# Conditional inference and advanced mathematical study

- 190 Downloads
- 7 Citations

## Abstract

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.

## Keywords

Advanced mathematical thinking Conditional inference Logic Reasoning Representation systems Theory of formal discipline## Notes

### Acknowledgements

We would like to thank Gary Davis, Paola Iannone and Keith Weber for their helpful comments on earlier drafts of this work.

## References

- 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 - Cheng, P. W., Holyoak, K. J., Nisbett, R. E., & Oliver, L. M. (1986). Pragmatic versus syntactic approaches to training deductive reasoning.
*Cognitive Psychology*,*18*, 293–328.CrossRefGoogle Scholar - Damarin, S. K. (1977). The interpretation of statements in standard logical form by preservice elementary teachers.
*Journal for Research in Mathematics Education*,*8*, 123–131.CrossRefGoogle 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 - Durand-Guerrier, V. (2003). Which notion of implication is the right one? From logical considerations to a didactic perspective.
*Educational Studies in Mathematics*,*53*, 5–34.CrossRefGoogle Scholar - Eisenberg, T. A., & McGinty, R. L. (1974). On comparing error patterns and the effect of maturation in a unit on sentential logic.
*Journal for Research in Mathematics Education*,*5*, 225–237.CrossRefGoogle Scholar - Evans, J. St. B. T. (2002). Matching bias and set sizes: A discussion of Yama (2001).
*Thinking and Reasoning*,*8*, 153–163.CrossRefGoogle 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 - Evans, J. St. B. T., & Handley, S. J. (1999). The role of negation in conditional inference.
*Quarterly Journal of Experimental Psychology,**52A*, 739–769.CrossRefGoogle Scholar - Hoyles, C., Küchemann, D. (2002). Students’ understanding of logical implication.
*Educational Studies in Mathematics, 51*, 193–223.CrossRefGoogle 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 - Kirshner, D., & Awtry, T. (2004). Visual salience of algebraic transformations.
*Journal for Research in Mathematics Education, 35*, 224–257.CrossRefGoogle Scholar - Landy, D., Goldstone, R. (2007). How abstract is symbolic thought?
*Journal of Experimental Psychology: Learning, Memory and Cognition, 33*, 720–733.CrossRefGoogle Scholar - Lehman, D. R., & Nisbett, R. E. (1990). A longitudinal study of the effects of undergraduate training on reasoning.
*Developmental Psychology, 26*, 952–960.CrossRefGoogle 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 - Newstead, S. E., Charles Ellis, M., Evans, J. St. B. T., & Dennis, I. (1997). Conditional reasoning with realistic material.
*Thinking and Reasoning, 3*(1), 49–76.CrossRefGoogle Scholar - Nisbett, R. E., & Wilson, T. D. (1977). Telling more than we can know: Verbal reports on mental processes.
*Psychological Review, 84*, 231–295.CrossRefGoogle Scholar - Oaksford, M. (2002). Contrast classes and matching bias as explanations of the effects of negation on conditional reasoning.
*Thinking and Reasoning, 8*, 135–151.CrossRefGoogle Scholar - Oaksford, M., & Chater, N. (2007).
*Bayesian rationality: The probabilistic approach to human reasoning*. Oxford: OUP.Google Scholar - Oaksford, M., Chater, N., & Larkin, J. (2000). Probabilities and polarity biases in conditional inference.
*Journal of Experimental Psychology: Learning, Memory and Cognition*,*26*, 883–899.CrossRefGoogle Scholar - O’Brien, T. C. (1972). Logical thinking in adolescents.
*Educational Studies in Mathematics*,*4*, 401–428.CrossRefGoogle Scholar - O’Brien, T. C. (1973). Logical thinking in college students.
*Educational Studies in Mathematics*,*5*, 71–79.CrossRefGoogle Scholar - O’Brien, T. C., Shapiro, B. J., & Reali, N. C. (1971). Logical thinking – language and context.
*Educational Studies in Mathematics, 4*, 201–219.CrossRefGoogle Scholar - Pollard, P., & Evans, J. St. B. T. (1980). The influence of logic on conditional reasoning performance.
*Quarterly Journal of Experimental Psychology*,*32*, 605–624.CrossRefGoogle 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.
- Schooler, J. W., Ohlsson, S., & Brooks, K. (1993). Thoughts beyond words: When language overshadows insight.
*Journal of Experimental Psychology: General*,*122*, 166–183.CrossRefGoogle Scholar - Schroyens, W., Schaeken, W., & d’Ydewalle, G. (2001). The processing on negations in conditional reasoning: A meta-analytical case study in mental model and/or mental logic theory.
*Thinking and Reasoning*,*7*, 121–172.CrossRefGoogle Scholar - Schroyens, W., Schaeken, W., Fias, W. & d’Ydewalle, G. (2000). Heuristic and analytic processes in propositional reasoning with negatives.
*Journal of Experimental Psychology: Learning, Memory and Cognition*,*26*, 1713–1734.CrossRefGoogle Scholar - 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 - Stanic, G. M. A. (1986). The growing crisis in mathematics education in the early twentieth century.
*Journal for Research in Mathematics Education*,*17*, 190–205.CrossRefGoogle Scholar - Stylianides, A. J., Stylianides, G. J., & Philippou, G. N. (2004). Undergraduate Students’ understanding of the contraposition equivalence rule in symbolic and verbal contexts.
*Educational Studies in Mathematics*,*55*, 133–162.CrossRefGoogle Scholar - Thorndike, E. L. (1924). Mental discipline in high school studies.
*Journal of Educational Psychology*,*15*, 1–22.CrossRefGoogle Scholar - Thorndike, E. L., Woodworth, R. S. (1901). The influence of improvement in one mental function upon the efficiency of other functions.
*Psychological Review*,*8*, 247–261.CrossRefGoogle Scholar - Wason, P. C., & Evans, J. St. B. T. (1975). Dual processes in reasoning?
*Cognition*,*3*, 141–154.CrossRefGoogle Scholar - Yama, H. (2001). Matching versus optimal data selection in the Wason Selection Task.
*Thinking and Reasoning*,*7*, 295–311.CrossRefGoogle 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