Educational Studies in Mathematics

, Volume 56, Issue 2–3, pp 209–234 | Cite as

Semantic and Syntactic Proof Productions

  • Keith Weber
  • Lara Alcock


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.

abstract algebra advanced mathematical thinking advanced mathematical concepts convergent sequence formal reasoning isomorphism limits proofs real analysis 


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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Keith Weber
    • 1
  • Lara Alcock
    • 1
  1. 1.Graduate School of EducationRutgers UniversityNew BrunswickUSA

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