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The Common Core State Standards
Art of Problem Solving courses are not designed to align to the Common Core State Standards (CCSS). However, we have many students who have easily replaced CCSS-aligned classes using our coursework. Our curriculum covers all of the Common Core Practices. Each of our classes also covers most or all of the Standards covered in the corresponding class from a CCSS-aligned school (and a good deal more!).
We've compiled some information below to help students and their schools choose the best possible education plan.
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.
Standards for Mathematical Practice
The Standards for Mathematical Practice (Practices) are eight general pedagogical goals suggested for all mathematical education:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with Mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
These practices closely match the educational objectives that Art of Problem Solving set out to promote when it was founded. For example:
- Make sense of problems and persevere in solving them: AoPS does not believe that math is memorizing Trick A to solve Problem A and Trick B for Problem B. Problem solving is about understanding the problem, weighing the possible approaches, and deciding which is the best strategy for solving it. Sometimes the path is not apparent, and sometimes what at first seemed like a promising start needs to be rethought. A good problem solver has the tenacity to work through these obstacles.
- Reason abstractly and quantitatively and Model with Mathematics: Part of problem solving is the ability to translate the problem description into mathematical symbols. At the same time, it is important for students to not treat equations as a series of symbols to be manipulated but understand what their equations mean in the context of the mathematical concepts and the problem. To understand problems deeply, a mathematician needs to be able to follow paths of context and abstraction simultaneously.
- Construct viable arguments and critique the reasoning of others, Use appropriate tools strategically and Attend to precision: Many traditional schools only discuss proofs in Geometry class, and only in simplistic terms. AoPS integrates proof-writing—having the students construct rigorous logical arguments and communicate their ideas clearly—starting from our Prealgebra courses. Each week in our subject series courses, students have one or more free response writing problems. Student responses are human-graded and receive feedback on their solution, including their ability to communicate mathematically. Students learn to construct and analyze mathematical arguments in our classroom as well, where we make the students do all the work.
- Look for and make use of structure and Look for and express regularity in repeated reasoning: Recognizing patterns is an important problem solving strategy. AoPS students learn by studying relevant problems and generalizing their ideas into the underlying principles. This requires dissecting each problem into its constituent pieces and their connective structures. This also requires finding common principles inside different problems and finding ways to extract those principles and use them on the current problem or on problems down the road.
Standards for Mathematical Content
The Standards for Mathematical Content (Standards) are a list of specific mathematical topics that should be addressed in a mathematical curriculum such as "Work with numbers 11-19 to gain foundations for place value." The Common Core standards can be split into the K-8 grade level standards and the high school standards. The AoPS Beast Academy series and Prealgebra courses correspond to standards for grades 2-8 while the AoPS courses Introduction to Algebra A, Introduction to Algebra B, Intermediate Algebra, Precalculus, and Introduction to Geometry cover the Common Core high school standards (and considerably more). The Art of Problem Solving curriculum covers almost all Common Core standards, allowing it to serve as a full curriculum for students. We outline the specific standards our courses cover at the bottom of this page.
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.
|Number and Quantity||Intro Alg A||Intro Alg B||Intro NT||Intro CP||Intro Geo||Interm Alg||Precalc||The Real Number System: Extend The Properties Of Exponents To Rational Exponents, Use Properties Of Rational And Irrational Numbers|
|RN.2||RN.2||Quantities: Reason Quantitatively And Use Units To Solve Problems|
|Q .1||Q .1|
|Q .2||Q .2||The Complex Number System: Perform Arithmetic Operations With Complex Numbers, Represent Complex Numbers And Their Operations On The Complex Plane, Use Complex Numbers In Polynomial Identities And Equations|
|CN.9||CN.9||CN.9||CN.9||Vector And Matrix Quantities: Represent And Model With Vector Quantities, Perform Operations On Vectors, Perform Operations On Matrices And Use Matrices In Applications|
|Algebra||Intro Alg A||Intro Alg B||Intro NT||Intro CP||Intro Geo||Interm Alg||Precalc||The Real Number System: Interpret The Structure Of Expressions, Write Expressions In Equivalent Forms To Solve Problems|
|SSE.4||Arithmetic With Polynomials And Rational Expressions: Perform Arithmetic Operations On Polynomials, Understand The Relationship Between Zeros And Factors Of Polynomials, Use Polynomial Identities To Solve Problems, Rewrite Rational Expressions|
|APR.7||Creating Equations: Create Equations That Describe Numbers Or Relationships|
|CED.4||CED.4||Reasoning With Equations And Inequalities: Understand Solving Equations As A Process Of Reasoning And Explain The Reasoning, Solve Equations And Inequalities In One Variable, Solve Systems Of Equations, Represent And Solve Equations And Inequalities Graphically|
|Functions||Intro Alg A||Intro Alg B||Intro NT||Intro CP||Intro Geo||Interm Alg||Precalc||The Real Number System: Understand The Concept Of A Function And Use Function Notation, Interpret Functions That Arise In Applications In Terms Of The Context, Analyze Functions Using Different Representations|
|IF.9||IF.9||IF.9||Building Functions: Build A Function That Models A Relationship Between Two Quantities, Build New Functions From Existing Functions|
|BF.5||Linear, Quadratic, And Exponential Models: Construct And Compare Linear, Quadratic, And Exponential Models And Solve Problems, Interpret Expressions For Functions In Terms Of The Situation They Model|
|LE.5||LE.5||LE.5||Trigonometric Functions: Extend The Domain Of Trigonometric Functions Using The Unit Circle, Model Periodic Phenomena With Trigonometric Functions, Prove And Apply Trigonometric Identities|
|Geometry||Intro Alg A||Intro Alg B||Intro NT||Intro CP||Intro Geo||Interm Alg||Precalc||The Real Number System: Experiment With Transformations In The Plane, Understand Congruence In Terms Of Rigid Motions, Prove Geometric Theorems, Make Geometric Constructions|
|CO.6||Similarity, Right Triangles, And Trigonometry: Understand Similarity In Terms Of Similarity Transformations, Prove Theorems Involving Similarity, Define Trigonometric Ratios And Solve Problems Involving Right Triangles, Apply Trigonometry To General Triangles|
|SRT.11||Expressing Geometric Properties With Equations: Translate Between The Geometric Description And The Equation For A Conic Section, Use Coordinates To Prove Simple Geometric Theorems Algebraically|
|GPE.6||GPE.6||Modeling With Geometry: Apply Geometric Concepts In Modeling Situations|
|Statistics and Probability||Intro Alg A||Intro Alg B||Intro NT||Intro CP||Intro Geo||Interm Alg||Precalc||Conditional Probability And The Rules Of Probability: Understand Independence And Conditional Probability And Use Them To Interpret Data, Use The Rules Of Probability To Compute Probabilities Of Compound Events In A Uniform Probability Model|
|CP.9||Using Probability To Make Decisions: Calculate Expected Values And Use Them To Solve Problems, Use Probability To Evaluate Outcomes Of Decisions|