The issue with this is it requires a fairly mature metacognition from the student. We've all fooled ourselves into thinking we understand something, only to be shown a question we can't answer later on. With some maturity one gets into the practice of asking the right illuminating questions, but it's a painful journey at times. It's also made harder of there's a lot of time pressure, eg if you are studying a bunch of courses at the same time.
How to learn mathematics: the asterisk method
21–30 of 79 posts
Re: How to learn mathematics: the asterisk method
#22Earlier quoted context omitted.
It seems like proof-based maths and applied maths are different fields. The latter is more closely related to engineering. 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.
Especially in abstract mathematics, you must practice computations with basic examples (e.g. computing some homotopy groups of spheres). If you cannot do this, you haven't actually grasped the mathematical content of the reading. The abstractions exist precisely because they are concise, powerful ways to deal with various examples.
I agree with you, but I think you might misunderstand what GuB-42 means by 'practical'?
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?
Re: How to learn mathematics: the asterisk method
#23Really great method! I think the hardest thing for me is to look back at the notes I have written and study the asterisks. I haven't been able to develop good habits to "re-read" already written notes.
Re: How to learn mathematics: the asterisk method
#24The issue with this is it requires a fairly mature metacognition from the student. We've all fooled ourselves into thinking we understand something, only to be shown a question we can't answer later on. With some maturity one gets into the practice of asking the right illuminating questions, but it's a painful journey at times. It's also made harder of there's a lot of time pressure, eg if you are studying a bunch of…
This is the so-called "Illusion of competence/learning" [1] and I have become fascinated with this topic (and learning in general). The only way out of this trap is to self-test.
[1] https://staciechoice1010.wordpress.com/2014/08/15/illusions-...
Re: How to learn mathematics: the asterisk method
#25I did this during university, but made it much more efficient. I wrote a program that would work like this: - 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 sta…
This is a good method for memorizing facts about some material that you understand. The asterisk method is about getting a grasp of material that is hard to understand. Your method is typically what is needed in studying subject like (foreign) languages and history, while the asterisk method is for subject where the primary problem is to understand the material, such as mathematics and physics.
Re: How to learn mathematics: the asterisk method
#26> If you read something which is difficult to understand, stop and think about it until you understand it clearly. Ahh, ok.
This is the essence of the method described in the link, and the ability to pull it off is called "mathematical maturity". How one acquires mathematical maturity is the matter of some debate.
Re: How to learn mathematics: the asterisk method
#27Re: How to learn mathematics: the asterisk method
#28This approach seems to be quite innovative.
Re: How to learn mathematics: the asterisk method
#29Earlier quoted context omitted.
Especially in abstract mathematics, you must practice computations with basic examples (e.g. computing some homotopy groups of spheres). If you cannot do this, you haven't actually grasped the mathematical content of the reading. The abstractions exist precisely because they are concise, powerful ways to deal with various examples.
I agree with you, but I think you might misunderstand what GuB-42 means by 'practical'? 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?
It is a little hard for me to interpret their comment. I took it as meaning that the learning strategies should differ for pure and applied mathematics, hence my response.
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...).
Re: How to learn mathematics: the asterisk method
#30Students should not assume that all mathematicians try to learn all the material thoroughly.
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.