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. Request Inspection Copy Contents: Preface to Students Preface to Professors Pedagogical Notes for Professors Brain Growth Team Building Setting Goals Logic Problem Solving Study Techniques Pre-proofs Direct Proofs (Even, Odd, & Divides) Direct Proofs (Rational, Prime, & Composite) Direct Proofs (Square Numbers & Absolute Value) Direct Proofs (GCD & Relatively Prime) Proof by Division into Cases Proof by Division into Cases (Quotient Remainder Theorem) Forward-Backward Proofs Proof by Contraposition Proof by Contradiction Proof by Induction Proof by Induction Part II Calculus Proofs Mixed Review Appendices: 100# Task Activity Sheet Answers for Hiking Activity Escape Room Proof for Exercise 17.11 Selected Proofs from all Chapters Proof Methods Proof Template Homework Log Bibliography Index