Mathematics majors with limited writing skills often struggle with the double hurdle of learning to formulate logical arguments and learning to write precisely and concisely to communicate mathematics effectively. I will introduce a pedagogical tool (which I call a Proof Outline) that provides a tabular format for logical arguments. The purpose of the tool is to separate the task of formulating the logical argument from the task of writing that argument in paragraph form, allowing students to make progress on logical reasoning and argument construction regardless of their skill level with mathematical writing. I will share examples that highlight the tool's purpose and lessons learned from using the tool in upper-level mathematics courses for several years.
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Logic and formal reasoning are foundational to mathematics, yet students often struggle to connect symbolic notation with conceptual meaning. This presentation explores the pedagogical value of semantic tableaux as an integrated approach that reinforces truth tables, normal forms, and tautology analysis while deepening conceptual mastery of propositional logic. Semantic tableaux offer a visual and algorithmic method for decomposing well-formed formulas into subformulas and atoms. This tree-based representation complements traditional truth-table reasoning by applying the rules for logical connectives to determine whether a statement is satisfiable. By making the logical structure visually explicit through a branching tree, this method reveals the conditions for tautologies and contradictions while naturally bridging to normal forms (CNF and DNF). In doing so, it also reinforces case-based reasoning that underpins probability models and proofs by cases. Ultimately, semantic tableaux move students beyond procedural symbolic manipulation and table-reading toward a deeper understanding of logical form, laying a rigorous foundation for advanced reasoning in mathematics and computer science.
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At Belmont University, our Global Honors Program curriculum includes a Mathematical Inquiry Seminar as the only required mathematics course for students in this program. Since its inception, this course has included a unit on logic, argumentation, and critical thinking. We share recent course activities that attempt to bridge the gap between mathematical approaches to logic and real-world argument analysis. These include discussing authentic debates hosted by the nonpartisan, nonprofit organization Open to Debate and arguments created with the assistance of generative AI tools.
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A course to introduce mathematics majors to upper level mathematics courses requires the students to develop familiarity with and the ability to use many mathematical tools. Because of this, the course lends itself to the use of specifications grading. This talk will discuss one attempt to implement this method of grading and describes some of the benefits and the drawbacks found will teaching the courses with this approach.
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