Transitioning to Proof

This paper describes some strategies used in a `transition’ course. Such courses help undergraduate mathematics majors move from learning procedures to learning to function as critical mathematicians in order to understand and work with abstract concepts. One of the co-authors of this paper was a student of Leon Henkin. His influence on her helped shape the strategies used in the course, and is described at the end of the paper.

💡 Research Summary

The paper “Transitioning to Proof” reports on a specially designed “transition” course intended to help undergraduate mathematics majors move from a procedural, computation‑focused learning style to the more abstract, proof‑oriented thinking required in higher‑level mathematics. The authors, Diane Resek and Dan Fendel, describe four interlocking pedagogical techniques that they implemented in the course and its accompanying textbook, “Foundations of Higher Mathematics: Exploration and Proof.”

1. Exploration – Before introducing formal definitions, students are asked to generate and investigate questions about a new topic. The course opens with a simple game whose rules are given, and students must discover a winning strategy, thereby confronting the need to justify a claim. As the semester proceeds, each new concept (sets, integers, quantifiers, functions, sequences) is first explored through concrete examples and counter‑example questions (e.g., comparing ((∃x∈S)(p(x)∧q(x))) with ((∃x∈S)p(x) ∧ (∃x∈S)q(x))). This stage cultivates a habit of asking “why” and “what if” rather than simply “how to prove.”

2. Get Your Hands Dirty (GYHD) – Short, hands‑on exercises are interspersed throughout the text. After a definition is presented, students immediately apply it to small, concrete instances (e.g., computing power sets of tiny sets, formulating the negation of a subset relation using quantifiers). The purpose is to force students to manipulate definitions directly, turning abstract symbols into tangible objects they can experiment with. By the final exam, students are asked to decode a novel definition of a connected set of real numbers, produce examples and non‑examples, and prove a straightforward theorem derived from that definition.

3. Logic Without Truth Tables – Recognizing that many students resist the conventional truth‑table treatment of conditionals (especially the “vacuously true” case), the authors replace truth tables with a counter‑example perspective: a conditional is true exactly when no counter‑example exists. Proofs of conditionals are therefore framed as demonstrations that any object satisfying the hypothesis cannot falsify the conclusion. The same reasoning underlies proofs by contrapositive and contradiction. An appendix supplies the formal logical background for instructors who need it, but the main text stays grounded in the intuitive counter‑example idea.

4. Proof Evaluation – Students are given “potential proofs” and asked to assess their validity. The task requires two steps: (a) verify that the stated theorem is indeed true, and (b) examine whether the argument correctly establishes the theorem. The authors provide a sample involving power sets, where the proof mistakenly treats a non‑empty intersection as evidence of an element in the intersection. Students must locate the logical flaw, distinguish between a mere omission of detail and a genuine error, and articulate why the proof fails. This activity sharpens students’ ability to critique proofs, spot hidden assumptions, and understand the role of definitions in each inference.

The paper concludes with a personal reflection by Resek on the influence of her mentor, logician Leon Henkin. Henkin’s emphasis on concrete examples, intuitive understanding, and making mathematics accessible to a broad audience shaped each of the four techniques. Resek recounts how Henkin encouraged her to write out examples on separate sheets of paper, a habit that later inspired the “hands‑dirty” exercises. His belief that the level of mathematics required for a major should be reachable by most students motivated the authors to design a course that demystifies proof‑writing and prepares a larger, more diverse cohort for advanced courses.

Overall, the authors argue that the transition course successfully bridges the gap between procedural competence and proof‑oriented reasoning. By foregrounding exploration, immediate application of definitions, an intuitive counter‑example view of logic, and rigorous proof evaluation, the course cultivates the habits of mind that professional mathematicians use daily. The reported strategies provide a concrete, replicable model for other institutions seeking to improve proof instruction and to make higher‑level mathematics more inclusive.