Mathematics Majors' Perceptions of Conviction, Validity, and Proof

Abstract

In this paper, 28 mathematics majors who completed a transition-to-proof course were given 10 mathematical arguments. For each argument, they were asked to judge how convincing they found the argument and whether they thought the argument constituted a mathematical proof. The key findings from this data were (a) most participants did not find the empirical argument in the study to be convincing or to meet the standards of proof, (b) the majority of participants found a diagrammatic argument to be both convincing and a proof, (c) participants evaluated deductive arguments not by their form but by their content, but (d) participants often judged invalid deductive arguments to be convincing proofs because they did not recognize their logical flaws. These findings suggest improving undergraduates' comprehension of mathematical arguments does not depend on making undergraduates aware of the limitations of empirical arguments but instead on improving the ways in which they process the arguments that they read.

The research in this paper was funded by a grant from the National Science Foundation (DRL0643734). The views expressed in this paper are not necessarily those of the National Science Foundation. I would like to thank Eric Knuth, Guershon Harel, and Michael Smith (at Temple University) for useful advice on the design of this study, as well as Pablo Mejia-Ramos, Matthew Inglis, and the anonymous reviewers for helpful comments on earlier drafts of this manuscript. Finally, I am indebted to Anna Brophy, Iuliana Radu, and Robert Search for their help in transcribing the data.

Notes

¹Throughout this paper, the term empirical argument refers to an argument in support of a general assertion by verifying the assertion for one or several elements of the set. This differs from Harel and Sowder (1998), who referred to such arguments as being indicative of inductive proof schemes. To Harel and Sowder (1998), empirical proof schemes include both the tendency to gain conviction via examples and conviction by perceptual evidence.

² Harel (2001) noted that generic examples are instances of the desirable transformational proof scheme, not the empirical proof scheme. Rowland (2001) suggested that it may be appropriate to use generic proofs in lieu of formal proofs because they can be equally convincing, or perhaps more so.

³Of course individuals may have their own idiosyncratic reasons for gaining conviction from a diagram. I am simply describing here conditions that I believe to be common and important.

⁴ Inglis and Mejia-Ramos (2009b) do not endorse this common view in their paper.

⁵In my personal judgment, conditions (a), (b), and (c) would be useful criteria for determining if a diagrammatic argument should be classified as an informal proof.

⁶This does not work, of course. As a counterexample, 13 and 19 are two primes that add to 32. But adding two to either will produce a composite number. Hence one cannot follow P6's procedure to use 13 + 19 to find two primes adding to 34.

⁷The proofs explored by these mathematicians were much more sophisticated. However, the point is that mathematicians would spend a significant amount of time studying a single assertion, while the participants in this study would spend little time on this task.

⁸It is possible, for instance, that the table in argument 5 suggested to participants that the proof was invalid or that participants would have been convinced if a large number of examples were verified.