Sometimes, it can be overwhelming to even think about how to approach a subject. Where do you start?
Here, I list some pointers and resources that have helped me most effectively learn math. I also point out some resources which I may not have used myself, but that other people I know have had positive experiences with.
Before I list them, I will say that here I prioritize better coverage of introductory topics over comprehensive coverage of more topics. The reason for this is that there are already many, many resources for math online now. I contend that if people understand the basic topics more clearly, then they will have a better framework for approaching anything that builds on top. However, without that strong framework, they won’t be able to effectively use all the resources out there. Besides, the subject of math is so staggeringly vast that it is impossible for any one course, textbook, or resource to be “completely” comprehensive anyway.
Also, I believe different situations may require different resources, and no one list may be able to cover all cases, so I may end up more closely targeting the situations that are closer to my experiences (since I may be able to understand those better off the bat.)
I now make some general points to keep in mind while learning math.
- Even if your goal is not necessarily math itself but a subject that applies math, I would still suggest that learning more math is always helpful, and advanced math tends to eventually show up even when it doesn’t seem necessary at first. (For example, this happens often in physics, which is actually a main point of one of the most famous articles of 20th century physics, written by a top theoretical physicist of the time.)
- It is important to become familiar with the notation. Remember that it is just symbolism to denote concepts that you already know when they are written out in words. Learn and practice how to “translate” words into mathematical notation and vice versa.
- Just as we use vocabulary words in human language to describe ideas and concepts more concisely than we could without those words, so we have many new terms in math that give more meaning to mathematical statements. Don’t be put off by all the new terminology: once you parse them and break them all apart, the statements will make sense.
- If you see a daunting block of math in front of you, don’t give up. Rather, take it a step at a time. Pause as much as you need to on each sentence, or even each term, and make sure you are comfortable with it before proceeding. You don’t need to finish an entire block of math in one sitting — you can always leave and come back to it later. As you study more math, you will become better and faster at getting through the material.
- Traditional math curricula neglect explaining why formulas and theorems are true, rather just expecting students to parrot them “because the teacher said so.” Instead, emphasize learning proofs as well, where you can understand the justification and logic behind the results. Then, you don’t have to take the teacher’s word for it, and you can be more confident in your work.
- On the flip side, one aspect of math that can be particularly challenging, especially as you advance further, is understanding why a particular statement isn’t actually as obvious as it may seem and in fact requires careful proof. It can become easy to oversimplify these subtler points and thus miss important concepts. To combat this, be meticulous in reading theorems and understanding what they are actually saying. Consider not just the typical use cases but also the more surprising scenarios in which a theorem would apply, to help you understand its true power.
- Many math practice problems have a format where you learn a procedure and then just repeat it blindly many times for different situations. Instead, look at “brainteasers” that challenge you to think more about how to solve them. Math contests and extracurricular programs typically feature more problems of the latter kind. For a more detailed argument explaining the benefits of this approach, check out this post.
- If you find that a certain topic is difficult, try a multiple-pass approach. First, go through and learn the definitions, along with their motivations, and understand what the connections between them are by learning the theorem statements, without the proofs. Next, go back and learn the proofs, focusing more on the high-level methods and ideas behind them. Finally, go back and focus on technical details, making sure you understand every subtlety.
- You can use a multiple-pass approach for homework and practice problems too, especially “brainteaser” problems like I mentioned earlier. First, try to figure out an outline of your solution, without getting too bogged down in the details. Then, in successive passes, solidify those details so that it all holds up together. At the end, write down your solution from start to finish to make sure you didn’t miss any spots.
- Challenge yourself to come up with some of the math yourself. For example, start with some motivation, and write down a mathematical definition from it; then, compare your definition to the standard one. Or start with a standard definition, and try to prove one of the standard theorems about it. Sometimes this is very hard to do, so don’t spend too much time on it, but often times it’s actually not too bad once you try it, and in those cases, nothing compares to the feeling of discovering something yourself, as opposed to seeing someone else lecture about it.
- A subject called discrete math is often neglected in pre-collegiate math curricula, but it ends up being very important to much of later math. Emphasize learning discrete math sooner rather than later. While it is typically covered way after algebra, you can actually get started with it right after algebra.
I now make some suggestions for resources. But first, shameless plug: I have written expository notes on specific math subjects here and there, collected here. For those subjects, my suggestion would be to use my notes, as there I have tried to combine the best of all the resources I have seen, along with my ideas, to create what I believe are the best materials. Of course, I would love your feedback there, and you should certainly consider other resources in addition to mine — studying a subject with multiple resources enhances the learning experience greatly.
Beyond my notes, here are my suggestions. First, I list resources that have complete curricula, including clearly described paths for going through them:
- Grades K-12
- Beast Academy: I did not use this myself, but it’s from the people behind a math program called Art of Problem Solving, which I have positive experiences with. From the website, this is meant for elementary and middle school math. Note that it’s not free (pricing is on their site.)
- Art of Problem Solving: I have positive experiences here. This is one of the resources that helped increase my interest in math, and it also influenced my philosophy around studies in general. Furthermore, they integrate proofs and discrete math early on. (Look at my pointers above.) This is for middle and high school math, but they also go into math that is traditionally covered in college. Note that some of their materials are free, but others are not (pricing is on their site.)
- Khan Academy: I did not use this myself, but I have heard good things about it from others. I did use Khan Academy’s videos on chemistry, which I liked. From the website, this covers math up to and including an associate’s degree. This resource is completely free.
- Beyond K-12
- I was originally going to include resources on specific math subjects beyond K-12 here, but there are so many subjects within math that one page wouldn’t be enough to collect resources on even a good fraction of them. Instead, I’ll suggest the general resources below for this.
- I will address one subject that seems to be viewed as notoriously difficult, though the resource I used made it much more palatable: for abstract algebra, I suggest Charles Pinter’s A Book of Abstract Algebra. This is readable after just K-12 algebra. This book is not free (pricing depends on where you buy from.)
There is also in general a lot of information and knowledge out there on the Internet that may not always be well-organized, which may be very useful but may also require you to “piece it together.” Here are some resources that can fall in this bucket:
- MIT OpenCourseWare (and other free online courseware): Some universities have initiatives where they publish many of their course materials (lecture notes, homework, etc.) online for anyone to use. I have used MIT’s most extensively. Even without an explicitly named initiative, many college courses have freely accessible course websites which contain their materials. Since these are college materials, this is best explored after a high-school program. Also, the availability of materials is generally inconsistent among courses; some course websites may be fully-featured, while others may have a lot less. This is completely free.
- Wikipedia is actually a great resource for math, and it is incredibly comprehensive too, at least in terms of breadth of topics at an introductory level. This is completely free.
- MathWorld is another great encyclopedia-style resource for math, and it is a great jump-off point for exploring math further. This is completely free.
Here are some points to keep in mind when using these “general” resources.
- Sometimes the material may be more “fully featured” (in terms of including background, proofs, etc.), but other times it may not be.
- As already discussed, sometimes you may need to figure out the prerequisites, the logical organization, etc. of the material yourself.
- Because of the last point, these resources might not always provide the best introduction and might instead be better after seeing an introduction elsewhere.
- Also related to these points, if you are studying something and you encounter a part that you don’t have the background for, while you may be able to skip just that part and come back to it later, this may not always work since math is ultimately cumulative (later knowledge builds on top of previous.)
Despite those caveats, I have found MIT OpenCourseWare pages and Wikipedia articles that have been well-written and comprehensive. It varies depending on the subject you’re looking at, so definitely give these resources a try!
First written June 2022, with additions and small revisions for clarity later on.
