How to Teach Yourself Mathematics
FAQs
How do I teach myself mathematics?
What resources are there to learn these topics?
How should I structure my learning?
What is a typical sequence of subjects to learn undergraduate mathematics?
What are some recommended books for these subjects?
FAQs’As
How do I teach myself mathematics?
Start by being very clear about what you want to learn and why — this will strongly influence every other decision you should make.
Are you doing this out of career needs? Then you need to be clear about what, exactly, your particular career requires. Are you going into quantitative finance, and already know calculus? Then research what should come next, specifically for where you are now and where you’re ultimately trying to go. Are you interested in AI but your algebra skills are rusty? Well, you definitely need to work on algebra, and there are a lot of resources for that. But you can already start planning the next several topics that you’re going to want to study after that.
Are you doing this not for a career but just out of interest? Then be very clear about what you’re interested in, or at least, what you are most interested in right now which you could learn right now. If you’re interested in Bayesian statistics, but your calculus is not strong, then Bayesian statistics is out of reach at the moment — so try to become interested in calculus for the next year, until you can move on!
Once you pick a topic, next is to pick the exact resources that you will use for your studies. I’ll describe how to do that below.
After you pick some resources, make progress every day. Don’t skip days — if you’re too busy with other things to carve out at least a half-hour each day, then you’re probably too busy to do this at all.
Progress doesn’t always mean moving on to the next topic, or even solving an exercise. Just thinking, struggling, and making an attempt at a problem is progress. You might even feel like you’re making progress when you spend a day on a problem and don’t figure it out.
But there is progress, I promise you, even when it seems invisible. I cannot tell you the number of times I’ve worked on a problem for a day, seemingly making no progress at all — but I go to sleep, wake up the next day, and have the solution in my head! If you’re really focusing on this material, your brain is doing things to store and process the information that you aren’t even aware of.
But progress does not mean reading. Reading is what you do just to be able to get to the exercises. But the exercises are where the real learning happens. Exercises are not in-between the reading, the reading is in-between exercises. The exercises are the “main dish”. If you skip exercises, you are fooling yourself about how much you’re really learning. Even when you think they’re tedious, boring, or whatever — do the exercises and do not fool yourself about how well you understand what you’re reading.
From here there is a bit of a question about how you structure your studies. I’ll address that in more detail below.
2. What resources are there to learn these topics?
Schools
It’s not for nothing that we have schools; they’re pretty great for learning! They’re not perfect, but if you can afford them and make the time for classes, I honestly don’t think there’s anything better.
Still, either because of time or money, or other circumstances, it’s not always possible to attend. Therefore, other resources are below.
Other people
If you don’t attend a school, I think one of the biggest things that disadvantages you as a non-traditional learner, is the lack of interaction with other people learning the same subject, at the same time, as you.
Let me make a bit of a case here: I have found, as a tutor, that I never learn a subject more deeply than when I have to explain it. The act of explaining forces your brain to do something that you don’t have to when you’re by yourself. You have to integrate this understanding into language, meaning, intuition, technical expertise, and just about every other part of how you think in order to communicate. Otherwise you know that the person you’re talking to will not understand, and probably (correctly) believe that you do not understand the topic either.
Also being social with your learning has other benefits. From just feeling better than being alone, to having an “accountability partner”, to sometimes letting the other person explain things that you were not able to figure out. Ideally, a learning partner is a symbiotic relationship, where both people gain some value from the alliance.
Reddit has at least one or two communities where you can try to connect with other people for studying. Probably there are Discord channels or other resources too.
Online resources
YouTube (do a lot of searching, there’s a lot available)
Khan Academy
IXL Learning
Coursera
EdX
MIT Open Courseware
CMU OLI
StackOverflow - Mathematics
Discord channels dedicated to math and math help
Offline resources
Textbooks are of course a great resource. And of course you can search Amazon or local bookstores.
There is also libgen, which you may need to research in order to use and find what you need.
Tutoring
You can also always hire a tutor!
You may find that it is especially cheap to hire a tutor over the summer when we are much, much less busy than during the regular school season.
3. How should I structure my studies?
Find an online course or retired course web page
I recommend finding a course syllabus for an old, retired course on the subject that you want to learn. For example, here is a retired course page for abstract algebra:
https://math.berkeley.edu/~ianagol/113.F10/
This is nice because it tells you exactly what to read, in what order, paired with a schedule of exercises. Sometimes you can even find retired course pages for your particular subject, which have links to YouTube videos for lectures and so on.
So the ideal is to find a retired course web page with as much helpful stuff — essentially, this way the professor structures your studies for you!
Almost as good, is to find a free online course, like at Coursera or EdX. These often do the work of structuring your studies for you.
Go it alone
If all else fails, you can either try to just read the book and do the exercises at the end of the chapter — here you will have to just rely on your instincts about how much exercise is enough to move on. You never want to do no exercises, but also, rarely should you do all of the exercises at the end of a chapter. That can cause a lot of wasted time that you could have more productively spent moving to the next subject. Picking the right balance is an art you can hone with lots of experience, but there is no generally applicable rule for how to strike the balance.
Tutor
Alternately, again, you can try to find a tutor who can give you a schedule of reading and exercises.
4. What is a typical sequence of subjects to learn undergraduate mathematics?
Below is a bare-bones sequence, which would be common to anyone who has finished a mathematics degree.
It is broken into levels. Inside of any level, the order of the courses does not matter, so they can all be studied in parallel. But in general, you should finish a level before moving on to higher-numbered levels.
It also doesn’t include classes that could be helpful but aren’t strictly necessary, like statistics or pre-calculus. The point is to list the least possible number of classes that would get you to the common core material that all math undergraduates should be exposed to.
You should feel invited to add into the sequence extra things that are special to your interests, or which might help or smooth over the transitions between levels.
I also list a subject at the earliest possible level that it would be reasonable to study it. For example, in America it is very unusual for a student to study real analysis before studying the full calculus sequence. Still, after calculus 1, there is really no reason why you can’t go ahead and start on real analysis. Hence I’ve put this subject at a lower level that one might expect given that it’s usually only studied by a relatively more mature student than one who takes calc 2.
Level 1
Algebra, geometry, symbolic logic.
Level 2
Calculus 1, linear algebra, discrete mathematics.
Level 3
Calculus 2, differential equations, abstract algebra, real analysis.
Level 4
Calculus 3, complex analysis, topology.
5. What are some recommended books for these subjects?
Rather than recommend specific books, I would probably encourage anyone to first pick a resource and syllabus as I described above for structuring your studies.