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On 11/30/2016 06:56 PM, Philip Keller wrote:
Even if it is true, I am not sure I want to show it to my students. I'm
afraid they will use it mindlessly,
There's an interesting physics context behind that, but let's
put that on the back burner for a moment, and instead start
by talking about the pedagogical issues.
We agree that mindless equation-hunting is a Bad Thing. As
the saying goes, the problem with mindless equation-hunting
isn't the equation, it's the mindlessness.
Now from a pedagogical point of view, I'm not convinced it is
good practice to withhold an idea for fear that it will be
abused. In fact, to my way of looking at it, that makes the
underlying problem worse, insofar as it leaves students with
the idea that in any particular situation there is only one
way to solve the problem.
Heller&Heller wrote an entire book on the fine art of withholding
information, so as to "compel" (their word) students to solve
each problem in a particular way. I find this whole idea to
be abhorrent from start to finish.
In the real world, equation-hunting is astoundingly inefficient
precisely because there are so many equations, and there is no
robotic mindless rule that tells you which one to pick.
Typical textbooks make the problem worse, because all-too-many
of the end-of-chapter problems for chapter 39 can be solved
using the methods of chapter 39 alone. That invites mindless
equation-hunting, by artificially reducing the list of equations
that might apply. In contrast, if the goal is to teach students
to think, and to solve real-world problems, it would be much
better to have each problem set contain a goodly number of
"review" problems and "synthesis" problems that mix in ideas
from all previous chapters. This would cause equation-hunting
to fail for the proper, natural reason, namely that it's not
obvious which equations to apply.
Note that the review problems and synthesis problems
should not be labeled as such, because that would defeat
the purpose, insofar as it tells the students when and
where to hunt. Real-world problems rarely come with
such labels.
When students ask what method they should use, remind
them of Pólya's dictum:
Try /something/.
If that doesn't work, try something else.
Better yet -- better in theory anyway -- one could have
a book of exercises, separate from the textbook, so there
is no typographical reason for students to associate a
particular exercise with a particular chapter.
In practice, though, you could never sell a pair of
books like that. The students and teachers would
scream bloody murder. They /want/ exercises that are
keyed to a particular chapter, because it makes teaching
and learning "easier". I say it makes the job easier,
but at the expense of teaching and learning the wrong
things!
Withholding an equation means that students won't apply
mindless processes to /that particular/ equation ... but
at a terrible cost, since it disproportionately increases
the odds that they apply mindless processes to some other
equation. Again I say, if a student can solve a homework
problem by mindlessly hunting up an equation, there is
nothing wrong with the student and there is nothing wrong
with the equation, but there is something desperately
wrong with the question.
Robert Cohen understands all this particularly well, and has
taken the constructive step of designing exercises where
the mindless approach fails for all the right reasons. It
fails in the natural way, not because the teacher forbade
it, but because it gives old-fashioned wrong answers.
Here's a somewhat-related pedagogical trick: Assign an
exercise, and then a couple of days later assign the same
exercise, with instructions to solve it by a different,
independent method. This underscores the idea that there
are multiple ways of solving a problem. It also reinforces
the idea of /checking the work/ which is fundamental to
any notion of critical reasoning.
This has to be done very carefully. The risk is that
the student will infer from the re-assignment that the
previous solution was wrong, which is definitely not
the correct inference. It helps to give notice in
advance that the exercise will be re-assigned, so
it doesn't come as a surprise, so it clearly isn't
punishment for a "wrong" first answer. Explain how
redundancy can lead to greatly increased reliability
and confidence.
I'll admit I am sometimes tempted to ask students to
solve a problem using a particular method ... but I
try really hard to resist the temptation. If the
method is as valuable as it's cracked up to be,
opportunities to use it should come up naturally.
One sometimes hears a counterargument, namely that there
isn't enough time in the schedule to allow assigning the
same exercise twice. To that I say, it depends on what
you're trying to teach. For homework, the bottom-line
answer was never what's important; you can look that up
in the back of the book, or online. It is the method of
solution -- and the process of finding the solution --
that are important. Solving real-world problems requires
using a complex combination of tools, not just mindlessly
applying the methods of chapter 39.
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