# Students' understandings of logical implication

- 270 Downloads
- 29 Citations

## Abstract

We report results from an analysis of responses to a written question in which high-attaining students in English schools, who formed part of a longitudinal nation-wide survey on proof conceptions, were asked to assess the equivalence of two statements about elementary number theory, one a logical implication and the other its converse, to evaluate the truth of the statements and to justify their conclusions. We present an overview of responses at the end of Year 8 (age 13 years) and an analysis of the approaches taken, and follow this with an analysis of the data collected from students who answered the question again in Year 9 (age 14 years) in order to distinguish learning trajectories. From these analyses, we distinguished three strategies, empirical, focussed-empirical and focussed-deductive, that represent shifts in attention from an inductive to a deductive approach. We noted some progress from Year 8 to Year 9 in the use of the focussed strategies but this was modest at best. The most marked progress was in recognition of the logical necessity of a conclusion of an implication when the antecedent was assumed to be true. Finally we present some theoretical categories to capture different types of meanings students assign to logical implication and the rationale underpinning these meanings. The categories distinguish responses where a statement of logical implication is (or is not)interpreted as equivalent to its converse, where the antecedent and consequent are (or are not) seen as interchangeable, and where conclusions are (or are not)influenced by specific data.

## Preview

Unable to display preview. Download preview PDF.

## REFERENCES

- Anderson, R.C., Chinn, C., Chang, J., Waggoner, M. and Yi, H.: 1997, ‘On the logical integrity of children's arguments’,
*Cognition and Instruction*15(2), 135–167.CrossRefGoogle Scholar - Balacheff, N.: 1988, ‘Aspects of proof in pupils’ practice of school mathematics’, in D. Pimm (ed.),
*Mathematics, Teachers and Children*, Hodder and Stoughton, London, pp. 216–235.Google Scholar - Bell, A.W.: 1976, ‘A study of pupils’ proof-explanations in mathematical situations’,
*Educational Studies in Mathematics*7, 23–40.CrossRefGoogle Scholar - Ceci, Stephen J.: 1990,
*On Intelligence - More or Less. A Bio-Ecological Treatise of Intellectual Development*, Century Psychology Series, Prentice Hall, USA.Google Scholar - Cheng, P.W. and Holyoak, K.J.: 1985, ‘Pragmatic reasoning schemas’,
*Cognitive Psychology*17, 391–416.CrossRefGoogle Scholar - Coe, R. and Ruthven, K.: 1994, ‘Proof practices and constructs of advanced mathematics students’,
*British Educational Research Journal*20(1), 41–53.Google Scholar - Deloustal-Jorrand, V.: 2002, ‘Implication and mathematical reasoning’,
*Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education*26(2), Norwich, England, pp. 281–288.Google Scholar - De Villiers, M.: 1990, ‘The role and function of proof in mathematics’,
*Pythagoras*24, 17–24.Google Scholar - Fischbein, E.: 1982, ‘Intuition and proof’,
*For the Learning of Mathematics*3(2), 9–18, 24.Google Scholar - Godino, J.D. and Recio, A.M.: 1997, ‘Meaning of proofs in mathematics education’,
*Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education*, 2, Lahti, Finland, pp. 313–320.Google Scholar - Hanna, G.: 1989, ‘Proofs that prove and proofs that explain’,
*Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education*, G.R. Didactique CNRS Paris, pp. 45–51.Google Scholar - Hanna, G.: 1995, ‘Challenges to the importance of proof’,
*For the Learning of Mathematics*15(3), 42–49.Google Scholar - Hanna, G.: 2000, ‘Proof, explanation and exploration: an overview’,
*Educational Studies in Mathematics*44, 5–23.CrossRefGoogle Scholar - Healy, L. and Hoyles, C.: 2000, ‘A study of proof conceptions in algebra’,
*Journal for Research in Mathematics Education*31(4), 396–428.CrossRefGoogle Scholar - Hewitt, D.: 1992, ‘Train spotters’ paradise’,
*Mathematics Teaching*140, 6–8.Google Scholar - Hoyles, C.: 1997, ‘The curricular shaping of students’ approaches to proof’,
*For the Learning of Mathematics*17(1), 7–16.Google Scholar - Inhelder, B. and Piaget, J.: 1958,
*The Growth of Logical Thinking*, Routledge and Kegan Paul, London.Google Scholar - Krummheuer, G.: 1995, ‘The ethnology of argumentation’, in P. Cobb and H. Bauersfeld (eds.),
*The Emergence of Mathematical Meaning: Interaction in Classroom Cultures*, Erlbaum, Hillsdale, NJ, pp. 229–269.Google Scholar - Mitchell, D.: 1962,
*An Introduction to Logic*, Hutchinson, London.Google Scholar - O'Brien, T.C., Shapiro, B.J. and Reali, N.C.: 1971, ‘Logical thinking - language and context’,
*Educational Studies in Mathematics*4, 201–219.CrossRefGoogle Scholar - Quine, W.V.: 1974,
*Methods of Logic*, (Third edition; first edition published in 1950) Routledge & Kegan Paul, London.Google Scholar - Reid, D.A.: 2002, ‘Conjectures and refutations in Grade 5 mathematics’,
*Journal for Research in Mathematics Education*33(1), 5–29.Google Scholar - Rodd, M.: 2000, ‘On mathematical warrants’,
*Mathematical Thinking and Learning*3, 22–244.Google Scholar - Simon, M.A.: 2000, ‘Reconsidering mathematical validation in the classroom’,
*Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education*4, Hiroshima, Japan, pp. 161–168.Google Scholar - Sowder, L. and Harel, G.: 1998, ‘Types of students’ justifications’,
*The Mathematics Teacher*91(8), 670–675.Google Scholar - Toulmin, S.: 1958,
*The Uses of Argument*, Cambridge University Press, Cambridge, UK.Google Scholar - Van Dormolen, J.: 1977, ‘Learning to understand what a proof really means’,
*Educational Studies in Mathematics*8, 27–34.CrossRefGoogle Scholar - Vinner, S.: 1983, ‘The notion of proof - some aspects of students’ views at the senior high school level’,
*Proceedings of the 7th Conference of the International Group for the Psychology of Mathematics Education*, pp. 289–294.Google Scholar - Wason, P.C.: 1960, ‘On the failure to eliminate hypotheses in a conceptual task’,
*Quarterly Journal of Experimental Psychology*12, 129–140.CrossRefGoogle Scholar - Wason, P.C. and Shapiro, D.: 1971, ‘Natural and contrived experience in a reasoning task’,
*Quarterly Journal of Experimental Psychology*23, 63–71.Google Scholar - Yackel, E.: 2001, ‘Explanation, justification and argumentation in mathematics classrooms’,
*Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education*, 1, Freudenthal Institute, Utrecht, Holland, pp. 9–23.Google Scholar - Yackel, E. and Cobb, P.: 1996, ‘Sociomathematical norms, argumentation, and autonomy in mathematics’,
*Journal for Research in Mathematics Education*27, 458–477.CrossRefGoogle Scholar - Zack, V. and Graves, B.: 2001, ‘Making mathematical meaning through dialogue: “Once you think of it, the Z minus three seems pretty weird”’,
*Educational Studies in Mathematics*46(1-3), 229–271.CrossRefGoogle Scholar