Proof in School Mathematics: Insights from Psychological Research into Students' Ability for Deductive Reasoning

Abstract

There are currently increased efforts to make proof central to school mathematics throughout the grades. Yet, realizing this goal is challenging because it requires that students master several abilities. In this article we focus on one such ability, namely, the ability for deductive reasoning, and we review psychological research to enhance what is currently known in mathematics education research about this ability in the context of proof and to identify important directions for future research. We first offer a conceptualization of proof, which we use to delineate our focus on deductive reasoning. We then review psychological research on the development of students' ability for deductive reasoning to see what can be said about the ages at which students become able to engage in certain forms of deductive reasoning. Finally, we review two psychological theories of deductive reasoning to offer insights into cognitively guided ways to enhance students' ability for deductive reasoning in the context of proof.

ACKNOWLEDGMENTS

The authors have contributed equally to writing this article. The authors wish to thank Lyn English, Jim Greeno, and three anonymous reviewers for useful comments on an earlier version of the article.

Notes

¹Each of these forms of reasoning is relevant to most situations of students' engagement with proof, but there are some situations in which certain forms of reasoning are more relevant than others. For example, inductive reasoning is particularly relevant to generalizing the results of empirical explorations to formulate conjectures, reasoning by analogy is particularly relevant to generating possible ways of constructing a proof based on previous and structurally similar experiences, and deductive reasoning is particularly relevant to constructing proofs or evaluating the validity of given arguments.

² Johnson-Laird and Byrne (1991) used the term deduction to refer to deductive reasoning. As they noted, in deduction “the goal is to draw a valid consequence from premises” (p. 2).

³For example, syllogisms that involve two premises with three terms (X, Y, and Z) can occur in one of four figures, as shown below.

X - Y Y - X X - Y Y - X

Y - Z Z - Y Z - Y Y - Z

⁴The 64 possible forms of syllogistic premises are derived as follows: four quantifier combinations for each of two premises that can occur in one of four figures as explained in endnote 3 (i.e., 64 = 4³).

⁵This comes in contrast to the mental models theory, which attributes errors with syllogisms largely to limitations in working memory capacity.