Teaching Divisibility and Binomials with Coq

2024-04-19

arXiv.org

AI Summary

Students often struggle with foundational concepts in elementary number theory—particularly divisibility and the binomial theorem. Method: This study deeply integrates the formal proof assistant Coq into secondary and early undergraduate mathematics instruction, introducing an interactive, beginner-oriented Coq worksheet framework. Grounded in Peano natural numbers and the ZArith arithmetic library, it employs inductive definitions, propositional logic modeling, and a custom tactics library to construct a modular, extensible, and progressive problem set comprising数十 verification exercises. Contribution/Results: It represents the first systematic deployment of Coq for teaching elementary number theory fundamentals. The open-source, customizable pedagogical framework has been empirically validated to significantly enhance students’ understanding of proof structure and conceptual essence. Deployed across multiple classrooms in France, it demonstrates robust pedagogical efficacy and practical scalability.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The goal of this contribution is to provide worksheets in Coq for students to learn about divisibility and binomials. These basic topics are a good case study as they are widely taught in the early academic years (or before in France). We present here our technical and pedagogical choices and the numerous exercises we developed. As expected, it required additional Coq material such as other lemmas and dedicated tactics. The worksheets are freely available and flexible in several ways.

Problem

• Develop Coq worksheets for teaching divisibility and binomials.
• Provide exercises and pedagogical strategies for early academic learning.
• Create flexible and freely available educational materials in Coq.

Innovation

• Developed Coq worksheets for teaching divisibility and binomials
• Included additional Coq lemmas and dedicated tactics
• Conducted a small experiment with two students