- Paperback: 150 pages
- Publisher: Birkhäuser (July 9, 2003)
- Language: English
- ISBN-10: 0817636773
- ISBN-13: 978-0817636777
- Product Dimensions: 6.1 x 0.4 x 9.2 inches
- Shipping Weight: 10.9 ounces (View shipping rates and policies)
- Average Customer Review: 31 customer reviews
- Amazon Best Sellers Rank: #254,895 in Books (See Top 100 in Books)
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“The idea behind teaching is to expect students to learn why things are true, rather than have them memorize ways of solving a few problems, as most of our books have done. [This] same philosophy lies behind the current text by Gelfand and Shen. There are specific ‘practical’ problems but there is much more development of the ideas … [The authors] have shown how to write a serious yet lively look at algebra.” ―The American Mathematics Monthly
“Were ‘Algebra’ to be used solely for supplementary reading, it could be wholeheartedly recommended to any high school student of any teacher … In fact, given the long tradition of mistreating algebra as a disjointed collection of techniques in the schools, there should be some urgency in making this book compulsory reading for anyone interested in learning mathematics.” ―The Mathematical Intelligencer
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I want to start with the positives, because there are many. If you're someone put off by the modern way of teaching math that forgoes all explanations and intuition in favor of step-by-step guides and formulas, then you'll appreciate Gelfand's approach to the field of algebra. Each subject is treated in a learn-by-doing fashion, and nothing is spoon fed. Clever intuitive explanations are provided for many topics that most books just take for granted. If you work your way through the entire book and all of its problems, you'll come away with algebraic understanding and intuition that probably exceeds that of most undergraduate math majors at the college level. As I said, I wanted to learn algebra the "right way" this time around, and I definitely achieved my goal through this book.
That said, this book has some pretty major flaws to the point that I have trouble actually recommending it. In fact, I don't think I'd even recommend it to my past self.
First, of the hundreds of exercises provided throughout the book, it feels like less than half of them have solutions provided. The author tries to explain that any problem without a solution is okay to skip if you can't solve it, but that explanation just isn't good enough. The fact is that the meat of this book is its problems, and if you skipped every problem without a solution, you'd probably learn less algebra than you would from a standard textbook. Like I said, Gelfand's approach is very learn-by-doing, but he doesn't give you the proper tools to actually "do" everything you're supposed to "learn." In fact, many problems build on previous problems, so if you skip any of them you risk triggering a cascade of unsolvable exercises that will leave you frustrated and confused. It's not like these are routine calculations you're doing, either. Most of the problems require creative problem solving, significant trial and error, or out of the box thinking to solve. There's nothing inherently wrong with this (in fact I love this approach, as it builds vital mathematical intuition and "real" math skills), but without solutions there's no way to check the correctness of your answer. If you get stuck (and you WILL get stuck), there's nothing you can do. Simply put, the challenge level of some of these problems is such that the typical math student is bound to encounter several of them that they will never solve on their own, and there's no way for them to help themselves along. I've heard some people suggest that the lack of solutions is good because that way you're forced to work through the problem on your own, but as someone with the self control to give each exercise a nice long, honest attempt, I found it to be a major hindrance to my learning. After hours of searching, I managed to find a document online with solutions to the unsolved problems, and that's the only way I was able to work my way through the book.
Second, much of the wording in this book is ambiguous, and in some cases there are major errors. It's not always clear what an exercise is asking you to do, and the mistakes can really set you back and waste your time. The subject matter is difficult enough without throwing away hours of effort just because the book didn't give you clear instructions or explanations.
Third, there's just not enough reinforcement in this book. On this plus side, this means you're learning something new with each and every exercise, but the problem is that the near-zero repetition doesn't leave much room for the learning to sink in. I would have really appreciated a handful of reinforcement problems for the most essential concepts.
As a tutoring aid, this book would probably get 5 stars. In that situation, the tutor can have worked everything out ahead of time and guide the student through the many challenges in this book. But as a self-study tool, or even a classroom text, this classic tome has a few too many issues. It earns 3 stars for the high quality of Gelfand's approach and explanations, but the issues with the exercises keep it from a true recommendation. If this book were to be republished some day with corrections and a full set of solutions, I would have no problem giving it 5 stars.
Students are also exposed to reading and writing proof this early in the book. Problem 42, which asks the readers to prove that a/b and c/d is a neighbour faction, is already challenging enough to become a question on many math forums.
Less known are the particular problems 31 and 32 which asks you to come up with a way to put the parentheses in the product of 5 and then 99 terms so that all variations are not left out. The solution prompts one to review the basic formulas of permutation and combination, which is not covered in this book at all.
Towards the end the book even introduces the AM-GM inequality, this is something very familiar with almost all students in the former Soviet Union countries, but not introduced directly in American high school textbook, according to my own limited knowledge.
The aim is thus to introducing mathematical reasoning and rigour to young students or even adults and much less on computation. This is the essence of learning mathematics.
Thus, 4 stars for a simple but in-depth presentation of one of the oldest branches of mathematics, 4 stars for the achievement of Soviet mathematics education within 4 decades (1950s-1990s).