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Re: Ask Marilyn "geometry test"



Carl Mungan wrote:

From today's Marily von Savant:

Q: "On a geometry test, Mary devises a set of steps to solve a
problem. Her solution is shorter and more elegant than the method
that she was taught in class. If you were her teacher, how would you
score Mary's answer?"

1) This is an excellent topic for discussion.

2) The only thing I can say for sure is that there
are no pat answers. I "mostly" disagree with Marilyn's
answer and with each of the counterarguments made so
far, although each has some good elements.

===========

I find myself in situation like this all the time.
Some student comes up with a "creative" solution
and asks me if it's OK. Mostly but not necessarily
I'll say "OK, that's not the solution I was
expecting but I suppose it works at least as
well." Then, often but not always, the student
will come back and ask what I would have done,
and we can have a discussion about the pros
and cons.

I try to be clear about
-- which things I really want done my way
-- which things are matters of taste and style

As to "shutting off creativity" by requring a
particular method of solution, that depends
partly on how often you insist on a particular
method, and on the details of _how_ you go
about asking. If you usually tolerate eccentric
solutions, but occasionally say that for
purposes of exercise you want to see a
particular method, then it's unlikely that
any budding Mozarts will be fatally stifled.

======

There are many shades of gray along the continuum
-- utterly conventional
-- slightly unconventional
-- eccentric
-- creative
-- smart-alec
-- perverse
and it's impossible to lay down hard-and-fast rules
about how much deviation from the beaten path you
should tolerate. At some point your instructional
professional judgement comes into play.

For instance, if you ask somebody to prove that for
a right triangle, a^2 + b^2 = c^2, then the simplest
and most-elegant solution is to invoke the Pythagorean
theorem. But if the point of the question was to
elicit a derivation of the Pythagorean theorem,
you are certainly within your rights to reject
the "most-elegant" solution as non-responsive to
the intent of the question.

To be clear: I'm not suggesting that Marilyn's
answer is right in general, because it's not, but
there are more than a few cases where the approach
she suggests is correct or nearly so.

=======

There are other details that affect the handling
of such issues.
a) If you're grading a quiz, commonly Marilyn's
method isn't applicable: you don't have the
opportunity to re-ask the question; you have
to grade the solution on the page, one way or
another. My tendency would be to accept almost
anything that's not toooo far out on the smart-alec
or perverse dimension. If the answer isn't what I
was expecting, then it's my problem because I didn't
ask a sufficiently restrictive question.
b) There's much less of a dilemma in interactive
instructional situations; if you get an unexpected
solution you can praise it for what it is and then
give your reasons for wanting to see the solution
you're fishing for.

Example: I'm the flight instructor, giving
somebody a checkout for night-flying privileges.
Near the end of the flight I will turn off
essentially all the electricity in the airplane
(so that the gauges are inoperative and/or
unreadable) and require the customer to land
the airplane. I ask 'em how they would handle
the situation. The smart ones say they would
start by turning things back on. I say fine,
but suppose you smelled burning phenolic, so
there was a reason for leaving things turned
off. Next they try turning on their flashlight,
but I take that away from them. Flashlights
can fail, too. They may protest that they've
got three or four backup flashlights (and they
know I've got six or seven more) but eventually
I explain that it's an excercise, and one of
the points is to land the thing without the use
of any gauges. There are lots of scenarios that
could deprive you of your favorite gauges
even in broad daylight; nighttime is just a
convenient way to set up a demonstration of
the situation. So eventually they get the
idea that I'm not kidding, I _really_ want 'em
to land it with no gauges, and that "creative"
ideas for bringing the gauges back to life are
not going to be accepted. The punchline of this
story is that the resulting landing is usually
the best landing they've ever made, because
this is a really good exercise. It forces
them to concentrate on the most-important
things. Secondly it gives 'em a tremendous
confidence-boost to know they can land the
thing even if all sorts of things are going
wrong.

As part of the debrief I tell them that the
song-and-dance about flashlights and debugging
the electrical system was part of the lesson,
too, and that their efforts in those directions
were 100% good. The point is that they need
to have multiple ways of solving every problem.
I want 'em to practice Plan A and practice Plan
B and practice Plan C. Sometimes this requires
artificially restricting the scope of their
creativity.

But ... you will notice that I didn't give
them the cut-and-dried ("rote") assignment
of using Plan C. I just took away Plan A
and Plan B and forced them into a situation
where they eventually figured out that Plan
C was the best remaining option.

So you see I can have my cake and eat it too:
-- I cultivate creativity by not telling 'em
what method to use. Instead (very different!)
I take away selected methods, leaving 'em full
scope to explore the remaining alternatives.
Then I take those away, one by one. The longer
this process goes on the happier I am.
-- Eventually each of the methods I want
to see demonstrated does get demonstrated.