Mathematics learning benefits

The following are some general reasons for learning mathematics well.

1. Hierarchical: Later material in the curriculum depends heavily on earlier material (with a few important exceptions, such as many parts of geometry, plus advanced techniques within individual sections).
2. Unidirectional transfer to many other domains: Mathematics, at both the computational and conceptual level, gets used extensively in physics, chemistry, economics, and the social sciences. Mathematics, up to and including multivariable calculus and linear algebra, is particularly important for understanding statistics rigorously.
3. Importance for college gateway examinations (such as SAT and ACT): A strong background in mathematics helps with the math (quantitative) part of the SAT, as well as with the SAT Math Subject Test. It also helps with the math part of the ACT if you choose to take that test. The situation is likely similar in many other countries.
4. General reasoning ability: Mathematics, if understood well, can help with general reasoning.

Hierarchical nature of the curriculum

Later material in the mathematics curriculum relies heavily on earlier material, both at a computational and a conceptual level:

• At a computational level, techniques for solving problems for later parts of the curriculum typically involve reducing those problems to problem types seen earlier in the curriculum. Some elementary examples of this are that implementing the division algorithm requires knowledge of subtraction, and that adding fractions written in decimal notation requires knowledge of addition with carrying. This trend persists throughout the high school and early college curricula.
• At a conceptual level, it is hard to understand ideas in later parts of the curriculum if one does not understand earlier ideas well. It is difficult to understand basic algebraic notation without familiarity with arithmetic.
• Even the techniques in a part of a mathematics curriculum that do not get used later may be helpful for overlearning important techniques that are a part of the curriculum and get used a lot in later parts of the curriculum. For instance, the long division algorithm helps review key skills of subtraction, multiplication, number sense, estimation, and place value, and may be useful to master even if you never need to do long division by hand in the future.

One exception to the generally hierarchical nature of the curriculum is high school geometry. A bare minimum of geometry, including lines, circles, and angles, is essential for understanding coordinate geometry. However, many theorems about triangles and circles are not used in the rest of mathematics. Geometry may be worthwhile as a way of improving one's general reasoning and proof discovery skills, but it is not the only way to develop such skills, and it is not specifically important to master material about triangles and circles. That said, many people may deeply enjoy learning geometry and it may ignite their passion for mathematics, so this should not be construed as a recommendation against learning geometry. The point here is simply that the general arguments for learning math well do not apply with the same force to geometry.

Note that the arguments in favor of learning math well do not apply with the same force to other subjects such as physics, chemistry, economics, history, biology, or English literature. In most other subjects, there is a relatively small base of foundational knowledge and skills, and a large variety of other topics within the subject that build on those. The foundational base is important to understand, but dependencies between later topics are minimal.

Unidirectional transfer to many other domains

Concepts as well as computational skills learned in math classes are used in other subjects. The usefulness of math for other subjects is greater than the usefulness of other subjects for each other, or the usefulness of other subjects for math.

• Usefulness for Advanced Placement and college prep coursework: Advanced Placement classes (or equivalents) in physics, chemistry, and economics rely on basic knowledge of the precalculus math curriculum. Some parts of these classes benefit from a knowledge of calculus. It is possible to do well in these classes without knowing the math, but doing so may require considerably more rote memorization. Math knowledge can also help with SAT subject tests in these subjects.
• Usefulness for undergraduate coursework: The mathematics curriculum leading up to calculus, as well as multivariable calculus and linear algebra, is useful for physics, chemistry, biology, economics, and other natural and social sciences at the beginning and intermediate undergraduate level. More advanced mathematics is useful for these subjects at the graduate level. This means that whether you want to major in the natural or the social sciences, learning math well is useful. Even humanities majors can benefit by mastering some college-level social science material, and knowledge of mathematics can help with such mastery.
• Usefulness via statistics: The mathematics curriculum leading up to calculus, as well as multivariable calculus and linear algebra, is crucial for a deep understanding of statistics, which in turn is widely used in the natural and social sciences.
• Usefulness in programming: Although advanced mathematical concepts are not directly used in programming, mathematical thinking is closely related to the type of thinking needed for programming. Some mathematical concepts that have closely related concepts in programming include basic algebra (the use of variables for unknowns), functions, piecewise definitions of functions, indexed summations, and logic (use of and, or, and quantifiers). Learning the concepts in mathematics is neither necessary nor optimal for learning them in the programming context, but this does count as an additional advantage.
• Usefulness for graduate coursework: Graduate coursework in most of physics and in some parts of chemistry, economics, and biology require even more math than the above. In particular, parts of mathematics that are widely used in other subjects include differential equations, differential geometry, linear programming and optimization, etc.
• Usefulness in the real world: Functions and their graphical representation are crucial for understanding real-world data. Computational formulas are rarely used, but a conceptual understanding of derivatives and integrals can occasionally be useful in data analysis. The main utility of mathematics in the real world may arise from its role in giving a number sense and basic intuition for statistics.

Another related point is that topics in mathematics are often taught in a manner that is more general than equivalent topics in other subjects, so it is easier for students to transfer learning from mathematics to other subjects. There is some research that demonstrates greater transfer of learning from the algebra of arithmetic progressions to the physics of constant acceleration than in the opposite direction.