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On the Importance of Set-Based Meanings for Categories and Connectives in Mathematical Logic

Abstract

Based on data from a series of teaching experiments on standard tools of mathematical logic, this paper characterizes a range of 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 paper also provides accounts of 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 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.

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Notes

  1. 1.

    Thompson et al. (2014) further distinguish whether a meaning is in-the-moment or stable based on whether the student is likely or not to assimilate other instances of an idea to the same meaning. Since we intend to set forth constructs for further study, we focus only on stable meanings we observed among our study participants.

  2. 2.

    All names are pseudonyms

  3. 3.

    Each turn is numbered for reference. “I” stands for the interviewer, who was the author. “…” denotes a pause while “[…]” denotes an omission.

  4. 4.

    According to formal logic, “∀x ∈ S , P(xor Q(x)” shares truth-values with “If not P(x) for x ∈ S, then Q(x).”

  5. 5.

    Horn and others dispute whether any principled distinction can be made between positive and negative claims, but this exceeds the scope of the present discussion.

  6. 6.

    I thank one anonymous reviewer for pointing out that this condition is necessary since “intersecting” and “parallel” are contraries in 3-dimensional space. As the reviewer noted, sometimes universal sets are implied rather than stated, leading to some ambiguity about negations and complements.

  7. 7.

    Teaching logic through guided reinvention was, at least in part, a research-oriented decision. By learning what meanings students need to construct to reinvent logic, one can better understand what learning all logic instruction should promote. I do not endorse that logic always be taught through guided reinvention, though I think there are some rich affordances there. In particular I think this lens forced me to consider “What is the mathematical activity that underlies logic?” I think future research on logic should attend more closely to grounding logic in students’ mathematical activity. I generally endorse teaching logic using meaningful mathematical language (see Dawkins and Cook 2017) rather than everyday language or pure logical syntax, but well-chosen everyday analogies can also be useful (Dawkins and Roh 2016).

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Dawkins, P.C. On the Importance of Set-Based Meanings for Categories and Connectives in Mathematical Logic. Int. J. Res. Undergrad. Math. Ed. 3, 496–522 (2017). https://doi.org/10.1007/s40753-017-0055-4

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Keywords

  • Mathematical logic
  • Guided reinvention
  • Disjunctions
  • Quantification
  • Mathematical properties