College students find it difficult to make the transition from viewing mathematics as a subject based on calculations to one based on problem-solving. This book outlines a method for teaching the introduction to proofs course that gradually introduces students to this new style of thinking. Recent discoveries in neuroscience about how the brain learns best are used in this introduction. Students are taught how to adjust their mentality toward learning, persevere through challenging issues, work effectively in a group, and reflect on their learning before moving on to proof. Students then study reasoning and problem-solving as an additional foundation using these tools. The introduction of numerous proof methods follows, including mathematical induction, proof by contradiction, proof by contraposition, and direct proofs. The context of number theory is used to introduce these proof methods. In the last chapter, students use the proof strategies they have learned using Calculus.