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- Sales Rank: #413862 in Books
- Published on: 1965
- Binding: Paperback
Most helpful customer reviews
11 of 11 people found the following review helpful.
probably the best modern geometry book available
By null_space
Edit: I noticed that there are third-party sellers on Amazon charging ridiculous prices (in excess of $500) for this book. Buy it on AoPS' website instead.
The Art of Problem Solving (AoPS) curriculum is absolutely revolutionary in my eyes. What differentiates this book from the others is the fact that nothing is spoon-fed to you. This book uses a radically different approach: Problems first, then explanations. Major results are broken down piece-by-piece and discovered through the problems in the beginning of each section. Of course, although it is recommended to try the problems first, it is not necessary as the solutions are revealed on the following pages. These problems serve as the lesson AND the examples, and they have traditionally served better to familiarize students with the main concepts than most other methods. The exposition is fully rigorous and motivated, with an emphasis on WHY the results exist and how they relate to each other. Nothing is cluttered or disorganized, and there are no stupid pictures or horrendous colors taking up all the space. Then, the section ends with 3-8 exercises. This may seem like too few, but I will touch on that in a moment.
Another aspect of the AoPS book is the problems. The pages are not filled with endless rote-drill. The problem sets are fantastic. There are 4 problem categories: The beginning-of-section problem, the exercises, the end-of-chapter review problems, and the challenge problems. The beginning-of-section problems, as I have mentioned, are the main teaching tools. Every main result is discovered through these problems. For example, in the first section of chapter 6, the Pythagorean theorem is discovered via similar triangles, and the following problems are example applications of the Pythagorean theorem in finding missing side lengths. These are usually very straightforward. After you go through those, the exercises at the end of each section help to solidify your understanding of the material. The first couple of exercises tend to be very trivial applications, but they get difficult really fast. The hardest exercises are marked with a star. There are relatively few exercises in each section, but they are presented in such a way that endless repetition is not necessary; the exercises in general will test your actual understanding of the concepts rather than simply how to mechanically apply procedures (although the easiest exercises can be used for that purpose). After going through the chapter, there are the review problems, which are the most numerous. These problems are only meant to serve as an assessment of basic familiarity with the concepts, so they are largely straightforward. If you find yourself stuck on a review problem, then go back and reread the chapter. And finally, there are the challenge problems, which are the ones you want to be spending most of your time on. There are usually 10-30 challenge problems at the end of each chapter, and they are the hardest problems in the book. These problems are meant to expand your understanding of the material, and are anything but straightforward. They can get incredibly challenging. Like the exercises, the most challenging challenge problems are marked with a star. There are challenging problems from major mathematics competitions scattered throughout the text, and there are plenty of proof-based problems as well. No two-column BS, either. Rote memorization is ideally the lowest priority when learning math, and the folks over at AoPS put much emphasis on "understanding over memorization," where this text succeeds.
Also, one very subtle thing that was done when writing this book was peppering in cute little cameos from more advanced topics. For example, in a chapter 7 challenge problem, the excircle, excenter, and exradius of a triangle are briefly defined and you are asked to prove a formula for the exradius of a triangle in terms of the triangle's area, semiperimeter, and side lengths. There are also interesting "extras" that occasionally appear in the margins of the pages, which contain anything from relevant quotes to explorations into interesting math concepts. Fagnago's problem, the nine-point circle of a triangle, the eight-point circle of a quadrilateral, a snippet from Archimedes' astoundingly brilliant argument for the volume and surface area of a sphere, and other classical results are shown to inspire and intrigue curious students of mathematics (unlike many other texts, which will emphasize contrived and oversimplified "applications" in these margins while ignoring the actual creativity and imagination involved in real mathematical discovery). A lot of these extras are discovery-based as well, and the reader is encouraged to prove some of those interesting properties. Some of these cameos were meant to be teasers for the topics found in the upcoming "Intermediate Geometry" book, but unfortunately, it seems that AoPS has postponed that project indefinitely.
Again, the organization of this text is amazing. The problems are ordered appropriately by difficulty, and nothing seems out-of-place. There may not be 50 exercises per section, but trust me when I say that the problems in all are more than sufficient to gain a mastery and deep understanding of the material. I cannot think of a better text. Even the acclaimed geometry text by Jurgensen and Brown that was used in honors geometry courses in the 90's cannot begin to compare to this text.
One warning, though: This text is designed to challenge high-performing students. Many students will have an incredibly difficult time without a good instructor.
4 of 4 people found the following review helpful.
Love these books!
By Connecticut Mom
My daughter and I are on Chapter 14 - Three-Dimensional Geometry - so I thought it was time for a review.
All my laudatory comments from my Amazon reviews (q.v.) of AoPS's Pre-Algebra and Introduction to Algebra books also apply to this book and the solutions manual. To recap: Very well-written expository material which presents the concepts clearly, and in many instances in insight-giving ways that make your jaw drop at the sheer beauty of mathematics. Brilliant problems, often drawn from math competitions, that are all thriller and no filler. Occasional fascinating sidebars as a reward. Super-rigorous, super-thorough, goes much, much deeper than the typical honors curriculum in schools.
Geometry is the math class that supposedly introduces kids to proof writing, which is what the practice of higher level mathematics looks like. Unfortunately, for reasons that I can only imagine have to do with ease of grading for teachers who aren't very math-literate, "proof" in most high school geometry classes involve making silly tables with statements on one side and reasons on the other. For instance, you might have to say on one side that angle A is equal to itself (duh), and then write "Reflexive Property" on the other side. Because every little step in the reasoning is written out, of necessity the problems can't involve too long a chain of reasoning, and so are limited in their sophistication. These mind-numbing proofs have little in common with what mathematicians do, and they are mercifully absent from AoPS. Instead, the proof problems are some of the most subtle and nuanced in the book.
These books are aimed at the top 5% of students, in terms of raw ability and personal motivation. My own children would NEVER work through these on their own, but luckily I'm here to work by their side. If you're willing to do so (the parent's skill level doesn't have to be top 5%, but it helps not to be math-phobic), I have some tips that may help:
(1) We are not organized enough for a dedicated notebook, and instead work the problems on random bits of paper. It's a good idea, when you're done with a session, to make a note of the next problem to be solved and stick it in as a bookmark. That way if you "forget" to do math for a week or three, you can pick right up where you left off.
(2) Don't pencil the figures! Make a photocopy of the page and write on that instead. Better yet, practice drawing the figure on a separate page. All you really need is a straightedge and a compass, but I like having a circle-drawing template on hand.
(3) Better a lot of short sessions than a few long ones. We aim for about 20 minutes, two to four times a week (and every day during vacations). It might take more than a year to get through the book, but so what? Every bit you do will cause your child's (and your) understanding to skyrocket. Also, the more regular your sessions, the easier it is to recover the proper math mindset. Long breaks in between make returning to math more painful than it needs to be. However, it is important to keep sessions short and have time in between sessions (at least a day) to allow the concepts to be digested.
Geometry is a wonderful subject, much of which was discovered/invented by the ancient Greeks, and studying it at this level is at once a math, history and philosophy lesson.
2 of 2 people found the following review helpful.
This is a great book. Some of problems are little bit difficult ...
By Garry Wu
This is a great book. Some of problems are little bit difficult but they really challenge my son to think. I like the math languages used in the book that are very easy for kids to understand so they can read by themselves.
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