Customer Reviews

Number Theory Through Inquiry (Maa Textbooks) (Mathematical Association of America Textbooks)

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If you were to use this book as a text in a number theory course, you first must have made the decision to teach it in a nonstandard manner. The approach used in the presentation of number theory is not the traditional listing of the fundamental theorems with their proofs. Concepts are stated as theorems but in no case is a proof offered.
There are several groups of specific exercises such as

Illustrate the division algorithm m = nq + r for m = 25, n = 7; m = 277, n = 4; m = 33, n = 22; m = 33, n = 45.

Questions such as

Do every two integers have at least one common divisor?
What other numbers can you show to be irrational? Make and prove the most general conjecture you can.
Which natural numbers can be written as the sum of two squares of natural numbers? State and prove the most general theorem possible about which natural numbers can be written as the sum of two squares of natural numbers, and prove it.

At several points in the text, there are exercises called "Blank paper exercises" which have the following structure.

After not looking at the material in this chapter for a day or two, take a blank piece of paper and outline the development of that material in as much detail as you can without referring to the text or to notes. Places where you get stuck or can't remember highlight areas that may call for further study.

The coverage is generally what is found in an introductory course in number theory. However, the lack of proofs means that either the students must derive them on their own, look them up in another reference or have the instructor provide them. While this does not preclude the use of this book, it will require extra effort on the part of the student and/or the instructor.

Published in Journal of Recreational Mathematics, reprinted with permission

I used this book to teach a semester course to talented high school students all of whom had at least two semesters of calculus before entering the course, so they were approximately at the college sophomore level. The students really learned how to prove things in a way that a standard approach did not. Because the students had to prove all the theorems, we covered less material than a standard lecture approach. My purpose, however, was not a huge amount of material, but rather to get the students to be able to write proofs, read proofs, and be critical of what they wrote themselves and what others wrote. It was a great experience for me and for them. At the end of the semester the students spontaneously broke into applause for the course. There were a few times when I helped them do proofs or gave some of the proofs as extra credit (if they were not central to the development of the material). I collected homework for almost every class and read it sometimes grading sometimes commenting. The book is well organized and has standard topics for such a course. I added the explicit statement of the division algorithm at the beginning of the course which helped in several instances later. I am using this book and this approach again.

The information in this book was clear and very helpful for my coursework. Expensive, but useful!

I had a teacher use this book to teach (well she really didn't give lectures, everything was left to the students) for an introductory number theory course.

This textbook gives a cursory, ambiguous review of number theory. The instructor used the "Moore Method" a.k.a the "let the students give your lectures for you" method. Therefore, the students trying to learn the material cannot depend on the veracity of the information the other students give in the lecture (yes you can do proofs, but of course you can do proofs INCORRECTLY). Therefore the only information that the student can depend on is feedback (if any) given by the instructure, which using the "Moore Method" is done through returned assignments. So, if the corrections on the assignments are done INCORRECTLY, through error or MALICE, then the student has NO WAY to verify the results or the methods used to obtain them. This is where a proper textbook would help, and where THIS TEXTBOOK UTTERLY FAILS. In a proper text book, a principle is demonstrated through theorem, and mabye a couple of examples. This book has non of this, only a vague hint on how something MIGHT work, and then it leaves you to "intuit" the answer from thin air. Well if I could "intuit" complex mathematical relationships out of thin air, I sure wouldn't need this book or even to take a course of study at a University, because I'd be rainman, or good will hunting, or something like that. This textbook and the "Moore Method" are THE WORSE way I have ever encountered for teaching mathematics. Like I said, if the student does not have any metric by which to measure how he is being evaluated and graded in a course of study, then how on earth is he to know that he is being treated fairly, and not with personal bias? Since there is no standard by which the student can review grades on coursework, it cannot be determined. You might say "get another textbook" but if the student does not know anything about the material, that is not a reasonable solution, since he cannot evaluate material about which he does not know independently. So, what you end up with is a radical feminist professor, whose Vita has more things on it that have to do with feminism than mathematics, who uses the "Moore Method", with its lack of an objective measure from which to form a grade, who uses it as a weapon to sandbag men. The feminist "professor" doesn't even have one published work in mathematics that is entirely her own work, so its very likely that she has just ridden gender politics through her entire career and relied on others to carry her water academically.