I'm currently self studying a Linear Algebra text book due to the fact that I forgot the vast majority of it since I took the course 10 years ago. My general strategy is to take every single example, practice problem, and exercise and do it myself. If I get it wrong, either I try to find where I made a mistake or I basically do the Feynman Method so that I find where the flaw in my understanding is. Generally this approach has been agreeable with me, however I've hit an exercise set where of the 37 exercises given, I only got 17 of them correct in total. So I got about a 46%, which got me wondering, what should my minimum threshold be for 'passing' a section of work when self studying a math text with no outside assistance sans a solution sheet to check my answers against?
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4$\begingroup$ I think it's important which problems you don't get. If you get only the easy questions, or only the computational questions and not the more theoretical ones, that's a problem. $\endgroup$– Alexander WooOct 13, 2020 at 4:31
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2$\begingroup$ What is your goal? Why are you revisiting linear algebra? What do you intend to do with the knowledge - is it a stepping stone to applications or more advanced theoretical courses? Are you studying it for its own sake? $\endgroup$– J WOct 13, 2020 at 5:10
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1$\begingroup$ What's the Feynman Method? $\endgroup$– user507Oct 13, 2020 at 14:00
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$\begingroup$ @JW My goal is to eventually be able to start studying advanced theoretical courses, though there are other subjects within the Linear Algebra text itself I'm interested in for their own sake like Convex Combinations $\endgroup$– RuinedOct 13, 2020 at 15:18
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$\begingroup$ @BenCrowell Here is a link to a University of Colorado page on the Feynman Method $\endgroup$– RuinedOct 13, 2020 at 15:22
2 Answers
My biggest advice is not just to "see where you went wrong" or "do the FM". But redo the entire missed problems as if you had not seen the answer. Very important!
IOW, after finding out your mistake and correcting it, put the old work aside, look only at the problem statement and work it fresh. (I.e. in your mind, treat it as a fresh problem, write the entire calculation out. Don't do it as sort of a regurgitation of the previous writing, but as if you were seeing it fresh. This sounds like a silly game, but I find it really drives the ability to crack such problems in the future. If you don't do this, you won't really master the material.)
In terms of numbers, of course "everything is relative". But to be responsive. 46% is obviously pretty concerning. As a general rule, you should shoot for 90%. There will be times, you fall short, especially on initial drill, so I wouldn't be upset at all about 80%+ (provided you DO force yourself to redo all missed problems, as I discuss above). Anything below 60-70, I would seriously re-read the chapter, work all the examples ("fresh"), and, if possible, find additional drill problems (like from a competing text).
Note: the small "punishment" of having to redo all problems can be used to gamify the process. This helps with getting some feedback to allow you to persist in the work and not get bored or give up. So, you are shooting for 100% each time. You won't get it much, but it can become a little apple you are chasing. Similarly, you should "grade yourself" and assign a number grade and write it at the top of your drill work. A for 90%, B for 80%, etc. It sounds juvenile, but self study can be incredibly boring and monastic. Having some element of play, even silly like this, can make it more endurable. Like dieting or quitting smoking, the big danger with self study, is not the methodology (they all pretty much work), but that you give up.
I would maybe give yourself a little bit of a lower hurdle if the problems are all ballbuster headscratchers (from one of these people who falsely deprecates the learning value of simple drill). But if that's the case, you need to find a text with some simpler drill.
You don't learn to do difficult sports by only practicing elite movements. You need drill on the basics. People are not never-forgetting silicon machines, they are animals, one step above P's dogs. They need and benefit from drill. This doesn't mean they can only do drill...they are capable of great things as well. But they do benefit from automaticity of drill material, and the only way to get that is to work simple drill problems until acing them.
I don't agree that there should be a passing score when it comes to studying using problem sets if you wish to simply grind through all the problems. Should it be acceptable to have one wrong answer? If so, why would having two wrong answers not be acceptable? Three? Out of how many problems?
Assuming that your problem sets all feature the same type of problem, below could be a good strategy:
Randomize the problems and answer them one at a time. Your goal is to correctly answer some number of problems consecutively -- ten to twelve problems sounds about right, but you can adjust to your own goals. If/when you make a mistake, start the count over and review the material to see where you had gone wrong. When you get those ten to twelve problems correct in a row, end your session.
The next session, repeat what was done above. Then, if you are able to get three sessions in a row without making mistakes, you can be pretty confident you have a good handle on the material.
If your problem sets feature different types of problems (e.g. computation, proof writing, etc.), then first separate your problems into categories, then repeat the above for each category.
