Open-source course materials for introduction to proof and abstract algebra

The speaker has written two open-source inquiry-based learning textbooks. One of the books is intended to be used for an introduction to proof course while the second book is designed for an undergraduate abstract algebra course. Both books are available as free downloads via GitHub. The initial development of the abstract algebra book was funded by a Small Grant from the Academy of Inquiry-Based Learning. In addition to saving students the cost of a textbook, the open-source model allows faculty members to adapt the books to their specific purposes. To our knowledge, roughly 30 different instructors have utilized at least one of the textbooks. In this talk, we will take a tour of both books, discuss their pedagogical approach, and communicate some of the design decisions that were made when creating the books.

This talk was given on January 12, 2018 in the "The Advancement of Open Educational Resources" contributed paper session at the 2018 Joint Mathematics Meetings in San Diego, CA.

Transcript

1. 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
   

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2. 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.
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3. Inquiry-Based Learning
   

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4. 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.
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5. 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.
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6. An Introduction to Proof via
   Inquiry-Based Learning
   

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7. 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)
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8. 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
   

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9. 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
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10. 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
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11. An Inquiry-Based Approach to
    Abstract Algebra
    

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12. 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.
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13. 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
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14. 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
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15. 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
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16. An Inquiry-Based Approach to Abstract Algebra
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17. An Inquiry-Based Approach to Abstract Algebra
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18. An Inquiry-Based Approach to Abstract Algebra
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19. An Inquiry-Based Approach to Abstract Algebra
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20. An Inquiry-Based Approach to Abstract Algebra
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21. Wish List
    

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22. 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.
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23. Licensing
    

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24. 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.
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25. GitHub
    

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26. GitHub
    Fork, pull requests, issues…
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