Learning mathematics

Math can be seen as a powerful tool that enables discoveries and advancements in all kinds of science and engineering. Modern computers, the internet, lifesaving technology such as pacemakers and MRIs, frontiers of technology such as in silico models of cells and biological processes, and the creation of robots that can navigate through unknown environments or cars that can drive themselves---none of these would be possible without advanced math. Mathematics has the ability to benefit mankind, to save lives and enhance the quality of life, as much as or more than any other field.

Math is also something that can be done purely for enjoyment---our mathematical knowledge is one of mankind's highest intellectual achievements, to be treasured as much as our great works of art or literature, and understanding and discovering math can provide a kind of pure intellectual pleasure analogous to what one might feel when reading poetry. Many people enjoy working on puzzles. That's an example, on a small scale, of the kind of intellectual enjoyment you can get from doing math.

Learning math used to be a lonely activity: a great mathematician a few centuries ago may have been isolated by geography from any other serious mathematicians, who were few in number anyway because not many people had access to good education. Now the situation is different. There are entire armies of serious mathematicians loose in the world, well-educated and inspired, pushing back the frontiers of knowledge in every direction, and they are all connected in a global community by the internet. I hope you're as excited about this as we are.

If you want to learn math, here's some advice.

Advice for learning math
1) Strive to UNDERSTAND what you are learning, rather than just memorize.  Don't just memorize a formula---instead, know how to derive it from scratch, understand how it was thought of in the first place.  Check it and make sure it's not wrong.  Einstein realized that classical physics was wrong, and he fixed it.  Maybe you will do something similar.  How can you figure out how to derive a new formula or theorem, if you don't know how to derive the old formulas or theorems?  How can you convince yourself that a new theorem or fact is true, if you can't convince yourself that the old theorems or facts are false?

This approach to learning math can be quite time consuming, and unfortunately you may not always have time to understand things as well as you would like. Sometimes, because of lack of time, and the desire to learn other things instead, you may need to accept a theorem or formula or fact without derivation or proof. That's okay. But the most basic thing to know about learning math is that it's supposed to be understood, not memorized. That's the only way to enjoy it.

2) Look at lots of specific examples.  Especially in more advanced math, things can get quite abstract.  Textbooks don't emphasize this enough, but the key to understanding abstract ideas is to look at a lot of specific, concrete, often simple examples.

3) Work on your problem solving skills.  You can do this by trying to derive formulas or prove theorems before reading the answer in a book or hearing the answer from someone.  You can do this by working on the wonderful contest math problems that the world's contest math community has provided for us.  Check out  or books like The Art of Problem Solving or The Art and Craft of Problem Solving for more on this.  Do contest math, at whatever level is appropriate for you.

4) Make friends with other people who are learning math.  Study with them, share your knowledge with each other, and learn from each other.  Everyone learns different things, and it's fun (and efficient) for us to teach each other what we've learned.

5) Read about the history of math.  Morris Kline has a three volume history of math that is quite good.  Isaac Asimov has great histories of science.  I Want to Be a Mathematician, by the 20th century mathematician Paul Halmos, is a great book.

6) Find the great internet resources out there, such as MIT OpenCourseware and the extraordinary community of problem solvers at.

7) Do math. Don't read math books like a novel. Math is not a spectator sport; apply your knowledge, either by solving problems, or even contributing to Mathematics articles here.

Well, that is some advice for how to learn math. Now, what should you learn?

Math subjects to learn in order to become a mathematician or scientist
1) Arithmetic.  Counting.  Addition and multiplication tables.  How to use the basic properties of arithmetic, together with the addition and multiplication tables, to add and multiply LARGER numbers.  (Rather than make children to memorize the long multiplication algorithm, I propose we teach them to break a large multiplication problem up into a bunch of smaller ones by using the distributive property. This will be slower, but they will UNDERSTAND it and therefore enjoy it. Speed is of very limited value, now that we have calculators and computers.) How to divide numbers and get a remainder.  Fractions, equivalent fractions, reducing fractions, adding and multiplying fractions, converting a mixed number to a fraction and vice versa.  The decimal system of representing fractions.  Arithmetic with negative numbers.  A minus times a minus is a plus.  Dividing a fraction by a fraction.  Stuff like that.

2) Algebra

3) Geometry

4) Pre-Calculus

5) Calculus

At this point, there are several options for what to learn next. (In fact, there were options earlier too. For example, it is possible to learn Number Theory before Calculus, and this happens at several excellent summer math programs for high school students.)  The following list of subjects might be appropriate for someone who wants to become an applied mathematician.

6) Multi-variable Calculus and Vector Calculus

7) Ordinary Differential Equations

8) Linear Algebra

9) Real Analysis

10) Partial Differential Equations

11) Fourier Analysis

(Numbers 10 and 11 should probably be woven together, as PDEs can motivate Fourier Analysis, and Fourier Analysis helps to solve PDEs. It's important to understand Fourier Analysis from a Linear Algebra point of view, and indeed to learn Discrete Fourier Analysis--which is some great Linear Algebra--as well as continuous Fourier Analysis.  In general, it's important to understand the analogy between the continous and the discrete.)

12) Numerical Analysis

13) Other more advanced topics.  Look at the undergrad and graduate curriculums for math majors at good universities, like MIT or Berkeley.