Online courses and math notes

Up until recently I was skeptical about online courses: they seemed like a waste of time, trying to maintain archaic traditions, roughly the same as with talks accompanied by slides. At first it takes plenty of effort to prepare a video lecture, and then it may be considerably more complicated to extract information out of it, comparing to, say, LaTeX. A bottleneck, and a created one.

And maybe it is, but there are good parts as well. Particularly about the methods they use – such as frequent tests mixed with new material. When working through a textbook alone, often there's just no puzzles, or no easy way to check whether your results are correct. The structure doesn't suggest where to take a break, and varies from book to book; while in courses, there's usually a plan that includes both timing and exercises. Perhaps it would eliminate the bottleneck if the courses were in LaTeX (or programming languages and theorem provers when appropriate) instead of video, while preserving the good parts.

"Learning how to learn"

After stumbling upon "Ask HN: How to learn new things better?", I've decided to try the "Learning How to Learn" course. Before that, I thought that the courses are restricted in time (with start/end dates), but apparently not all of them are: one can just jump into it and go through a course anytime.

That particular course was quite interesting; the somewhat scary 4 weeks took a day to complete, though that's the opposite of what that course suggests doing (i.e., not cramming the material, spacing, interleaving, etc).

But the expected unpleasant bits are also there: sound level varies among videos, it's rather hard to find a particular part of it if one would want to review it later, Coursera UI is broken because they are reinventing even hyperlinks and page loading (that failing reinvention of the wheel is really annoying). Some bits of that course are scripted, which apparently took more of the lecturer time than it should have, while in others you just watch a man being interviewed, trying to put his thoughts together: probably it would be less stressful for him, would take less of his time (assuming that the interview requires some preparation), and would lead to a better structured message if he just wrote it down instead of replying in real time.

Nevertheless, the course was also funny. And somewhat cute and sad in the end (there is a spoiler further in this sentence), where a man from a video record concludes the course with "We've learned from you as we hoped you've learned from us".

Calculus One

Inspired by that course, I decided to try something more serious, yet basic still – so it wouldn't take too much time. Chose Calculus One, and it turned out to be fun, though not without its quirks and errors. But the teacher there really tries to make it fun, and often succeeds; that seems to be useful for students to stay engaged. That's one of the aspects in which a video may be better than a book.

The course includes plenty of nice examples (applications), which seems to be quite useful, but often not present.

Its 16 "weeks" took 11-12 days and about 2500 lines of LaTeX to complete, so apparently one could do a whole bunch of those courses simultaneously to stay on their intended schedules – at least, of basic ones.

The separation of weeks, as well as the way it resembles a game by assigning internet points, requires less discipline to maintain perseverance, and to actually work through all the tasks.

By the way, taking a calculus course now has surprised me with something I didn't notice before: all the continuous functions and approximations point to a non-discrete, pre-digital age; reminds me of steampunk, in short.

Math notes

I like to do things using a computer, and some math problems can be worked on using a programming language (one with dependent types for certain proofs, or one with handy functions for calculations in a particular domain, or pretty much any if it's about implementing algorithms), or just by writing things down in a text file, but for others (and for analysis in particular) a rather complex notation without much of computations is required, and though it's just a convention, it's much easier to read it once one is used to it.

There is LaTeX, but I've always assumed that it would be cumbersome to use for actual problem solving, yet it turned out to be pretty handy with preview-latex, at least while touch typing and having some previous experience with LaTeX. Maybe it would have been faster with handwriting though.

I've also used R for occasional computations and plots, but drawings were missing. Inkscape would be too cumbersome for that, so I've tried to use TikZ (what required to add \PreviewEnvironment{tikzpicture} into the preview-latex default preamble, btw). It works, but seems to take too long to compose a drawing; maybe just requires more practice.

There also is asymptote, and org-mode may be more handy to embed LaTeX, R, and other things, along with results. But I haven't tried to use it for working through math problems yet.

I'd add a screenshot, but it's not nice to publish solutions, and I'm too tired/lazy to make up (or find) and solve a problem just for that. Anyway, it looks really nice, and I'm happy about this setup: I just regret that I didn't start using it years ago.

Linear algebra

A bit off-topic, but I've continued to re-learn and refresh maths basics with linear algebra, trying to apply what's described above; used a free (libre, even) Linear Algebra book (that provides exercises, answers to those, and even some kind of a plan), in combination with videos on essentce of linear algebra, and it seems to be useful to combine: the book lacks visualizations and geometric interpretations (which make much more sense than plain definitions, and make things clearer), while the videos lack exact definitions, proofs, etc, only aiming to provide insights, so they complement each other well.

I've noticed it in the past though, particularly with topology: the books are often bad in helping to get insights, but in that case it was harder to find helpful visualizations – though the few I did find were indeed helpful.

On the topic of libre books, there also are Abstract Algebra book, an Online Mathematics Textbooks collection, a Graphical Linear Algebra blog.

Other video courses

There's a youtube "crashcourse" channel with various not-quite-techy but fun courses. I guess there's more resources like that around.