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.

摘要

在数学教育中，研究人员通常根据他们提出的理由来推断学生的信念标准。具体来说，如果学生用经验论证来证明数学陈述的合理性，研究人员通常会推断基于实例的论证为学生提供确定性，即一般的数学陈述是正确的。在本文中，我们提供了一个理论框架来解释与上述解释相抵触的个人与证据相关的行为。根据预期价值理论的构想，我们认为个人是寻求演绎证明还是为经验辩护而定，取决于几个因素，包括他们在了解所考虑的数学陈述的准确性时所具有的价值，搜索证据所需的时间和精力，以及他们发现能够成功找到证据的可能性。我们证明了数学家在确定他们将尝试证明的陈述时会考虑价值，成本和努力，因此让学生进行同样的考虑不会是非理性或非数学的。我们通过在解决问题的过程中研究11名职前和在职中学数学教师的正当行为来说明我们框架的解释力。尽管这些人经常凭经验证明数学陈述是合理的，但这些人意识到经验依据的局限性，因此他们通常无法从这些依据中获得确定性。价值，成本，