# Making the transition to formal proof

- 710 Downloads
- 95 Citations

## Abstract

This study examined the cognitive difficulties that university students experience in learning to do formal mathematical proofs. Two preliminary studies and the main study were conducted in undergraduate mathematics courses at the University of Georgia in 1989. The students in these courses were majoring in mathematics or mathematics education. The data were collected primarily through daily nonparticipant observation of class, tutorial sessions with the students, and interviews with the professor and the students. An inductive analysis of the data revealed three major sources of the students' difficulties: (a) concept understanding, (b) mathematical language and notation, and (c) getting started on a proof. Also, the students' perceptions of mathematics and proof influenced their proof writing. Their difficulties with concept understanding are discussed in terms of a concept-understanding scheme involving concept definitions, concept images, and concept usage. The other major sources of difficulty are discussed in relation to this scheme.

## Preview

Unable to display preview. Download preview PDF.

### References

- 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. 216–230.Google Scholar - Bell, A. W.: 1976, ‘A study of pupils' proof-explanations in mathematical situations’,
*Educational Studies in Mathematics***7**, 23–40.Google Scholar - Bittinger, M. L.: 1969, ‘The effect of a unit in mathematical proof on the performance of college mathematics majors in future mathematics courses’,
*Dissertation Abstracts***29**, 3906A. (University Microfilms No. 69-7421)Google Scholar - Bittinger, M. L.: 1982,
*Logic, Proof, and Sets*, Addison-Wesley, Reading, MA.Google Scholar - Dreyfus, T.: 1990, ‘Advanced mathematical thinking’, in P. Nesher and J. Kilpatrick (eds.),
*Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education*, Cambridge University Press, Cambridge, pp. 113–134.Google Scholar - Dubinsky, D. and Lewin, P.: 1986, ‘Reflective abstraction and mathematics education: The genetic decomposition of induction and compactness’,
*Journal of Mathematical Behavior***5**, 55–92.Google Scholar - Duval, R.: 1991, ‘Structure du raisonnement d éductif et apprentissage de la démonstration’,
*Educational Studies in Mathematics***22**, 233–261.Google Scholar - Fletcher, P. and Patty, C. W.: 1988,
*Foundations of Higher Mathematics*, PWS-Kent, Boston.Google Scholar - Galbraith, P. L.: 1981, ‘Aspects of proving: A clinical investigation of process’,
*Educational Studies in Mathematics***12**, 1–28.Google Scholar - Glaser, B. G. and Strauss, A. L.: 1967,
*The Discovery of Grounded Theory*, Aldine, New York.Google Scholar - Goldberg, D. J.: 1975, ‘The effects of training in heuristic methods on the ability to write proofs in number theory’,
*Dissertation Abstracts International***35**, 4989B.Google Scholar - Hart, E. W.: 1987, ‘An exploratory study of the proof-writing performance of college students in elementary group theory’,
*Dissertation Abstracts International***47**, 4313A. (University Microfilms No. 8707982)Google Scholar - Laborde, C.: 1990, ‘Language and mathematics’, in P. Nesher and J. Kilpatrick (eds.),
*Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education*, Cambridge University Press, Cambridge, pp. 51–69.Google Scholar - Leron, U.: 1985, ‘Heuristic presentations: The role of structuring’,
*For the Learning of Mathematics***5**(3), 7–13.Google Scholar - Lewis, S. M.: 1987, ‘University mathematics students' perception of proof and its relationship to achievement’,
*Dissertation Abstracts International***47**, 3345A. (University Microfilms No. 8629607)Google Scholar - Moore, R. C.: 1990,
*College Students' Difficulties in Learning to Do Mathematical Proofs*, unpublished doctoral dissertation, University of Georgia, Athens.Google Scholar - Morash, R. P.: 1987,
*Bridge to Abstract Mathematics: Mathematical Proof and Structures*, Random House, New York.Google Scholar - Morgan, W. H.: 1972, ‘A study of the abilities of college mathematics students in proof-related logic’,
*Dissertation Abstracts International***32**, 4081B.Google Scholar - Rin, H.: 1983, ‘Linguistic barriers to students' understanding of definitions’, in R. Hershkowitz (ed.),
*Proceedings of the Seventh International Conference for the Psychology of Mathematics Education*, Weizmann Institute of Science, Rehovot, Israel, pp. 295–300.Google Scholar - Schoenfeld, A. H.: 1985,
*Mathematical Problem Solving*, Academic Press, Orlando, FL.Google Scholar - Smith, D., Eggen, M., and St. Andre, R.: 1990,
*A Transition to Advanced Mathematics*(*3rd ed.*, Brooks/Cole, Pacific Grove, CA.Google Scholar - Solow, D.: 1990,
*How to Read and Do Proofs: An Introduction to Mathematical Thought Processes*(*2nd ed.*, Wiley, New York.Google Scholar - Tall, D. and Vinner, S.: 1981, ‘Concept image and concept definition in mathematics with particular reference to limits and continuity’,
*Educational Studies in Mathematics***12**, 151–169.Google Scholar - Vinner, S.: 1983, ‘Concept definition, concept image and the notion of function’,
*International Journal of Mathematics Education in Science and Technology***14**, 293–305.Google Scholar - Vinner, S. and Dreyfus, T.: 1989, ‘Images and definitions for the concept of function’,
*Journal for Research in Mathematics Education***20**, 356–366.Google Scholar