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. Key Features: o This book introduces a new way of structuring the typical "intro to proofs" course by utilizing current neuroscience research on how the brain works and the psychology of how people learn best o This book integrates the ideas presented in it as best ways to learn. It doesn’t just highlight the concepts; they will also be put into practice utilizing exercises and homework o This book is meant to be used as a textbook as well as a journal that students can and should write in

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