Mathematical Thinking: Problem-Solving and Proofs, 2/e
John P. D'Angelo, University of Illinois
Douglas B. West, University of Illinois
Coming December, 1999 by Prentice Hall Engineering/Science/Mathematics
Copyright 2000, 480 pp.
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Transition to Advanced Mathematics-Mathematics
For one/two-term courses in Transition to Advanced Mathematics
or Introduction to Proofs. Also suitable for courses in Analysis or
This text is designed to prepare students thoroughly
in the logical thinking skills necessary to understand and communicate
fundamental ideas and proofs in mathematicsskills vital for success
throughout the upperclass mathematics curriculum. The text offers
both discrete and continuous mathematics, allowing instructors to
emphasize one or to present the fundamentals of both. It begins by
discussing mathematical language and proof techniques (including induction),
applies them to easily-understood questions in elementary number theory
and counting, and then develops additional techniques of proof via
important topics in discrete and continuous mathematics. The stimulating
exercises are acclaimed for their exceptional quality.
NEWA clearly outlined transition courseRearranges
material to facilitate a clearly defined and more accessible transition
course using Chs. 1-5, initial parts of Chs. 6,8 and Chs. 13-14.
NEWApproaches to ProblemsIn
selected chapters. Summarizes key points and presents problem-solving
strategies relevant to exercises.
- By narrowing the focus, makes it easy to present a course
with rich content to beginning students in a transition course without
NEWA clearly outlined analysis courseNow
contains an excellent course in analysis using Part I as background,
touching briefly on Ch. 8, and covering Part IV in depth.
- In the transition course, helps students organize their
understanding of the chapter, avoid typical pitfalls, and learn ways
to approach problems of moderate difficulty.
NEWExpanded and improved selection of
exercisesNew, easier exercises check mastery of concepts;
some difficult exercises are clarified.
- Provides review reading on proof methods in Part I while
being as thorough and accessible as introductory texts in Part IV.
NEWDefinitions in boldTerms being
defined are in bold type with almost all definitions in numbered terms.
- Enlarged selection of easier exercises provides greater
encouragement for beginning students; clarifications make other exercises
NEWReorganization of materialProvides
smoother development and clearer focus on essential material.
- Makes it easier for students to follow the mathematical
development and how to know what assumption can be used when working
NEWMore accessible presentationSome
terse discussions expanded, examples added, and more computations
placed in displays.
- Makes definitions easier for students to find.
Engaging examplesInteresting applications introduce
and motivate the underlying mathematics.
- Makes material easier for students to comprehend and
conveys a greater sense of progress by making pages less dense.
Logical OrganizationIntroduces concepts as needed
with each item carefully selected. Distinguishes between Lemmas/Theorems
(the mathematical development) and Examples/Solutions (the illustration
or application of mathematical results).
- Engage student interest and commitment from the beginning
and keep the material lively.
Flexibility in course designMathematical background
in Part I can be treated quickly with strong students or in detail
for beginners. Rich variety of subsequent topics permits a broad introduction
to mathematics or a focus on discrete mathematics (Part III) or Analysis
- Enables students to find fundamental results easily and
learn more efficiently. Helps students understand the difference between
mathematical tools and their application in problem-solving.
Emphasis on understanding rather than manipulationStresses
full comprehension rather than rote symbolic manipulation for mastery
of proof techniques and mathematical ideas.
- Enables instructors to design courses appropriate to
students' abilities; a two-semester treatment offers a thorough introduction
Emphasis on clear communicationDiscusses the
use of language and requires written arguments in many exercises.
- Helps students develop critical thinking skills and appreciation
of coherent arguments, preparing them both for later courses in mathematics
and for problem-solving situations outside school.
Richness of TopicsAfter the elementary material,
provides a wealth of intellectual highs from diophantine
equations to Fermat's Little Theorem to Pythagorean triples, Bertrand's
Ballot Problem, the pigeonhole principle, the Euler totient, Hall's
Marriage Condition, the theory of calculus, interchange of limiting
operations, series of functions, the existence of continuous nowhere
differentiable functions, complex numbers, and the Fundamental Theorem
- Helps students develop or remediate their written communication
skills, both in forming English sentences and in presenting coherent
arguments. Supports the efforts of instructors to apply such skills
in a mathematical context.
Superior exercise setsOffers over 850 exercises
ranging from relatively straightforward applications of ideas in the
text to subtle problems requiring some ingenuity.
- Permits the inclusion of topics to enrich the mathematical
experience. Depending on the talents of students, these can be presented
in class or left for outside reading.
Hints for selected exercisesProvides immediate
hints for some exercises and hints for others in an appendix.
- Gives students the flexibility to learn at their own
pace; weaker students have more opportunity to be successful, and
stronger students have more opportunity to be stimulated.
Gradation of exercisesDistinguishes easier exercises
by (), harder by (+), and particularly valuable or instructive
exercises by (!).
- Helps students at all levels to understand the ideas
of the course and to broaden their mathematical interests.
Instructor's ManualContains solutions to exercises
and pedagogical suggestions. Only available directly through editor.
- Aids instructor in selecting appropriate exercises and
students in practicing for tests.
I. ELEMENTARY CONCEPTS.
1. Numbers, Sets and Functions.
2. Language and Proofs.
3. Properties of Functions.
II. PROPERTIES OF NUMBERS.
5. Cardinality and Counting.
III. DISCRETE MATHEMATICS.
7. Modular Arithmetic.
8. The Rational Numbers.
9. Combinatorial Reasoning.
IV. CONTINUOUS MATHEMATICS.
10. Two Principles of Counting.
11. Graph Theory.
12. Recurrence Relations.
13. The Real Numbers.
14. Sequences and Series.
18. The Complex Numbers.