Teaching Linear Algebra
bentilly.blogspot.com
bentilly.blogspot.com
So every few weeks, the four of us in the class would spend days trying to prove the continuum hypothesis or something, and going over the homework, he'd get to the question and ask casually if anyone solved it. We'd all say no and he'd tell us no one could, and show us proof. But we all learned a hell of a lot trying. Sometimes an avenue we tried ended up proving something interesting on the way, and it sharpened our thinking.
Of course, we also got into the habit trying to prove something was provable before we started (which turned out helpful when, at the end of the year, we collectively recreated [with help] much of the groundwork for Gödel's big theorems).
Thanks for obliging.
However, I will gripe about his messianic tone. As far as I can tell from his blog post, he's taught only one class! Even if he taught a few others as a grad student, my point holds.
When I first started teaching nine years ago, as a grad student, I had miraculous success with my classes. I didn't have any grand scheme like btilly did; all I had were some commonsense observations from my father, a lifelong teacher.
But as a new teacher, I was really learning the material for the first time (that's the case with virtually all new teachers -- it has to do with the process of teaching), and so I could carefully observe the students' progress at every step. If they stepped into a pothole, I stepped into it with them: I felt that I had failed to explain something clearly if the students didn't get it. (btilly says the same.) The students loved the class, they became very competent, and their enthusiasm for the subject attracted attention. I was a star teacher.
However, as the years have ground on, I have taught the same subjects over and over again. I'm no longer on a journey of discovery along with the students. Instead, I know where every curve and pitfall lies. I can see a pothole a mile before we get there. You might think that this makes me a more effective teacher, because I can make the journey a smooth arc from beginning to end. But it doesn't: intellectual discovery is about stepping into potholes, not sailing by on a cruise ship.
I'm still sort of a star teacher (evals are very high, and my classes are always full, in part because word gets around), but I've never had a class with the same enthusiasm and brilliance as that first one.
(I'm an adjunct professor, so I get paid shit wages, and I have no incentive to do a good job, but that's a different story.)
So I gripe about the messianic tone in btilly's post because it sounds like precisely the sort of thing a successful new teacher would say. (BTW, not all new teachers are successful.) I wonder if with a decade of teaching experience he would see his miraculous method as the distinguishing factor in his early success. In fact, his early success may have been due primarily to the advantages of a neophyte, which are hard to sustain.
The system would be improved if all such positive classroom teaching experiences were recorded, distributed, tested, etc.
That said, I was pushing students to use learning strategies that I have personally found to be incredibly valuable. I firmly believe that it is a good thing to read ahead before class, pay close attention during class, ask questions promptly and follow a regular review schedule later. This is both effective and takes surprisingly little time to do.
On a final note, I sympathize with your pain about not being rewarded for your teaching. The academic system rewards professors for research, not teaching, and teaching suffers greatly for it. Why the Professor Can't Teach is as relevant today as it was when it was 30 years ago. (You can find it online at http://www.marco-learningsystems.com/pages/kline/prof.html.) That was not the reason that I left academia, but it made my decision easier.
The perverse incentives this creates are massive. Universities hire scientists rather than teachers in order to get their hands on half the scientist's grants [1]. Scientists waste their time masquerading as teachers because they can't get grants if they don't work for a university [2]. This is harmful both to science (I'm not doing research in class) and students. Of course, they don't put much effort into teaching because they are judged on their ability to get grants. Actual teachers are squeezed out, since there is no room for them.
My proposed solution? Completely decouple research and teaching. The NSF can subsidize science, the DOE can subsidize universities, and teaching institutions will no longer have an incentive to hire scientists as teachers.
[1] If a scientist gets a grant, the university will take about half as "overhead".
[2] Not strictly true, due to a few national labs, but close enough.
But everything else you said: pretty much, yeah. Once, I was in a meet-and-greet between grad students an interviewing faculty candidate when it dawned on me that as a second year grad student, I had as much teaching experience as he did.
I guess what i am saying is this: if all research moves to national labs and private institutes you will discover graduate education migrating there as well.
You already know the obvious stuff; you have a framework to slot the lecture content into; and are primed to be interested in the parts that you didn't get.
* Sometimes you'll read the material... and then the lecture will be a regurgitation of the material. For me, this means doodling in notebooks or falling asleep
* Other times you'll read the material... and it will be wildly different from what the Professor decides to cover. So now you've read something and you're just thinking about what the hell you're talking about
* Some teachers don't actually provide material
* Sometimes lecture is just about going over homework problems (this was mainly in intro courses, but it was SEVERELY lame if you got most of the answers right).
I think it's best when there's material to be covered and the teacher has extra insight to add or confusing elements that they can somehow explain better.
I abhor the attitude of the students. Simply working hard is not always enough. Results matter. This is a problem in the corporate world as well.
The best I ever got was "do these odd-numbered problems from the book", which is not quite as beneficial.
The ones with the solutions in the back are for studying for the exam.
(I do find it annoying when I buy textbooks for my personal use that I don't get the answers for all the questions though. Lazy authors!)
If people just put up the answers they do work out somewhere, this would help others. This is happening with SICP for example.
But there's extra money to be made by selling the solutions manual!
I might try assigning the next section reading before class to see how they respond. I don't think they will read ahead...
I once had a psychology class where the professor, who worked on learning theory, went so far as to work with some software company to develop Palm OS software which could be used to make students answer questions. He would put up a question on a projection screen in front of the class, and everyone would tap out an answer on a Wi-Fi enabled Treo PDA, which would send the students' answers to a server. Then the right answer would show up on the projection screen, so students get immediate feedback.
Maybe this shows that active involvement is essential, and it does not matter if students ask or answer questions. A little common sense might have done better than high technology.
The method that I used wouldn't scale to a large class. Your psychology professor's piece of software would.
http://www.ecae.net/category/saturday-school/
immediately. I also like asking lots of questions in class, and routinely do that whenever I teach anything. (This probably comes from being a language major as an undergraduate--language classes have to be interactive.) The one advantage supplementary classes after school hours have over school classes is not having to issue grades, and thus not evoking disputes about grading policies.
I agree that having read the material helps, but if you haven't ( either because you were lazy/busy or the material was not available beforehand ) taking notes surely does help.
The approach described would probably be great for many other subjects besides math, of course.
Wouldn't it be interesting if we taught people how to learn instead of just teaching them the material?
As for notes, that would end the moment math teachers used slides you could download after class. I wish more people would do that (in my experience only the math teachers do this, all my CS teachers don't).
Asking questions during the lecture is pretty good, but I wonder how you would prevent the students from feeling embarrassed because they could not answer?
The apparent harshness of the policy was softened by the fact that only 3/4 of your homework sets counted. Therefore not being allowed to turn in a homework set or three wasn't a big deal.
As for preventing students from feeling embarrassed, my strategy was to try really hard to make sure that they succeeded on the questions. I would guess I averaged perhaps one missed question per week.
Of course, some students won't really understand exactly what the teacher is asking, might have problems talking in front of people, etc. What I would suggest, then, is to make sure that if someone doesn't know a problem, you immediately move on to another student (i.e. don't badger one student if they don't know the answer).
If you find that a student frequently passes on answering questions, that would be a great indication that they might not be understanding the issues, and that would be a good time to schedule a meeting with them to get it sorted out (via email or something, to avoid singling that student out in front of his/her peers).
No, what's harsh is showing up late, which is a distraction to everyone else.
how you would prevent the students from feeling embarrassed because they could not answer?
If they don't want to feel embarrassed, they should be prepared. And even if they are embarrassed, who cares? This is college, not some encounter group. Think this is tough, just wait til you see what comes after college.
It's not really a distraction. If I or someone else comes in late usually no one even looks, even if it's a tiny room. Then I can turn in whatever homework after class. If I am doing the homework during class and not paying attention? Harsh, I guess I'll fail then.
It's not really a distraction. If I or someone else comes in late usually no one even looks...
According to whom? I guarantee you, the person in front of the room notices when someone comes in late. Others in the room notice as well, even if you don't think so.
Showing up, being prepared, and having consideration for others are half the battle. If you can't at least do that much, why bother?
What's harsh or an embarassment in college is a disaster in the real world. Many people I know lock the doors when the meeting starts. Show up unprepared and be sent away. People are depending on you to do the minimum. That's the least anyone could learn in college.
Though in the OP's case he did make up for the harshness by allowing some missed homeworks.
Classes are different from meetings. They shouldn't simulate real world meetings. People pay for classes and if they want to come in late, they reap the consequences. Otherwise why pay? When you work for someone it becomes your duty to perform the job, attending meetings on time is one such duty.
That being said there's no need for college professors to try and simulate what you'll face in the real world by being harsh. A little leniency goes a long way and frankly in college people have the understanding that you know how to manage your time and automatically have a good reason for being late.
George bush locks people out of meetings if they're late. I highly doubt Obama does that. So it's not necessarily real life. Some people are understanding in real life too.
There's no reason to propagate harshness.