Course Notes – J.S. Milne
jmilne.org
jmilne.org
I don't think you can. Sure there are a select few insanely motivated geniuses who could do it. But learning math is just really, really hard. Harder than anything most people ever come in contact with. It's kind of like being able to punch yourself in the face. You can try to, but you'll never be able to do it with the same force as someone else. Or like being on a sports team, where a good coach will push you far beyond the limits of what you could have possibly set for yourself. This is coming from someone who taught themselves to code completely solo from nothing and built a successful career as a software engineer. I've recently gone back to school because I simply could not teach myself even remedial high school math with sufficient rigor to prepare for upper division science/engineering courses.
I think the issue is not how hard it is, it's just that most people have never done anything like learning advanced mathematics before. If you're on your own, you can't learn math by autopilot and reading and regurgitating and allowing the info to naturally propagate through your brain. You have to learn to focus and interactively refine your own version of mathematics.
Students often don't fully grasp one concept before the teacher moves onto the next topic. Once this happens, the student is missing a crucial building block and tends to just get stuck with all the later things that build on it. This is a common experience in schools and I suspect is the origin of the "math is hard" trope.
It can be hard for a student to self-evaluate whether he/she has full mastery of a topic. This is exacerbated by situations where a struggling student will go to office hours or work with classmates and "solve" a problem, without having acquired the skills to solve slightly different permutations of that same problem on their own.
In programming, we don't have this problem. You can write HelloWorld without understanding console IO, terminal protocols, window buffers. You can write a webapp without knowing the OSI stack. Software is built upon abstractions.
I'm not an educator so I can only speak from my experience and the experiences of people that I know.
I've found the opposite around foundational concepts -- very often you simply learn how to use them mechanically with no real understanding, and then move on. Only much later do you get the epiphany that lets you unlock the understanding.
I found this first with trigonometric functions; I had "SOH-CAH-TOA" memorized, and the double-angle and a+b things memorized, and could solve basic problems about triangles. It wasn't until much later on (in calculus) that I broke through and really understood what the hell was going on.
Limits I learned early in calculus, and found them incomprehensible -- I could mechanically complete epsilon-delta proofs and intuitively apply them to things like single-point singularities or rational function evaluation, and even follow the sin(x)/x proof, but it wasn't until much later (after differential calculus, and maybe even later than that) that it just clicked and I had trouble remembering what I found difficult about the concept.
Same with quotient groups in abstract algebra; it wasn't until field theory and Galois theory that I realized why my textbook used the Z/nZ notation instead of just saying Z_n like everyone else did.
Learning about asymptotes in rational functions was something that my high school did in pre-calc; that was a concept that I was perfectly capable of mechanically executing without any understanding. There are a million examples of this kind of thing for me.
And so on -- through differential geometry and topology and algebraic topology, complexity theory (formal computer science notation is made even worse if you've actually used computer programming languages, because it seems like an alien language -- it's always stuff like "take the machine S(sigma, alpha, q, r_alpha, delta) over the language M_gamma" -- what? In what sense is this a machine?). I can even feel echoes of this in terms of learning addition and multiplication vs. understanding what multiplication is -- why is multiplication associative is an intuition that came late to me, after I was able to see the geometric analog of multiplication.
For a long time my thought was that this could be fixed by introducing clarifying concepts earlier -- I feel like the concept of a limit as you go to infinity is easier to understand than local ones (something like, if the limit is infinity, a game, "hey, I say infinity, so you say, 'bet it's not bigger than 1,000!', and I say 'if I plug in 10, then the function evaluates to 2,000', and you say 'bet it's not bigger than a million'" -- if I can prove that I will always win this game, then I've proven the limit).
But I think now that the progression from rote and memorization to deep understanding is something that can't be hurried -- as a student, you just have to have the dedication to keep learning more even when the foundation is weak.
I'll accept the premise, but I still wonder if there are things that can be done to make it easier for someone. In my case, I've been trying to learn some more mathematics recently, and one of the most annoying things is coming across notation that isn't defined in a paper, presumably because "everyone" who can read the paper is familiar with the context and knows what the "skinny long arrow" means (good luck with that internet search). I wonder if there could be a wiki-like / forum / stackoverflowish site, which people could use to discuss and provide running commentary on a paper/book. Especially useful would be the ability for people to be able to annotate the paper by translating the formulas in to a formal language where you could track down the definition of the various operators, and try to figure out why the author used both of → and ↦ in the paper, when they both appear to be for functions/maps. (Just to preempt the easy objections, I'm not trying to suggest that each paper be formalized and proven in something like Isabelle/Coq).
In the ideal form, this website would allow you to see the paper or book page in question, and then see all the people who commented or had questions on each particular sentence (in the margin?). There could be filtering and voting so that experts could bypass the newbie commentary, etc..
I suppose part of my problem would be solved by getting a book like:
https://www.amazon.com/Mathematical-Notation-Guide-Engineers...
...(which I just came across when composing this message).
Maybe someone has a other suggestions for something like this? Maybe a site similar to this already exists?
And on a slightly related note to making things easier to learn, I think learning programming is much easier than math, because even though both are abstract, at least with programming you get a tangible, concrete thing (the program) that you can run and modify and extend, and the computer will tell you when you went wrong (e.g. won't compile, output result is unexpected, etc.).
You wouldn't try to learn math from papers until you've built that foundation (unless you have access to a tutor/mentor), at which point the notation usually shouldn't be an issue.
http://www.shapeoperator.com/2016/12/12/sunset-geometry/
...might be easier for me to understand if I could use Haskell to implement the wedge and geometric product operators on an algebraic data type describing the scalar/vector/bi-vector thingy. There is probably an applied vs. pure thing here as well. My motivations for investigating geometric algebra is to see if geometric algebra makes synthesizing mechanical linkages easier, whereas maybe most expositions on geometric algebra are focused on teaching geometric algebra to advance the state of geometric algebra. That's probably a long winded way of saying that mathematicans are writing for mathematicians (whether by design or accident). I suppose I should re-read Mindstorms again, but this time in the context of adult learning.
Unlike mathoverflow, it is meant for every kind of math question below research level.
(Regarding $\to$ vs $\mapsto$, I think of it as type-level vs lambda expression. I think you can find it in any introductory abstract algebra book that assumes you still need to learn a thing or two about functions.)
In my experience, it seems the usual way people in the math community resolve these issues is to ask an expert, or at least a knowledgeable grad student.
In other words how good a person becomes at the task is essentially monotonically increasing (as long as you don't take a long break and forget it all), the difference between individuals is the slope. The curve doesn't seem to have any upper limit, though at times it seems to have a very slow rate of growth. But you'll only reach a desired level if you invest enough time on it. In school you have a fixed time-constraint, which is why, together with the mental model I described, some people fail (as well as not investing enough time on it of course).
Talent or genius only affects how rapidly a person learns, but as long as you're motivated enough to keep going and your time-constraints are wide enough I don't see a rational reason you couldn't do it. But I also think the growth-rate could be bumped up for everyone if we talked more about the ins-and-outs of how to go about teaching this all to ourselves.
I am interested in Machine Learning and there is just so. much. to. learn. I am currently a student, but i really don't know how to master the life-long learning in these more theoretic aspects. Universities are just made for this. From analysis, probability theory, statistics to information theory there is an abundance in things that are relevant to this field.
Places to start would be Barbara Oakley's Learning How to Learn and Cal Newport's Study Hacks blog, which has actionable advice like the following: http://calnewport.com/blog/2008/11/25/case-study-how-i-got-t...
You would also likely benefit from scheduling review of stuff you already understand with tools like Anki.
Beyond how to study any one thing, the most important thing is studying the right things in the right order starting at a level that is perhaps too easy so as to make sure your foundation is strong and not the source of your difficulties. You can more or less recreate this on your own through online resources and following along typical undergraduate mathematics curricula. There are often dedicated problem books with solutions for most common subjects even at the advanced undergraduate level.
http://www.jmilne.org/math/Books/iag.html
I've found his notes most useful when, because of the medium, he is able to provide more background, detail and examples compared to the standard texts.