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

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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 P �lya'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.

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

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.

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