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A Case Study of Undergraduates’ Proving Behaviors and Uses of Visual Representations in Identification of Key Ideas in Topology

Abstract

Visual representations, such as diagrams, are known to be valuable tools in problem solving and proof construction. However, previous studies have shown that simply having access to a diagram is not sufficient to improve students’ performance on mathematical tasks. Rather, students must actively extract information about the problem scenario from their diagrams for them to be useful. Furthermore, several studies have described the behaviors of mathematicians and students when solving problems and writing proofs, but few have discussed students’ behaviors in the context of proof writing in introductory point-set topology. We present a case study of an undergraduate, Stacey, enrolled in a general topology course. Throughout a semester, we presented Stacey with several proof-related tasks and examined how and why Stacey used diagrams when working on these tasks. Based on our analysis, we concluded that Stacey’s diagram creation and subsequent use during the construction of a given proof was an effort to identify the key idea of the proof. We describe Stacey’s overall proving behaviors through the lens of a problem-solving framework and present Stacey’s use of diagrams as an aid to discovering the key ideas of proofs in topology.

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Correspondence to Keith Gallagher.

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Appendices

Appendix

Proof Tasks

The proof tasks given in each session are listed below, in the order they were presented to the participants.

Session 1

  1. 1.

    Prove: Let f : S → T, and let {Ui}i ∈ I be a family of subsets of T. Prove that \( {f}^{-1}\left(\bigcap \limits_{i\in I}{U}_i\right)={\cap}_{i\in I}{f}^{-1}\left({U}_i\right) \).

  2. 2.

    Disprove: If f : S → T and A and B are subsets of T, then f−1(A ∪ B) ⊆ f−1(A).

Session 2

  1. 1.

    Disprove: Every relation C that is both symmetric and transitive must be reflexive.

  2. 2.

    Prove: Let f : A → B be a function. Define a relation ~ on A by setting a0~a1 if f(a0) = f(a1). Show that ~ is an equivalence relation.

Session 3

  1. 1.

    Disprove: Let (x1, y1), (x2, y2) ∈ 2. Then d((x1, y1), (x2, y2)) = min {|x1 − x2|, |y1 − y2|} is a metric on 2.

  2. 2.

    Prove: Let f : (X, d) → (Y, d) be a function such that there exists a y ∈ Y such that for all x ∈ X, f(x) = y. Prove that f is continuous.

Session 4

  1. 1.

    Prove: Let \( \left(X,\mathcal{T}\right) \) be a topological space. Prove that ∅, X are closed sets, that a finite union of closed sets is a closed set, and an arbitrary intersection of closed sets is a closed set.

Session 5

  1. 1.

    Disprove: Let \( \left(X,\mathcal{T}\right) \) be a topological space, and let A ⊆ X. Define the boundary of A, Bdry(A), by \( Bdry(A)=\overline{A}\cap \overline{C(A)} \). Then Bdry(A) is both open and closed in X.

  2. 2.

    Prove: A subset A of a topological space \( \left(X,\mathcal{T}\right) \) is said to be dense in X if \( \overline{A}=X \). Prove that if for each open set \( O\in \mathcal{T} \) we have A ∩ O ≠ ∅, then A is dense in X.

Session 6

  1. 1.

    Disprove: Let \( \left({X}_1,{\mathcal{T}}_1\right),\left({X}_2,{\mathcal{T}}_2\right),\dots, \left({X}_n,{\mathcal{T}}_n\right) \) be topological spaces. Then the set \( \prod \limits_{i=1}^n{X}_i \), together with the collection \( \mathcal{T} \) of all subsets of \( \prod \limits_{i=1}^n{X}_i \) of the form O1 × O2 × … × On, where each Oi is open in Xi, is a topological space.

  2. 2.

    Prove: Let \( \left(X,\mathcal{T}\right),\left(Y,\mathcal{T}^{\prime}\right) \) be topological spaces. If A is closed in X and B is closed in Y, then A × B is closed in X × Y.

Session 7

  1. 1.

    Disprove: Let \( \left(X,\mathcal{T}\right) \) be a topological space. Let A ⊆ X. If D is a connected subspace of X that intersects both A and CX(A), then D ∩ Bdry(A) = ∅.

  2. 2.

    Prove: Let \( \left(X,\mathcal{T}\right) \) be a topological space. A separation of X is a pair U, V of disjoint open subsets of X whose union is X. X is connected if no separation of X exists. If the sets C, D form a separation of X and if Y is a connected subspace of X, then either Y ⊆ C or Y ⊆ D.

Session 8

  1. 1.

    Disprove: in the cofinite topology is Hausdorff.

Session 9

  1. 1.

    Prove: If Y is a compact subspace of the Hausdorff space X, and if x0 is a point of X that is not in Y, then there exist disjoint open sets U and V containing x0 and Y, respectively.

Appendix B

Demographic Survey

On Stacey’s demographic survey, she self-reported that she was a Mathematics major with minors in Arabic Studies and in English. Stacey indicated that she had completed the following mathematics courses prior to enrolling in the current course in general topology:

  • Plane Trigonometry

  • Pre-Calculus with Trigonometry

  • Calculus I (Limits and Differential Calculus)

  • Calculus II (Integral Calculus, Sequences, and Series)

  • Multivariable Calculus

  • Introduction to Proof

  • Introduction to Linear Algebra

Appendix C

Coded Transcript Excerpt From Session 5.

The transcript excerpt below is the Prove task from Session 5. Speaker identifiers were abbreviated in the SPEAKER column: I is the Facilitator (Interviewer), S is Stacey, and T is Tom.

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Gallagher, K., Infante, N.E. A Case Study of Undergraduates’ Proving Behaviors and Uses of Visual Representations in Identification of Key Ideas in Topology. Int. J. Res. Undergrad. Math. Ed. (2021). https://doi.org/10.1007/s40753-021-00149-6

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Keywords

  • Topology
  • Key ideas
  • Proof
  • Visual representation
  • Diagrams