Using Expectancy Value Theory to Account for Individuals’ Mathematical Justifications

Abstract

In mathematics education, researchers commonly infer students’ standards of conviction from the justifications that they produce. Specifically, if students justify a mathematical statement with an empirical justification, researchers often infer that example-based justifications provide the students with certainty that a general mathematical statement is true. In this article, we present a theoretical framework for interpreting individuals’ proof-related behavior that challenges the aforementioned interpretations. Adapting constructs from expectancy value theory, we argue that whether an individual will seek a deductive proof or settle for an empirical justification depends on several factors, including the value they place on knowing the veracity of the mathematical statement being considered, the cost in terms of time and effort in searching for a proof, and their perceived likelihood of success of being able to find a proof. We demonstrate that mathematicians consider value, cost, and effort in deciding what statements they will try to prove, so it would not be irrational or unmathematical for students to make the same considerations. We illustrate the explanatory power of our framework by studying the justification behavior of 11 preservice and in-service secondary mathematics teachers in a problem-solving course. Although these individuals frequently justified mathematical statements empirically, these individuals were aware of the limitations of empirical justifications and they usually did not obtain certainty from these justifications. The notions of value, cost, and likelihood of success could explain why they settled for empirical justifications and ceased seeking proofs.

Notes

Acknowledgments

We are grateful for the extensive and very helpful feedback that Amanda Jansen, Andreas Stylianides, and the anonymous reviewers provided on earlier drafts of this article.

Notes

1 Stylianides et al. (2017) noted that there is disagreement amongst researchers as to whom a proof should convince, with some proposing an enemy or skeptic (Mason, Burton, & Stacey, 1982; Volminik, 1990), a mathematician (Davis & Hersh, 1981), a particular community (Balacheff, 1987), or an individual student (Harel & Sowder, 1998).

2 For the remainder of this article, by proof, we follow Stylianides’ (2007) characterization. This is in contrast to Harel and Sowder’s (1998) student-centered notion of proof as a justification that is convincing to the student. However, it is consistent with their notion of mathematical proof, which is a justification that would be convincing to, and labeled as a proof by, mathematicians.

3 Harel and Sowder (1998) would refer to this as an inductive empirical proof scheme.

4 For instance, the claim that n² − n + 41 is always a prime number for all natural numbers is true when n is between 1 and 40, but is false for 41.

5 Of the studies cited in this section, only Martin and Harel (1989), Morris (2002, 2007), and Tapan and Arslan (2009) studied prospective teachers, the main population investigated in this article. However, we emphasize again that our focus is on how mathematics educators infer the proof schemes that individuals hold, rather than to make claims about the proof schemes of any specific population.

6 Implicit in the argument that follows in this section is that because prominent mathematicians such as Andrew Wiles and David Hilbert engaged in a particular type of mathematical practice, then this type of practice is a desirable practice or at least a legitimate one. However, we acknowledge that there are grounds to challenge this presumption, as surely some prominent mathematicians have engaged in unscrupulous behavior that we would not want students to emulate. Consequently, we have not ruled out the following criticism to our article: “Yes, your participants behaved similarly to Wiles and Hilbert, but the participants, Wiles, and Hilbert all failed to satisfy the standards of mathematical practice and they ought to have exhibited more normative behavior.” Although we disagree with this critique, we have not supplied a good argument against it.

7 By “getting nothing,” Wiles was not just referring by failing to prove the theorem, but also failure to generate other mathematical contributions resulting in the search for the theorem. However, the theme we wish to draw—that mathematicians consider costs, values, and likelihood of success when deciding what to prove—remains.

8 We did not consider one student, Dennis, because he differed from our population in that he was a school administrator who never taught mathematics and he had not taken college mathematics courses. He also sometimes refused to state how confident that he was in his answers.

9 For instance, to show all multiples of 10 could be written as the sum of consecutive integers, Group C showed that any sequence of integers of the form 10k, 10k + 1, 10k + 2, 10k + 3, and 10k + 4 would sum to a multiple of 10. This is not sufficient to prove the claim, as one would also need to show that any multiple of 10 can be found with this sequence (which is not the case). In the mathematics education literature, we would say that students used an impermissible proof framework (Selden & Selden, 1995).

10 The transcripts represent streamlined conversations in which we omitted parts of the conversations between the interviewer and students discussing ideas that were not germane to the phenomena that we are attempting to illustrate. The reason for this decision is that this article contains over 100 speaking turns in our transcripts, which to us is pushing the limit of what we can reasonably expect a reader to consider. At no point did we add or alter the words that were spoken and we do not believe that we have changed the intended meaning of the conversation.

11 For Tables 2–5, we counted the number of interviews in which a comment of this type was made. We did not count the number of students who made this comment, as participants would often express agreement to what their group mates said, but we could not determine which comment (or even which group mate) to which the participant was agreeing.