worked-out examples +- principles (was: Pedagogy)

[John Denker <jsd@AV8N.COM>]

I've been following the 35+ msgs in the 'pedagogy' thread,
trying to find out what the thread is about.  I think
I've identified four main themes:
 -- The role of the instructor and/or school in the
    learning process.
 -- The role of the student in the learning process.
 -- The role of worked-out examples (in class or in books).
 -- The role of textbooks.

Michael Edmiston wrote:
> We maintain a fairly long bookshelf in our library of "alternative
> textbooks."  ... I often refer students to these alternative texts so
> they can find more worked-out examples than their text contains.
>
> Those students who heed this advice usually benefit greatly. However,
> I find that those who do not heed this advice typically are not
> reading the "required" textbook either.

Yeah, that's one extreme.  If the students aren't sufficiently
motivated to use the available resources, making more resources
available isn't going to help.

I've seen abuse at the other extreme, too.  In my first year
of grad school, I knew quite a few folks who did their QM
homework by going to the library.  It turns out there are
only a smallish number of intro-level QM problems.  Each text
works out one of them and assigns the others as homework.  If
you look in enough books, you can find one that works out the
one you need.

I was mystified by this behavior.  I had not seen anything
like it at undergrad school.  I wondered whether these
folks wanted to get a PhD in physics, or whether they wanted
a master's degree in librarianship.  I thought the purpose
of homework was to noodle out how to solve the problem and
to learn the principles involved.  Rote copying of a worked-out
example wasn't going to teach me much.

In part because I declined to copy the worked-out problems, I
got lousy grades that first year.  The other students and some
of the professors thought the low grades indicated that I
wasn't very bright.

The next year it was their turn to be mystified.  We were
working in the lab, and we faced a real-life problem, about
ten times harder than any homework problem.  I was able to
figure it out, by combining some of the principles "we" had
been taught the previous year.  They couldn't work it out,
and they were astonished that anybody could.

============

I've told this story many times.

I tell students that the point is to learn the principles.

In real life, there are something like 10^10 different
instances of the problem.  That number arises if you
have 10 variables each of which can take on 10 different
values.

There is not enough time in school, indeed not enough time
in the whole world, to learn all the instances by rote.
Looking at ten worked-out examples is only a very small
drop in a very large bucket.  But if you can see past the
example to see the principle involved, learning 10 principles
might well give you a basis that spans the entire 10-
dimensional space, thereby empowerint you to handle all
10^10 instances.

Examples are part of the means to an end, but they are
definitely not an end unto themselves.