Open-source course materials for introduction to proof and abstract algebra The Advancement of Open Educational Resources Dana C. Ernst Northern Arizona University January 12, 2018

Overview Two open-source textbooks • An Introduction to Proof via Inquiry-Based Learning https://github.com/dcernst/IBL-IntroToProof • “An Inquiry-Based Approach to Abstract Algebra” https://github.com/dcernst/IBL-AbstractAlgebra Comments • As titles suggest, both books are designed to be used with an inquiry-based learning (IBL) approach. • Source written in LaTeX. • Source housed on GitHub, which is a web-based Git version control repository hosting service. 1

Inquiry-Based Learning

What is IBL? The Three Pillars 1. Deep engagement in rich mathematics. 2. Opportunities to collaborate. 3. Instructor use of student thinking. Laursen et al. Rasmussen et al. Common Vehicles to IBL 1. Student presentations. 2. Small group work. This is not an “either-or” choice. Most IBL instructors implement some combination. 2

What do IBL course materials look like? • Unlike traditional textbook, exercises and problems are interspersed among the prose. • Students are given tasks requiring them to solve problems, conjecture, experiment, explore, and create as they read. • These sense-making tasks are meant to guide students through a process of mathematical discovery. • Students are responsible for proving the key theorems. • It is not the intention that most students will independently succeed in accomplishing most of the IBL tasks on 1st attempt. • Instead, students collaborate with peers and instructor to iteratively hone in on rigorous arguments. 3

An Introduction to Proof via Inquiry-Based Learning

An Introduction to Proof via Inquiry-Based Learning • IBL task sequence for an introduction to proof course. • 1st half is an adaptation of notes written by Stan Yoshinobu (Cal Poly) and Matthew Jones (CSU, Dominguez Hills). • ≈ 25 confirmed instructors have used or about to use book. • Current version is 78-page PDF consisting of 8 chapters and 3 appendices. • Additional content & suggestions for improvement from: • Paul Ellis (Manhattanville College) • Jason Grout (Bloomberg, L.P.) • Anders Hendrickson (St. Norbert College) • Rebecca Jayne (Hampden-Sydney College) • Jane Long (Stephen F. Austin State University) • Kyle Petersen and Bridget Tenner (DePaul University) • Dave Richeson (Dickinson College) 4

An Introduction to Proof via Inquiry-Based Learning Chapter 1: Introduction 1.1 What is This Course All About? 1.2 An Inquiry-Based Approach 1.3 Your Toolbox, Questions, and Observations 1.4 Rules of the Game 1.5 Structure of the Notes 1.6 Some Minimal Guidance Chapter 2: Mathematics and Logic 2.1 A Taste of Number Theory 2.2 Introduction to Logic 2.3 Negating Implications and Proof by Contradiction 2.4 Introduction to Quantification 2.5 More About Quantification Chapter 3: Set Theory and Topology 3.1 Sets 3.2 Power Sets and Paradoxes 3.3 Indexing Sets 3.4 Topology of R 5

An Introduction to Proof via Inquiry-Based Learning Chapter 4: Three Famous Theorems 4.1 The Fundamental Theorem of Arithmetic 4.2 The Irrationality of √ 2 4.3 The Infinitude of Primes Chapter 5: Induction 5.1 Introduction to Induction 5.2 More on Induction 5.3 Complete Induction Chapter 6: Relations 6.1 Relations 6.2 Equivalence Relations 6.3 Partitions Chapter 7: Functions 7.1 Introduction to Functions 7.2 Compositions and Inverses 6

An Introduction to Proof via Inquiry-Based Learning Chapter 8: Cardinality 8.1 Introduction to Cardinality 8.2 Finite Sets 8.3 Infinite Sets 8.4 Countable Sets 8.5 Uncountable Sets Appendix A: Elements of Style for Proofs Appendix B: Fancy Mathematical Terms Appendix C: Definitions in Mathematics 7

An Inquiry-Based Approach to Abstract Algebra

An Inquiry-Based Approach to Abstract Algebra • IBL task sequence for a one-semester undergraduate abstract algebra course. • Groups (up to 1st Isomorphism Theorem) then rings (up to quotients by maximal and prime ideals). • Emphasizes visualization and inspired by Visual Group Theory by Nathan Carter (Bentley University). • Meant to be playful with a focus on developing intuition. • Initial development funded by grant from the Academy of Inquiry-Based Learning. • ≈ 10 confirmed instructors have used or about to use book. • Battle-tested Fall 2017 version is 125-page PDF with lots of figures. • Currently rewriting first few chapters to speed up intuition-building phase. 8

An Inquiry-Based Approach to Abstract Algebra Chapter 1: Introduction 1.1 What is Abstract Algebra? 1.2 An Inquiry-Based Approach 1.3 Rules of the Game 1.4 Structure of the Notes 1.5 Some Minimal Guidance Chapter 2: An Intuitive Approach to Groups Chapter 3: Cayley Diagrams Chapter 4: An Introduction to Subgroups and Isomorphisms 4.1 Subgroups 4.2 Isomorphisms 9

An Inquiry-Based Approach to Abstract Algebra Chapter 5: A Formal Approach to Groups 5.1 Binary Operations 5.2 Groups 5.3 Group Tables 5.4 Revisiting Cayley Diagrams & Our Original Definition of a Group 5.5 Revisiting Subgroups 5.6 Revisiting Isomorphisms Chapter 6: Families of Groups 6.1 Cyclic Groups 6.2 Dihedral Groups 6.3 Symmetric Groups 6.4 Alternating Groups Chapter 7: Cosets, Lagrange’s Theorem, & Normal Subgroups 7.1 Cosets 7.2 Lagrange’s Theorem 7.3 Normal Subgroups 10

An Inquiry-Based Approach to Abstract Algebra Chapter 8: Products and Quotients of Groups 8.1 Products of Groups 8.2 Quotients of Groups Chapter 9: Homomorphisms and the Isomorphism Theorems 9.1 Homomorphisms 9.2 The Isomorphism Theorems Chapter 10: An Introduction to Rings 10.1 Definitions and Examples 10.2 Ring Homomorphisms 10.3 Ideals and Quotient Rings 10.4 Maximal and Prime Ideals Appendix A: Prerequisites Appendix B: Elements of Style for Proofs Appendix C: Fancy Mathematical Terms Appendix D: Definitions in Mathematics 11

An Inquiry-Based Approach to Abstract Algebra 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 12

An Inquiry-Based Approach to Abstract Algebra 1 i k j −1 −i −k −j e s Q8 S2 13

An Inquiry-Based Approach to Abstract Algebra ∗ e s sr r2 rs r e e s sr r2 rs r s s e r rs r2 sr sr sr r2 e s r rs r2 r2 sr rs r s e rs rs r r2 sr e s r r rs s e sr r2 14

An Inquiry-Based Approach to Abstract Algebra 1 i k j −1 −i −k −j K iK kK jK 15

An Inquiry-Based Approach to Abstract Algebra g2 H g3 H g1 H • • • • • • • • −→ g1 H g2 H g3 H g2 H g1 H • • • • • • • • −→ g1 H g2 H 16

Wish List

Wish List Introduction to Proof • Sections on graph theory (maybe to replace baby number theory) and Schröder–Bernstein. • Discussions about Axiom of Choice and continuum hypothesis. Abstract Algebra • Revamp early chapters to speed up intuition-building phase. • Incorporate GAP or Sage exercises. Both • Intersperse more historical tidbits. • Retool source into PreTeXt or R Markdown/Bookdown or Ximera. • Deploy as HTML, ePub, and PDF. 17

Licensing

Licensing Both books are licensed under a Creative Commons Attribution-ShareAlike 4.0 license. You are free to: • Share: copy, distribute, and transmit the work, • Remix: adapt the work Under the following conditions: • Attribution: You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). • Share Alike: If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one. 18

GitHub

GitHub Fork, pull requests, issues… 19