What is Art of Problem Solving
Founded in 1993 by former USA Math Olympiad winner Richard Rusczyk, Art of Problem Solving (AoPS) is a company that produces rigorous math instruction courses and products that can help outstanding math students develop a more thorough understanding of math concepts, as well as help prepare them for success in math competitions.
From textbooks to online classes to physical learning centers, AoPS offers a variety of educational products and services that can help challenge kids, deepening their knowledge and strengthening their mathematical thinking.
AoPS Math textbooks
Art of Problem Solving has created a series of textbooks for middle and high school math textbooks that are designed to give outstanding math students a deeper and more rigorous curriculum in math.
Originally designed to help talented math students prepare for competitions, over the years AoPS’s textbook line has expanded to offer full curriculums in middle and high school math courses, and their problem-based and rigorous approach to math has made them very popular with parents across the world as a top enrichment option.
What Grades and Math Subjects does AoPS Math cover?
Art of Problem Solving textbooks cover middle and high school math, as well as competition prep.
Generally speaking, the AoPS math textbooks can be broken down into two curricula- introductory and advanced – that roughly correspond to most middle and high school math programs (in terms of overall scope, that is).
Parents of younger math enthusiasts should note that Art of Problem Solving covers elementary school math (Grades 1-6) in their Beast Academy series, which you can read about in our review.
Introductory Curriculum (Middle School)
|Book Subject||Examples of Topics Covered|
|Arithmetic properties, exponents, primes/ divisors, fractions, equations and inequalities, decimals, ratios and proportions, unit conversions and rates, percents, square roots, some geometry, statistics, counting and probability|
|Introduction to Algebra||Linear equations, quadratic equations, ratios, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities, functions, polynomials, exponents and logarithms, absolute value, sequences and series|
|Introduction to Counting & Probability||Combinations, permutations, Pascal’s Triangle, basic combinatorial identities, expected value, fundamentals of probability, geometric probability, Binomial Theorem|
|Introduction to Geometry||Similar and congruent triangles, quadrilaterals, polygons, circles, areas, power of a point, elementary plane geometry, translations and rotations, three-dimensional geometry, transformation, introductory trig, analytic geometry|
|Introduction to Number Theory||Number sense, primes and composites, multiples and divisors, palindromes, prime factorization, base numbers and their manipulation, modular arithmetic, perfect, abundant and deficient numbers, divisibility rules, linear congruences|
Advanced Curriculum (High School)
|Book Subject||Examples of Topics Covered|
|Intermediate Algebra||Complex numbers, quadratics and conic sections, inverse functions, polynomials and polynomial roots, multivariable expressions, sequences and series, recursive sequences, identities, inequalities, rearrangements, exponents and logs, functional equations, absolute values and piecewise defined functions|
|Intermediate Counting & Probability||Sets and Logic, Inclusion-exclusion, constructive counting and 1-1 correspondences, the Pigeonhole Principle, constructive expectation, Fibonacci and Catalan numbers, recursion, conditional probability, generating functions, graph theory|
|Pre-Calculus||trigonometry, trigonometric identities, parameterization and coordinates, geometry, complex numbers, vectors, and matrices|
|Calculus||Sets and functions, limits, derivatives, integrals, power series, plane curves, and differential equations|
When taken as a whole, Art of Problem Solving’s math textbooks cover the topics included in most US Math curricula, as well as touching on a few topics that aren’t usually covered in most public high school programs.
That said, the point isn’t really to get kids learning college level math or a curriculum beyond high school math, but instead to get students to develop their problem solving skills and develop more creative and flexible mathematical thinking, to get them to recognize and appreciate different approaches to problem solving, as well as getting a better understanding of the why of math, rather than just focusing on how to compute problems.
As such, AoPS’s curricula tend to go deeper into your typical middle and high school math topics, letting kids examine concepts more rigorously, more thoroughly and with more challenging problems than they would otherwise be able to do in other math courses.
Art of Problem Solving Contest Prep
In addition to their more academically-focused textbooks, Art of Problem Solving also offers a variety of books designed to further enrich exceptional students or help with preparing for math contests and Olympiads.
These books generally tend to work on developing stronger problem solving skills, going far deeper into various concepts and exploring far more challenging questions and problems, while introducing various approaches for understanding and solving them quickly and effectively.
|Book||Examples of Topics Covered|
|Art of Problem Solving, Volume 1: the Basics||Exponents and logarithms, complex numbers, linear equations, proportions, quadratic equations and more|
|Art of Problem Solving, Volume 2: and Beyond||Diophantine equations, linear and quadratic congruences, combinatorics, geometry and inequalities, analytic geometry|
|Competition Math for Middle School||Algebra, counting, probability, number theory, and geometry|
Geared more for gifted enrichment and contests preparation, each of these books tend to go over a greater variety of concepts and topics, touching on concepts in Geometry, Algebra, Number theory and more, and aren’t really bound to any linear curriculum.
In addition, the problem sets, geared as they are to helping students prepare for national tournaments and contests, are far more challenging and in-depth than would be expected of even an advanced middle or high school course.
For these reasons we don’t usually think this series is where parents should necessarily start off when working on math at home, but in our experience we do feel they are great supplements to the main textbooks and can be excellent for enrichment purposes and preparing for contests.
How Art of Problem Solving Teaches Math
AoPS pedagogical approach
Art of Problem solving is a big believer in teaching through solving problems.
The books consequently include a wide variety of problems, many of which kids will have never encountered before.
In fact, some come directly from various math competitions such as:
- The American Mathematics Competitions (AMC)
- The Harvard-MIT Math Tournament
- And more
The general idea is that by getting kids to work through problems themselves, and more importantly discovering how to solve certain problems, kids will develop a deeper understanding of the material.
As a result, AoPS Math textbooks are quite problem set heavy.
Explanations of each concept are quite short and to the point and are followed by a good deal of exercises for students to try out on their own.
When introducing these textbooks, parents should expect that kids will have to think things through a bit more and work out the answers themselves without a lot of hand holding or spoon feeding, and that there will be a heavier emphasis on logic and proof than other curricula.
All this really drives home Art of Problem Solving’s place as a resource for outstanding or talented math students who don’t need a lot of time or explanation to grasp the material.
Consequently, students who are less adept at math may find the instructions a little too short and too quick and may need extra help in order prevent getting frustrated by skill and knowledge gaps as the exercises come rolling in
Regardless of the book in question, Age of Learning’s lessons tend to follow a particular format.
The books are made up of several chapters, each of which covers a particular topic within the subject and contains several sections.
Each section is then typically broken down into various related concepts, an overview of the types of problems kids may come across (both common and uncommon) and often the various factors that can affect outcomes.
In Introduction to Algebra, for example, when discussing multivariable linear equations, the chapter is divided up into an introduction, a discussion of substitution, elimination, some word problems, common and uncommon problem sets, different variables and so on.
As kids go through their lessons, they are given lots of examples to try and lessons tend to work through some of them step-by step in a fairly in-depth and rigorous manner to demonstrate concepts.
Sections typically end with a variety of exercises for that section and, at the end of each chapter, there are review and challenge problems.
Review problems go over and test what the student has learned with similar problems, while challenge problems go a step further and test mastery of the material with far more challenging questions.
If kids get stuck, there are always hints and solutions that are helpfully included in the back of the book (no cheating!)
Look and feel
As you might expect from a problem solving and word problem-heavy methodology, these textbooks contain lots of typical math diagrams and pictures floating about to go along with and illustrate the word problems.
AoPS textbooks also tend to have a lot of floating boxes that highlight important information for kids, including:
- Pointing out various strategies they can take on given concepts or problems
- Offering extra work
- Giving extra information
- Even offering “bogus” solutions that point out the most common mistakes made by students when solving a problem
Despite its rigor, Art of Problem Solving does its best to keep its material from becoming too dry and boring, which we appreciate.
The books are written in a very casual tone, which makes it feel as if a math-whiz friend were explaining the material rather than a textbook.
There are also a good deal of amusing and interesting examples and concept demonstrations sprinkled throughout, sometimes even involving sly pop culture references (some of which may go over kids heads, but parents will appreciate).
Does this approach really work?
Due to its philosophy and the way it teaches, we feel the Art of Problem solving takes more or less a constructivist/Problem Based Learning approach to teaching math where, instead of receiving formal lectures about math, students build up their own knowledge and skill by working through and solving various problems.
This learner-centric approach to teaching math and science actually has been linked to positive outcomes when teaching math and science, fostering greater problem-solving skills, improving self-motivation and encouraging creative and critical thinking skills as they relate to mathematics.
Past customers have also reported that the series challenges their students pretty thoroughly, increasing the depth of their knowledge on relevant subjects and increasing their speed at solving difficult-math problems, sometimes dramatically.
It is perhaps unsurprising, then, that the Art of Problem Solving curriculum is often used in honors math classes across the US.
Some Drawbacks to Art of Problem Solving Textbooks and Curriculum
Can be time consuming
Due to its focus on doing exercises, exploring concepts and working through problems to gain a better understanding of the subject matter, Art of Problem Solving can take a little more time to work with than some other programs.
This can be particularly true as AoPS tends to use far more challenging questions than kids are used to, some of which are in formats they haven’t seen before.
While great for learning, this approach isn’t exactly a time saver. It’s not uncommon, for example, for parents to report spending up to 45 min (or more) each day on math (in addition to other homework).
Can be tricky to jump into from another curriculum
With its particular approach and pedagogy, as well as its more rigorous approach to mathematics and problem solving (including the use of proofs), Art of Problem Solving can be somewhat tricky to get used to if you jump into it from another curriculum.
Because math is a cumulative process, kids who begin Art of Problem solving without having at least reviewed some of the foundational material in previous books can find themselves lost or slowed down by skill and knowledge gaps they didn’t realize they had.
Helpfully, the AoPS website does have free, printable diagnostic assessments for each book to help parents determine if their kids are at the right skill level.
Discovery approach can frustrate some learners
Despite the fact that Age of Problem Solving’s approach has been shown to get results and improve the mathematical thinking and skills of talented math students, sometimes it just isn’t the right approach for the student.
AoPS often requires students to play around with numbers and concepts and discover missing information themselves.
Some students, even really talented students, can get frustrated by this approach and may prefer a more straightforward, traditional math course where they can get down to computation and see their results more quickly.
Who is Art of Problem Solving For?
Overall, we think Art of Problem Solving is a great resource for parents and kids looking for a far more thorough, challenging and enriched math program.
It is an ideal course for students who demonstrate an aptitude for math and are looking to deepen and strengthen their math skills with more challenging grade-level material.
We think AoPS textbooks can be particularly good for students interested for more rigorous preparation for math-heavy STEM subjects in university, where their greater focus on problem solving, proofs and logic skills will be a strong asset, such as with physics, engineering and even computer science,
We also think that Art of Problem solving’s textbooks and methodology can be an excellent base material for students interested in or preparing for math contests and olympiads (AMC 10, AMC 12, MATHCOUNTS and the like), particularly their Contest Math Prep Series, as they promote creative approaches to problem solving and strengthen mathematical thinking that kids can use when faced with new problems.
Who is Art of Problem Solving Not Great For?
That said, Art of Problem Solving textbooks are obviously not for every student.
These books are not the best curriculum for kids who are struggling with math concepts as AoPS math is primarily aimed at enriching math study.
AoPS math goes far deeper into the material with far more rigor, exploring various high school and middle school math topics at a more advanced level and with more challenging problem sets, while emphasizing multiple approaches to problem solving and flexibility when approaching new math problems.
Struggling students, while they often can benefit from learning the why’s behind math, can usually spend their time better by reviewing the fundamentals and practicing basic strategies, as well as by working on more targeted skill development with programs like IXL and Khan Academy.
Similarly, we don’t feel that AoPS textbooks are really the best resource for preparing for the SAT and other timed standardized tests where answering speed and efficiency (and test taking strategies) can be far more effective when it comes to success than gaining a deep understanding of concepts and working through problems.
In these instances, kids are better served through specific standardized prep programs that will work with them on developing their proficiency at solving very particular types of questions.
Finally, AoPS textbooks are also not the best solution for kids looking to explore college level math as, despite its more challenging nature, AoPS math goes deeper into middle school and high school math topics (algebra, geometry, number theory, single variable calculus), rather than beyond it.
Price: How much do AoPS Textbooks Cost?
The price of AoPS math textbooks really depends on the particular book and subject you’re interested in.
Generally speaking, though, each book costs between $45 and $70, which is roughly the same as the average middle or high school textbook.
The length of each book varies, however, from just under 300 pages of instructional material in some cases to well over 700 in others.
Unlike many other middle and high school textbooks, however, these are designed to serve as a complete curriculum for each topic as every book contains instructional material as well as hundreds of practice problems, hints, and a step-by-step solution guide that itself is usually a couple hundred pages long as well.
If you have a talented middle or high school math student and you’re looking for ways to nurture their excellence, Art of Problem Solving’s math textbooks might be right for you.
Although certainly not for everyone, with their challenging curriculum and in-depth exploration of math concepts, AoPS can foster better problem solving skills, stronger analytical ability and improved creative and critical math thinking, all of which can help students take their math skills to the next level.
About the Author
David Belenky is a freelance writer, former science and math tutor and a tech enthusiast. When he’s not writing about educational tech, he likes to chill out with his family and dog at home.