2. Teaching
3. MAT431
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An Inquiry-Based Approach to Real Analysis

We will not be using a traditional textbook this semester, but rather a problem sequence adopted for an inquiry-based learning (IBL) approach to real analysis. The problem sequence that we are using is an adaptation of the analysis notes by Karl-Dieter Crisman, which are a modified version of notes by W. Ted Mahavier. Both authors have been gracious enough to grant me access to the source of these notes, so that we can modify and tweak for our needs. You can find a link to our textbook below. The textbook is subject to change and improvements.

Table of Contents

• Preface
• Chapter 1: Introduction
  • 1.1 What is This Course All About?
  • 1.2 An Inquiry-Based Approach
  • 1.3 Structure of the Notes
  • 1.4 Rules of the Game
  • 1.5 Rights of the Learner
  • 1.6 Some Minimal Guidance
• Chapter 2: Preliminaries
  • 2.1 Sets
  • 2.2 Induction and the Well-Ordering Principle
  • 2.3 Functions
• Chapter 3: The Real Numbers
  • 3.1 The Field Axioms
  • 3.2 The Order Axioms
  • 3.3 Absolute Value and the Triangle Inequality
  • 3.4 Suprema, Infima, and the Completeness Axiom
  • 3.5 The Archimedean Property
• Chapter 4: Standard Topology of the Real Line
  • 4.1 Open Sets
  • 4.2 Accumulation Points and Closed Sets
  • 4.3 Compact and Connected Sets
• Chapter 5: Sequences
  • 5.1 Introduction to Sequences
  • 5.2 Properties of Convergent Sequences
  • 5.3 Monotone Convergence Theorem
  • 5.4 Subsequences and the Bolzano–Weierstrass Theorem
• Chapter 6: Continuity
  • 6.1 Introduction to Continuity
  • 6.2 Additional Characterizations of Continuity
  • 6.3 Extreme Value Theorem
  • 6.4 Intermediate Value Theorem
  • 6.5 Uniform Continuity
• Chapter 7: Limits
  • 7.1 Introduction to Limits
  • 7.2 Limit Laws
• Chapter 8: Differentiation
  • 8.1 Introduction to Differentiation
  • 8.2 Derivative Rules
  • 8.3 Mean Value Theorem
• Chapter 9: Integration
  • 9.1 Introduction to Integration
  • 9.2 Properties of Integrals
  • 9.3 Fundamental Theorem of Calculus
• Appendix A: Elements of Style for Proofs
• Appendix B: Fancy Mathematical Terms
• Appendix C: Definitions in Mathematics

I will not be covering every detail of the notes and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the notes whenever necessary by asking questions. Here’s one of my favorite quotes about reading mathematics.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

Exams

Below are links to the take-home portions of each exam. If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam.

• Exam 1 (Part 2) (PDF) (Due Friday, October 20)
• Exam 2 (Part 2) (PDF) (Due Wednesday, November 22)
• Final Exam (Part 2) (PDF) (Due Thursday, December 14)

Miscellaneous Materials

• Pictures of board work that we ran out of time to discuss. Note: You should not assume that the solutions/proofs posted below are valid or sufficient.
  • Problem 3.55
  • Problem 3.56
• Setting the Stage (PDF)
• Living Proof
• The Secret to Raising Smart Kids | Carol Dweck (article on Scientific American)
• 7 Things Growth Mindset is Not (article on Turn Around for Children)
• Videos on Growth Mindset and Productive Failure
  • Grit: the power of passion and perseverance | Angela Lee Duckworth (6:12 min)
  • How to Become a Better Learner | Learner Lab (13:30 min)
  • Learning Like a Jungle Tiger | Trevor Ragan (5:23 min, similar to previous video)
  • Productive Failure/Struggle Skateboard Video 1 (1:46 min)
  • Productive Failure/Struggle Skateboard Video 2 (0:40 min)
  • Productive Failure/Struggle BMX Video (1:39 min)
  • Michael Jordan Failure Commercial (0:32 min)
  • Mindsets: Fixed Versus Growth (2:19 min)
  • Growth Mindset Animation (3:50 min)
  • KhanAcademy interview with Carol Dweck about growth mindset (3:06 min)
  • Make Mistakes | Michael Starbird (2:11 min)
  • Ira Glass on the Creative Process (1:54 min)
  • Productive Failure | Manu Kapur | TEDxLugano (17:28 min)
• The State of Being Stuck | Ben Orlin (blog post on Math with Bad Drawings)
• The Chinese Farmer | Alan Watts (2:26 min)
• Do Schools Kill Creativity? | Ken Robinson (19:22 min)
• The Learner Lab Podcast
• The Subtle Art of the Mathematical Conjecture (article on Quanta Magazine)
• Lessons from My Math Degree That Have Nothing to Do with Math (6 min read on Medium)
• Is Mathematics Invented or Discovered? | Roger Penrose (4:14 min)
• What is Mathematics? The Most Misunderstood Subject (short article by Dr. Robert H. Lewis, Professor of Mathematics, Fordham University)