The Science of Learning Mathematical Proofs

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.Contents: Preface to StudentsPreface to ProfessorsPedagogical Notes for ProfessorsBrain GrowthTeam BuildingSetting GoalsLogicProblem SolvingStudy TechniquesPre-proofsDirect Proofs (Even, Odd, & Divides)Direct Proofs (Rational, Prime, & Composite)Direct Proofs (Square Numbers & Absolute Value)Direct Proofs (GCD & Relatively Prime)Proof by Division into CasesProof by Division into Cases (Quotient Remainder Theorem)Forward-Backward ProofsProof by ContrapositionProof by ContradictionProof by InductionProof by Induction Part IICalculus ProofsMixed ReviewAppendices:100# Task Activity SheetAnswers for Hiking ActivityEscape RoomProof for Exercise 17.11Selected Proofs from all ChaptersProof MethodsProof TemplateHomework LogBibliographyIndex
Readership: Undergraduate in mathematics majors, for use in an undergraduate introduction to mathematical proofs course.Mathematics;Mathematical Proofs;Growth Mindset;Mathematical Logic;Problem Solving;Number Theory0Key Features: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 bestThis 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 homeworkThis book is meant to be used as a textbook as well as a journal that students can and should write in