The Effect of Authority on the Persuasiveness of Mathematical Arguments

Abstract

Three experiments are reported that investigate the extent to which an authority figure influences the level of persuasion undergraduate students and research-active mathematicians invest in mathematical arguments. We demonstrate that, in some situations, both students and researchers rate arguments as being more persuasive when they are associated with an expert mathematician than when the author is anonymous. We develop a model that accounts for these data by suggesting that, for both students and researchers, an authority figure only plays a role when there is already some uncertainty about the argument's mathematical status. Implications for pedagogy, and for future research, are discussed.

Notes

¹Some authors (e.g., Martin & Harel, 1989; Healy & Hoyles, 2000) have used the term “proof” in its everyday sense, as an argument that persuades. Others (e.g., de Villiers, 1990) retain the term for deductive arguments. We have attempted to adopt the more neutral term “argument” unless specifically referring to an argument that would universally be referred to by mathematicians as a proof (a more satisfactory characterization of what constitutes a proof is a difficult and current philosophical topic, and is well beyond the scope of this article; see, for example, Azzouni, 2005; Bundy, 2005; Thurston, 1994).

²We prefer the term “gaining conviction” to more linguistically satisfying alternatives such as “becoming convinced,” as we wish to emphasize the continuous nature of one's level of conviction. In particular, conviction in our sense does not imply a threshold level at which one moves from being unconvinced to being convinced.

³All three universities typically required that their prospective mathematics undergraduates obtain grades of AAA—the highest possible—in their A Level examinations (the qualification taken by school leavers in England and Wales).

⁴Both named mathematicians in Experiment 1 are very highly regarded in the mathematical community. Gowers received the 1998 Fields Medal (the mathematical equivalent of the Nobel Prize) for his work on functional analysis and combinatorics. Littlewood is well known for his prolific collaboration with Hardy in the fields of number theory and classical analysis; among other awards, he received the LMS De Morgan medal, the Royal Medal of the Royal Society of London, and the LMS Senior Berwick Prize.

⁵Due to an initial typographical error in the presentation of the induction argument, the first 18 responses from the researcher group were removed from the analysis of this argument.

⁶The issue of whether or not a picture can be a proof is a controversial topic in the philosophy of mathematics literature (e.g., Brown, 1997; Folina, 1999).

⁷Throughout the study we approached research-intensive universities from English-speaking countries to participate. In practice, this meant selecting universities from the Association of Commonwealth Universities membership list (for non-U.S. universities) and the USNews.com “Best Graduate Schools” list of “top mathematics programs” (for U.S. universities).

⁸One-tailed tests were used for the heuristic and visual comparisons, as predictions of the direction of the difference in means existed from Experiment 1 (these comparisons constituted direct replications of Experiment 1).

⁹Note that in those comparisons where no between-conditions differences existed in Experiments 1 and 2 (the induction argument, the Pólya argument and the visual argument for students) we might expect that the two different interpretations of the instructions discussed earlier would lead to similar ratings. For both formal proofs we would expect high ratings on both admissibility and persuasiveness; and, as discussed earlier, we might expect students to give low ratings to the visual argument on both admissibility and persuasiveness.