A Transition to Mathematics with Proofs書封皮有一小條輕微刮痕

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Developed for the "transition" course for mathematics majors moving beyond the primarily procedural methods of their calculus courses toward a more abstract and conceptual environment found in more advanced courses, A Transition to Mathematics with Proofs emphasizes mathematical rigor and helps students learn how to develop and write mathematical proofs. The author takes great care to develop a text that is accessible and readable for students at all levels. It addresses standard topics such as set theory, number system, logic, relations, functions, and induction in at a pace appropriate for a wide range of readers. Throughout early chapters students gradually become aware of the need for rigor, proof, and precision, and mathematical ideas are motivated through examples. Proof techniques and strategies are thoroughly discussed and the underlying logic behind them is made transparent. Each chapter section begins with a set of guided reading questions intended to help students to identify the most significant points made within the section. Practice problems are embedded within chapters so that students can actively work with a key idea that has just been introduced. Each chapter also includes a collection of problems, ranging in level of difficulty, which are perfect for in-class discussion or homework assignments.

The Science of Learning Mathematical Proofs

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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