Yeah, I think we're talking past each other, but some of it probably has to do with applied math vs. pure math. At least half of the math I did, and definitely the harder half, and definitely the only half that was really interesting to me, was proofs, and that is where I would consistently spend days thinking about how to arrive at the answer.
Problem sets...I mean, those classes tended to be much easier, and I took them because they were required for the math degree, not because they were the kind of math I wanted to do. But even when I was taught all the necessary math, in physics or in math, there was a very good chance that I had to think about how to do the homework assignments incrementally. In applied math, I'm pretty sure I had to make incremental progress in calculus and linear algebra, at least. Differential equations maybe not, I remember not understanding a thing that went on in that class and still being able to solve the problems effortlessly--I made almost a 100% in the class and felt like I never actually learned differential equations. But with the majority of the classes, I did not consistently just sit down and apply a skill I knew. I often had to think about how to apply what I'd learned and come back later, making incremental progress.
I also thought about this last night and came to the conclusion that cognitively, the difference between solving a problem incrementally over a week and solving a problem incrementally over the course of an hour during a timed exam...felt to me like a difference in degree, not in kind. It was just a question of how many times I had to set the problem on the back burner mentally, let it simmer, come back, add a little that I'd thought of, and then go off again. I don't feel like I would have been especially ill prepared for a problem that took months.
Years is qualitatively different, because then you have to think about whether you've chosen your problem well, and no, classwork doesn't prepare you for that. You don't get to choose your problems!
But if I'd gone to grad school and tackled hard problems, like for a thesis, I feel like the throwing myself at a hard problem I didn't know how to solve and making incremental progress on it would have been the one part I was prepared for! That was my life!
Thinking about it, one hard part of the transition from coursework to research would have been the shift from a textbook to academic journals. When you're not throwing yourself incrementally at a problem using material that you know is in the 150 pages you've covered so far this semester, and all you have to do is flip back through the book and hope you recognize what you need, but when someone out there has probably written something useful that hopefully you will find. That is radically different, and classwork doesn't prepare you for it. But up until you write your master's thesis, at least at my university, you're doing classwork, and as mentioned, much of the same classwork that I did as an undergrad.
Now, how much of the fact that I had to throw myself at problems I didn't know how to solve was because of poor teaching? I don't even know what "poor" or "good" means by current standards (as opposed to my imaginary reforms); I know that every teacher I had for math taught pretty much the same way, and the hardness of the class was just a function of how fast the teacher covered the material and how they graded. (And how familiar you already were with the material.) The teacher began at the beginning of the textbook, lectured on a chapter, gave the students a homework assignment testing them on that chapter, and then went on to the next chapter. That's the same way history was taught, and physics, and chemistry, and French, and almost everything else I took.
And that is the *wrong way* to teach, imo. Me, at least (and as I keep observing, there's a reason we're not doing that in salon--I don't think it's the right way to teach many people).
I know you mentioned in one of these discussions that you feel strongly about problem sets and pedagogy, and I would like to hear your thoughts. I can tell you that forcing me to do a problem set as soon as I learned a new concept and then moving on to a new concept with a new problem set was responsible for both 1) why problem sets were hard when they were hard (often they were easy), 2) why I never went beyond the undergraduate level even when I aced the individual classes and they were too easy. And the same thing is true for proofs, where maybe I had a better conceptual grasp than with applied math, but the work was orders of magnitude harder, and certainly harder than it needed to be.
Chapter-by-chapter, test-as-you-go ruined math for me in the long term. I didn't figure out what I should have been doing until several years after I had given up on advancing in math and finished grad school in the humanities, having forgotten all I learned of math.
I knew at the time I was missing a good grasp of the concepts, but I didn't know how to acquire them except by doing more of the same thing, working harder when what I needed was to work smarter.
plus maybe I just knew a lot more people who procrastinated a lot so we did problem sets the night before??
Yeah, I would start immediately on my own, and then meet up with people well before it was due. And at the end of the study group, there would frequently be unsolved problems that you would then go off and think about on your own again. Starting the night before, I think I would have just failed everything (barring the too-easy classes that I complained about). ;)
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Re: Grad school
Date: 2023-01-15 10:55 pm (UTC)Problem sets...I mean, those classes tended to be much easier, and I took them because they were required for the math degree, not because they were the kind of math I wanted to do. But even when I was taught all the necessary math, in physics or in math, there was a very good chance that I had to think about how to do the homework assignments incrementally. In applied math, I'm pretty sure I had to make incremental progress in calculus and linear algebra, at least. Differential equations maybe not, I remember not understanding a thing that went on in that class and still being able to solve the problems effortlessly--I made almost a 100% in the class and felt like I never actually learned differential equations. But with the majority of the classes, I did not consistently just sit down and apply a skill I knew. I often had to think about how to apply what I'd learned and come back later, making incremental progress.
I also thought about this last night and came to the conclusion that cognitively, the difference between solving a problem incrementally over a week and solving a problem incrementally over the course of an hour during a timed exam...felt to me like a difference in degree, not in kind. It was just a question of how many times I had to set the problem on the back burner mentally, let it simmer, come back, add a little that I'd thought of, and then go off again. I don't feel like I would have been especially ill prepared for a problem that took months.
Years is qualitatively different, because then you have to think about whether you've chosen your problem well, and no, classwork doesn't prepare you for that. You don't get to choose your problems!
But if I'd gone to grad school and tackled hard problems, like for a thesis, I feel like the throwing myself at a hard problem I didn't know how to solve and making incremental progress on it would have been the one part I was prepared for! That was my life!
Thinking about it, one hard part of the transition from coursework to research would have been the shift from a textbook to academic journals. When you're not throwing yourself incrementally at a problem using material that you know is in the 150 pages you've covered so far this semester, and all you have to do is flip back through the book and hope you recognize what you need, but when someone out there has probably written something useful that hopefully you will find. That is radically different, and classwork doesn't prepare you for it. But up until you write your master's thesis, at least at my university, you're doing classwork, and as mentioned, much of the same classwork that I did as an undergrad.
Now, how much of the fact that I had to throw myself at problems I didn't know how to solve was because of poor teaching? I don't even know what "poor" or "good" means by current standards (as opposed to my imaginary reforms); I know that every teacher I had for math taught pretty much the same way, and the hardness of the class was just a function of how fast the teacher covered the material and how they graded. (And how familiar you already were with the material.) The teacher began at the beginning of the textbook, lectured on a chapter, gave the students a homework assignment testing them on that chapter, and then went on to the next chapter. That's the same way history was taught, and physics, and chemistry, and French, and almost everything else I took.
And that is the *wrong way* to teach, imo. Me, at least (and as I keep observing, there's a reason we're not doing that in salon--I don't think it's the right way to teach many people).
I know you mentioned in one of these discussions that you feel strongly about problem sets and pedagogy, and I would like to hear your thoughts. I can tell you that forcing me to do a problem set as soon as I learned a new concept and then moving on to a new concept with a new problem set was responsible for both 1) why problem sets were hard when they were hard (often they were easy), 2) why I never went beyond the undergraduate level even when I aced the individual classes and they were too easy. And the same thing is true for proofs, where maybe I had a better conceptual grasp than with applied math, but the work was orders of magnitude harder, and certainly harder than it needed to be.
Chapter-by-chapter, test-as-you-go ruined math for me in the long term. I didn't figure out what I should have been doing until several years after I had given up on advancing in math and finished grad school in the humanities, having forgotten all I learned of math.
I knew at the time I was missing a good grasp of the concepts, but I didn't know how to acquire them except by doing more of the same thing, working harder when what I needed was to work smarter.
plus maybe I just knew a lot more people who procrastinated a lot so we did problem sets the night before??
Yeah, I would start immediately on my own, and then meet up with people well before it was due. And at the end of the study group, there would frequently be unsolved problems that you would then go off and think about on your own again. Starting the night before, I think I would have just failed everything (barring the too-easy classes that I complained about). ;)