Lessons Learned About Incorporating High-Leverage Teaching Practices in the Undergraduate Proof Classroom to Promote Authentic and Equitable Participation

Abstract

In recent years, professional organizations in the United States have suggested undergraduate mathematics shift away from pure lecture format. Transitioning to a student-centered class is a complex instructional undertaking especially in the proof-based context. In this paper, we share lessons learned from a design-based research project centering instructional elements as objects of design. We focus on how three high leverage teaching practices (HLTP; established in the K-12 literature) can be adapted to the proof context to promote student engagement in authentic proof activity with attention to issues of access and equity of participation. In general, we found that HLTPs translated well to the proof setting, but required increased attention to navigating between formal and informal mathematics, developing precision around mathematical objects, supporting competencies beyond formal proof construction, and structuring group work. We position this paper as complementary to existing research on instructional innovation by focusing not on task trajectories, but on concrete teaching practices that can support successful adaption of student-centered approaches.

Appendix

	Lesson 1: Structural Property Task	Lesson 2: Lagrange’s Theorem Task	Lesson 3: First Isomorphism Theorem Task
Primary Proof Activity	Proof Validating	Proof Constructing	Proof Comprehending
Proof Learning	Role of conclusion in proof framework	Using a diagram exploration to identify a key idea	Attending to global and local aspects when reading a proof
Outline of Lesson Structure	Opening Discussion • Public record of assumptions and conclusion • Discuss key definitions Discussion of Proof Approaches in Small Groups • Develop expertise around one of two student proofs • Explain proof to partner who is prompted to state one thing that makes sense and one question Public Discussion of Two Approaches • Presentation • Identifying similarities and differences (think-pair-share) • Public Record of similarities and differences Proof and Statement Analysis • Conjecture what assumptions are needed based on the existing proofs • Public Record of conjectured statements (with varying properties: 1–1 and onto) • Testing statements use proofs and examples to determine what properties are needed Counterexample to Identify the Necessity of Onto (Visual Representation) • Function diagram discussion showing the role of onto Summary and Conclusion • Finalizing of revised statement (onto, but not 1–1) • Discussion of patching the proof that did not use onto • Discussion of the role of conclusion in structuring proofs (proof framework)	Opening Discussion and Exploration • Exploring examples groups and the order of their subgroups to generate a conjecture (in small groups) • Creating a public record of conjectures from different groups • Connecting conjecture to formal Lagrange Statement • Formally defining divisibility and exploring the meaning of “multiplication” on boards and coming to class consensus (to anticipate proof structure) Creating Cosets in Small Groups • Each group works with a different group (and subgroup) to create cosets in a form that can be reasoned with diagrammatically Conjecture Discussion • Discussion of noticings and conjectures about the structure of the cosets to arrive at key lemmas for the proof of Lagrange’s Theorem Matching Class Lemmas to their Formalizations in Small Groups • A set of six formal statements to identify as the translation of an informal lemma or a tool to prove one of the lemmas Proving Lagrange’s Theorem • Small group and whole class discussion of structuring Lagrange’s Proof using the lemmas Summary and Conclusion • Discussing the role of the key idea from the coset diagram examples • Wrap-up on the implications of Lagrange’s Theorem	Opening Discussion • Discussion of key concepts and definitions in the theorem Small Group Exploration of Specific Examples • Each small group works at board space to connect the theorem to a specific example using a function diagram (and assigned roles) Class Discussion of Defining the Isomorphism Map • Identifying a map in symbolic form that will describe the input (cosets) and outputs (image of the coset representative) that is consistent across the examples Discussion (small group and whole class) of Proof Structure • Identifying what needs to be proven • Subdividing the proof to find the sections of what needs to be proven Making sense of a subsection of a proof • Each small group is responsible for one of four sections. Each member of a group has a question for leading discussion Class Presentations • A representative from each small group explains their section to the class Summary and Conclusion • Summarizing the proof at a high level • Discussing the practice of proof comprehension