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ERIC Number: ED518763
Record Type: Non-Journal
Publication Date: 1999-Feb
Pages: 14
Abstractor: As Provided
Reference Count: 23
ISBN: N/A
ISSN: N/A
The Role of Logic in the Validation of Mathematical Proofs. Technical Report. No. 1999-1
Selden, Annie; Selden, John
Online Submission
Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., abstract algebra and real analysis). What accounts for this seeming neglect of an essential ingredient of deductive reasoning? We will suggest a partial answer by comparing the contents of traditional logic courses with the kinds of reasoning used in proof validation, our name for the process by which proofs are read and checked. First, we will discuss the style in which mathematical proofs are traditionally written and its apparent utility for reducing validation errors. We will then examine the relationship between the need for logic in validating proofs and the contents of traditional logic courses. Some topics emphasized in logic courses do not seem to be called upon very often during proof validation, whereas other kinds of reasoning, not often emphasized in such courses, are frequently used. In addition, the rather automatic way in which logic, such as modus ponens, needs to be used during proof validation does not appear to be improved by traditional teaching, which often emphasizes truth tables, valid arguments, and decontextualized exercises. Finally, we will illustrate these ideas with a proof validation, in which we explicitly point out the uses of logic. We will not discuss proof construction, a much more complex process than validation. However, constructing a proof includes validating it, and hence, during the validation phase, calls on the same kinds of reasoning. Throughout this paper we will refer to a number of ideas from both cognitive psychology and mathematics education research. We will find it useful to discuss short-term, long-term, and working memory, cognitive load, internalized speech and vision, and schemas, as well as reflection, unpacking the meaning of statements, and the distinction between procedural and conceptual knowledge. (Contains 11 footnotes.)
Publication Type: Reports - Evaluative
Education Level: N/A
Audience: N/A
Language: English