How do I approach studying what I don't really understand?
July 7, 2017 6:25 PM Subscribe
For better or worse, I've been one of those people who either intuitively gets something or doesn't persist with it. Usually I've been bright enough to get by. But I am faced with exams now where there are clear gaps in my knowledge, and I can find them hard to identify or fill in. What's a good method to identify what I -don't- know in a question?
Context: This is third-year undergraduate pure mathematics.
Working through problems only goes so far; I encounter difficulty when A: solutions seem to be working on assumed knowledge I don't have, or B: I simply have no idea how to approach and break down an idea. Looking at the answer and working backward doesn't seem to help me fully grasp the concept in the end, even if it seems logical at the time.
I would like to know how people break problems down when they may be missing information, and how to get a sense of what information you should then be looking for.
Context: This is third-year undergraduate pure mathematics.
Working through problems only goes so far; I encounter difficulty when A: solutions seem to be working on assumed knowledge I don't have, or B: I simply have no idea how to approach and break down an idea. Looking at the answer and working backward doesn't seem to help me fully grasp the concept in the end, even if it seems logical at the time.
I would like to know how people break problems down when they may be missing information, and how to get a sense of what information you should then be looking for.
This was a way I thought about certain subjects for a long time.
"I get most stuff, but I don't get this, and I never learned to get something I didn't get in the first place," or whatever.
But I don't think there's on or off switches in your brain like that -- I think it's just a matter of taking more or less time.
Things I didn't get immediately, it turned out, I did pick up and understand eventually. I just wasn't used to waiting to understand something intuitively -- that usually happened real quick for me.
There's no magic spell to intuitively grasp concepts faster. That's the territory of weird self-improvement cults and stuff, you know? There's just taking the time and effort to learn something, and that's going to be slower or faster depending on the topics and on the people learning them.
posted by Rinku at 7:24 PM on July 7, 2017 [2 favorites]
"I get most stuff, but I don't get this, and I never learned to get something I didn't get in the first place," or whatever.
But I don't think there's on or off switches in your brain like that -- I think it's just a matter of taking more or less time.
Things I didn't get immediately, it turned out, I did pick up and understand eventually. I just wasn't used to waiting to understand something intuitively -- that usually happened real quick for me.
There's no magic spell to intuitively grasp concepts faster. That's the territory of weird self-improvement cults and stuff, you know? There's just taking the time and effort to learn something, and that's going to be slower or faster depending on the topics and on the people learning them.
posted by Rinku at 7:24 PM on July 7, 2017 [2 favorites]
The usual advice in Math is Do The Problems.
However, for you specifically, my advice is to to look at specific examples of the things that you are working with theoretically. Too much abstraction and generality leads to mushy thinking.
posted by SemiSalt at 7:42 PM on July 7, 2017
However, for you specifically, my advice is to to look at specific examples of the things that you are working with theoretically. Too much abstraction and generality leads to mushy thinking.
posted by SemiSalt at 7:42 PM on July 7, 2017
This - both what you've done so far, the act of noticing you don't understand, and what you're doing now, the act of trying to find the specific areas you don't understand - is one of the key skills for really advancing and learning, and making your studying work efficiently for you. My experience in studying math and in physics has been that the greatest benefit comes from:
1. Doing the difficult, complicated problems, slowly, skipping no steps. When the book says something is left as an exercise for the reader, do the exercise. Write it down and figure out how they get from A to Z.
2. When you are doing the problems, ask yourself how you know what step is next. What are some techniques that might be relevant? What assumptions am I making if I use this particular equation or technique? Are those applicable here, and how do I know that?
Even though I hated the algebra teachers who would insist we show "x+2 = 7 --> x+2-2 = 7-2 --> x = 5" when it was so obvious x=5, that becomes a useful technique when you are dealing with problems that you can't see the entirety of at once. The principles are applicable for more advanced areas of the subject too, and the habit of keeping all the details tidy and straight becomes critical at higher levels.
For higher-level math you're going to have to also build up just some familiarity with techniques, shortcuts, and shapes of certain techniques - a piece of the equation looks like a particular type of transform, or the set of equations might be reduceable to a special case that has a particular trick for it, etc. Nothing for that but practice and examples.
posted by Lady Li at 10:12 PM on July 7, 2017 [1 favorite]
1. Doing the difficult, complicated problems, slowly, skipping no steps. When the book says something is left as an exercise for the reader, do the exercise. Write it down and figure out how they get from A to Z.
2. When you are doing the problems, ask yourself how you know what step is next. What are some techniques that might be relevant? What assumptions am I making if I use this particular equation or technique? Are those applicable here, and how do I know that?
Even though I hated the algebra teachers who would insist we show "x+2 = 7 --> x+2-2 = 7-2 --> x = 5" when it was so obvious x=5, that becomes a useful technique when you are dealing with problems that you can't see the entirety of at once. The principles are applicable for more advanced areas of the subject too, and the habit of keeping all the details tidy and straight becomes critical at higher levels.
For higher-level math you're going to have to also build up just some familiarity with techniques, shortcuts, and shapes of certain techniques - a piece of the equation looks like a particular type of transform, or the set of equations might be reduceable to a special case that has a particular trick for it, etc. Nothing for that but practice and examples.
posted by Lady Li at 10:12 PM on July 7, 2017 [1 favorite]
Can you set up a study group to discuss the material with classmates? Maybe you understand aspects they don't.
Does your instructor hold office hours? I resisted for attending office hours for the obvious reasons, but... they're great.
Get a tutor? If you can't afford a tutor, perhaps you have a skill/ability/resource you can offer in barter.
posted by at at 6:32 PM on July 9, 2017
Does your instructor hold office hours? I resisted for attending office hours for the obvious reasons, but... they're great.
Get a tutor? If you can't afford a tutor, perhaps you have a skill/ability/resource you can offer in barter.
posted by at at 6:32 PM on July 9, 2017
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I started to tackle this problem by reading proofs. I would go through each line of a proof and make sure I could understand the logic that connected each one. It was an agonizing process, but I made sure that I could understand each step of the proof. That helped me become more familiar with the type of thinking that was required in developing basic concepts. I also separately made note of first principles that I came across in more advanced proofs. I was able to eventually build up a list of basic facts that I think other students took for granted, but that weren't obvious to me.
Whenever I started to work on a new problem I would first check my list of first principles to see whether any of them might apply. By repeating this process, I begin to memorize which first principles and theories applied to various types of problems. It can also be helpful examining problems in a class setting where the problem has been introduced by the professor. You then know that the theories introduced by the professor before the problems have a direct connection to understanding how to solve the problem.
It might feel like you're not getting anywhere at first, but give the process time. Don't allow yourself to gloss over gaps in understanding the logic of proofs. They give you insight into the type of thinking you need to apply to problems.
Good luck!
posted by bkpiano at 7:11 PM on July 7, 2017