Conditional inference and advanced mathematical study
- 190 Downloads
Many mathematicians and curriculum bodies have argued in favour of the theory of formal discipline: that studying advanced mathematics develops one’s ability to reason logically. In this paper we explore this view by directly comparing the inferences drawn from abstract conditional statements by advanced mathematics students and well-educated arts students. The mathematics students in the study were found to endorse fewer invalid conditional inferences than the arts students, but they did not endorse significantly more valid inferences. We establish that both groups tended to endorse more inferences which led to negated conclusions than inferences which led to affirmative conclusions (a phenomenon known as the negative conclusion effect). In contrast, however, we demonstrate that, unlike the arts students, the mathematics students did not exhibit the affirmative premise effect: the tendency to endorse more inferences with affirmative premises than with negated premises. We speculate that this latter result may be due to an increased ability for successful mathematics students to be able to ‘see through’ opaque representations. Overall, our data are consistent with a version of the formal discipline view. However, there are important caveats; in particular, we demonstrate that there is no simplistic relationship between the study of advanced mathematics and conditional inference behaviour.
KeywordsAdvanced mathematical thinking Conditional inference Logic Reasoning Representation systems Theory of formal discipline
We would like to thank Gary Davis, Paola Iannone and Keith Weber for their helpful comments on earlier drafts of this work.
- Barnard, A. D. (1995). The impact of ‘meaning’ on students’ ability to negate statements. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th international conference on the psychology of mathematics education (Vol. 2, pp. 3–10). Recife, Brazil, IGPME.Google Scholar
- Braine, M. D. S., & O’Brien, D. P. (1998). Mental logic. Mahwah, NJ: Erlbaum.Google Scholar
- Davis, C. (1850/1970). The logic and utility of mathematics. In J. K. Bidwell & R. G. Clason (Eds.), Readings in the history of mathematics education (pp. 39–62). Washington DC: NCTM.Google Scholar
- Dubinsky, E., Elterman, F., & Gong, C. (1988). The student’s construction of quantification. For the Learning of Mathematics, 8(2), 44–51.Google Scholar
- Evans, J. St. B. T. (2007). Hypothetical thinking: Dual processes in reasoning and judgement. Hove, UK: Psychology Press.Google Scholar
- Evans, J. St. B. T., Clibbens, J., & Rood, B. (1995). Bias in conditional inference: Implications for mental models and mental logic. Quarterly Journal of Experimental Psychology, 48A, 644–670.Google Scholar
- Johnson, D. L. (1998). Elements of logic via numbers and sets. London: Springer.Google Scholar
- Johnson-Laird, P. N. (2006). How we reason. Oxford: OUP.Google Scholar
- Lesh, R., Behr, M., & Post, T. (1987). Rational number relations and proportions. In C. Janiver (Ed.), Problems of representations in the teaching and learning of mathematics (pp. 41–58). Hillsdale, N.J.: Lawrence Erlbaum.Google Scholar
- Oaksford, M., & Chater, N. (2007). Bayesian rationality: The probabilistic approach to human reasoning. Oxford: OUP.Google Scholar
- QAA (2002). Mathematics, statistics and operational research subject benchmark standards. Online article [accessed 15/07/2005]: http://www.qaa.ac.uk/academicinfrastructure/benchmark/honours/mathematics.pdf.
- Smith, A. (2004). Making mathematics count: The report of Professor Adrian Smith’s Inquiry into Post-14 Mathematics Education. London: The Stationery Office.Google Scholar
- Zazkis, R., & Gadowsky, K. (2001). Attending to transparent features of opaque representations of natural numbers. In A. Cuoco (Ed.), The roles of representation in school mathematics (pp. 146–165). Reston, VA: NCTM.Google Scholar
- Zazkis, R., & Liljedahl, P. (2004). Understanding primes: The role of representation. Journal for Research in Mathematics Education, 35, 164–186.Google Scholar