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.