What are good habits in learning mathematics well?

Question

Please bear with me while I provide some background and then ask my question, which I hope you will be able to answer and thus aid my capacity to learn mathematics more efficiently and thoroughly.

I am 30 years old and I have always desired to learn mathematics. Until about a year ago, I have believed that I am not clever enough to grasp the foundations of the subject. In school, while I have done well in sciences, my ADHD has proved it difficult to focus for a considerable amount of time and dedicate and drill the math foundations necessary for progress onto more challenging aspects of the field. However, the desire to be able to explain the phenomena around me in mathematical terms has never left and, as mentioned, one year ago I have decided to put the effort in and started investing a considerable amount of time learning the very foundations. I am currently on Algebra 2 on Khan Academy and I am supporting the learning through other online sources. I am an autodidact.

However, my time is limited. I have other responsibilities and other fields of inquiry which demand my attention. To cut it short, in about 5 years I wish to make a transition into computational neuroscience (masters and onward to PhD.). I am going back to university next year to do my second degree in neuroscience and wish to study mathematics in parallel.

I am asking you, veteran mathematicians to provide me with advice on good habits. On which aspects of mathematics should I focus on past Algebra 2. Which calculus should I learn? What textbooks do you recommend? Any online lectures? Am I still able to grasp advanced mathematics at 30? How much time, on average per day, should I dedicate in order to make progress as efficient as possible. What habits should I culture? I normally do 2-3 hours per day at the moment but I find that it is not enough; the progression is steady but lacks thorough understanding that is based on underlying concepts which would make my knowledge of mathematics less superficial.

Any advice which you can give me will be deeply appreciated, considered and applied.

Answer 1

Unfortunately online learning isn't the best way to "grok it" for anything beyond the very simplest things that can be learned by rote.

To learn mathematics as to learn most other things you need to actually change your brain's wiring. See The Art of Changing the Brain by James E Zull for a run-down on the science behind it.

To do this you need two things: reinforcement and feedback. In a normal university course, in which there is interaction with the professor, the reinforcement is in the form of exercises, student written proofs, exams and such. The feedback is what you get from the professor who judges, and hopefully comments on your work. An opportunity is also provided for asking questions of the professor or a TA. The reinforcement is what wires the brain. The feedback is to help assure that you don't reinforce the wrong thing.

The goal is insight into the inner workings of some part of math. I once learned a lot about how "real valued rational functions" work by graphing (by hand) literally hundreds of examples using first and second (primarily) derivative information and some notions of asymptotes. This gave me insight, not only in to the specific topic (rational functions) but into the deep relationship between derivatives and integrals.

In an online course, the lecturer will provide correct (we hope) explanations of things and, perhaps, some exercises. If there is a text book recommended then you can probably find more exercises. You can always get a book in any case, bought or borrowed. But it is the feedback that is lacking. If I do an exercise, especially a complex one (like a proof) then I'd like someone with more knowledge than I have at the moment to comment on my work. This is true whether I think I've done it correctly or get stuck somewhere.

You can also take good notes, summarize those notes, and if the short summaries are on index cards, carry them with you easily for constant review. But that only helps with memorization, not with gaining competence through working with the material and discovering the consequences. Memorizing the definition of the derivative (in some future course) will tell you almost nothing about how functions behave.

So, seek to find ways to reinforce what you are "seeing and hearing" and try to find a way to get feedback. This can also come from peers if the professor is just an online presence to you. Study groups that meet occasionally and comment on each other's work, looking for errors in the other person's work and insight into your own.

An important hint: Reading solutions of exercises/problems is nothing like developing your own.

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You learn to swim by swimming. If you want to do it at a high level (competition), you swim a lot and you need a coach; reinforcement and feedback. You don't learn it by watching.