How to learn mathematics: the asterisk method
geometry.org
geometry.org
- while reading, instead of copying, the software would ask me to enter "facts" in form of questions with the most important piece of knowledge being the answer. I would type the question and the answer into the software - much faster than writing anyway
- after I have gone through all the material, I would start the Q&A part of the software, which would ask me all the questions in either random order or sequentially (it was an option).
- at first, it would only show the question and wait for a key press to show the answer. After the answer was shown, I could mark whether I knew the answer or not. If not, it would mark that question to be asked in the next loop. This is basically the same as the asterisk method.
- once I got through all the questions, it would go into next pass, asking only those questions that I didn't know the answer to. And then filter out the remaining ones, and loop again and again until the all answers are known.
- then I would restart the whole system with all the questions to check.
What I learned after using this for about 2 years, is that there's a short term memory problem. Often I would know some of the answers on the first pass, but a week later I might forget it.
I found a way that works much better: Even if you mark that you know the answer, it will come up once again in the next pass. If you mark that you know it twice in a row, only then it would be removed. For some reason, this made the knowledge stick much better.
[1] https://www.gwern.net/Nicotine
Lots of great Flashcard apps out there. Definitely a great way to learn - especially things that fade quickly if not using them regularly (looking at you, Powershell ;-p ).
Parallel adaptation of probability estimation?
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You can derive that back from the Pythagorean theorem, but only if you have the sine and cosine definition in mind (SohCahToa).
I was also better at solving problems involving trigonometry than anyone I knew - including my professors.
You could, of course, argue that if you needed to use it every week, you would know all the identities merely by using them so much. Don't assume this is correct, though. One of the reasons I used them so much is because I had memorized them. My colleagues who used them as often as I did and who didn't memorize them did not, in fact, have them burnt into memory after so much usage.
Atleast two methods I still recall:
1. Look up the SOH CAH TOA method.
2. Knowing that Sin(0) = 0 and Cos(0) = 1 can give a hint which one to use. You may know an equation should have a Sin or Cos but unsure which one so you pick the one that gives the expected results at 0 degrees. Works great for engineering type of problems.
(I am not affiliated in any way apart from being a happy user)
But, it's only half the story. After you learn the definitions and theorems, you have to learn how to apply them to do computations and solve problems. This means working at least a few "easy" problems to learn how to crank through rote computations, and a few harder ones to learn how to think through novel applications.
If you can't solve problems with the material you've supposedly learned, you haven't actually learned it. (Otherwise, what did reading all that stuff really accomplish? You picked up some cool vocabulary words?)
I am the kind of person who likes to solve more practical problems but it is more of a hindrance in advanced maths, where you manipulate concepts that are too abstract for practical use, like infinities.
Computing some homotopy groups of spheres is a good simple exercise for the right kind of abstract math. But probably not 'practical' by GuB-42's standards?
Maybe another way to say this is: pure math is just a kind of applied math where the applications are resolving theoretical problems. Essentially all of the big mathematical programs/fields/whatever were created to solve or understand some Big Central Theoretical Problem(s), and prove their worth by continuing to be useful in solving other problems. And generally these problems can be understood in terms of concrete examples.
Even Grothendieck, perhaps the canonical example of a "theory builder," had resolving the Weil conjectures firmly in mind while writing his famous texts (and then got annoyed at Deligne for doing it the "wrong way"; see https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircl...).
I suspect that a pure mathematician would look scornfully at that as a waste of time, as the whole point is the math doesn't depend on the particular instance. An applied mathematician / engineer on the other hand sees no point in the mathematics unless it is manifested in the form of physical problems.
I'm thinking in particular of a lot of stuff in mathematical physics and PDEs.
Proper abstract mathematicians view applications not with scorn, but as a challenge to come up with even more abstract math. :)
Though just like libraries in programming mean that you don't have to worry about how to implement a hash table when using Python's dicts, advances in math mean that you don't have to worry about infinities and infinitesimals when you are constructing a bridge using differential equations.
That's progress!
I was going to say this article was a bad advice.
I second the solving problems approach. It's pretty mainstream opinion really, at least among math/phys students. You can try to follow the article and feel you "understand" the topic. And still unable to solve any of the homework problems, let alone exams.
When reading textbooks, I would often try and prove theorems myself before looking at the proof - and when I looked at it because I was stuck I would just try and see what the next insight or step and go from there.
I suppose I am thinking somewhat of reading maths papers here which don't come with a nice set of exercises, which probably isn't what we are talking about - but mathematics does move gradually in that direction from areas where the routine computations are laid out to areas where you have to work out for yourself how to make the abstract seem less abstract and where there are assumptions that you will fill in lots of details yourself.
When I started learning to code. I remember people would buy the famous books and do an intellectual dive into them.
As for me I bought 'SAM's Learn C in 21 Days.' Then for the next one month like a dumb bot I would just read the code and type it out on my desktop. Religiously. Like wake up everyday spend significant part of the day just to read and write code into the editor and run it. Most of the time I had a hazy understand of what I was doing.
But surprisingly this approach worked better than the intellectual fantasising exercise(Most of my friends quit after a while). I've used this technique of using dumbest possible method to study anything useful in everything I've touched since. Including workout and fitness.
Over the years I've also wondered why it works. One reason is when you just wake up everyday and do this ritual, you are basically spending time with the subject. Add a month or two to this journey it's now a habit. Even though your understanding is relatively poor, you haven't quit at a point most people quit. You are comfortable doing work in the subject and therefore you are already scoring wins everyday. This not only builds confidence, it will eventually take you to higher levels.
Turns out the hardest part of learning and doing anything is sticking with a subject for long. Once you have practice the hard parts become easy.
Why avoid spiders, to prevent bots stealing your posts? This also prevents search engines crawling them, which is a major downside, no?
Instead, in practice, even among good mathematicians, there is a fairly wide range of how carefully they study and how well they learn some material.
So, it's possible and common (1) to get mostly just an overview, and even the overview can be at various levels of thoroughness, (2) try to get the main ideas of the most important points, (3) think about the material mostly just intuitively to build good intuitive models that can be the basis of more in learning, applications, research, (4) deliberately go over the material more than once with only the later passes quite thorough. In short there is more than one way to slice an onion.
Here is what did me the most good: First get an overview, i.e., what is the material really about? Second understand the details, say, after reading a definition, theorem, or proof, be able to write it down. Third, look back and get a relatively succinct, intuitive overview, model, that keeps all or nearly all the important content.
Uh, of the five Ph.D. qualifying exams, I got the best in the class on four of them. For my research, (a) for a paper I published and (b) for my dissertation, I did all the work with essentially no faculty direction. For the research, sure, needed to understand enough low level details of some material, but the real key was intuitive models that led to, permitted guessing, the original math with theorems and proofs.
[1] https://staciechoice1010.wordpress.com/2014/08/15/illusions-...
Programming was easier for me to learn and now helps me to understand math a bit better.
Coding a real project is the equivalent to math's proofs, I think.
The book "Badass - Making Users Awesome" says, learning something just requires two steps. Perceptual exposure (of hundreds of correct examples) and deliberate practice.
I think, math falls short in the first step, and I don't know why, but somehow mathematicians often see much part of syntax/grammar as a given, and use different ways to describe the same thing (sqrt and power of 1/2, for example).
Surprisingly, this turns out to be true: https://en.m.wikipedia.org/wiki/Curry%E2%80%93Howard_corresp...
You can use for (...) { ... }, you can use while (...) { ... }, and you can use do { .... } while ( ... ).
Of course, which one you pick depends on context. But, then, which one of sqrt or "raised to 1/2" you choose also depends on what's more convenient at the time.
Good points, thanks.
This is probably true. I always used to try to understand to avoid the bother of rote learning... but the truth is, I probably examined it so intensively while trying to understand, that I rote learned without realizing it.
Yes, our mind has a 'backdoor' which is imho a slow write/read/breathe manifold we all can summon and it's free (both as a freedom, and as a beer).
This technique is often criticized for teaching only the writing and rough meaning of characters, while ignoring actual usage and pronunciation. However, for me it was an absolute revelation that allowed me to finally break through and eventually achieve something close to fluency in the language (passing the highest level of proficiency exam, the JLPT N1).
Later, I began to comprehend how the method itself--which is kind of abstract and involves quite a bit of raw repetition--achieved its amazing results by building a rock-solid scaffolding in my brain upon which I could hang all of my Japanese language knowledge. If learning is the process of building new knowledge structures in the brain, then understanding is the process of linking existing structures into tightly integrated patterns which can support higher-level reasoning. The stronger the base structures, the deeper the understanding that one can develop on top of them.
As for the "backdoor", I believe the original comment was referring to the way that the brain seems to naturally (and sometimes effortlessly) work to strengthen the connections between knowledge structures that are sufficiently "exercised". So, one can use conscious efforts to simply reinforce the raw knowledge as much as possible, and then trust in the "backdoor" to reveal the important insights and connections as they are uncovered.
But what do I do if i don't have the time necessary to do this always?
At my university i have 4 graded assignments i have to hand in every week and they always consume a lot of time. Between working on those there is barely any time to even study, except on the weekend, which by this method might be enough to study two subjects max. I feel like this works but not for the pace of a college degree.
(Basically it's a consequence of ADHD. While it's a novelty it's easy to focus, connect it to other things, grasp the rough outline of the concepts, mash them together, be creative, talk about it, barf up ideas about it ... but when it comes to memorizing a proof glyph by glyph reading does not help, because it just makes me feel like reading something that I completely understand and remember the 1000th time. Of course it turns out without actually writing it down from memory it's hard to judge how well I could write it down from memory based on what I feel while reading it...)
I tried taking a trigonometry couse. Something I aced in college. Some parts were easy. But I kept running into references that I could not remember.
Eventually I just had to back to pre-algebra. Most of it is trivial but I do keep finding things I had completely forgotten.
Ahh, ok.
The asterisk method just reminded me about this (how to draw an owl):
https://www.reddit.com/r/funny/comments/eccj2/how_to_draw_an...
Killer quote.
What a throwback to the early days of Internet explorer when it asked you if you wanted to allow a cookie for every cookie the website asked you. I think that was available later as an option, but it became unbearable to browse with that activated later on...