As several people have mentioned, it pays to
give the students a steady stream of exercises
that require conceptual thinking, insight,
out-of-the-box thinking, whatever you want to
call it ... as opposed to mere plug-and-chug.
One slight refinement to that idea: Don't
separate the exercises into N problems of
one type and M problems of the other type.
Solving typical real-world problems requires
layers and layers of conceptualizing mixed
with number-crunching.
Here's a modest example of what I mean. This
is based on a real-world calculation I needed
to do the other day (having to do with the
performance of a refrigerator at low temperatures)
but simplified for pedagogical purposes.
The theoretical calculation says that the
quantity of interest is
y(x) = cosh(x) - sinh(x).
(The real formula is a lot messier than that,
but this suffices to illustrate the point.)
The assignment is to calculate y(5), y(10),
y(15), and y(20), each accurate to 1% or better.
For extra credit, plot the function on appropriate
axes.
Hint: If you evaluate y(x) directly using a
spreadsheet program or almost any other computer
program, or typical calculators, or anything else
that uses IEEE floating point, the calculation of
y(20) will fail miserably. Try it. Then see if
you can come up with a Plan B that produces a
better result.
True story: I tried using a spreadsheet program
to create a graph of y(x). It was fine for small
x and garbage for large x. It was pretty obvious
that I had a problem. From there, figuring out the
etiology of the problem didn't take long, either.
So the actual process of "setting up the problem"
involved
-- a tentative numerical step
-- a teeny bit of skepticism and common sense
-- an analytical step
-- a final numerical step.
I tell this story to refute *both* extremes of the
argument; that is:
*) starting with a quick&dirty calculation is not
always good, but it's not always terrible, either.
*) starting with a deep theoretical analysis is
sometimes good, but it is sometimes unnecessary
and indeed sometimes unhelpful. For small x
the formula given above works fine.
=====================
Here's another example I cooked up a while back
that illustrates the same point: http://www.av8n.com/physics/wig-eck.htm
There is a calculation to be done, which is not
entirely trivial no matter how clever you are,
but a little bit of cleverness and insight will
change the calculation from being truly horrific
to routine and straightforward.
=====================
Story problems in general and ill-posed problems
in particular are hard. They require initiative,
resourcefulness, judgement, a certain breadth of
knowledge, and other things that don't grow on
trees. http://www.av8n.com/physics/ill-posed.htm
=====================
The "Hell's Library" cartoon can be found on page 52
of: Gary Larson, _Night of the Crash Test Dummies_.
Mr. Larson does not want people to scan his cartoons
and put them on the web. So don't do it. The book
lists for $5.95.