Typical. I don't get an idea for an Ada Lovelace Day post until well after the day itself has passed. "But I just don't know any women scientists other than the obvious ones!" Not an excuse. How many women scientists was I taught by in elementary and middle school, completely oblivious to their prior lives? Of course, I was taught by men scientists too, and they have equally interesting stories. I'm just going to focus on one particular woman for the moment.
Molly was the science specialist at my elementary school. She gave the impression of a powerful but friendly science warlock, introducing us grade-school apprentices to a world of wonderful things. Yes, it sounds cheesy, but there you are. I don't have any data, but I think I must credit Molly with giving me that initial (strong) inclination toward science that tipped and boiled over when I ran into a copy of The Cartoon Guide to Genetics (Gonick & Wheelis). It's been more than a decade since then, and I've never looked back.
I can only remember a few fragments of specific things that Molly taught us, but I distinctly remember some of the things that we did. My school had a lot of forested property around the buildings, some of which had been invaded by the vinca that someone decided to plant around the parking lots. We spent many an hour tramping around the woods, of course, and not only identifying trees and noticing erosion, but waging war on the vinca. We marked out test plots and poured mulch over some, staked black plastic over others, and wielded picks, shovels, and good ol' pulling on the rest. I lost track of the project once I was in middle school, so I don't know how successful the ongoing efforts were, but it was pretty awesome at the time.
We also -- this is kind of surprising, in retrospect -- had a strong focus on communication. At some level, every third grader is going to remember "communicating" (writing the occasional couple of paragraphs). But we wrote full-length letters to lawmakers about environmental issues (not skimping on either the science or the rhetoric), and drew a cartoon illustrating a forest fire. The final exhibition of everyone's cartoons was a sight to see.
The other day, I happened to idly wonder what Molly was up to now, and so I googled her. I found out little about what she's up to since her teaching career, but there was quite a bit about what she had been up to before taking up the overhead projector. She published several papers in the sixties about Blepharisma and the recovery of cells from X-ray irradiation. She was at Stanford! Doing real science! Back in the day, before sequencing and so on were invented! Why was I not informed?
She also had an active life outside school, running marathons and working with the San Francisco Sidewalk Astronomers to build DIY Dobsonian telescopes. Man, how cool is that?
Molly Lusignan, scientist, scholar, teacher, and all-around awesome woman, I salute you. The moral of the story for the rest of us is to find out what our teachers (and our children's teachers) do outside of the classroom, about their history, their hobbies, and their dreams. These days, it's of course made easy to stalk people via Google (and Google Scholar! <3), but there's a lot more to be said for connecting in person. I wish I could have found her current contact information...
Showing posts with label education. Show all posts
Showing posts with label education. Show all posts
Monday, March 29, 2010
Wednesday, January 6, 2010
The sovereign importance of mentorship
A few days ago I met up with the professor from the lab at Stanford where I worked in high school. I had a pretty rough fall semester, and I felt totally unmotivated and lost, so I wanted to hear what an Actual Successful Scientist had to say on the subject.
We spent nearly two whole hours talking, and his message boiled down to: "Mentorship is the most important thing."
This Ur-message applies, or has applied, or will apply, to my life in many different ways. I'll do my best to summarize here. (Context: I'm currently aiming for grad school and academic research, but considering alternate careers. In particular I have half-formed intentions of taking a gap year after I finish undergrad, and working in industry to see what it's like.)
If you have good grades, lab experience matters more than improving your grades. I have a very good GPA, mostly because I still have the overachieving high school student complex. I'm a perfectionist and I prioritize classwork above everything else. Last semester, I prioritized classwork above continuing my research from the summer, which was a big mistake. I was trapped by the subconscious impression that I absolutely had to maintain my very good GPA or else... what?
Really good lab experience, including good mentorship, is harder to come by than well-taught or interesting classes, and much harder to come by than an A+. It also yields greater benefits later on. I had to take this part on faith a little, because I haven't been to grad school, but it seems very plausible. "Lab hands" are very hard to teach, but a small gap in knowledge is easily corrected.
This is going to be really difficult for me to implement, because I'm so well trained at Being A Student that it's difficult to even imagine doing anything else as a primary occupation. But it has to happen eventually, and it may as well happen now.
The best mentors are those with proven track records in both scholarship and mentorship. It's often said that there's a tradeoff involved in choosing an older or younger advisor for your Ph.D.: the older profs are better mentors, but the younger ones are doing all the exciting new stuff. From what I remember of our conversation, and also a little bit from my own experience, I understand that well-established advisors beat young flashy advisors hands down.
A lot of this apparently comes down to having observed the most common failure modes in one's students. My prof was able to advise me particularly aptly because he's seen many people go through similar struggles to my own. It wasn't that he had seen my exact set of problems before; it was that he had seen parts of my problems in many other people and was able to synthesize them. He had something cogent to say about gap years, about engineering vs science, about depression, about grades, about anything I cared to ask.
A diverse lab environment is better for everyone. This goes hand in hand with the previous point: the greater variety of problems a mentor has seen before, the bigger the library of memories they can draw on when advising someone with the same problem or a new problem. (Not to mention issues of sexism etc, which are also very important.)
Finally, never hesitate to ask for help. People who can give good advice love to do so, because it makes them feel helpful and all warm and fuzzy inside. People who don't wish to give you advice may ignore you or brush you off, so there's basically no harm in asking very widely. It took me years to accept that my parents might occasionally have something wise to say, but it's true. Now that I've reached the point where my parents' academic/career advice is no longer particularly helpful, I have to cast a wider net -- but it's crucially important that I actually do so, and not rely on assumptions or PhD Comics.
We spent nearly two whole hours talking, and his message boiled down to: "Mentorship is the most important thing."
This Ur-message applies, or has applied, or will apply, to my life in many different ways. I'll do my best to summarize here. (Context: I'm currently aiming for grad school and academic research, but considering alternate careers. In particular I have half-formed intentions of taking a gap year after I finish undergrad, and working in industry to see what it's like.)
If you have good grades, lab experience matters more than improving your grades. I have a very good GPA, mostly because I still have the overachieving high school student complex. I'm a perfectionist and I prioritize classwork above everything else. Last semester, I prioritized classwork above continuing my research from the summer, which was a big mistake. I was trapped by the subconscious impression that I absolutely had to maintain my very good GPA or else... what?
Really good lab experience, including good mentorship, is harder to come by than well-taught or interesting classes, and much harder to come by than an A+. It also yields greater benefits later on. I had to take this part on faith a little, because I haven't been to grad school, but it seems very plausible. "Lab hands" are very hard to teach, but a small gap in knowledge is easily corrected.
This is going to be really difficult for me to implement, because I'm so well trained at Being A Student that it's difficult to even imagine doing anything else as a primary occupation. But it has to happen eventually, and it may as well happen now.
The best mentors are those with proven track records in both scholarship and mentorship. It's often said that there's a tradeoff involved in choosing an older or younger advisor for your Ph.D.: the older profs are better mentors, but the younger ones are doing all the exciting new stuff. From what I remember of our conversation, and also a little bit from my own experience, I understand that well-established advisors beat young flashy advisors hands down.
A lot of this apparently comes down to having observed the most common failure modes in one's students. My prof was able to advise me particularly aptly because he's seen many people go through similar struggles to my own. It wasn't that he had seen my exact set of problems before; it was that he had seen parts of my problems in many other people and was able to synthesize them. He had something cogent to say about gap years, about engineering vs science, about depression, about grades, about anything I cared to ask.
A diverse lab environment is better for everyone. This goes hand in hand with the previous point: the greater variety of problems a mentor has seen before, the bigger the library of memories they can draw on when advising someone with the same problem or a new problem. (Not to mention issues of sexism etc, which are also very important.)
Finally, never hesitate to ask for help. People who can give good advice love to do so, because it makes them feel helpful and all warm and fuzzy inside. People who don't wish to give you advice may ignore you or brush you off, so there's basically no harm in asking very widely. It took me years to accept that my parents might occasionally have something wise to say, but it's true. Now that I've reached the point where my parents' academic/career advice is no longer particularly helpful, I have to cast a wider net -- but it's crucially important that I actually do so, and not rely on assumptions or PhD Comics.
Monday, September 29, 2008
Open Courseware is not the be-all and end-all
MIT's Open Courseware gets a lot of press. And a lot of it is well deserved -- it's fairly comprehensive, and does a good job managing the technical side of throwing vast amounts of material up on the internets (hooray for PDF). But it's not everything.
The past few days I've been stressing over various topics from my genetics class, mostly yeast tetrad analysis. Going to lecture hardly helps; for this class, it's been generally true that the more confusing topics are very poorly explained. Usually the textbook does a good job of explaining things, but it doesn't cover yeast tetrads hardly at all. And I have a test in two days.
(Yeast tetrad analysis is pretty interesting, by the way. I might write a post about it if I ever actually understand it.)
Turning to OCW didn't help, because more or less the same professors have been teaching the class for N years, so the lecture notes are exactly the same. I can get practice problems (old homeworks and exams), which are very useful once I actually understand something, but not in this case.
So I turned to Google. And lo and behold, googling {yeast tetrad analysis} brings up several universities' intro-genetics webpages about yeast tetrad analysis! With clear, coherent explanations and well-drawn diagrams! UC Berkeley (pdf), Indiana U (pdf), U of Saskatchewan, and U of Rochester, just to name a few.
(Even better, some of those are straight-up webpages. Better than everything being PDF, although not better than everything being .doc, which sometimes happens.)
I often hear, second or third hand, how people from other schools use MIT OCW all the time. And I'm not denying that OCW is awesome. But it's not perfect, and there are other places to go if the OCW explanation of a topic happens to be consistently lousy.
The past few days I've been stressing over various topics from my genetics class, mostly yeast tetrad analysis. Going to lecture hardly helps; for this class, it's been generally true that the more confusing topics are very poorly explained. Usually the textbook does a good job of explaining things, but it doesn't cover yeast tetrads hardly at all. And I have a test in two days.
(Yeast tetrad analysis is pretty interesting, by the way. I might write a post about it if I ever actually understand it.)
Turning to OCW didn't help, because more or less the same professors have been teaching the class for N years, so the lecture notes are exactly the same. I can get practice problems (old homeworks and exams), which are very useful once I actually understand something, but not in this case.
So I turned to Google. And lo and behold, googling {yeast tetrad analysis} brings up several universities' intro-genetics webpages about yeast tetrad analysis! With clear, coherent explanations and well-drawn diagrams! UC Berkeley (pdf), Indiana U (pdf), U of Saskatchewan, and U of Rochester, just to name a few.
(Even better, some of those are straight-up webpages. Better than everything being PDF, although not better than everything being .doc, which sometimes happens.)
I often hear, second or third hand, how people from other schools use MIT OCW all the time. And I'm not denying that OCW is awesome. But it's not perfect, and there are other places to go if the OCW explanation of a topic happens to be consistently lousy.
Friday, April 25, 2008
Trivia-heap syndrome
Part of the reason I think a lot of hard scientists look down on biology is that introductory biology is so often poorly taught in a particular way. Intro physics is almost entirely problem-solving, and intro chemistry is similar, with maybe a bit more memorization (Quick! How many valence electrons does aluminum have?). But biology frequently ends up being taught as a large heap of random terms and facts, almost entirely without any unifying themes or methods of thought -- just a disconnected jumble.
I've dubbed this "trivia-heap syndrome".
Any biologist (or, hell, anyone who's gotten past the required intro course) can tell you that biology is not about memorizing terms and facts, any more than physics is about blocks sliding down inclined planes. But as Chad Orzel says in that post: "To some degree, this is inescapable-- those repeated exercises are used to establish a pattern of thought that is a necessary prerequisite for moving on to more interesting material." A similar thing could be said of biology: to some degree, it's necessary to memorize a lot of terms and random facts and unconnected processes and so on, before you can get to the interesting work of studying how they interact and how they can be manipulated.
But this is actually true of every field, not just biology (or other fields prone to trivia-heap syndrome); it's just not as apparent. Consider intro mechanics again: all about things falling, hitting each other, rotating, etc etc. In order to do interesting things, you first have to know what balls, rods, strings, pulleys, blocks, inclined planes, and gravity are. You also have to understand the basic types of things they can do: move, rotate, accelerate, come into contact, break, exert forces on each other. Of course these are trivial things to know, because we've all been exposed to simple objects and their motions since we were born -- and this is why intro mechanics courses don't begin with a couple weeks of definitions and memorization. The only difference between that and biology is that we're not exposed from birth to genes and proteins and cells and their interactions. The world teaches us the trivia-heap for physics, but we have to be taught about biology's trivia-heap.
All right, so the trivia-heap is an unavoidable evil whenever you're starting in a new field. Fine. But never fear, there are still ways to get around trivia-heap syndrome. As soon as you know a very few things, you can start thinking about them in terms of experiments to be done and puzzles to be solved, instead of facts and descriptions. What would happen if this particular thing were mutated in this way? What effect would that have on the cell? How could you (the experimenter) tell that this was in fact the case? What if you got the opposite observation -- what might have gone wrong? Here's a simple system you're interested in; outline an experiment to find out whether this particular part of it works this way or that way. This is both interesting and a lot more like what actual biologists do; certainly much more so than the typical dreck of "Define a gene", "Outline how a gene gets translated to protein" that most high-schoolers get shoved down their throats. That's lazy teaching for you (or the creeping horror of bad standardized curricula for things like AP tests, which reduces to the same thing).
I learned a decent bit of biology just by reading random things. I can practically recite The Cartoon Guide to Genetics, by Gonick & Wheelis (that link is to the updated edition, not the old-school edition I read). I also took an introductory class at one of those academic summer camps. The net result was that I came into my high school biology class already knowing about half the stuff we would cover, which just made the trivia-heap syndrome that much more painful. A lot of my classmates struggled with memorizing things. Very few people did well on the "lab practical", in which we had to plan and carry out experiments to identify a mystery substance -- precisely because we spent so much time on trivia-heaping and so little time on problem solving / sensible experiment design. MIT's intro biology course (7.013), by contrast, is like a breath of fresh air. It's not entirely free of trivia-heap syndrome, but there's a gigantic emphasis on solving puzzles, considering what-ifs, proposing experiments, and interpreting results. (`Gigantic' emphasis relative to other intro biologies, of course.)
Granted, even MIT's intro biology problem-solving is simplistic, and occasionally feels like doing an inclined-plane problem in physics (especially when I know a decent bit about a particular system, and I can tell the problem vastly oversimplifies the situation, even if I don't know exactly how). But now we're back to Chad Orzel, inclined planes, and "establishing patterns of thought". There is a certain intuition of biological ways and means, of how cells/genes/proteins work in broad strokes, that is immensely valuable but hard to obtain. This intuition is something like a toolkit of abstractions over specific examples, but I'd venture to say it's not something that can be taught explicitly in the abstract, the way math can. The analogous thing in physics, again, is something the world teaches us from birth: unsupported objects fall, if you push something its speed changes; that sort of general idea about how things operate. I've developed my biological intuition somewhat, through reading and studying (and sometimes working directly with) boatloads of examples, and I think that's just about the only way it can be developed. But it's something every experimenter (or bioengineer!) needs in spades, and it's something the general public could also really use.
So, to sum up: the trivia-heap is sometimes a necessary evil, but trivia-heap syndrome is eminently avoidable. Emphasize puzzle-solving and experiments, instead of facts and definitions, and it'll do everyone a heap of good.
I've dubbed this "trivia-heap syndrome".
Any biologist (or, hell, anyone who's gotten past the required intro course) can tell you that biology is not about memorizing terms and facts, any more than physics is about blocks sliding down inclined planes. But as Chad Orzel says in that post: "To some degree, this is inescapable-- those repeated exercises are used to establish a pattern of thought that is a necessary prerequisite for moving on to more interesting material." A similar thing could be said of biology: to some degree, it's necessary to memorize a lot of terms and random facts and unconnected processes and so on, before you can get to the interesting work of studying how they interact and how they can be manipulated.
But this is actually true of every field, not just biology (or other fields prone to trivia-heap syndrome); it's just not as apparent. Consider intro mechanics again: all about things falling, hitting each other, rotating, etc etc. In order to do interesting things, you first have to know what balls, rods, strings, pulleys, blocks, inclined planes, and gravity are. You also have to understand the basic types of things they can do: move, rotate, accelerate, come into contact, break, exert forces on each other. Of course these are trivial things to know, because we've all been exposed to simple objects and their motions since we were born -- and this is why intro mechanics courses don't begin with a couple weeks of definitions and memorization. The only difference between that and biology is that we're not exposed from birth to genes and proteins and cells and their interactions. The world teaches us the trivia-heap for physics, but we have to be taught about biology's trivia-heap.
All right, so the trivia-heap is an unavoidable evil whenever you're starting in a new field. Fine. But never fear, there are still ways to get around trivia-heap syndrome. As soon as you know a very few things, you can start thinking about them in terms of experiments to be done and puzzles to be solved, instead of facts and descriptions. What would happen if this particular thing were mutated in this way? What effect would that have on the cell? How could you (the experimenter) tell that this was in fact the case? What if you got the opposite observation -- what might have gone wrong? Here's a simple system you're interested in; outline an experiment to find out whether this particular part of it works this way or that way. This is both interesting and a lot more like what actual biologists do; certainly much more so than the typical dreck of "Define a gene", "Outline how a gene gets translated to protein" that most high-schoolers get shoved down their throats. That's lazy teaching for you (or the creeping horror of bad standardized curricula for things like AP tests, which reduces to the same thing).
I learned a decent bit of biology just by reading random things. I can practically recite The Cartoon Guide to Genetics, by Gonick & Wheelis (that link is to the updated edition, not the old-school edition I read). I also took an introductory class at one of those academic summer camps. The net result was that I came into my high school biology class already knowing about half the stuff we would cover, which just made the trivia-heap syndrome that much more painful. A lot of my classmates struggled with memorizing things. Very few people did well on the "lab practical", in which we had to plan and carry out experiments to identify a mystery substance -- precisely because we spent so much time on trivia-heaping and so little time on problem solving / sensible experiment design. MIT's intro biology course (7.013), by contrast, is like a breath of fresh air. It's not entirely free of trivia-heap syndrome, but there's a gigantic emphasis on solving puzzles, considering what-ifs, proposing experiments, and interpreting results. (`Gigantic' emphasis relative to other intro biologies, of course.)
Granted, even MIT's intro biology problem-solving is simplistic, and occasionally feels like doing an inclined-plane problem in physics (especially when I know a decent bit about a particular system, and I can tell the problem vastly oversimplifies the situation, even if I don't know exactly how). But now we're back to Chad Orzel, inclined planes, and "establishing patterns of thought". There is a certain intuition of biological ways and means, of how cells/genes/proteins work in broad strokes, that is immensely valuable but hard to obtain. This intuition is something like a toolkit of abstractions over specific examples, but I'd venture to say it's not something that can be taught explicitly in the abstract, the way math can. The analogous thing in physics, again, is something the world teaches us from birth: unsupported objects fall, if you push something its speed changes; that sort of general idea about how things operate. I've developed my biological intuition somewhat, through reading and studying (and sometimes working directly with) boatloads of examples, and I think that's just about the only way it can be developed. But it's something every experimenter (or bioengineer!) needs in spades, and it's something the general public could also really use.
So, to sum up: the trivia-heap is sometimes a necessary evil, but trivia-heap syndrome is eminently avoidable. Emphasize puzzle-solving and experiments, instead of facts and definitions, and it'll do everyone a heap of good.
Monday, January 21, 2008
Math lagging (or leading) physics
Senior year of high school, I took AP Physics C and AP Calculus BC concurrently: allowable, but not advisable, the admins said. For a good part of the year, I struggled with physics because I hadn't `really' learned the math yet. Reading the textbook, I'd just skip over derivations that I didn't understand, and hope that the end result was something I could memorize, if not comprehend. Mr. T held a session or two of `physics math', which helped a little, but didn't make me feel less intimidated.
Last semester, I was in pretty much the same situation, only with line and surface integrals instead of all integrals. Oh, we never had to evaluate any -- they were always of the simple kind that reduce to multiplication problems -- but it took several tries to be able to recognize those situations, and remember how to reduce them to multiplication problems. Didn't solve the intimidation problem, either. On the final exam, even though I knew what to do and how the problems worked, I still felt like a trespasser for writing down those fancy-schmancy closed-surface double integral signs, knowing I didn't really know how they worked.
At one point Prof. Hudson said something (can't recall the exact words) that gave the impression this math-lagging-physics business was a dreadful intractable plague, oh me oh my, whatever shall we do. And at the time, struggling to understand everything, I was rather inclined to agree with him. But since then I've somewhat reversed position, or at least become neutral.
Back in high school, after I'd learned to fudge my way through integration, I had a much, much easier time with it when we `officially' learned it in math class. This was especially noticeable in the case of integrating (still in a single variable, though!) over 3D objects. We'd been doing this routinely in physics, to find the moments of inertia of various shapes. So while the rest of the calculus class was struggling to put together the relatively new concept of integration with the relatively new concept of bizarre-shaped objects in 3-space, I could relax.
Recently (within the past two weeks), the same thing happened for me in multivariable calculus. I'm in 18.01A/18.02A, which is a blend of two courses: a quick six-week review of single-variable, and then the regular multivariable curriculum, finishing over the January term. Because of the way things are timed, I never saw a line integral in 18.02A until just now, well after the end of my physics class. And the same thing happened: I'd learned to fudge my way through easy line integrals, and that gave me a leg up on people who'd never seen them before. I had a well-developed intuition about what line integrals meant in terms of the real world, which kept me from getting confused. That in turn allowed me to focus on the complicated aspects that we hadn't worked with in physics.
Given these experiences, I don't think it's a very bad thing at all that math education tends to lag physics education. It can often be a good thing: physics is leading math, to frame it positively. When you tackle something complicated and abstract, having `live-fire' experience with simple cases and workarounds can hardly do anything but help.
This comes with some caveats. It's a classic example of delayed gratification: I sure didn't feel happy about math lagging physics when I was actually struggling with the physics, and it was only a month later that I was able to grin to myself in calculus recitation, watching the people around me struggle with a problem I'd found obvious. (Schadenfreude, too, is not necessarily a good thing. In my defense, I do help my peers if I can, and if they're not already being helped.) And naturally, it's a tradeoff; who can say if knowing the math beforehand would have brought my physics grade up a little?
But most importantly, if physics is going to lead math, physics has to be taught right. I approve very strongly of Prof. Hudson's philosophy here: conceptually difficult problems with easy math = win. If you absolutely must be bashing your head against the wall at midnight the night before the problem set is due, it shouldn't be because you can't solve a differential equation. It should be because you can't figure out which way the induced current goes. Physics can be taught adequately without using too much complicated math (though of course the definition of `complicated' will vary with the physics). You learn just as much integrating (read: multiplying) over a sphere as you do integrating (read: flailing) over the surface of some crazy-shaped thing.
You wouldn't expect it, but knowing too much math can be a problem. Last semester's physics class was taught in a rather unusual format, about which more later. The relevant thing is that I spent the whole time in a group of three. There was me, there was S, and there was L. S had excellent intuition, and tended to approach problems by thinking them through qualitatively first, and then seeing if the math bore him out. This was really a very effective strategy -- a lot of the time the math looked right, but his intuition cried foul, and then we found the sign error twenty steps back. L, on the other hand, was advanced in math, and already pretty far beyond what the physics course used. She approached problems very mathematically, instead of with intuition like S. Even when something made complete physical sense, she wasn't satisfied until she justified it mathematically -- even if it was a qualitative problem and didn't have any equations in it to begin with. and because of how every other thing is backwards in E&M, she tied herself into an awful lot of knots. (Not that she wasn't brilliant; I think she outscored both S and me on every test.) L and S complemented each other, and they were a great group to work with. But I have to say that S's intuitive method was the more effective. To the same question, S might answer "because the charge is over here so the potential is higher here, and then this happens"; L might answer "because the potential is [insert equation here] and the electric field is the negative gradient of the potential."
I admit that this is based on my own experience, and what's better for different students will vary. I said that it's easier to learn complicated abstract math if you've first grounded it in physics; you could just as well say that it's easier to learn physics if you're well practiced with the mathematical tools already. But I think the first argument is more valid than the second. It's easier to abstract from the concrete than it is to learn the abstract first and only then get concrete examples. You see the same thing in computer science classes: pick up a copy of the blue wizard book and on every other page you'll find a discussion of how to abstract some instance, not how to instantiate some abstraction. Even though physics was made genuinely more difficult by not knowing the math, I'd still rather have done it this way around.
Last semester, I was in pretty much the same situation, only with line and surface integrals instead of all integrals. Oh, we never had to evaluate any -- they were always of the simple kind that reduce to multiplication problems -- but it took several tries to be able to recognize those situations, and remember how to reduce them to multiplication problems. Didn't solve the intimidation problem, either. On the final exam, even though I knew what to do and how the problems worked, I still felt like a trespasser for writing down those fancy-schmancy closed-surface double integral signs, knowing I didn't really know how they worked.
At one point Prof. Hudson said something (can't recall the exact words) that gave the impression this math-lagging-physics business was a dreadful intractable plague, oh me oh my, whatever shall we do. And at the time, struggling to understand everything, I was rather inclined to agree with him. But since then I've somewhat reversed position, or at least become neutral.
Back in high school, after I'd learned to fudge my way through integration, I had a much, much easier time with it when we `officially' learned it in math class. This was especially noticeable in the case of integrating (still in a single variable, though!) over 3D objects. We'd been doing this routinely in physics, to find the moments of inertia of various shapes. So while the rest of the calculus class was struggling to put together the relatively new concept of integration with the relatively new concept of bizarre-shaped objects in 3-space, I could relax.
Recently (within the past two weeks), the same thing happened for me in multivariable calculus. I'm in 18.01A/18.02A, which is a blend of two courses: a quick six-week review of single-variable, and then the regular multivariable curriculum, finishing over the January term. Because of the way things are timed, I never saw a line integral in 18.02A until just now, well after the end of my physics class. And the same thing happened: I'd learned to fudge my way through easy line integrals, and that gave me a leg up on people who'd never seen them before. I had a well-developed intuition about what line integrals meant in terms of the real world, which kept me from getting confused. That in turn allowed me to focus on the complicated aspects that we hadn't worked with in physics.
Given these experiences, I don't think it's a very bad thing at all that math education tends to lag physics education. It can often be a good thing: physics is leading math, to frame it positively. When you tackle something complicated and abstract, having `live-fire' experience with simple cases and workarounds can hardly do anything but help.
This comes with some caveats. It's a classic example of delayed gratification: I sure didn't feel happy about math lagging physics when I was actually struggling with the physics, and it was only a month later that I was able to grin to myself in calculus recitation, watching the people around me struggle with a problem I'd found obvious. (Schadenfreude, too, is not necessarily a good thing. In my defense, I do help my peers if I can, and if they're not already being helped.) And naturally, it's a tradeoff; who can say if knowing the math beforehand would have brought my physics grade up a little?
But most importantly, if physics is going to lead math, physics has to be taught right. I approve very strongly of Prof. Hudson's philosophy here: conceptually difficult problems with easy math = win. If you absolutely must be bashing your head against the wall at midnight the night before the problem set is due, it shouldn't be because you can't solve a differential equation. It should be because you can't figure out which way the induced current goes. Physics can be taught adequately without using too much complicated math (though of course the definition of `complicated' will vary with the physics). You learn just as much integrating (read: multiplying) over a sphere as you do integrating (read: flailing) over the surface of some crazy-shaped thing.
You wouldn't expect it, but knowing too much math can be a problem. Last semester's physics class was taught in a rather unusual format, about which more later. The relevant thing is that I spent the whole time in a group of three. There was me, there was S, and there was L. S had excellent intuition, and tended to approach problems by thinking them through qualitatively first, and then seeing if the math bore him out. This was really a very effective strategy -- a lot of the time the math looked right, but his intuition cried foul, and then we found the sign error twenty steps back. L, on the other hand, was advanced in math, and already pretty far beyond what the physics course used. She approached problems very mathematically, instead of with intuition like S. Even when something made complete physical sense, she wasn't satisfied until she justified it mathematically -- even if it was a qualitative problem and didn't have any equations in it to begin with. and because of how every other thing is backwards in E&M, she tied herself into an awful lot of knots. (Not that she wasn't brilliant; I think she outscored both S and me on every test.) L and S complemented each other, and they were a great group to work with. But I have to say that S's intuitive method was the more effective. To the same question, S might answer "because the charge is over here so the potential is higher here, and then this happens"; L might answer "because the potential is [insert equation here] and the electric field is the negative gradient of the potential."
I admit that this is based on my own experience, and what's better for different students will vary. I said that it's easier to learn complicated abstract math if you've first grounded it in physics; you could just as well say that it's easier to learn physics if you're well practiced with the mathematical tools already. But I think the first argument is more valid than the second. It's easier to abstract from the concrete than it is to learn the abstract first and only then get concrete examples. You see the same thing in computer science classes: pick up a copy of the blue wizard book and on every other page you'll find a discussion of how to abstract some instance, not how to instantiate some abstraction. Even though physics was made genuinely more difficult by not knowing the math, I'd still rather have done it this way around.
Friday, January 18, 2008
Feeling a teacher's love
In high school, we had a teacher, Mr. T, who taught computer science and some of the advanced math courses. And he was wonderful. Everybody loved him. I've never encountered so much infectious enthusiasm, even at the ungodly hour of 8am on a Monday morning. He could light us on fire by drawing box-and-pointer diagrams. We had a party the day we learned the Fundamental Theorem of Calculus. Mr. T was a mentor, a friend, a father, a rock; his room was a haven for geeklings. Everybody loved him, and he loved us too.
I understood his love in a rather abstract sense back then, and up until this morning. But this afternoon, I understood it from the inside.
The research I'm doing at the moment involves teaching pairs of people a constructed language by immersion, and then having them take a test. (More details on this (very interesting) paradigm in an upcoming post!) Today was only the second time this experiment has ever been run, so I'm also happy just because it worked. We ran two pairs of people.
I had an awfully hard time teaching the first pair. They just seemed to fundamentally not get a lot of the grammar. It was very difficult to keep myself from grabbing one guy and telling him, loudly and in English, that "the way to turn a sentence into a question is NOT just to say the sentence with a rising intonation at the end!!!". And so on, and so on. Teaching them was frustrating, occasionally painful.
And then the second pair came in, a pair of undergrads, and they more than made up for the first pair. It only took them twenty minutes to get to more than adequate proficiency, where it had taken the first pair almost an hour to get to less than adequate proficiency. They seemed to pick everything up right away. It was like watching a rose bloom in time-lapse. My favorite moment was when I taught the question form, using only one verb in my examples. When I prompted one of the participants, she immediately came up with several correct question-form sentences, and she generalized to all the verbs we'd learned, not just the one I demonstrated with. If we hadn't been in an experiment, I might have proposed to her on the spot.
I graded the four tests just now. The first pair had decidedly mediocre scores, no surprise there. But the second pair did extremely well, and they both did just about perfectly on the part we're most interested in. I was overjoyed to see them using linguistic terminology correctly to explain their answers (even though their knowledge of terms had nothing to do with how well they acquired the language). When I was done grading, I spontaneously picked up the last test and kissed it.
Tonight, I think I experienced a little of what made Mr. T such a good teacher, what made him love his students and his work. It's not an easy feeling to articulate. Pride and joy in students, enthusiasm for the subject, and a little bit of self-satisfaction that, yeah, I taught them that. The feeling was very powerful, even though the teaching I did today wasn't very real. What if I were teaching material that was beautiful and meaningful, instead of an arbitrary constructed language? What if I could teach for an entire semester and witness long-term progress and synthesis, instead of bidding goodbye after an hour? What if I were teaching students who learned for love of the material, not because they were paid to be subjects in a study?
For the record, this is not the first time I've taught, nor the first time I've found it satisfying. For a couple of years I tutored 6th graders in math and Japanese, and it was always great when they'd get a flash of insight after a long hard slog. (They were remedial students, so it wasn't often.) And in the Karate club, it's traditional that you help teach the people who rank below you. I was co- leading brown belt my senior year, so that meant I taught just about everybody. I always enjoyed seeing some yellow-belts perform a technique really well, and thinking "Yay, they remembered X subtle point I taught them about!". But I've never felt a teacher's love as strongly as I did today; certainly not strongly enough to merit using the word `love'.
If I follow the track I'm planning to follow, I'll end up a professor. I understand that teaching is mostly tedious, frustrating, and difficult, not full of brilliant-student-love. But it's the possibility, the hope that springs eternal, and when it's fulfilled it makes up for everything else.
I understood his love in a rather abstract sense back then, and up until this morning. But this afternoon, I understood it from the inside.
The research I'm doing at the moment involves teaching pairs of people a constructed language by immersion, and then having them take a test. (More details on this (very interesting) paradigm in an upcoming post!) Today was only the second time this experiment has ever been run, so I'm also happy just because it worked. We ran two pairs of people.
I had an awfully hard time teaching the first pair. They just seemed to fundamentally not get a lot of the grammar. It was very difficult to keep myself from grabbing one guy and telling him, loudly and in English, that "the way to turn a sentence into a question is NOT just to say the sentence with a rising intonation at the end!!!". And so on, and so on. Teaching them was frustrating, occasionally painful.
And then the second pair came in, a pair of undergrads, and they more than made up for the first pair. It only took them twenty minutes to get to more than adequate proficiency, where it had taken the first pair almost an hour to get to less than adequate proficiency. They seemed to pick everything up right away. It was like watching a rose bloom in time-lapse. My favorite moment was when I taught the question form, using only one verb in my examples. When I prompted one of the participants, she immediately came up with several correct question-form sentences, and she generalized to all the verbs we'd learned, not just the one I demonstrated with. If we hadn't been in an experiment, I might have proposed to her on the spot.
I graded the four tests just now. The first pair had decidedly mediocre scores, no surprise there. But the second pair did extremely well, and they both did just about perfectly on the part we're most interested in. I was overjoyed to see them using linguistic terminology correctly to explain their answers (even though their knowledge of terms had nothing to do with how well they acquired the language). When I was done grading, I spontaneously picked up the last test and kissed it.
Tonight, I think I experienced a little of what made Mr. T such a good teacher, what made him love his students and his work. It's not an easy feeling to articulate. Pride and joy in students, enthusiasm for the subject, and a little bit of self-satisfaction that, yeah, I taught them that. The feeling was very powerful, even though the teaching I did today wasn't very real. What if I were teaching material that was beautiful and meaningful, instead of an arbitrary constructed language? What if I could teach for an entire semester and witness long-term progress and synthesis, instead of bidding goodbye after an hour? What if I were teaching students who learned for love of the material, not because they were paid to be subjects in a study?
For the record, this is not the first time I've taught, nor the first time I've found it satisfying. For a couple of years I tutored 6th graders in math and Japanese, and it was always great when they'd get a flash of insight after a long hard slog. (They were remedial students, so it wasn't often.) And in the Karate club, it's traditional that you help teach the people who rank below you. I was co- leading brown belt my senior year, so that meant I taught just about everybody. I always enjoyed seeing some yellow-belts perform a technique really well, and thinking "Yay, they remembered X subtle point I taught them about!". But I've never felt a teacher's love as strongly as I did today; certainly not strongly enough to merit using the word `love'.
If I follow the track I'm planning to follow, I'll end up a professor. I understand that teaching is mostly tedious, frustrating, and difficult, not full of brilliant-student-love. But it's the possibility, the hope that springs eternal, and when it's fulfilled it makes up for everything else.
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