• School Mathematics
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• v.14 no.4
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• pp.469-493
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• 2012

The purpose of this study is find the relation between students' concept and types of proof construction. For this, four undergraduate students majored in mathematics education were evaluated to examine how they understand mathematical concepts and apply their concepts to their proving. Investigating students' proof with their concepts would be important to find implications for how students have to understand formal concepts to success in proving. The participants' proof productions were classified into syntactic proof productions and semantic proof productions. By comparing syntactic provers and semantic provers, we could reveal that the approaches to find idea for proof were different for two groups. The syntactic provers utilized procedural knowledges which had been accumulated from their proving experiences. On the other hand, the semantic provers made use of their concept images to understand why the given statements were true and to get a key idea for proof during this process. The distinctions of approaches to proving between two groups were related to students' concepts. Both two types of provers had accurate formal concepts. But the syntactic provers also knew how they applied formal concepts in proving. On the other hand, the semantic provers had concept images which contained the details and meaning of formal concept well. So they were able to use their concept images to get an idea of proving and to express their idea in formal mathematical language. This study leads us to two suggestions for helping students prove. First, undergraduate students should develop their concept images which contain meanings and details of formal concepts in order to produce a meaningful proof. Second, formal concepts with procedural knowledge could be essential to develop informal reasoning into mathematical proof.

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1521-1536
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• 2017

This paper introduces a representation style of variable binding using dependent types when formalizing meta-theoretic properties. The style we present is a variation of the Coquand-McKinna-Pollack's locally-named representation. The main characteristic is the use of dependent families in defining expressions such as terms and formulas. In this manner, we can handle many syntactic elements, among which wellformedness, provability, soundness, and completeness are critical, in a compact manner. Another point of our paper is to investigate the roles of free variables and constants. Our idea is that fresh constants can entirely play the role of free variables in formalizing meta-theories of first-order predicate logic. In order to show the feasibility of our idea, we formalized the soundness and completeness of LJT with respect to Kripke semantics using the proof assistant Coq, where LJT is the intuitionistic first-order predicate calculus. The proof assistant Coq supports all the functionalities we need: intentional type theory, dependent types, inductive families, and simultaneous substitution.