.

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## Data availability

This is a theoretical paper that is not based on data, so there is no data to share.

1. Throughout this paper, we treat “mathematical assertion” and “mathematical statement” as synonymous.

2. For instance, Simon (1996) and Harel and Sowder (1998) note that inferences can be made in a proof by asking the reader to anticipate the outcome of a transformational process.

3. Not all textbooks present the Bolzano Weierstrass Theorem in the same way, and some may use fewer or different instructions. We are treating this as an illustrative example, but not necessarily a representative one. Whether and when mathematicians should use instruction-based proofs is an interesting didactical question. The effects it has on student cognition is a worthwhile research question.

4. The B-W Theorem does not depend on the Axiom of Choice, and the proof could have been amended to avoid its use. For instance, if the instructions had been to choose the left half-interval if there were an infinite number of terms of the sequence in it and the right interval otherwise, and if nk+1 had been chosen to be the least index greater than nk such that xn(k+1) was in that interval, no choice function would be necessary.

5. In the philosophical literature, Azzouni (2004) has used the recipe analogy in a different way: A proof is a recipe to produce a formal derivation that a formally stated version of the theorem is true. Elsewhere, each author has argued against that position (Tanswell, 2015; Weber, 2021), and we are not using that analogy here.

6. To avoid misinterpretation, “proving activities” in this case refers to deductively justifying within a proof, not broader activities associated with proof like conjecturing or generalizing.

7. Consider the sequence defined by an = 1 if n = 1 or if there is an odd perfect number less than n, and an = 0 otherwise. We cannot determine the tail of this sequence because we do not know if an odd perfect number exists.

8. A similar suggestion is made by Hamami and Morris (2020).

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## Acknowledgements

We would like to thank Paul Dawkins, Anderson Norton, and the anonymous reviewers for useful feedback on an earlier version of this manuscript. The research of the second author is funded by the Research Foundation – Flanders (FWO) - Project G083620N - The Epistemology of Data Science: Mathematics and the Critical Research Agenda on Data Practices.

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Correspondence to Keith Weber.

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