
Teaching linear algebra - btilly
http://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html
======
jasonkester
Nice approach. And wow, that makes me realize that I was doing it right after
all as a student.

I used to take the following approach, which maps surprisingly well to the
author's teaching method:

    
    
      - Pay attention, but don't really take notes during class
    
      - Do the homework, but don't sweat getting stuck.  
        Skip stuff that I hadn't picked up in class.
    
      - Take notes the next day when they went over the homework.
        Pay extra attention to the stuff I didn't get the night before.
    
      - The night before a test, redo (or actually do) all the homework.
    
      - Ace all the tests
    
      - Don't really study for the final
    
      - Ace the final
    
    

I feel like I learned pretty much everything from every Engineering class I
took in school. When it came time to do the EIT exams at the end of undergrad,
I took them cold, walked out with half the time remaining (and nearly
everybody else still in their seats) and passed by a comfortable margin.

Naturally, grades suffered a bit. You can map my grades to how a given prof
weighed homework in his grading process. Usually you can drop one test score,
and occasionally they'd let you count homework as one of those droppable
scores. Those were my A's.

The rest were B's and C's, but that doesn't matter even one bit now, 20 years
later. I could still pass that Professional Engineering test again, deriving
pretty much all of Mechanical Engineering from F=MA using differential
equaitions today. And that's the whole point, right?

Glad to see a professor who gets it.

~~~
wiredfool
My method was:

    
    
      - Show up. Pay Attention. Sit in the front of the class
      - Take notes. 1 page is slacking, 4 sides is about right.  
      - Do the problem sets. 
      - Studying? What's that?  
      - Rock the tests.
    

If I sat in the first couple rows, my grade would be an A. Back row, I'd be
lucky with a B. Back row in a large lecture at lunchtime was a C. Then again,
I was known to be able to answer questions when obviously snoozing in the
front row of an 8am class. (Steel design, IIRC. "What's wrong with all you? He
can answer the questions and he's asleep")

I needed the notes, and specifically the ear - brain - hand - eye loop to make
sure that the info got in my head and processed. If I slacked on notes, then
the slippery slope started and I'd end up losing the thread of the class for
minutes at a time. I didn't really need the notes later, I might go over them
before a test, but not often. Mostly when it was an open note or 'one sheet of
notes' test. Though, one time my one sheet of notes was "Don't Panic" written
in letters large enough to be be seen by anyone glancing at the sheet.

Problem sets were key. As were the bigger design projects and the labs. You
could fake getting the problem sets done, but you couldn't fake understanding
them.

------
btilly
Note, I resubmitted this because I was hoping that it would find
<http://news.ycombinator.com/item?id=850485> for me. But it didn't. (I later
found it through my user name.)

I would delete it, but it has been several years, and so it does not seem bad
to have the dupe.

~~~
jleader
Actually, you just waited until we'd almost forgotten your earlier post...

------
thebigshane
The "spaced repetition" theory is similar to something I read in Wired[0] long
ago, of a guy and his note-taking "SuperMemo"[1] app. The article drifts
aimlessly about the guy, but the theory behind the app is very interesting.
And the chart is excellent[2].

    
    
      For example, say you're studying Spanish. Your chance of
      recalling a given word when you need it declines over 
      time according to a predictable pattern. SuperMemo tracks 
      this so-called forgetting curve and reminds you to 
      rehearse your knowledge when your chance of recalling it 
      has dropped to, say, 90 percent. When you first learn a 
      new vocabulary word, your chance of recalling it will 
      drop quickly. But after SuperMemo reminds you of the 
      word, the rate of forgetting levels out. The program 
      tracks this new decline and waits longer to quiz you the 
      next time. 
    

[0]:
[http://www.wired.com/medtech/health/magazine/16-05/ff_woznia...](http://www.wired.com/medtech/health/magazine/16-05/ff_wozniak?currentPage=all)

[1]: <http://en.wikipedia.org/wiki/SuperMemo>

[2]:
[http://www.wired.com/images/article/magazine/1605/ff_wozniak...](http://www.wired.com/images/article/magazine/1605/ff_wozniak_graph_f.jpg)

~~~
goldfeld
SuperMemo has been around for quite some time and IIRC it was indeed the first
piece of software to employ the theory of spaced repetition. So it's not
coincidentally similar, SuperMemo is immediately associated with spaced
repetition. That Wired article is a great read, moreso IMHO because it drifts
purposely about the guy, who's as eccentric as he is fascinating in his
extreme lifestyle choices and devotion to his life's work.

p.s. Something dies within me when I hear good ole' software being referred to
as an 'app'.

------
jmduke
(Disclaimer: I am a current college student.)

I took a seminar on Film Studies my freshman year (all freshmen are required
to take a seminar outside of their prospective major). We watched exactly one
film -- Psycho -- which I greatly admired. That being said, I can't tell you
much about the theory behind Psycho; but I can tell you pages upon pages about
the evolution and use of 90mm film.

The results in the blog post speak for themselves; the final exam scores were
phenomenal.

That being said, I feel the author didn't spend enough time dwelling on the
repercussions of his approach to the course. If you're in a two- or three-
credit class, you take that class with the implicit understanding course that
the time commitment is going to be similar to that of other classes; if your
students are spending all night working on an introductory class, then they
might master that material -- but at the expense of other classes.

I took a Linear Algebra course last semester that I did well in. It was a
conventional class. I definitely couldn't answer that final bonus question
(and I bet I couldn't answer most of the questions on that exam) but is that
an issue? Introductory courses are meant to be breadth-based, not depth-based;
my class wasn't filled with Math majors but CS majors, Chem majors, Stats
majors, etc. etc. -- I recognize the huge role tenets of L.A. play in
programming, but I'd absolutely resent a professor who essentially uses false
advertising in his course.

The metric for a successful college course, I'd argue, is not 'amount learned'
but 'amount learned with respect to time and respect to the goals of the
course.'

~~~
naner
_my class wasn't filled with Math majors but CS majors, Chem majors, Stats
majors_

I don't know much about chemistry, but for CS and stats people linear algebra
is an applicable and important course. Some of these "introductary" courses
are foundational to your degree.

~~~
jmduke
Absolutely! I remember the basics of linear algebra very well, and can
reproduce them easily -- proofs and more arcane aspects of the curriculum, on
the other hand, not so much. (Furthermore, I'd argue that the average non-Math
major is not going to have to apply such aspects, and thus an introductory
course should be relatively cursory regarding them.)

------
fferen
I know this isn't the point of the post, but I'm learning me some linear
algebra from online video lectures now and wanted to check my answer for the
bonus question, part b. Is it [[-3/2 2 -1/2] [-1/2 0 1/2] [1/2 -2 3/2]]?

My method was to represent f(x) as ax^2+bx+c, giving v = [c, a + b + c, 4a +
2b + c] as the vector in the coordinate system and dv/dx = [b, 2a + b, 4a + b]
as the derivative. Then d/dx * v = dv/dx. Then split both vectors into a
product of a 3x3 number matrix and [a b c], cancel the [a b c] from both
sides, find the inverse of the left matrix, and multiply it by the right. I
did the inverse by hand so I might have made an arithmetic mistake. Am I on
the right track or did I totally misunderstand the question?

~~~
btilly
That is correct.

Your double check is to take the vectors for the polynomials 1, x, and x^2
(namely [1, 1, 1], [0, 1, 2], [0, 1, 4]) and multiply out to see that you get
the vectors for the polynomials 0, 1, 2x.

Your verbal description does not make sense to me. But you appear to have done
the right operations.

The approach that my class universally took was to realize that if we take the
coordinate system whose basis is (1, x, x^2) then the answer is easy to write
down. After that you apply change of basis matrices to each side to get the
answer in the requested basis.

~~~
fferen
Cool. I haven't gotten to the change of basis lecture video yet, or I might
have tried that :)

~~~
btilly
If you do the linear function/notation thing right, the idea of a change of
basis comes for free. The basic idea is this. Suppose that we have a linear
function F between two vector spaces (which might be the same), and we have
coordinate systems on the vector spaces (which might be the same). Then we can
take the elements of the first basis, v1, v2, ..., vn, and write out a matrix
whose columns are F(v1), F(v2), ... F(vn) in the second coordinate system.

This matrix uniquely represents F. The function F can be applied to a vector
by writing out the vector in the first coordinate system, putting that on the
right of the matrix, then doing matrix multiplication. You get an answer in
the second coordinate system. And the operation of matrix multiplication turns
out (if the coordinate systems match up) to be exactly the same as composition
of functions.

(This is no coincidence, this relationship is the motivation for matrix
multiplication being defined as it is.)

OK, with all of that mess, what is a change of basis matrix? It is simply the
matrix you get for the identity function going from one coordinate system to
another.

Now in this case the matrix A = [[1 0 0] [1 1 1] [1 2 4]] is the matrix to
change from the basis where a+bx+cx^2 is represented by [a b c] to the basis
where it is represented by [a a+b+c a+2b+4c] (ie [p(0) p(1) p(2)]). Its
inverse is the change of basis the other way. And B = [[0 1 0] [0 0 2] [0 0
0]] is, of course, in the [a b c] basis the representation of the function
d/dx.

Then A B A^(-1) represents change from pointwise coordinates to coefficients,
then differentiate in coefficient coordinates, then change from coefficient
coordinates back to pointwise coordinates. (Remember, the matrices are applied
right to left, so you do A^(-1) first.)

If you can keep that straight, you now understand change of basis matrices.

------
gwern
One of my favorite real-world example uses of
<http://www.gwern.net/Spaced%20repetition>

------
trinovantes
As a 2nd year undergrad, I wish he taught my first year linear algebra course.
I remember just scrambling to keep up with the prof writing the notes on the
board. For my final I just crammed the steps (without knowing why they work or
why we do it) for matrix multiplication, diagonalization, orthogonal
diagonalization, etc.

I'm going to regret this in my 3rd year...

------
wiredfool
20 years ago, I had a professor who would ask lots of questions in class. I
_still_ remember that Lindsey's designated answer was "Static Electric
Charge". I'm not in the field anymore, but I still remember a bunch about that
class, and I count the professor as one of my two or three best.

I only wish that I had a linalg professor who was that good.

------
mikevm
I've been thinking of using spaced repetition in the upcoming semester to help
me study (computer science). Does anyone have any tips on when to review, and
what kind of material should I be preparing for review? Do I just re-do parts
from previous homework assignments?

~~~
btilly
The study schedule that I was taught is to work in half-hour sections. Review
for 5 min, study for 20, take a 5 min break. Start the next block with a
review of the previous.

After the first day, review on a schedule. For instance the following day,
several days later, a week later, a month later. Each day for your review look
back in your notes the appropriate time.

When I was a student I was never that organized. Instead I would sit down and
try to think through sketches of everything that I was supposed to have
learned. Just pretend that I'm teaching it, and start sketching out my
knowledge. Wherever I ran into trouble remembering details, that was generally
a spot that I needed to review and I'd look it up in the text.

This is unorganized, but I'd walk into follow-up courses a year or two later
and still know the previous course cold. So it worked for me.

You could also do it by just redoing parts of previous homework assignments.
But that feels to me like a lot more work (unless the professor assists).

------
eru
Yeah, I read that a few years ago, too.

Looking back, the authors approach mirrors very well what I tried to set up
for myself as a student. (Asking questions, no notes, repetition schedule..)

------
bbg
btilly has come full circle! It's still a great post.

<http://news.ycombinator.com/item?id=850485>

Coincidence that I should see this; I'm not really active on HN anymore, and
just happened to click.

~~~
btilly
I'm curious. The first time it was posted, you indicated that you were trying
the thirds homework idea.

How did that work out?

------
pnathan
As a former TA, this sounds like an amazing approach.

