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Historical Characters, Including Frederick the Great, Discussion Post 40
I'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
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
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
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
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
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
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
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
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
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!