Oh, is that what you meant by not knowing how to solve a problem? Having a longer time to solve it, as opposed to solving or not solving it and then handing it in? I mean, I and my classmates definitely took an entire week to solve homework problems, making incremental progress, but I don't think we were ever given weeks or months to solve a problem, that's true.
But I suppose if one isn't going to grad school in math or physics, maybe it's not so important to have those kinds of experiences.
I *wanted* to go to grad school in math or physics, but my education did not equip me to do this. I think if I'd had the math pedagogy I wanted, I might have been *able* to go to grad school in math! I was taught to do math in a way that meant I was a top student through my bachelor's degree, but I knew I was not equipped to survive at a graduate school level.
But somehow I also knew I had the intelligence and math ability to get at least a master's and maybe more, I was just missing something and I didn't know how to make up the lack. At the time I thought it would maybe help if my tuition waiver had been for 5 years instead of 4, that that extra year might have made a difference, but now I'm pretty sure I know what I was lacking: pedagogical alternatives to the received method.
I only knew how to study math the way I was taught, and I couldn't do it on my own. Starting in about 5th grade, I was always frustrated that I could never teach myself math on my own, and I never understood why. I knew that was going to be important at some point, and that real mathematically successful people could do it, and also that it would relieve so much of my intellectual frustration if I could...and yet I never could. Not until after my PhD did I figure out what went wrong.
I succeeded up through undergrad by being trainable when it came to solving problems and figuring out proofs as you put them in front of me, but there was a bit of a "trained monkey" and "studying for the exam, then forgetting" aspect. I frequently didn't have the concepts down as well as I needed to, and for that reason, as well as others, I struggled to retain math from one year to the next (another thing I knew I needed to be able to do to move to a more advanced level).
The whole problem was that I was taught to study math one page at a time, one sentence at a time, and never move on to the next thing until you understood the current thing. When understanding the next thing frequently makes the current thing easier to understand!
What I needed was to be able to do breadth before depth. I needed to understand how a bunch of concepts related and helped make sense of each other, and I needed to have a good grasp of the relevant concepts before I started getting bogged down in making sure I had remembered to carry the 1. If I had felt I was allowed to do that (you know, the thing they actually punish you for in school, and the opposite of the way books are written), I could have 1) taught myself math, 2) understood what I was doing at a deeper level, 3) retained concepts after the exam.
Once I had the ability to do all that, I suppose being given problems that took longer than a week to solve would have been useful too, but until I had the ability to do 1-3, there was no chance I was going to grad school in math or physics at all.
Also, if you're saying you never ever had to sit with a problem without knowing how to solve it (???)* and it never took you longer than a day to solve it incrementally? That is probably why you went to grad school in physics. I and everyone I knew, except possibly the people who went to grad school in math/physics/astronomy, had the experience on a regular basis. :P It sounds like we have different priorities pedagogically because graduate school selects for people who had no difficulty solving the assigned problems, and those of us who had the kind of difficulties you wanted were discouraged from attending grad school because it would be too hard.
* ETA: "???" because "I would have benefited from more experience with throwing myself against problems without knowing how to solve them" does not compute and never has, that's why that sentence has always confused me so much whenever you say that. :P
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Re: Grad school
Date: 2023-01-08 11:02 pm (UTC)But I suppose if one isn't going to grad school in math or physics, maybe it's not so important to have those kinds of experiences.
I *wanted* to go to grad school in math or physics, but my education did not equip me to do this. I think if I'd had the math pedagogy I wanted, I might have been *able* to go to grad school in math! I was taught to do math in a way that meant I was a top student through my bachelor's degree, but I knew I was not equipped to survive at a graduate school level.
But somehow I also knew I had the intelligence and math ability to get at least a master's and maybe more, I was just missing something and I didn't know how to make up the lack. At the time I thought it would maybe help if my tuition waiver had been for 5 years instead of 4, that that extra year might have made a difference, but now I'm pretty sure I know what I was lacking: pedagogical alternatives to the received method.
I only knew how to study math the way I was taught, and I couldn't do it on my own. Starting in about 5th grade, I was always frustrated that I could never teach myself math on my own, and I never understood why. I knew that was going to be important at some point, and that real mathematically successful people could do it, and also that it would relieve so much of my intellectual frustration if I could...and yet I never could. Not until after my PhD did I figure out what went wrong.
I succeeded up through undergrad by being trainable when it came to solving problems and figuring out proofs as you put them in front of me, but there was a bit of a "trained monkey" and "studying for the exam, then forgetting" aspect. I frequently didn't have the concepts down as well as I needed to, and for that reason, as well as others, I struggled to retain math from one year to the next (another thing I knew I needed to be able to do to move to a more advanced level).
The whole problem was that I was taught to study math one page at a time, one sentence at a time, and never move on to the next thing until you understood the current thing. When understanding the next thing frequently makes the current thing easier to understand!
What I needed was to be able to do breadth before depth. I needed to understand how a bunch of concepts related and helped make sense of each other, and I needed to have a good grasp of the relevant concepts before I started getting bogged down in making sure I had remembered to carry the 1. If I had felt I was allowed to do that (you know, the thing they actually punish you for in school, and the opposite of the way books are written), I could have 1) taught myself math, 2) understood what I was doing at a deeper level, 3) retained concepts after the exam.
Once I had the ability to do all that, I suppose being given problems that took longer than a week to solve would have been useful too, but until I had the ability to do 1-3, there was no chance I was going to grad school in math or physics at all.
Also, if you're saying you never ever had to sit with a problem without knowing how to solve it (???)* and it never took you longer than a day to solve it incrementally? That is probably why you went to grad school in physics. I and everyone I knew, except possibly the people who went to grad school in math/physics/astronomy, had the experience on a regular basis. :P It sounds like we have different priorities pedagogically because graduate school selects for people who had no difficulty solving the assigned problems, and those of us who had the kind of difficulties you wanted were discouraged from attending grad school because it would be too hard.
* ETA: "???" because "I would have benefited from more experience with throwing myself against problems without knowing how to solve them" does not compute and never has, that's why that sentence has always confused me so much whenever you say that. :P