The Art of Problem Solving (AoPS) math courses for grades five through twelve were designed for high-performing math students. The publisher says on their website,
We present a much broader and deeper exploration of challenging mathematics than a typical math curriculum and show students how to apply their knowledge and problem-solving skills to difficult problems. We help students learn the critical problem-solving skills necessary for success at mathematics competitions (such as MATHCOUNTS and the AMC), top universities, and competitive careers.
Their courses cover much more than typical math courses for middle through high school. They have courses that cover the standard sequence at advanced levels, plus other courses that take students deeper into the math needed for physics, engineering, computer science, and other math-based careers.
AoPS students often work a few years ahead of other students, which means that capable fifth or sixth graders might start with Prealgebra. Note that AoPS is the publisher of Beast Academy math courses for grades one through five, and those courses prepare students to move right into AoPS Prealgebra.
Courses and Format Options
AoPS lists five courses as part of their “Introductory Curriculum” for students up through tenth grade: Prealgebra, Introduction to Algebra, Introduction to Counting & Probability, Introduction to Geometry, and Introduction to Number Theory. Their “Intermediate Curriculum” for advanced high school students includes Intermediate Algebra, Intermediate Counting & Probability, Precalculus, and Calculus. Even so, students in a traditional program can still use these courses following a more typical timeline and concentrating on the required courses.
The website page for each course has two free diagnostic tests (PDFs) that help determine whether a student has the prerequisites for the course or whether they already have mastered what the course covers. These tests are accessed by clicking on “Are You Ready?” and “Do You Need This?” on each course’s description page.
Students do not need to complete all books in the series, but if they start the series in sixth grade, they should be able to complete most of them. Students who want to participate in math competitions might also be interested in AoPS books written specifically for that purpose: Competition Math for Middle School; the Art of Problem Solving, Volume 1: the Basics; and the Art of Problem Solving, Volume 2: and Beyond. (The titles of the last two books do not begin with capital T.)
AoPS sells printed or online books or a combination of both. They also offer live, online options for all courses and a self-paced-online option for Prealgebra and Introductory Algebra A. (The online courses might be a great way for eager students to find the community support they need to enter competitions.)
The printed textbooks have separate solutions manuals with worked-out solutions for every problem. The online books include the solutions, and they also integrate the textbooks with interactions with the AoPS community, Alcumus (described below), and the free videos (also described below). The textbooks vary in length; those for the standard courses (except Calculus) run from 528 to 720 pages, while other courses have from 256 to 400 pages.
Free videos are available online for Prealgebra, Introduction to Algebra, and Introduction to Counting & Probability. You can view these without having to pay or register. The videos do not replace the textbooks or online classes but supplement them. There are one or more videos for each lesson, all taught by Richard Rusczyk, a very engaging presenter as well as a former USA Mathematical Olympiad winner. I highly recommend watching them.
How the Courses Work
The courses divide the content into chapters, with several lessons within each chapter. Each lesson begins either with brief instruction or a set of three or more problems. Students should try to solve the problems on their own. The lesson continues with thorough explanations for how to solve each problem, and this is where most of the instruction is presented. This strategy very much reflects the title of the series, the Art of Problem Solving—students are focused on developing problem-solving skills as well as accuracy.
After studying the solutions, students have another set of problems to solve, a few of which are drawn from advanced math exams (no longer in use), such as the AHSME (American High School Mathematics Examination).
Lessons often use blue boxes to highlight key concepts, important ideas, and warnings about common mistakes.
There are Review Problems at the end of each chapter but no quizzes or tests for any of these courses. (The second diagnostic test for each course, titled “Do You Need This?,” could function as a final exam if needed.) The publisher’s explanation to me regarding this was: “Since our curriculum focuses on teaching students mathematical concepts and problem-solving skills, we believe that students who are using our textbooks have mastered the material if they can successfully solve the Review Problems at the end of each chapter.”
Alcumus, AoPS’s online learning system, is available for free to all students, even if they have not purchased any AoPS course. Alcumus adapts to the student’s performance, giving them problems to solve that are appropriate for their level—problems to solve in addition to those in their course. Alcumus can be used alongside the Introductory Curriculum courses, whether in print or online. (Students using courses from other publishers for pre-algebra through geometry should also find the program useful.) Alcumus provides ambitious students with work that will both reinforce and stretch their skills.
The AoPS website offers many other resources for advanced math students, including information about competitions, online forums, and training for competitions.
AoPS math courses should be fantastic for avid math students who are eager to learn and go deeper, but they also offer excellent and thorough instruction for the average student.