That said, I love how this article gives practical hints on how to replicate the insight and solve the question, rather than just the insight itself.
While it's frustrating and time consuming, self teaching a difficult subject is just like that unless you're a god amongst men (and few of us are). Sometimes you'll want to fight through the strange unproven thing by thinking hard for a couple weeks about it while googling intermittently to find key steps. Sometimes you'll have to give up on it and keep moving. If it's foundational then fighting through can be highly beneficial, but a lot more things are presented as foundational than actually are.
I'd recommend finding good books by searching relentlessly on reddit and other forums for opinions, dedicating the time necessary to self teach something difficult (it can take upwards of a year to get through a smaller textbook if you have other things in your life going on), and if you really want it then fight for it. Give it everything you have, really let the problem consume your thoughts because eventually you'll wake up at 3am and know exactly what to do. And finally, move on if you don't want to do that. Try just keeping moving. Review from time to time but don't let a hard first couple chapters prevent you from ever learning the concepts. Or you could find something you want to know and work backwards through every term that's used until you're at a concept and then attempt to apply it to the larger idea.
In general, self teaching math is extremely difficult, and only really works if you're willing to dedicate the time to fight through ideas.
> "Review from time to time but don't let a hard first couple chapters prevent you from ever learning the concepts."
This is a very good approach, and I wish I started doing this earlier. Even in my university math courses, the professors sometimes skipped ahead to have students focus on a few later chapters before coming back, or told the class to skip several pages in the book. I also found that working on later exercises in a textbook would sometimes help me better understand concepts introduced in earlier chapters.
Lastly—though this may not be completely relevant to studying mathematics—I've explicitly been taught in various language courses (explicitly for audio courses and implicitly for in-person university courses) that it's okay to move ahead if I know at least 80% of the material. The percentage may be higher for studying math topics, but especially for someone self-learning out of interest or for a specific application, it's much more preferable to move forward and revisit earlier exercises as needed, instead of quit the book. If you find yourself getting lost in later chapters, there is no problem with revisiting earlier chapters. You'd also likely be no worse off (possibly even better) than many undergraduates studying the textbook for a course for the first time.
The most important thing is just to not quit the habit of consistent study. Perfectionism in understanding is a pitfall for self-directed studies, which consistency in studying beats every time.
> it's okay to move ahead if I know at least 80% of the material.
One of the worst feelings is moving on in a textbook and realizing you are indeed totally lost.
Here’s my own list of necessary but not sufficient conditions for deep learning:
- Motivation. Easy to overlook; hard to get if you don’t already have it.
- Frequent experience of “I have no idea how to solve this,” followed by hours or days of playing with the problem, followed by a eureka moment. You can’t be sure you’ve learned the thing unless you’ve constructed the solution yourself. Builds confidence too.
- Seeing the same material in different contexts or presented in different ways. It’s like looking at an object from different angles.
And for bonus points:
- Teach the concept to a curious friend. Their questions will lead you to deeper understanding.
That said, I have been long thinking about a dependency graph for knowledge, where the nodes are great books on the topic.
Then find the syllabus for each course and look at the recommended textbooks.
You should be able to find the best ones that way.
The point is, many part of high school math is actually really “algorithmic”. I was one of the few in my class who absolutely loved coordinate geometry over “normal” geometry, because I simply felt really comfortable with equations — once you have it down, you can basically solve it, even if it is harder than the “notice this and that” elegant solution.
Most integration problems require this intuition-based solution which has a certain elegance to it.
It was especially humbling to me that Wolfram alpha fails most of the interesting calculus problems I encountered during my analysis classes, but after a while I managed to solve most of them. But it unfortunately does disappear after not using it for a time..
That's the Feynman method: write down the question, think really hard, then write down the answer. Only three simple steps!
Unfortunately, some of us are not Feynman.
Then for calculus, we learned concepts, like what a derivative is, and I understood that, and understood conceptually (as in, what everything "means" and what it tells you) but I could never take that concept knowledge to the practice problems with me.
I could follow along as the professor walked us through a problem, showing us what heuristics helped and what patterns to follow and how to manipulate the functions to get to something that followed one of the patterns to pull an answer out of your ass, but I could never commute those heuristics and patterns to novel examples. It's weird because I was great at doing the exact same thing for physics: Taking a novel and purposely opaque problem and finding which pattern it corresponds to.
Derivatives are friendly enough that I feel like a certain amount of grinding is justifiable, and it's justifiable as practice for symbolic manipulations in general. But we're leaving a lot of useful stuff on the table while we're jamming down how to integrate with trig identities and other such things.
But it's all pie in the sky anyhow. The Curriculum Must Not Be Changed. The Curriculum Is Perfect. Nothing Can Be Dropped From The Curriculum. I don't know what miracle would have to be worked to get people to reconsider the curriculum from some sensible perspective of what students should be taught rather than the way that question happened to be answered about 100 years ago when the curriculum froze into place, but it probably involves the total destruction of the school system at this point. I can't even get people to process the idea that shoving incomprehensible combinations of 450-year-old words in what is effectively another language at children and telling them this is High True Art is a bad idea, what chance is there of prying away the utterly vital fact that cos(θ/2) = SqRt((1 + cos(θ))/2) out of The Curriculum?
Maybe if colleges continue dropping the SAT and the ACT we can start actually fixing these curricula.
I have a blog article about this in progress.