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

Reading, Writing, and Proving: A Closer Look at Mathematics (Undergraduate Texts in Mathematics)
Ulrich Daepp,Pamela Gorkin | 2003-08-07 00:00:00 | Springer | 408 | Mathematics
The reader of this book is probably about to teach or take a "first course in proof techniques." Students are taking this course because they like mathematics, and the authors hope to keep it that way. At this point, they have an intuitive sense of why things are true, but not the exposure to detailed and critical thinking necessary to survive in the mathematical world. The authors have written this book to bridge the gap. Often, students beginning this course have little training in rigorous mathematical reasoning; they need guidance. At the end, they are where they should be; on their own. The authors aim is to teach the students to read, write and do mathematics independently, and to do it with clarity, precision, and care.
Reviews
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.
Reviews
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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