In the classic TV show Lost, there's a computer into which you must enter a random string of numbers every 108 minutes without fail. We don't know what the numbers mean, or why it matters, just that it must be done. That's a nice metaphor for how we often approach math instruction. We don't know why 2+2=4, we just know that it does. Right?
Richard Rusczyk, former math competition winner, has created a math curriculum to combat that approach. Unlike on Lost, there are very definite answers out there. In Art of Problem Solving students are taught why the rules are true rather than merely what they are. They get to learn math the way that mathematicians already know it; as a layered and rich discipline that isn't just about numbers, but about creative problem solving.
(Let that be your first warning that this is not "easy math.")
How It Works:
The books are fairly simple in format, but heavy on content. At the beginning of each chapter, students are asked to solve (or attempt to solve) an introductory problem set before they read about the concept being taught. They compare their solutions to the book's solution, and the book discusses where they may have gone wrong, and explores different paths to get to the solution. This type of "learning through mistakes" gives students an indepth perspective on the problem itself.
The opening of each chapter typically has an optional math puzzle for students to play with, and a mathy sort of quote. The book takes a laidback approach to solving the exercises at the end. Assuming you have been doing the practice problems throughout the whole lesson, it advises you to work on the review problems until you can solve most of them (proving that you understood the chapter), and then move on. If you can't, go back and review the chapter again.
The exercises also include "starred" challenge problems that are more difficult. These have hints, and students are expected to at least give them a try but not to stress too much about solving them right away. The text takes a mathematician's view of solving problems. Sometimes you have to let a problem sit and come back to it later with fresh eyes. Working on it progressively is acceptable.
It also recommends that you don't look at the solutions manual until you've absolutely exhausted your ability to solve the problem  but that you certainly read the solution after you find the answer in case the book solves it a different way.
The emphasis of the texts are on knowing why the rules are true rather than merely memorizing the rules. The tone is fairly conversational and shows a true interest for the subject matter. The design isn't too flashy, and will engage those already interested in math for math's sake, or at least those who need a challenge.
Our Honest Opinion:
Traditional American math education is flawed (a generally agreed upon fact). Students don't learn how to solve problems. They learn how to input formulas and output solutions. A student could tell you how to solve for X, but often couldn't tell you why it works. It just does. That's math. You don't question it. And the faster you are at identifying the formula and correctly applying it, the better you are at math, but in a purely robotic way.
Math, therefore becomes a memorization game instead of a uniquely creative activity. And the makers of Art of Problem Solving are fighting back against that mindset. Math is not about memorizing onesizefitsall formulas, but about creative problem solving with multiple paths towards an ending solution. The more ways you can find to solve a problem, the more deeply you'll understand the solution. This is true for any problem.
Creative problem solving is understandably difficult and will stretch students in ways they may be unused to being stretched. So though we are fans of this method, it's certainly not for everyone. Formulas do have their place, and most people will grow up using the math they need. But if your student is ready for a brain workout, ready to eschew traditional ideas on math proficiency, and can handle spending a long time on a single problem, then this just might be the program that will give them the tools to creatively solve problems not just in math, but in life.

Review by Lauren Shearer
Lauren Shearer writes words for fun and profit. She also makes films, but everyone knows you can't make a profit doing that. Her other hobby is consistently volunteering way too much of her time. You can read more of her reviews here.

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