The Science of Learning Mathematical Proofs: An Introductory Course

✍ Scribed by Elana Reiser

• Publisher: World Scientific Publishing
• Year: 2020
• Tongue: English
• Leaves: 243
• Category: Library

✦ Synopsis

College students struggle with the switch from thinking of mathematics as a calculation based subject to a problem solving based subject. This book describes how the introduction to proofs course can be taught in a way that gently introduces students to this new way of thinking. This introduction utilizes recent research in neuroscience regarding how the brain learns best. Rather than jumping right into proofs, students are first taught how to change their mindset about learning, how to persevere through difficult problems, how to work successfully in a group, and how to reflect on their learning. With these tools in place, students then learn logic and problem solving as a further foundation.

Next various proof techniques such as direct proofs, proof by contraposition, proof by contradiction, and mathematical induction are introduced. These proof techniques are introduced using the context of number theory. The last chapter uses Calculus as a way for students to apply the proof techniques they have learned.

✦ Table of Contents

Contents
Preface to Students
Preface to Professors
Chapter 0 Pedagogical Notes for Professors
0.1 Introduction
0.2 Homework
0.3 Chapters in book
0.4 Thinking classrooms
0.5 Classroom structures
0.6 Assessment
Chapter 1 Brain Growth
1.1 Mindsets
1.2 Struggling and making mistakes
1.3 Speed
1.4 What is math?
Chapter 1 Homework
Growth Mindset Pledge
Chapter 2 Team Building
2.1 History of cooperative learning
2.2 Components of cooperative learning
2.3 Group tasks
Chapter 3 Setting Goals
3.1 SMART goals
3.2 Motivation
Chapter 4 Logic
4.1 Statements
4.2 Number sets
4.3 Quantifiers
4.4 Negations
4.5 Logical arguments
Chapter 4 Homework
Chapter 5 Problem Solving
5.1 Metacognition
5.2 Creativity
5.3 Problem solving
5.4 Games
Chapter 5 Homework
Chapter 6 Study Techniques
6.1 Types of learning
6.2 Homework
Chapter 7 Pre-proofs
7.1 Math terms
7.2 Proof tips
Chapter 8 Direct Proofs (Even, Odd, & Divides)
8.1 Definitions
8.2 Proofs
Chapter 8 Homework
Chapter 9 Direct Proofs (Rational, Prime, & Composite)
9.1 Definitions
9.2 Proofs
Chapter 9 Homework
Chapter 10 Direct Proofs (Square Numbers & Absolute Value)
10.1 Definitions
10.2 Inequality properties
10.3 Proofs
Chapter 10 Homework
Chapter 11 Direct Proofs (GCD & Relatively Prime)
11.1 Definitions
11.2 Proofs
Chapter 11 Homework
Chapter 12 Proof by Division into Cases
12.1 Proof mistakes
12.2 Proof by division into cases
Chapter 12 Homework
Chapter 13 Proof by Division into Cases (Quotient Remainder Theorem)
13.1 Quotient remainder theorem
13.2 Proofs
Chapter 13 Homework
Chapter 14 Forward-Backward Proofs
14.1 Proofs
14.2 Concise proofs
Chapter 14 Homework
Chapter 15 Proof by Contraposition
15.1 Contrapositive review
15.2 Proof by contraposition
Chapter 15 Homework
Chapter 16 Proof by Contradiction
16.1 Negation review
16.2 Proof by contradiction
Chapter 16 Homework
Chapter 17 Proof by Induction
17.1 Motivation for induction
17.2 Induction definition
17.3 Proof by induction
Chapter 17 Homework
Chapter 18 Proof by Induction Part II
18.1 Helpful properties
18.2 Proof by induction with inequalities
Chapter 18 Homework
Chapter 19 Calculus Proofs
19.1 Limit proofs
19.2 Exercises
Chapter 20 Mixed Review
20.1 Review problems
20.2 Possible solutions
Appendix A 100# Task Activity Sheet
Appendix B Answers for Hiking Activity
Appendix C Escape Room
Appendix D Proof for Exercise 17.11
Appendix E Selected Proofs from all Chapters
Appendix F Proof Methods
Appendix G Proof Template
Appendix H Homework Log
Bibliography
Index