Semantic and Logical Negation: Students' Nonnormative Interpretations of Negative Mathematical Properties

• In Event: 70.066 - Roundtable Session 28
  In Roundtable Session: 70.066-2 - High School and University Students' Mathematical Thinking

Abstract

We present results from exploratory teaching experiments intended to foster undergraduate mathematics students’ meta-linguistic reflection and reinvention of truth-functional definitions for logical connectives (i.e. normative interpretations of mathematical parlance). We proffer meta-linguistic reflection as a novel instructional tool for helping students’ conform their mathematical reasoning to formal logical norms. Study participants spontaneously began attending to their linguistic interpretations before reinventing interpretations consistent with the truth-functional definition for non-quantified disjunctions. However, students struggled to systematize their reasoning about negative properties in quantified disjunctions because they consistently unpacked negative properties (e.g. “not a rectangle”) in terms of alternative semantic properties (“is a parallelogram”). We contrast such semantic negations with standard logical negations and discuss their implications for students’ transition to formal mathematics.

Authors

• Paul Christian Dawkins, Northern Illinois University

• John Paul Cook, University of Science & Arts of Oklahoma