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Prealgebra Paperback – August 10, 2011
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About the Author
Richard Rusczyk is the founder of Art of Problem Solving. He is co-author of the Art of Problem Solving, Volumes 1 and 2 and Intermediate Algebra, and author of Introduction to Algebra, Introduction to Geometry, and Precalculus. He was a national MATHCOUNTS participant in 1985, a three-time participant in the Math Olympiad Summer Program, and a USA Mathematical Olympiad winner in 1989. He is also the co-founder of the Mandelbrot Competition. Mr. Rusczyk is a graduate of Princeton University. David Patrick is an instructor and curriculum developer at Art of Problem Solving. He is the author of Art of Problem Solvings Introduction to Counting & Probability, Intermediate Counting & Probability, and Calculus textbooks. He had the sole perfect score in North America on the 1988 American High School Mathematics Examination, was a USA Mathematical Olympiad winner, and was a top-10 finisher on the William Lowell Putnam Mathematics Competition. Dr. Patrick is a graduate of Carnegie Mellon University and has a Ph.D. in Mathematics from MIT. He taught mathematics at the University of Washington before joining Art of Problem Solving in 2004. Ravi Boppana is the Director of Mathematics at Advantage Testing, as well as the co-founder and co-director of the national Math Prize for Girls competition, sponsored by the Advantage Testing Foundation. He was 27th in the country on the William Lowell Putnam Mathematics Competition. Dr. Boppana received a Ph.D. in Computer Science from MIT at the age of 22. He was a tenured professor at NYU, where he received the Presidential Young Investigator Award for Excellence in Research and the Golden Dozen Award for Excellence in Teaching. His daughter Meena finished second in New York State at MATHCOUNTS and was a USA Math Olympiad qualifier.
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This book has met and exceeded all my expectations, and I cannot recommend it highly enough: the philosophy behind the 'Art of Problem Solving' -- the folks who wrote this book and others in the series, is that you learn more when you wrestle with ideas and try to figure them out for yourself than having them spoon-fed to you. So their approach is to begin each unit with problems for the student to sit down and try to solve for herself, using what she already knows. The problems help lead the student to discover (for example,) *why* the product of the square of two numbers is equal to the square of their product.
After attempting those problems, the student reads on and compares her solution with the one in the book, and a lot of the learning happens from reading and really *understanding* how and why the authors solved the problems the way they did. Along the way, the text will 'lecture' a bit, and point out important rules the students have 'proven', as well as introduce any unfamiliar terms needed for the next section.
Next, there are exercises at the end of each unit, for the student to practice what has been learned. These are NOT 'worksheet'-style problems like in elementary texts, where you have pages of basically the same problem over and over. These exercises seem to be hand-crafted to guide the student from simpler to harder problems, step by step, and they do a great job. Finally, at the end of each chapter (there are 5 or so units per chapter) there are whole-chapter review exercises of increasing difficulty, followed by 'challenge problems'.
The main difference between 'exercises' and 'challenge problems' is that it is expected that most students will be able to get most for the 'exercises' correctly by working through the chapter and working through the other examples. The challenge problems require more 'thinking outside the box' and some are really very challenging, even for folks like me, who survived differential equations in college. NOBODY should expect to figure them all out, certainly not without a LOT of effort, but everybody can learn from *trying* to solve them. Nobody *ever* asked me questions like this in my middle school classes, even in the honors classes, and yet my son, by trying to solve them (and he certainly can't solve all of them,) is gaining a *much* deeper understanding of the material than I ever had at his age. (To be honest, I've had fun wrestling with the harder ones myself, after he goes to bed. But I'm weird like that.)
(1) If you are using this book as a private text -- as opposed to a middle-school classroom with an instructor -- YOU ABSOLUTELY NEED THE SOLUTIONS MANUAL; it's not just an answer key, it actually details *how* to solve every single problem in this book. Without it, there's no way for the student to check his work on the exercises, and they won't learn nearly as much; it's published as a separate volume mostly to let it be used as a classrom text, but most buyers should think of the two books as one indivisible item.
(2) THERE ARE OTHER, FREE SUPPLEMENTAL MATERIALS AVAILABLE -- the authors have a web site, artofproblemsolving.com, which also hosts literally *hundreds* of short youtube videos, many aligned to specific topics in the textbooks, and there's a pretty cool (currently free) tool called 'Alcumus' that challenges you with math problems of increasing difficulty in a quiz-game format. [solutions are shown after you answer, so you can learn even from the problem you don't solve.] They also host free online math competitions [quiz style] and paid online classes aligned with the text, but I can't comment on those because we haven't tried them - yet.
UPDATE: we tried the online PreAlgebra I class (covers the first half of the book) -- it is an excellent class, but be aware that it is generally aimed at kids who 'get math' and like it, and who are mature and disciplined enough to work hard at difficult things until they 'get it' -- if your kid requires a lot of hand-holding to get him to do the work, you will hate this class, but if he/she likes math and enjoys a challenge, they will learn a LOT. The format is more like a college class than a typical middle-school classroom; there's a through syllabus given out before the class starts, there are assigned reading and problems you are expected to solve, preferably before the weekly class meeting, and the content moves *fast* [generally one or at most two weeks per topic.] If your kid takes this seriously, I think you will get your money's worth out of the classes and then some, but it requires hard work and dedication.
(3) THIS BOOK IS NOT FOR EVERYBODY -- although I firmly believe this philosophy of learning through solving problems is a *better* way to teach math than what most kids get in school, the level and pacing in this textbook seems clearly and deliberately aimed at the mathematically inclined or gifted reader.
(3a) Kids who don't "get" math or don't like it will, in my opinion, be enormously frustrated by this book, and kids whose parents don't "get" math may be unable to help them; of course that's true for any textbook; you have to know the subject, at least a little, to help your kids with their homework, right? :^)
(3b) [This caveat is aimed at a small group of people with truly exceptional kids - the 99.9+ %-ers] Kids who are very bright sometimes get freaked out when they have to struggle, if they can't easily get 100% scores on everything without really trying. Your child should *expect* to struggle with parts of this book, and come out knowing more at the end, but I don't think *anybody* is going to breeze through this book and get all the challenge problems correct. I believe it is critical to expose kids like this to 'insolvable' challenges so they can *learn* how to struggle with, and master difficult material. But some young, exceptionally gifted kids who are homeschooling might get overly frustrated and quit if they lack the emotional maturity to deal with the setbacks and the challenges productively.
This book's approach is to teach you by having you solve problems. Apparently, brain studies have found that when you are just presented with information your brain doesn't grow as much as when you attempt to do something and make a mistake, then learn what the answer is. This books' approach takes advantage of that to accelerate learning. In each chapter, it gives a brief (perhaps one page) of introduction to the current topic, then gives you problems to solve, without telling you how to solve them. Then it gives step-by-step answers and explanations about how to get to those answers and why. At the end of the chapter, it gives extra questions as well as challenge questions that are more advanced. The solutions guide gives full explanations for each of the chapter end and challenge questions. So, you don't have to have the answer guide, but I've found it great extra practice and review. I also enjoy that this takes the approach that you should always do the least amount of work. Where other programs offer "shortcuts", this one teaches the shorter ways of doing things as a matter of course. When you are in a competition, taking a timed exam, or coming up on a deadline on your work, you don't have time to figure things out the long way--if there is a quicker way. This helps you look for a quicker way from the start and get used to doing it. Also, it becomes obvious right away that when you look for the quicker way, often it is easier to check to be sure your answer is correct also. For instance, is it easier to multiply 125 squared by 16 squared or multiply 125 by 16, then square the result? Try it both ways. Sometimes math doesn't have to be as tedious as we've been led to believe.
I started with prealgebra because I wanted to be sure I had a good foundation. I did Khan Academy (free online) up through prealgebra for review, but when I hit algebra, it just didn't give enough explanations (neither did Aleks online), and I wasn't retaining what I did learn. My daughter also used Khan an hour a day for months, but the explanations weren't enough for her either. I ended up moving over to Mathematical Reasoning by the Critical Thinking Company for her. I considered using their books for myself, but was not sure they go into as much depth as this one.
So, the main reason I chose this series of books (they have about 9 math books above this level, incl. number theory, and statistics), is because I don't just need to know math to pass a course. I need to know math well enough to use it for my future profession. I need to KNOW it. I don't find this book harder than any others, but it's not fancy. It gives you the facts with no fluff. It is easy enough to understand, but it asks you to think--a lot. I put a circle in my notebook next to the problems I get wrong, and before I begin each day, I re-do the problems I missed the day before, and some problems I missed a few days or a week before. It's amazing how fast I forget how it's done. This repetition gets it into my brain. I also write why I didn't get it correct under the circle. Often, when I write, "I still don't understand this", it clicks a few days later. So, I recommend review like this; it really helps. Sometimes, my comment is just "Arithmetic error, be more careful" or "Always reread the problem before moving on to be sure your answer is what the problem was actual asking for." I've also learned to not just use scratch paper for my work, but have realized how much neatness counts in making sure I don't make a mistake. It became a habit that just made sense very quickly. I feel like I'm unlearning all the bad habits I learned in school.
Just yesterday, I was presented with a programming problem in my studies, where I considered it for a few moments, then went "Oh, this requires adding all the numbers together up to a certain digit." And I knew exactly how to do it and was able focus on how to program it, rather than how to do that math. A couple months ago, I didn't know how to do that and now it's easy and makes perfect sense, so I'm unlikely to forget it. Awesome. Thank you Art of Problem Solving.
BTW, Beast Academy is produced by the same company. I ordered the second grade level for my daughter, and she LOVED the comic book format! My daughter was reading about math and was really enjoying it and understanding it! She is going into 6th grade and found the workbooks really challenging. I would have started her on the 1st grade level, but that isn't due to come out till the autumn of this year (2017). For now, we have paused, but might try again once the first level comes out. This prealgebra book is for after completing the five levels of Beast Academy (labeled grades 1 through 5). But I'd say it's rather challenging for a sixth grader unless they are gifted, but I may be wrong. My girl has always been in a Montessori school and they do things differently. But do keep in mind that Beast Academy and Art of Problem Solving are gifted curriculum. Prealgebra and up are designed to help middle school students to compete in math competitions. But again, if you are out of school already and finding you need to know math well for your profession, this is a great choice! I wouldn't say that Beast Academy is too simplistic if you find the prealgebra book hard and want to review foundational math. I found the workbook at the second grade level challenging! And just for reference, my IQ is well above average--it tries to give you harder questions than your current level to ones that mathematicians find challenging. And it's a fun way of learning. Only downside to Beast Academy is the price. Ouch! I can understand they need to pay for all the many hours of work that must have gone into writing, drawing, and designing the books, but it's a bit hard to justify buying the whole series for my daughter. I recommend buying only the first book and first workbook to see if it is a fit for you or your child, before buying a whole level as I did. I might just buy the rest of the comic books for my daughter and just skip the workbooks since the comic books are wonderful. We didn't get too far in the workbooks since frustration set in for my child and she really didn't like math to start with, so I can't really comment on them. This prealgebra book is well priced in my opinion, as it is about 600 pages, with no space wasted. If it had been more, I might not have taken the chance on it, but at this price and this quality, I hope to do many more in the series!
And, best yet, there are more practice problems, videos, and even an active discussion forum online--for free! If you just can't figure something out, you can post about it online or look to see if someone else has already asked the question. This is the next best thing to taking a course and you can do it at your own pace.