
Teaching Linear Algebra - btilly
http://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html
======
tierack
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).

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

Thanks for obliging.

~~~
btilly
You're welcome.

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pbz
"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.

------
bbg
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.

~~~
btilly
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
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.

~~~
joeyo
At some point though, you definitely want the teachers to be _experts_. This
may not be true for most undergraduate curricula, true, but are not most
undergraduate courses already taught by graduate students, lecturers and
adjunct faculty? At the graduate level, you really want people who are active
participants in their fields-- and this is probably true for the arts as well
as the sciences.

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.

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10ren
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.

~~~
DannoHung
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.

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boredguy8
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.

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absconditus
"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.

~~~
btilly
Results were there as well.

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

~~~
PebblesRox
I don't think they were disgruntled with the grades that they actually got.
From what btilly said, it seems that the frustration occurred before they
received their final grades because they didn't realize that the grading
system was going to be fair. They expected to be graded relative to each other
(on a curve), with A's going to a certain, pre-fixed percent of the class.
Instead, everyone who did well got an A for their good work. I think the
complaint was about the demoralizing effects of the erroneous assumptions
during the semester, not about the grades themselves.

~~~
scotty79
Grading on a curve is a strange idea. It completely destroys incentives for
pear to pear learning, that was very important part of my college experience.
I learned a lot by explaining material to my peers. It would be much harder if
there was something as discouraging as grading on a curve.

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dbz
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.

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jrockway
The homework sounds good.

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

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

~~~
jrockway
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!)

~~~
kingkawn
ducd, that's the point. By only giving you half the answers, they provide
enough material to learn from and test yourself against. Instructor's manual
has all of the answers, so its up to the teacher to provide them.

~~~
chrischen
I'd still rather have the solutions so I can go at them my own pace.

~~~
plinkplonk
well we could always start a "sharetextbookanswers.com" website and let people
(students or not) contribute.

If people just put up the answers they do work out somewhere, this would help
others. This is happening with SICP for example.

------
maximilian
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...

------
gcv
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.

~~~
btilly
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
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.

------
tokenadult
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.

------
mitko
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.

~~~
chrischen
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.

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fburnaby
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!

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marze
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.

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

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

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srn
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?

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tomjen2
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?

~~~
btilly
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.

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

