What is the Common Core?

The Common Core State Standards are a collection of Standards for Mathematical Practice combined with a collection of Standards for Mathematical Content. You can find the Common Core here.

Differences Between AoPS and the CCSS

Traditional schools and the Common Core are generally concerned with getting students through the typical four-year high school curriculum, in a large part to prepare them for college-level Calculus. Art of Problem Solving serves high-performing students who learn at a faster pace and can handle more complex cognitive tasks. These students are not stretched by the traditional curriculum, so they do not grow to their full potential. They need more intellectual stimulation.

One way AoPS addresses this is by including advanced topics in Algebra and Geometry that do not appear on the Common Core State Standards. The AoPS curriculum also includes courses in discrete math, namely counting and number theory, which also are not part of the Common Core or standard high school curricula. For example, our Introductory and Intermediate Counting & Probability and Number Theory courses do not contain many Standards because much of their material lies outside the Common Core. This omission in the CCSS is unfortunate since discrete math is a key part of college-level math and the mathematics of programming, and students who have a background in these fields will have a significant advantage in those majors and in the real world. These might be the most applicable skills our students learn in their high school careers.

Furthermore, counting and number theory are accessible fields where students can further their mathematical reasoning and proof techniques. The traditional curriculum generally teaches students many tools to apply to specific straightforward problems, again in large part because the school is in a rush to provide the student all the prerequisites to enroll in Calculus. But this is not what mathematics is about. Our philosophy is to teach students problem solving which, in contrast, is about taking the tools they have and applying them to solve complex problems. Real life will throw students many problems they have not specifically been trained to solve, and the students with stronger problem solving skills will be better positioned to tackle these problems. Learning discrete math, in addition to expanding the students' horizons, gives them the opportunity to develop problem solving skills before moving onto more complex topics.

The other major difference between the Common Core State Standards and the Art of Problem Solving curriculum is the treatment of Statistics. AoPS currently does not have a Statistics course. Several AoPS courses cover standards regarding probability, specifically using problems where student calculate probabilities to illustrate interesting applications of counting techniques and geometric probability. Many Common Core standards regarding interpreting data and justifying conclusions are not covered by AoPS courses at this time. If a student is using an AoPS course to skip a class in a traditional school, she may need to spend a couple of days in the school library reading through the statistics chapter of her local textbook.

There are also some minor differences in when and how AoPS and the Common Core teach the material. The three most apparent are:

• The Common Core grades 6-8 contain numerous standards involving 3-dimensional shapes. The Geometry component in AoPS Prealgebra 2 is closer to a traditional Euclidean plane geometry course, where students apply geometric properties to solve problems. AoPS discusses 3-dimensional geometry first in its Beast Academy 5A curriculum and then defers until Introduction to Geometry, where students can extend their mastery of 2-dimensional geometry to reason about 3-dimensional configurations. We reach the depth expected in the CCSS in our 5A explorations and then wait on the material until students have the tools to solve more complex problems instead of learning extra forumulas earlier.
• The Common Core uses geometric transformations—dilation, reflection, rotation, and translation—to define congruence and similarity. In contrast, AoPS uses an informal definition of congruence and similarity that utilizes the students' intuition, saving transformations for later in the Introduction to Geometry course. AoPS also discusses geometric transformations in Precalculus using the tools of complex numbers and vectors.
• Also, AoPS and the Common Core emphasize different aspects of the Expressions & Equations domains in grades 6-8. The Common Core uses the coordinate plane to connect concepts in ratios, expressions, and geometry that deal with linear equations and their graphs. AoPS does not cover the graphs of linear equations until the high school-level Introduction to Algebra A course. Similarly, the grade 8 Functions domain and standards regarding systems of linear equations are taught in Algebra, where we can provide a more rigorous treatment. Instead, the Common Core high school cluster A.SSE (Algebra: Seeing Structure in Expressions) is pushed forward into AoPS Prealgebra as it is an important problem solving skill when manipulating algebraic expressions.

Common Core Coverage

The following tables describe which AoPS courses cover the various Common Core content standards. You can find similar information for our courses in each Course Syllabus.