Resources and Pedagogical Techniques for Enhancing the Teaching of Proof-Writing Across the Curriculum

Description

The challenges students can face in the transition from computational mathematics to proof-writing lead many instructors to seek pedagogical techniques that extend beyond standard lecture. This Notes volume unites a wide variety of such techniques, along with resources to aid in incorporating them. Written with the busy instructor in mind, the articles present practical methods in a "nuts-and-bolts" fashion, for easy access to the details of each technique. Courses throughout the entire undergraduate mathematics curriculum are represented: this includes a variety of proof-based courses and also non-traditional ones such as calculus and mathematics for liberal arts Beyond Lecture should appeal to both novice and seasoned instructors, while also hopefully providing a springboard for experimentation in readers' own classrooms. Introduction 1 Jump-Starters and Other Activities Introductory Discussions of “What Constitutes a Proof?” 30 “Classical” Statements and What We’ve Learned Using a Proof-Scrambling Activity to Enhance Student Understanding of Mathematical Proof Dependency Trees Counterexamples and Correspondence: Two Techniques for Sense-Making Proof-Evaluation as a Step towards Proof Authorship Teaching with Two-Column 2 Small-Group Activities and Presentations Proving in the Right Circles Group Examinations in Introduction-to-Proof Courses Improving Proof-Writing Skills Through Weekly Student Presentations Individual Presentations of Group-Written Proofs 3 Whole Class Activities Use of Video Case-Studies in Learning to Prove Pass-the-Proof Engaging Students in Interactive Peer-Review Proof-Writing Activities Using Proof-Writing through Electronic Document Editing A Humanistic Reading Component for an Introduction-to-Proofs Course Teaching Proof-Writing by Public Grading 4 Portfolios, Journals and Peer Review Using Proof Portfolios in a Bridge Course Becoming Successful Proof-Writers Through Peer Review, Journals, and Portfolios Running a Class Journal Peer Grading of Exams 5 Inquiry-Based Learning and Flipped Classrooms An Inquiry-Based Introduction to Mathematical Proof Within epsilon of Independence Reading, ‘Riting, and Reals Introduction to Proofs Over Easy 6 Ideas Borrowed from English and Other Composition Courses Elements of Style for Proofs Learning Proofs via Composition Instruction Techniques Abstract Algebra, Part of Writing Across the Curriculum 7 Teaching Proofs in Non-Traditional Courses Discovering the Art of Mathematics Teaching Writing and Proof-Writing Together Teaching Reasoning Skills in Calculus I and Below Introducing Proofs to Calculus Students 8 Long-Term Activities Mastery Problems Promote Polished Proofs in Abstract Algebra From Formal to Expository Growing Proof-Writing Skills Throughout the Undergraduate (Majors’) Curriculum