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cahnI'm trying to use my other account at least occasionally so I posted about my Yuletide gifts there, including the salon-relevant 12k fic that features Fritz, Heinrich, Voltaire, Fredersdorf, Saint Germain, Caroline Daum (Fredersdorf's wife), and Groundhog Day tropes! (Don't need to know canon.)
Re: Grad school
Date: 2023-01-07 07:18 pm (UTC)Re: Grad school
Date: 2023-01-08 10:36 pm (UTC)YES. ALL OF THIS.
(Though,
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
Re: Grad school
Date: 2023-01-10 09:10 pm (UTC)That's definitely not how I read a math text, anyway! I start by identifying what the main theorem is, and reading whatever definitions and preliminaries I need in order to understand the theorem, and also the bits about why the theorem is important. Then I identify what theorems/lemmas are needed to prove the main theorem, often by drawing a diagram showing how the different theorems/lemmas hang together and lead to each other. And then after that I dive into the proofs (if I actually need to--sometimes you just need to use the theorem).
I don't know that I necessarily needed to have harder problems in high school, or early in my university education. It might have been good, but actually I think we got a fairly okay progression. The master thesis was a sort of mini-graduate project, after all, and before that was various smaller projects. I think I just hit...well, it was partly about my abilities (I don't think I have it in me to be a brilliant mathematician), but also about my interests. I'm often serially geeky, and I had moved on to other geeky interests.
Re: Grad school
Date: 2023-01-10 11:55 pm (UTC)That's now how anyone should read a math text, in my opinion. What I needed was someone to tell me I could look at later chapters at a high level to just grasp what they were about and what we were building up toward, instead of going, "No, Mildred, you're getting ahead! We must go line by line or it's not rigorous."
...Yes, if you're writing a proof. There is a difference between a rigorous proof and a rigorous pedagogy. The needs are different.
(My math and physics profs seemed to think that if they walked you through proofs line by line that was the same thing as teaching you, and you would then be able to 1) grasp the concepts, 2) apply the concepts to concrete problems without further effort. Maybe that works for the
actually I think we got a fairly okay progression
More evidence you went to school in Sweden!
Re: Grad school
Date: 2023-01-11 07:37 pm (UTC)Also useful when teaching theorems and proofs, is to let students try and find examples where one or more of the assumptions in the theorem is not met, and what then happens with the result. (Example theorem: a continuous function on a closed interval has a maximum and a minimum value. What happens if the function is not continuous, does it need to have a maximum and minimum value? Does it necessarily need NOT to have a maximum and minimum value? Where does the proof fail? Etc.)
Re: Grad school
Date: 2023-01-16 12:46 pm (UTC)Re: Grad school
Date: 2023-01-14 05:59 am (UTC)Well, not until college, certainly, except that I used to do the USAMTS which really did give you weeks to play around. But I got a different idea from what you've previously said: I got the impression that it took you a week to do the problems because you hadn't been taught the material you needed to solve the problem.
This is not what I'm talking about. I mean, yes, I've had plenty of problems like that too, especially in college, and, sure, they took me the entire week to solve, but that's not interesting or useful to me pedagogically because the solutions are either "beg the TA to teach us what we need to know to solve the problem" (which I have done) or "learn this thing by myself or with the help of B, College Partner in Crime" (which I have also done, for whole classes even, with varying degrees of success in actually learning and retaining things), but in both cases it's still "once I know the material then I can solve the problem." (I mean, I guess it's... helpful in some ways... that I got crash courses in teaching myself things?)
I'm talking about the kind of problem where you do know everything you need to know to solve the problem, you just don't know yet how to put it together, and you have to keep thinking about it and trying different things and playing with it, many of which might not work. Or where you might not see how to do the whole problem, but you might see how to do a little part, and you have to play around with that part, then maybe once you've done that part you can see how to do another part, but you won't be able to see it until then.
I had a little of that in college, but the pedagogical system of classwork isn't really set up to foster that (except maybe in Sweden :) )
Re: Grad school
Date: 2023-01-14 01:40 pm (UTC)That is exactly what I am talking about too! And it is *blowing my mind* that you never had this experience. But like I said, that's why you went to grad school in this and I never did. :P Though I wanted to.
The "not teaching us what we needed to know" was only two classes, physics in my freshman year of college, and that's exactly why I was unable to continue with physics. I didn't have the ability to make up for any gaps in what they'd taught us. You obviously did.
That kind of thing never happened in high school physics, or I wouldn't have been able to try majoring in physics in college, and it never happened in math, which is why I was successful in getting a math degree and being a top student (we had mixed graduate/undergrad classes in college, and I regularly outperformed the graduate students in the same class).
When my physics prof asked why I was switching my major to math when it was math I was complaining about in his class, I told him, "It's because in math, first they teach you the math, then they test you on it! They never ever test you on something they haven't taught you. I didn't even know that was a thing! I want no part in it and I'm switching to math." Okay, I didn't use those exact words, those last two sentences were my emotions, but I did explain the facts in the first part.
So, again: the rest of us had to incrementally approach a problem over the course of multiple days. Because we had been taught all the relevant material, but it took a while to figure out which bits applied and to connect the dots. (I mean, lots of people gave up the same day and handed in what they had, but those were not the people who were majoring/went on to major in math/physics. People like me were somewhere in between "giving up on the same day because who cares" and "but I already know how to solve every problem": we got degrees in these subjects but did not manage advanced degrees.)
I would walk around campus tackling math problems in my head, I would meet up with a study group to tackle the same homework assignment more than once over the course of a week, I would fall asleep and wake up with the answer.
Now, would I have needed so many days to solve these problems/write these proofs if I'd had a good grasp of the concepts before being asked to work through a problem/proof beginning to end? Probably yes for some of them, for at least things like number theory, where the whole concept of "I know how to prove everything as soon as I sit down" is, again, blowing my mind, but far fewer. I would have been able to go farther in math, both on my own and in a classroom setting, and you might be looking at someone with an advanced degree in math today.
I had a little of that in college, but the pedagogical system of classwork isn't really set up to foster that (except maybe in Sweden :) )
Actually, that might be your survivor bias at work again. ;) This was almost every single math homework assignment I had in college, minus some of the too-easy classes.
Now, I agree that they need to make it so that people like you *also* get challenging enough material before grad school, but that I think is a problem with the one-size-fits-all approach (which is a major thing that gets reformed in my imaginary quest to reform pedagogy).
Re: Grad school
Date: 2023-01-14 06:22 pm (UTC)The reason you hear about this disproportionately is that it's a big problem, it forced me to change my entire career plans and was mildly traumatic, and is something I would want to reform. You don't hear about all the times I solved math problems incrementally because 1) that was fun! that's why I have a degree in math! and 2) it is pedagogically correct, as you note, so I see no problem there.
Re: Grad school
Date: 2023-01-15 10:02 pm (UTC)Because we had been taught all the relevant material, but it took a while to figure out which bits applied and to connect the dots.
Yeah, this is me too.
...I think some of this may be that physics problems generally taught a skill, so the problems were generally speaking to make sure you figured out the skill and the problems were often a little more straightforward, plus maybe I just knew a lot more people who procrastinated a lot so we did problem sets the night before?? But now that we're talking about this I'm remembering my senior year class, a graduate class, where there were three undergrads and we all worked together -- and second semester two of us were totally lost, and the third always seemed to know what was going on and spent our problem set sessions trying patiently to explain to us other two what was going on. (I wonder what happened to Third Guy -- besides going on to grad school, which I know he did. I just googled him but it's a reasonably common name and nothing obvious came up.)
And now that I think about it, my college math classes tended to be poorly taught, for various reasons. Now that you mention it, my abstract algebra class did have problems where I couldn't sit down and do the problems and had to think about them during the course of the week. My other classes did too, but it was hard for me to separate the "this isn't challenging in a pedagogically interesting way" from "I have no idea what's going on here because the professor hasn't taught us anything."
(I think I've given you partial and therefore misleading information about how good my teaching was, and so it looks like I keep contradicting myself. Let me give a more complete picture:
High school math: in general quite good, had a very reasonable foundation
High school physics: best teacher I ever had; strong classes for two years; I had an extremely strong foundation going in to college
College physics: Freshman year: quite good teaching for college level, reinforced the strong foundation
Post-freshman physics: wide variety, some very poor teaching, some perfectly adequate, but the strong foundation from high school and freshman year was enough to make up for the poor teaching
College math: algebra, good; complex analysis, good but focused strongly on using it as a tool; everything else pretty terrible actually, for reasons ranging from "can't speak English as a second language very well" to "have clearly never thought about pedagogy in my life" to "am currently having an undiagnosed brain tumor," which I told you about
So when you mentioned me having poor teaching, you were right on the college level, but I was thinking about the high school part, where I had sufficiently strong teaching that it made up for a lot.)
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). ;)
Re: Grad school
Date: 2023-01-16 06:13 am (UTC)you feel strongly about problem sets and pedagogy, and I would like to hear your thoughts.
These thoughts are not very profound (although surprisingly controversial), it's that there seems to be this strain in elementary education of NOT giving problem sets at all, even for (especially for, it seems like) skill-building, which is just ludicrous. (I totally am on board, of course, with giving problems that are more interesting than extremely dry drill -- but you can't just not practice the skills.) My kids' school has tried to do it this way, though Good Math Teacher has always tried to push back, and it looks like in the last couple of years there has been more pushback.
ETA: Early elementary math, of course, is another beast entirely, where a lot of the applicable concepts you can practice in ways other than written problem sets.
Re: Grad school
Date: 2023-01-16 12:45 pm (UTC)Ah, yes, if we're talking about elementary school, then I'm on board with a problem-set oriented approach. For the simple reason that you're going to actually use this in real life.
Starting around middle school and definitely by high school, whether you're going to use this math is highly career- and interest-dependent. If you're not, at best you need the concepts. If you are--well, I submit that you need the concepts all the more.
So I would make problem sets a whole lot more optional at this stage, make it clear what skills are needed for what, and teach how to acquire these skills if you decide later in life that you're going to want them. (Much of my pedagogical reform is teaching students what information is out there, why you would need it, and how to go about learning it, over preselecting some random subset of information that may or may not be important for them, then forcing them to learn it when they're just going to forget it.)
The one branch of math I know I would make mandatory at the post-elementary math level is statistical concepts. Because at one point I made excellent grades based on my (promptly-forgotten) memorized ability to calculate sigma and whatnot, but I made it to almost the end of grad school without understanding what a standard deviation was, and most people still don't.
Number of times I've needed to calculate a standard deviation in my life: well, maybe for my dissertation, but other than that, 0.
Number of times I've needed to understand what a standard deviation is? A very, very large number.
People are going to encounter claims about science in the news/on social media, and statistical concepts are just not taught. You get problem sets on calculating Greek and Roman letters, but not taught how to evaluate claims, and then we end up with a very ill-informed population.
Trigonometry, which I had a whole year of in high school? It was fun because I was a math geek, but that's not the class I would make mandatory for all college-bound students.
Curriculum overhaul
Date: 2023-01-21 06:22 am (UTC)The one thing I would differ in may be terminology -- I would absolutely have problem sets! However, they would be problem sets of the "evaluating claims" type. (The following may be more middle-school than high-school kinds of problems, for reasons that should be obvious in a couple of paragraphs, but, you know, modify as needed for high school.) E.g., "Look at this graph, tell me what the author of the graph wants you to see, and whether they are using any tricks to try to get you to see that thing." [I have in mind things like, mucking about with the axes of the graph.] Or: Here's a claim about the average salary of a graduate in this field. Does this sound reasonable? How could an "average salary" number be manipulated? Or: I'm going to show you the studies that a company did. Based on the results of the studies that they reported, do you think that they've shown their product is successful or not? [How probable is it that they could have gotten that result by chance? A more technically-oriented class might directly compute the probability, but a nontechnical class might simply build their intuition about how often things occur by chance, which is not something that people generally have a good intuition about.] Or: A company is making these claims that they can do something spectacular. What kinds of questions/tests would you do to test whether they can actually do the thing? [Theranos! I am thinking of Theranos here!] Or: ~insert your favorite "correlation is not causation" test case here!~
I stand by my assertion that problem sets are really, really useful for building skills! But the skill I want to build here is NOT "can you calculate a standard deviation," it's "can you think critically about this claim/piece of data? Let's practice doing that." This is not a skill that kids are necessarily practicing!
(Honestly, if the majority of students got out of high school understanding that correlation is not causation, and that A implies B does NOT mean that B implies A, I would feel like this was a huge win!)
I actually did a "guest lecture" with E's 6th grade class where we talked about some of this stuff and I gave them (in-class) "problems" of this sort where they talked about them as a class, and they LOVED it. (I did a followup class about bias in articles, more of the "reading critically" thing above, which was mostly OK but didn't land nearly as well -- partially because I had to thread a fine line with not making the material inflammatory or controversial, but still interesting enough that I wanted to spend time thinking about it, which meant it probably wasn't so interesting to the sixth graders; partially because E's class is weighted towards technical types; and also partially because I don't have as much facility with it myself and haven't thought about it nearly as much, lol. I'll probably try this with the sixth graders again next year -- they're on a two-year curriculum cycle -- and see if I can hit the mark better the second time around.)
I also only thought about this this morning, but I'd also put in a LOT more estimation as something that was taught in the middle school/high school level. I don't care a fig whether kids get out of school knowing that you use the quadratic formula to solve a quadratic equation (much less knowing the formula itself) (unless you're a technical person, in which case, sure, you should understand this stuff) but I want everyone to get out of high school with some facility for being able to figure out, so, if X happens to one in 1000 people, and the US has about 300 million people, that means X will happen to about 300,000 people in the US. Or the other way around! Know what numbers mean! Be able to do order of magnitude calculations and Fermi type problems, or at least come out with a sense of what you'd need to know in order to do them!
Re: Curriculum overhaul
Date: 2023-01-23 12:31 am (UTC)if I could redesign the curriculum I would not require non-technically-oriented students to take advanced algebra/trig/etc. Nothing over, probably, Algebra I or so (and I'm not even totally sure about that)
YES PLEASE. I would keep enough algebra to teach at least the concepts of solving for a variable and when you might need to know it, and how to learn it, and then...most of the details can be left to the person to decide when they need to know it.
AND everyone should take statistics and probability (with a focus on "evaluating claims" as you say, more about that in a bit), and I would further make a "reading critically" class mandatory
Yes, these are two of my highest priorities!
The one thing I would differ in may be terminology -- I would absolutely have problem sets! However, they would be problem sets of the "evaluating claims" type.
Yeah, that's just a terminology difference: I've been using "problem set" to mean "working through a calculation" as opposed to other sorts of questions. Your "problem sets" I would just call "homework questions."
NOT "can you calculate a standard deviation," it's "can you think critically about this claim/piece of data? Let's practice doing that." This is not a skill that kids are necessarily practicing!
YES.
(Honestly, if the majority of students got out of high school understanding that correlation is not causation, and that A implies B does NOT mean that B implies A, I would feel like this was a huge win!)
THANK YOU.
I don't care a fig whether kids get out of school knowing that you use the quadratic formula to solve a quadratic equation (much less knowing the formula itself)
YES.
(unless you're a technical person, in which case, sure, you should understand this stuff)
Exactly!
Okay, we are on the same page more than I thought, I thought I was the only one with these opinions! Or, you know, not literally, but the only person I knew. I've encountered them online/in books, but only rarely.
Re: Grad school
Date: 2023-01-18 01:27 pm (UTC)Out of curiosity, is this part of the trend to not give elementary school kids homework at all, in your observation, or separate?
Re: Grad school
Date: 2023-01-21 06:16 am (UTC)But also there's been a definite shift towards "teaching concepts" rather than "teaching skills," which works for early elementary and can sometimes work for older students, as well, but there's a middle ground where it's actually important to build the skills!
Re: Grad school
Date: 2023-01-23 12:36 am (UTC)Huh. So in this model, are you having students practice their early elementary math skills at school and not at home? Because early elementary math is the only math where I think building up mental muscle memory/memorization for repetitive math skills is actually valuable across the general, non-technical population!
But also there's been a definite shift towards "teaching concepts" rather than "teaching skills," which works for early elementary and can sometimes work for older students, as well, but there's a middle ground where it's actually important to build the skills!
Yeah, I mean, my stance is that skills need to be taught:
1) Only when important, otherwise just teach students what skills are important for what, and how to learn more when they need it,
2) Not at the expense of concepts.
But I'm with you that some skills do need to be taught and taught well!