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I agree with you except for the very last point, that "students need to be guided on problems at the time they do them". I think this issue is fundamental to the tradeoffs of self studying that I've experienced. I want to point out: the guide doesn't necessarily have to be a human. And textbooks accomplish this by giving examples that walk through how to solve problems similar to the exercises. Unfortunately this is a useful secret about textbook and problem design that is not common knowledge to students. (I.e., the principle of charity, principle of relevance, Chekhov's gun, etc.) I agree with the point of your example, but maybe not the choice of example. For your example, Gallian (which imo is the standard intro algebra book) includes this property in the definition of a ring: "Property 6. a(b + c) = ab + ac and (b + c)a = ba + bc". (Probably in order to disclaim the confusion you're talking about.) Then he goes on to derive 6-7 other properties of groups that will (of course) be useful in the exercises. I definitely found it's really important when self-studying to pick the right textbook. Definitely you have to be an experienced educator to write a good book that anticipates most of these things. But I agree that it's impossible for any textbook to anticipate every place someone reading it might get stuck. Let's say for a given student it happens 5-6 times in a really high quality textbook like Gallian or Rudin PoMA. They have 3 options (1) ask on mathematics.stackexchange.com and probably get an answer because it's a great community, (2) try to figure it out themselves, or (3) pay $600 and invest 3h/wk to take an algebra class that might cover the first 1/2 of the textbook in 4 months. I think where we disagree is what are the constraints of the tradeoff between attending a class vs. reading the book. My experience has been that, if the subject is interesting enough, reading a good quality textbook is cheaper and 2-3x faster than taking a course, but at certain points it can be much more challenging. I think where it depends on which student is how much more challenging, i.e., will they be able to dig themselves out of those holes in a few minutes or a few hours. I was personally in the middle of these two extremes, but I still thought it was worth it to self study after taking 3-4 classes in the math department, then I skipped a bunch (7 semesters) of analysis/algebra/topology and came back and took 2-3 grad courses, where I didn't really understand the main goal of the subjects until I took the classes. And then I went into CS industry and never used any of it again. But I don't regret the experience; it was legitimately interesting to learn about math. I think you have to be legitimately interested in a subject to self study it successfully. But that's true about studying serious math in general though: you have to be unrelentingly into it, or else you're just really misguided and shouldn't be there, given the high competition, poor odds for any future in math, and no practical use for any of the theory. That's my experience with it and why I made the argument that I made. |
The two properties you quoted are about the fact the distributivity works whether you're multiplying on the left or on the right. That's one possible left-right confusion, but I would argue most weak students believe it holds even when it doesn't, so in a sense they're too permissive in their reasoning.
However, syzarian's example is about a different left-right confusion: whether you can read an equality both forwards and backwards. I've seen this confusion in students: they will readily believe that you can distribute a factor over a sum (going forward), but be very skeptical about the act of factoring out (going backward), even though it's justified by the same equation. In this case, the students aren't permissive enough in their reasoning.
This is the kind of misconception (that equality has a "direction") that's much easier to suss out with an in-person interaction, whether it's with a teacher or other students.