Experienced provers’ uses of contexts while engaging in combinatorial proof of binomial identities

Abstract

Combinatorial proofs of binomial identities involve establishing an identity by arguing that each side enumerates a certain set of outcomes. In this paper, we share results from interviews with experienced provers (mathematicians and upper-division undergraduate mathematics students) and examine one particular aspect of combinatorial proof, namely the kinds of contexts that experienced provers used to establish combinatorial proofs of binomial identities. Our findings show that overall, our participants used a variety of contexts in their work; we also demonstrate ways in which previous experiences influenced the contexts they chose, and we offer some instances in which features of a context supported their combinatorial proof production. We offer some theoretical implications of our work, and we conclude with a discussion of limitations and avenues for future work.

KEYWORDS:

• Combinatorics
• combinatorial proof
• proof
• contexts
• representation systems

MSC:

• 97K20
• 97N70

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Here, accessible refers to the fact that one does not require extensive mathematical prerequisite knowledge to understand many combinatorial proofs; tangible refers to the fact that many combinatorial proofs involve objects that can be more readily visualized than, say, abstract algebraic expressions.

2 We acknowledge that this is a wide range. Ash worked on two identities, while the other students all worked on at least six. Our interviews with Ash revealed that they did not have the same level of prior combinatorial and proving experience as some of the other participants, so we spent more time on preliminary problems with Ash. Thus, we did not get to as many identities with Ash as with other students.

3 Specifically, the combinatorial argument Riley used was that each side of the identity enumerated binary strings of length n containing exactly one 1 that is underlined. The left side of the identity does this by counting the number of ways to make a string containing i 1s and then selecting one of the i 1s to underline, and the right side does this by first selecting one of n positions to fill with an underlined 1, and then each of the remaining n – 1 positions in the string can be filled with a 1 or 0.

4 For example: each 1 in a binary string maps to an A, each 0 maps to a B, and the 1 that is underlined maps to a C.

5 These are differential forms, which are used in the mathematical fields of differential geometry and tensor calculus (Lee, 2013, p. 360). We discuss Robin’s work with these in more detail in Section 5.2.2.

6 The left side of the identity does this by choosing k people to be on the committee, and then picking one of those k people to serve as the chair. The right side also counts this set by first choosing one of the n people to be the chair, and then choosing k – 1 out of the remaining n – 1 people to fill the remaining positions on the committee.

Additional information

Funding

This material is based upon work supported by the National Science Foundation under grant number 1650943 and 1920538.