Reading, Writing, and Proving: A Closer Look at Mathematics (Undergraduate Texts in Mathematics) [Hardcover]

Ulrich Daepp (Author), Pamela Gorkin (Author)

Book Description

ISBN-10: 0387008349 | ISBN-13: 978-0387008349 | Publication Date: August 7, 2003 | Edition: 1

This book, based on Pólya's method of problem solving, aids students in their transition to higher-level mathematics. It begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics and ends by providing projects for independent study. Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them.

Editorial Reviews

Review

From the reviews:

U. Daepp and P. Gorkin

Reading, Writing, and Proving

A Closer Look at Mathematics

"Aids students in their transition from calculus (or precalculus) to higher-level mathematics . . . The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations."

—L'ENSEIGNEMENT MATHEMATIQUE

"Daepp and Gorkin (both, Bucknell Univ.) offer another in the growing genre of books designed to teach mathematics students the rigor required to write valid proofs … . The book is well written and should be easy for a first- or second- year college mathematics student to read. There are many ‘tips’ offered throughout, along with many examples and exercises … . A book worthy of serious consideration for courses whose goal is to prepare students for upper-division mathematics courses. Summing Up: Highly recommended." (J.R. Burke, CHOICE, 2003)

"The book Reading, Writing, and Proving … provides a fresh, interesting, and readable approach to the often-dreaded ‘Introduction to Proof’ class. … RWP contains more than enough material for a one-semester course … . I was charmed by this book and found it quite enticing. … My students found the overall style, the abundance of solved exercises, and the wealth of additional historical information and advice in the book exceptionally useful. … well-conceived, solidly executed, and very useful textbook." (Maria G. Fung, MAA online, December, 2004)

"The book is intended for undergraduate students beginning their mathematical career or attending their first course in calculus. … Throughout the book … students are encouraged to 1) learn to understand the problem, 2) devise a plan to solve the problem, 3) carry out that plan, and 4) look back and check what the results told them. This concept is very valuable. … The book is written in an informal way, which will please the beginner and not offend the more experienced reader." (EMS Newsletter, December, 2005)

From the Back Cover

 This book, which is based on Pólya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics. It ends by providing projects for independent study.

Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.

Product Details

• Hardcover: 408 pages
• Publisher: Springer; 1 edition (August 7, 2003)
• Language: English
• ISBN-10: 0387008349
• ISBN-13: 978-0387008349
• Product Dimensions: 9.2 x 6.4 x 0.9 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #791,948 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

15 of 15 people found the following review helpful:

4.0 out of 5 stars A good text for an introduction to proofs course, March 26, 2005

By 

Jerry D. Rosen (Los Angeles, CA United States) - See all my reviews
(REAL NAME)   

This review is from: Reading, Writing, and Proving: A Closer Look at Mathematics (Undergraduate Texts in Mathematics) (Hardcover)

I have used the Daepp, Gorkin text twice, for an introduction to proofs type of course. This course is usually taken by Math and Computer Science majors after Calculus and either with or after a course in Linear Algebra. This type of course was not in existence when I was a student, in the 70's. In those days, there was some proofs in Calculus (certainly some delta-epsilon type arguments were given) and Linear Algebra was much more proof oriented. Hence, most math majors picked up the ability to read and learn abstract mathematics during the first two years. These days, Linear Algebra has become a course in row-reducing matrices and very little abstraction takes place. Hence there is a real need for a course (and texts) to pave the way for courses in Analysis and Abstract Algebra. The Daepp, Gorkin text compares favorably to all similar texts I have looked at and it is priced reasonably. I passed on another text that I liked because it was $125, which is ridiculous for studenst who are not wealthy. On the plus side, this text covers all the material you would need in such a course and, in fact, there are several avenues open to the instructor of a one semester course. Besides the usual material on sets and mappings, there are chapters on cardinality issues, intoductory analysis ideas and slightly more advanced topics in number theory. The chapters are short and "digestable." There are some possible independent research topics at the end of the text. On the negative side, the examples given in the text are mostly all drawn from the standard number systems. This makes it harder to motivate basic concepts of sets and mappings. Why not give some examples from sets of mappings (e.g. the composite of two odd functions is odd, does a better job of teaching about composition than just composing two standard function), 2 by 2 matrices, and some examples from Calculus (e.g. the derivative, viewed as a function from degree n polynomials to degree n-1 polynomials is a non one-to-one, onto map)? The Division Algorithm is not mentioned until page 315 and it is not proved and there is no discussion on the Fundamental Theorem of Arithmetic. In fact, the number theory is a bit strange. While there are proofs of Fermat and Euler's Theorems, they omit more elementary number theory needed to get to these results. The Binomial Theorem is left as an exercise and no applications are given. As an application, I show my classes tha the sequence (1+1/n)^n is increasing and bounded above. Also, the exercises are a little strange. Too many are very easy and there are not enough basic practice types or challenge types. I would eliminate the logic chapters in these texts. They really don't help with the math and their elimination would create more room for math topics.
I would recommend this book, but warn the reader and/or teacher, that some supplementing would be needed.

12 of 12 people found the following review helpful:

4.0 out of 5 stars I was skeptical, but this is a n unsually good mathmatics textbook, March 4, 2006

By 

Elfan (Acton, MA United States) - See all my reviews

This review is from: Reading, Writing, and Proving: A Closer Look at Mathematics (Undergraduate Texts in Mathematics) (Hardcover)

First of all I am a student of Computer Science at a US university. This book was used for an intermediate level mathematics course intended to provide a bridge between Calculus and higher level courses. For that purpose this book was well suited.

Like many (probably most) students my high school and early college mathematics courses taught me a great deal about using a graphing calculator to guess-and-check but much less about the fundamentals of mathematics. The writing is generally clear and concise and avoids leaving "obvious" (read: very difficult) theorems as exercises to the reader. Like the reviewer Jerry D. Rosen I think some of the exercises are "odd", for lack of a better term. I think the authors tried to avoid the "question 1, parts a-f [trivial exercise]" format of many mathematics textbooks but the mix of questions did not always come out well. Also note that there are no answers provided so this book is not well suited to self-study.

The greatest virtue of this book is that that one gets the sense of *liking* mathematics, and that is contagious. I found this book well suited to the course I took it for and it has proven even more valuable for getting through Linear Algebra and Discrete math courses with truly horrible texts.