On the Importance of Set-Based Meanings for
Categories and Connectives in Mathematical Logic
Paul Dawkins
Department of Mathematical Sciences
Northern Illinois University
Based on data from a series of teaching experiments on
standard tools of mathematical logic, this talk will
characterize some key student meanings for mathematical
properties and logical connectives. Some observed meanings
inhibited students’ adoption of logical structure, while
others greatly facilitated it. Reasoning with predicates
refers to students’ propensity to coordinate properties
(e.g. "is a square" or "is not a square") with the set of
examples exhibiting the property (squares and
non-squares). The negation/complement relation refers to
students’ association of a negative property with the
complement of the set associated with the corresponding
positive property. These meanings afforded students
efficient ways to reason about mathematical disjunctions
in normative ways. The talk also will describe how
students who did not have these meanings reasoned about
mathematical categories in ways that precluded normative
logical structure. In particular, students frequently
substituted positive categories for negative ones though
they were not mathematically equivalent and overly relied
in familiar categories learned in school, both forms of
what I call reasoning about properties. I conclude that
proof-oriented instruction may need to help students
develop set-based meanings and interpret negative claims
in terms of set complements in order to appropriately
interpret statements in ways compatible with mathematical
logic.