Learn mathematics well

Key takeaways

• If you had sufficient time and resources, a case may be made that you should learn all subjects in the curriculum well. But that's not possible in the real world. We still strongly recommend that you consider learning mathematics well, in addition to the subjects that fascinate you.
• In general, a number of people do not learn math as well as their potential. Not everybody can become a genius at mathematics, but most people (including you) can get a lot better than they currently are. A lot of people mistakenly believe, or affirm, statements of the form "I'm not a math person" and these prevent them from achieving their potential.
• If the resources at your school are not helpful (for instance, you have a bad teacher or peer learning environment) there are still many ways for you to learn math well. Some strategies are described and linked on this page.
• Learning mathematics conceptually is hardest, but it generates greater benefits than learning mathematical techniques through rote memorization and practice. However, even the latter may be quite helpful to you relative to not learning the subject at all. If you are thorough with the computational processes, it would make it easier for you to learn math at a conceptual level later when you have access to quality teachers and resources.

Why learn math?

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 gateways 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. [SHOW MORE]

For instance, techniques for solving equations in high school algebra rely on basic algebraic manipulation and arithmetic computation. Techniques in calculus (such as finding local maxima and minima) rely on solving algebraic equations.

At a conceptual level, it is hard to understand ideas in later parts of the curriculum if one does not understand earlier ideas well. [SHOW MORE]

For instance, it is difficult to understand how differentiation without understanding limits. It is difficult to understand any of calculus without understanding functions and graphs of functions. It is difficult to understand functions without understanding basic algebraic notation. 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 do long division by hand.

One exception to the generally hierarchical nature of the curriculum is high school geometry. [SHOW MORE]

A bare minimum of geometry, including lines, circles, and angles, is essential for understanding coordinate geometry. However, many aspects of geometry, such as geometric facts 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.

How can one learn math well?

There are some general good study habits that help with learning any subject well. There are also some general online learning resources that help with all subjects. On this page, we describe some aspects of the strategies and resources that are specific to mathematics.

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