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# Mathematics learning benefits

This page lists benefits of learning the subject mathematics. In other words, it tries to answer the question Why should I learn mathematics? |See all pages on the benefits of learning specific subjects

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:

### Hierarchy at a computational level

The computational skills mastered in an earlier part of the mathematics curriculum are useful in later parts. Examples are below:

• The algorithms for multiplication of multi-digit numbers rely on algorithms for addition.
• The algorithm for long division relies on the algorithms for multiplication and subtraction.
• The algorithms for exponentiation rely on algorithms for multiplication.
• The algorithms for operations on fractions rely on algorithms for the corresponding operations on integers.
• Solving equations (in prealgebra and algebra) relies on a thorough understanding of arithmetic.
• Finding local extrema of functions in calculus relies on equation-solving (from algebra).
• Differentiation and integration in multivariable calculus rely on differentiation and integration in single variable calculus.

### Hierarchy at a conceptual level

• The concept of multiplication relies on the concept of addition (multiplication is repeated addition).
• Addition of fractions relies on a deep understanding of equivalence of fractions (i.e., you can multiply both the numerator and the denominator of a fraction by the same number and not affect what the fraction represents).
• Understanding the meaning of algebraic expressions (such as polynomials) requires a deep understanding of the arithmetic operations (addition, subtraction, and exponentiation) used to build those expressions.
• Understanding calculus requires a strong understanding of algebraic, graphical, verbal, and numerical interpretations of functions.

### Some apparently useless parts of the curriculum help with overlearning

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. Examples:

• 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.
• Integration by parts problems help review the basics of integration of easy functions, as well as give a qualitative sense for how the integrations of basic functions behave.

### Geometry as a partial exception

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.

### This is specific to mathematics!

This argument in favor of learning mathematics well does 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. In mathematics, on the other hand, the chain of dependencies is really long (stretching all the way from addition to calculus).

## 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.

### General point about greater generality

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.

### Elementary school level: usefulness of understanding arithmetic

Basic arithmetic knowledge, along with basic reading ability, can be crucial for young children to be able to make sense of the world around them. Mastery of arithmetic concepts makes it possible for them to read a lot of non-technical material targeted at adults, including fiction and popular nonfiction.

### Middle school level: usefulness of getting started early on algebra

The basic idea of algebra -- using variables for unknowns and reasoning with them -- is a powerful one. It opens up the way for the use of a wide variety of mathematical representations (such as graphs) that are found in discussions of real-world issues and data (such as election data, sports data, economic indicator data). It also lays the foundation for high school science subjects, such as physics and chemistry, where we often deal with formulas relating unknown physical quantities using the language of algebra.

### High school level: 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.

### College level: 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.