# Student difficulty in constructing proofs: The need for strategic knowledge

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## Abstract

The ability to construct proofs is an important skill for all mathematicians. Despite its importance, students have great difficulty with this task. In this paper, I first demonstrate that undergraduates often are aware of and able to apply the facts required to prove a statement but still fail to prove it. They thus fail to construct a proof because they could not use the syntactic knowledge that they had. By comparing doctoral students and undergraduates constructing proofs in abstract algebra, I have hypothesized four types of `strategic knowledge' – knowledge of how to choose which facts and theorems to apply – which the doctoral students appeared to possess and undergraduates did not. The doctoral students appeared to know the powerful proof techniques in abstract algebra, which theorems are most important, when particular facts and theorems are likely to be useful, and when one should or should not try and prove theorems using symbol manipulation.

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## REFERENCES

- Asiala, M., Dubinsky, E., Mathews, D., Morics, C. and Oktac, A.: 1997, ‘Student understanding of cosets, normality and quotient groups’,
*Journal of Mathematical Behavior*16, 241–309.CrossRefGoogle Scholar - Baylis, J.: 1983, ‘Proof - The essence of mathematics, part 1’,
*International Journal of Mathematics Education and Science Technology*14, 409–414.Google Scholar - Begle, E.G.: 1979,
*Critical Variables in Mathematics Education: Findings from a Survey of the Empirical Literature*, Mathematics Association of America and National Council of Teachers of Mathematics, Washington, D.C.Google Scholar - Cupillari, A.: 1989,
*The Nuts and Bolts of Proof*, Wadsworth, Belmont, CA.Google Scholar - DeFranco, T.: 1996, ‘A perspective on mathematical problem-solving expertise based on the performances of male Ph.D. mathematicians’,
*CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education*II, 195–213.Google Scholar - Ericsson, K.A. and Simon, H.A.: 1993,
*Protocol Analysis: Verbal Reports as Data*(2nd ed.), Bradford Books/MIT Press, Cambridge, MA.Google Scholar - Franklin, J. and Daoud, A.: 1988,
*Introduction to Proofs in Mathematics*, Prentice Hall, Sydney.Google Scholar - Frazier Sr., R.C.: 1970, ‘A comparison of an implicit and two explicit methods of teaching mathematical proof via abstract groups using selected rules of logic’,
*Dissertation Abstracts International*30, 5317A.Google Scholar - Greeno, J.G.: 1973, ‘The structure of memory and the process of solving problems’, in R.L. Solso (ed.),
*Contemporary Issues in Cognitive Psychology: The Loyola Symposium*, Winston, Washington, D.C.Google Scholar - Harel, G.: 1998, ‘Two dual assertions: The first on learning and the second on teaching (or vice versa)’,
*American Mathematical Monthly*105, 497–507.CrossRefGoogle Scholar - Harel, G. and Sowder, L.: 1998, ‘students' proof schemes’,
*CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education*III, 234–283.Google Scholar - Hart, E.W.: 1994, ‘A conceptual analysis of the proof-writing performance of expert and novice students in elementary group theory’, in J.J. Kaput and E. Dubinsky (eds.),
*Research Issues in Undergraduate Mathematics Learning: Preliminary Analysis and Results*, MAA Notes 33, Washington, D.C.Google Scholar - Hazzan, O.: 1999, ‘Reducing abstraction level when learning abstract algebra concepts’,
*Educational Studies in Mathematics*40, 71–90.CrossRefGoogle Scholar - Hazzan, O. and Leron, U.: 1994, ‘Students' use and misuse of mathematical theorems: The case of Lagrange's Theorem’,
*For the Learning of Mathematics*16, 23–26.Google Scholar - Knuth, E.J. and Elliot, R.L.: 1998, ‘Characterizing students' understandings of mathematical proof’,
*Mathematics Teacher*91, 14–17.Google Scholar - Lester, F.: 1994, ‘Musing about mathematical problem solving: 1970- 1994’,
*Journal for Research in Mathematics Education*25, 660–675.CrossRefGoogle Scholar - Martin, G.W. and Harel, G.: 1989, ‘Proof frames of preservice elementary teachers’,
*Journal for Research in Mathematics Education*20, 41–51.CrossRefGoogle Scholar - Moore, R.C.: 1994, ‘Making the transition to formal proof’,
*Educational Studies in Mathematics*27, 249–266.CrossRefGoogle Scholar - National Council of Teachers of Mathematics: 2000,
*Principles and Standards for School Mathematics*, NCTM, Reston, VA.Google Scholar - Reif, F.: 1995, ‘Millikan Lecture 1994: Understanding and teaching important scientific thought processes’,
*American Journal of Physics*63, 17–32.CrossRefGoogle Scholar - Schoenfeld, A.H.: 1978, ‘Presenting a strategy for indefinite integration’,
*American Mathematical Monthly*85, 673–678.CrossRefGoogle Scholar - Schoenfeld, A.H.: 1985,
*Mathematical Problem Solving*, Academic Press, Orlando.Google Scholar - Selden, A. and Selden, J.: 1987, ‘Errors and misconceptions in college level theorem proving’
*Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics*. Cornell UniversityGoogle Scholar - Selden, A., Selden, J., Hauk, S. and Mason, A.: 2001, ‘Why can't calculus students access their knowledge to solve nonroutine problems?’
*CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education*IV, 128–153.Google Scholar - Senk, S. L.: 1985, ‘How well do students write geometry proofs?’,
*Mathematics Teacher*78, 448–456.Google Scholar - Simon, D.P. and Simon H.A.: 1978, ‘Individual differences in solving physics problems’, in R.S. Siegler (ed.),
*Children's Thinking: What Develops?*, Erlbaum, Hillsdale, N.J.Google Scholar - Solow, D.: 1990,
*How to Read and Write Proofs*(2nd ed.), Wiley, New York.Google Scholar - Thompson, D.R.: 1996, ‘Learning and teaching indirect proof’,
*Mathematics Teacher*89, 474–482.Google Scholar - Watson, F.R.: 1978,
*Proof in Mathematics*, Institute of Education, University of Keele, Staffordshire.Google Scholar - Williams, E.: 1980, ‘An investigation of senior high school students' understanding of the nature of mathematical proof’,
*Journal for Research in Mathematics Education*11, 165–166.Google Scholar