Teaching Linear Algebra

http://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html

by btilly • 17 years ago

208 59 17 years ago

bentilly.blogspot.com

ananthrk• 17 years ago

I believe this is btilly's response to this request http://news.ycombinator.com/item?id=818527

Thanks for obliging.

brown9-2• 17 years ago

I'd like to second the thanks.

btilly5• 17 years ago

You're welcome.

tocomment• 17 years ago

This is great.If I ever teach a class I'm going to try to do this stuff.

dbz• 17 years ago

I went through a class with a similar teaching style. I loved it. I sat in the front row each day because I was learning so much. It was in fact the first math class I thoroughly enjoyed.

jrockway2• 17 years ago

The homework sounds good.

The best I ever got was "do these odd-numbered problems from the book", which is not quite as beneficial.

chrischen9• 17 years ago

Yea and they always pick the ones with no solutions in the back.

jrockway2• 17 years ago

You get the answers to the homework in class.

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!)

pbz• 17 years ago

"I treated failure to get the answer as my failures, not theirs." -- I wish more teachers would have this attitude. A while back I used to teach programming classes and asking them questions about what I was talking about was a very important metric for me. If I would notice any kind of hesitation I would focus more of that issue. I'd try to approach the same problem from different angles until I would be convinced they "got it" and were not just guessing. It's the teaching equivalent of debugging your programs and see if they run as expected.

tomjen2• 17 years ago

He seems harsh, I would properly let the first student hand in his homework.

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?

btilly5• 17 years ago

The whole point of the homework policy was to make sure that class started on time. That was critical given that the most important part of the class was the first 10 minutes. Accepting homework from students who walked in late would have ruined that.

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.

harpastum• 17 years ago

In terms of being embarrassed, if you tell your students that you are expecting them to read ahead and only ask questions on subjects you've already explained, everyone should know the answer.

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).

edw5192• 17 years ago

He seems harsh, I would properly let the first student hand in his homework.

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.

chrischen9• 17 years ago

> No, what's harsh is showing up late, which is a distraction to everyone else.

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.

chrischen9• 17 years ago

He does seem a little harsh. I've never encountered a professor that harsh.

tokenadult• 17 years ago

I really appreciate the detailed blog post description of the linear algebra class at Dartmouth a decade ago. There are ideas here worth applying to my nonprofit's supplementary math classes

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.

10ren• 17 years ago

Just on the notes: as a student, I found that reading the text ahead of class to be amazingly effective. Just skimming for 15 minutes.

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.

chrischen9• 17 years ago

I find it impossible to to follow along in class, so I just read the text. But it would be amazingly effective and instructors usually expect you to read the chapter ahead anyways, which might be why it's so much more difficult to understand if you haven't read it.

DannoHung• 17 years ago

Generally I agree, but there are some issues:

* 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.

tigerthink• 17 years ago

My favorite teacher sometimes lets us discover stuff for ourself just by repeatedly asking us questions about it, so it spoils the surprise to read ahead.

boredguy8• 17 years ago

I wish my linear algebra teacher was like this. I could barely understand the words he was saying, and linear algebra ultimately led to me dropping my Math major and moving to computer science.

gcv• 17 years ago

One curious difference between the author's approach and learning theory: he just made students ask questions, whereas learning theory wants students to actively answer them.

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.

btilly5• 17 years ago

Look at item 4 in the set of ideas that I tried. I asked a lot of questions, and did it in such a way that everyone figured out the answer before someone was asked to say it.

The method that I used wouldn't scale to a large class. Your psychology professor's piece of software would.

me2i81• 17 years ago

My son's high school physics teacher has something like this that appears to be a commercial product, with lots of IR remote controls that get pointed at an IR receiver connected to the teacher's laptop. He runs it sort of like Who Wants to be a Millionaire.

PebblesRox3• 17 years ago

Several of my high school classes, including math classes, used "clickers." http://telr.osu.edu/clickers/ They were better when the questions were interesting, but the competition to get the fastest response made even easy questions fun. I think there's room for improvement in the technology, though. For example, my geometry class figured out the code to reassign individual clickers, in order to make other people lose points by changing their answers.

kingkawn2• 17 years ago

Wonderful, I wish that my math teachers had been so thoughtful.

bbg• 17 years ago

btilly clearly expressed a fine method for effective teaching. I was impressed when I read the HN post a while back, and I'm even more impressed with the blog post. I am incorporating the idea about homework divided into thirds into my own teaching. Many of his other ideas are already there, if in slightly different form.

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.

marze2• 17 years ago

"messianic tone"? It read like a very levelheaded factual recount of his experience. His students seemed to learn the material well.

The system would be improved if all such positive classroom teaching experiences were recorded, distributed, tested, etc.

btilly5• 17 years ago

I absolutely agree. I actually taught several classes, but this was the only one where I had the freedom to choose the pacing, homework sets, and exams. I will never know whether my experience would have been easily replicated, or how important the factors you cite are. If anyone else tries something similar, I would be interested in hearing how it worked out for them.

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.

yummyfajitas• 17 years ago

The job description of "professor" is certainly a strange beast: teacher/scientist. It makes about as much sense as actor/programmer.

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.

chrischen9• 17 years ago

For the review schedule, while it is a good learning strategy, how many students coninue it after the course? It seems naturally as we get jobs we review by necessity, not out of habit. Do you think the students reAlly understood the importance of reviewing after taking your class or do they just forget it?

plinkplonk2• 17 years ago

So where and what do you (bbg) teach?

marze2• 17 years ago

The difference between the math is presently taught and the way it could ultimately be taught is immense. Given the problems math and reasoning illiteracy cause I think this is a much more important issue than most people assume.

The approach described would probably be great for many other subjects besides math, of course.

maximilian• 17 years ago

I was also quite inspired by his post. I am teaching precalculus as a grad student and I am absolutely going to try to incorporate the cumulative hw sets. For me, this is his most important discovery.

I might try assigning the next section reading before class to see how they respond. I don't think they will read ahead...

absconditus2• 17 years ago

"Since everyone worked hard and they thought that I was going to grade them on a curve, there was a lot frustration that they wouldn't properly be recognized for their work. (In fact I gave half of them A's in the end.)"

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.

btilly5• 17 years ago

Results were there as well.

absconditus2• 17 years ago

Correct results. I assume you didn't give some students lower grades just to hurt their feelings.

ilyak• 17 years ago

Most math is boring, and the homework he refers to would better be assigned to octave, not to humans.

mitko• 17 years ago

On the not taking notes part: on a lecture in an intro AI class the professor was giving the following advice to the audience: "I always take notes and never look at them. By taking notes the brain process the information at language level which increases remembering"(PHW).

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.

chrischen9• 17 years ago

Taking notes does not help me. I get distracted by the writing and can't pay attention. Then again, I have ADHD, but you can't claim taking notes will help everyone.

PebblesRox3• 17 years ago

I take notes in class because I feel that I have to record the information, forgetting that the information is almost always available in the textbook. Taking notes means I don't pay as much attention to the professor. When I miss something, I have to figure out what it means outside of class. It would be much easier to learn if I gave my full attention to the professor. On the other hand, I have found writing to be a great mnemonic tool. Perhaps a good compromise for me will be to take notes from the textbook instead of the professor, by summarizing each chapter as I read it before class. This would avoid distraction during class and allow me to devote my full attention to the valuable resource of my professor.

tierack• 17 years ago

In a slight twist, and one that may only work for small classes of highly motivated students, I took a set theory class where the professor would sneak an impossible question into the homework. He warned us during the very first class that he would do that occasionally.

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).

fburnaby• 17 years ago

To the author: That's too funny that you were doing this at Dartmouth college. I commented after your last post to HN on this topic that I'm helping (by teaching tutorial sessions with the prof) do much the same thing in a linear algebra class this year. This prof was attending Dartmouth college during the time you mentioned! No wonder this all sounded so familiar!

srn• 17 years ago

I'll have to try applying some of these study techniques, thanks.

Wouldn't it be interesting if we taught people how to learn instead of just teaching them the material?

Teaching Linear Algebra

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