Text: D'Angelo and West, Mathematical Thinking: Problem Solvings and Proofs, 2nd Edition, Prentice-Hall, 2000.

Math 247 introduces students to mathematics, improving their ability to absorb and communicate it via practice in problem-solving and writing proofs. Topics that prepare for upperclass courses include logical reasoning, induction, equivalence relations, elementary counting, and limits.

When taught from Mathematical Thinking: Problem-Solving and Proofs by D'Angelo and West (Second edition), the course samples discrete and continuous topics from Parts III-IV to accompany the basic material in Parts I-II. Chapters 1-4 and 13 are the most crucial but don't fill the course. One can also present Chapters 5-10 and 14-15. In these, the central material appears early, and some later material (including all optional sections) can be skipped. It is not necessary to present all the examples.

Part II is more fundamental than Chapters 9-10, which can be skipped if more time is needed in other chapters. Skip Chapters 11-12 (and 16-18) to sharpen the focus of the course. A worthy goal is to end with the Intermediate Value Theorem and its applications in Chapter 15.

The course demands effort from the students; they must read the text and work many exercises. En:courage them to read the sections on Approaches to Problems. Unlike most of their earlier math courses, this course requires independent thinking and understanding; students cannot expect all questions to be instances of procedures listed in class.

Part I §1 §2 §3 §4 Part II §5 §6 §7 §8 Part III §9 §10 §11 §12 Part IV §13 §14 §15 * *

Elementary Concepts Numbers, Sets and Functions Language and Proofs Induction Bijections and Cardinality Properties of Numbers Combinatorial Reasoning Divisibility Modular Arithmetic The Rational Numbers Discrete Mathematics Probability Two Principles of Counting Graph Theory (skip) Recurrence Relations (skip) Continuous Mathematics The Real Numbers Sequences and Series Continuous Functions Leeway and Exams Total

12 2.5 2.5 4 3 10 3 2 3 2 5 3 2 0 0 10 3 4 3 6 43

Notes:

Chapter 1: allude to but don't present "The Real Number System"; some other elementary definitions can also be left as background reading.

Chapter 2: treat lightly in class, emphasizing understanding rather than formality for quantifiers and conditionals - practice with logical statements comes throughout the course.

Chapter 3: 3.26 and 3.27 are not both necessary; 3.27 can be done simply for powers of 2.

Chapter 4: skip Schroeder-Bernstein.

Chapter 5: 5.30-31 optional.

Chapter 6: Dart Board Problem very appealing but optional; skip the section on polynomials.

Chapter 7: Newspaper Problem optional; skip "Congruence and Groups".

Chapter 8: "Pythagorean triples" is appealing but optional; omit "Further Properties".

Chapter 9: the ideas of conditional probability and expectation are the most important if the chapter is covered; "Multinomial Coefficients" optional.

Chapter 10: choose a few applications as time permits.

Chapter 13: cover completely.

Chapter 14: the proofs of convergence tests apply Cauchy sequences but can be treated lightly; Exercise 14.58 is a valuable addition.

Chapter 15: stress that the results on sequences imply the results on continuity; don't mention uniform continuity.

Chapter 16: if reached in honors sections, treat lightly; state definitions, assume basic properties, perhaps prove chain rule and Rolle/MVT, skip Newton's method and convexity, aim to convey the idea of a continuous nowhere-differentiable function.