# Semantic and Syntactic Proof Productions

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## Abstract

In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical concepts to guide the formal inferences that he or she draws. We present two independent exploratory case studies from group theory and real analysis that illustrate both types of proofs. We conclude by discussing what types of concept understanding are required for each type of proof production and by illustrating the weaknesses of syntactic proof productions.

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## REFERENCES

- Alcock, L.J. and Simpson, A.P.: 2001, 'The Warwick Analysis project: Practice and theory', in D. Holton (ed.),
*The Teaching and Learning of Mathematics at University Level*, Dordrecht, Kluwer, pp. 99–111.Google Scholar - Alcock, L.J. and Simpson, A.P.: 2002, 'Definitions: Dealing with categories mathemati-cally',
*For the Learning of Mathematics*22(2), 28–34.Google Scholar - Burn, R.P.: 1992,
*Numbers and Functions: Steps into Analysis*, Cambridge, Cambridge University Press.Google Scholar - Cornu, B.: 1992, 'Limit', in D.O. Tall (ed.),
*Advanced Mathematical Thinking*, Dordrecht, Kluwer, pp. 153–166.Google Scholar - Davis, P.J. and Hersh, R.: 1981,
*The Mathematical Experience*Viking Penguin Inc., New York.Google Scholar - de Villiers, M.: 1990, 'The role and function of proof in mathematic',
*Pythagoras*23, 17–24.Google Scholar - Fischbein, E.: 1982, 'Intuition and proof',
*For the Learning of Mathematics*3(2), 9–24.Google Scholar - Hadamard, J.: 1945,
*An Essay on the Psychology of Invention in the Mathematical Field*, New York, Dover.Google Scholar - Hanna, G.: 1990, 'Some pedagogical aspects of proof',
*Interchange*21(1), 6–13.CrossRefGoogle Scholar - Hazzan, O.: 1999, 'Reducing abstraction level when learning abstract algebra concept',
*Educational Studies in Mathematics*40, 71–90.CrossRefGoogle Scholar - Leron, U., Hazzan, O., and Zazkis, R.: 1995, 'Learning group isomorphism: A crossroad of many concept',
*Educational Studies in Mathematics*29, 153–174.CrossRefGoogle Scholar - Monaghan, J.: 1991, 'Problems with the language of limit',
*For the Learning of Mathe-matics*11(3), 20–24.Google Scholar - Newell, A. and Simon, H.A.: 1972,
*Human Problem Solving*. Prentice Hall, Englewood Cliffs, NJ.Google Scholar - Pinto, M.M.F. and Tall, D.O.: 1999, 'Student construction of formal theories: Giving and extracting meaning', in O. Zaslavsky (ed.)
*Proceedings of the 23rd Meeting of the Inter-national Group for the Psychology of Mathematics Education*, 1,281–288, Haifa.Google Scholar - Poincaré, H.: 1913,
*The Foundations of Science*(translated by Halsted, J.B.). The Science Press, New York.Google Scholar - Raman, M.: 2003, 'Key ideas: What are they and how can they help us understand how people view proof?',
*Educational Studies in Mathematics*52, 319–325.CrossRefGoogle Scholar - Skemp, R.R.: 1976, 'Relational and instrumental understanding',
*Mathematics Teaching*77, 20–26.Google Scholar - Skemp, R.R.: 1987,
*The Psychology of Learning Mathematics*. Lawrence Erlbaum Associates, Hillsdale, NJ.Google Scholar - Tall, D.O.: 1989, 'The nature of mathematical proof',
*Mathematics Thinking*127, 28–32.Google Scholar - Tall, D.O. and Vinner, S.: 1981, 'Concept image and concept definition in mathematics, with special reference to limits and continuity',
*Educational Studies in Mathematics*12, 151–169.CrossRefGoogle Scholar - Thurston, W.P.: 1994, 'On proof and progress in mathematic',
*Bulletin of the American Mathematical Society*30, 161–177.Google Scholar - Vinner, S.: 1991, 'The role of definitions in teaching and learning', in D. Tall (ed.),
*Advanced Mathematical Thinking*. Kluwer, Dordrecht, pp. 65–81.Google Scholar - Vinner, S. and Dreyfus, T.: 1989, 'Images and definitions for the concept of function',
*Journal for Research in Mathematics Education*20(4), 356–366.Google Scholar - Weber, K.: 2001, 'Student difficulty in constructing proof: The need for strategic knowl-edge',
*Educational Studies in Mathematics*48(1), 101–119.CrossRefGoogle Scholar - Weber, K.: 2002a, 'Beyond proving and explaining: Proofs that justify the use of defini-tions and axiomatic structures and proofs that illustrate technique',
*For the Learning of Mathematics*22(3), 14–17.Google Scholar - Weber, K.: 2002b, 'The role of instrumental and relational understanding in proofs about group isomorphism', in
*Proceedings from the 2nd International Conference for the Teaching of Mathematics*, Hersonisoss.Google Scholar