Re: Ask Marilyn "geometry test"

[John Clement <clement@HAL-PC.ORG>]

>  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?"
>
> Marilyn's A: "I'd ask her to solve the problem by the method that she
> was taught. If she could, I would give her full credit plus extra
> credit for the extra solution. If she could not, I would give her no
> credit at all: She doesn't understand what was taught in class.
> Methods of teaching are not necessarily the shortest and most
> elegant. Instead, they may simply be a good way for students to learn
> the principles of the subject."

This is a typical answer for someone who thinks that learning is
memorization rather than learning to think.  Remember there are teachers who
would mark the following question and answer wrong.
Q: What causes things to fall
A: The earth
Because they expect the word gravity, instead of the correct "thing".
I would give Mary a bonus or better yet praise her for inventiveness and
give full credit.  There have been studies that show that praise works much
better than grades at motivating students.

> My comments: I found this interesting and have talked about this with
> my classes before, encouraging them to think out of the box and
> emphasizing with actual examples that there is often more than one
> way to solve a problem. So I don't fully agree with Marilyn. I think
> alternate solutions are as good if not better than "my way." However,
> I would keep the following in mind:
>
> (a) Marilyn doesn't say what level class it is. I might agree more
> with her if it is middle school but not if it is an undergraduate
> class or higher.

Unfortunately when you force MS and elementary school children to follow
fixed algorithms you shut off their thinking processes.  It is just as
important to have students come up with their own solutions at all levels.
In Holland they use a style of teaching where students must come up with
algorithms for subtraction and addition and they are not taught the US
standard method of carrying or borrowing.  As a result they can do mental
arithmetic much faster and more accurately than US students.  For
information about this go to
http://educationupdate.com/archives/2002/dec02/issue/spot_mathtorrence.html

> (b) I would look to see if the alternate method really has some
> generality to it, or is just a fluke, a memory trick, or right for
> the wrong reasons.
>
> (c) It would probably also depend on how the question was worded. If
> I said, "Use Kepler's laws to ..." I might still accept an alternate
> solution, but only if it were tied to Kepler's three laws.

Yes of course if you ask for a particular method or the usage of a
particular principle then certainly the student should be able to do it
using the method you asked for.

> Anyhow, I throw this out to see what interesting comments you
> might have. Carl
> --
> Carl E. Mungan, Asst. Prof. of Physics  410-293-6680 (O) -3729 (F)
>        U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
> mungan@usna.edu    http://physics.usna.edu/physics/faculty/mungan/

I always get students to do problems by a variety of methods.  But they also
have to think their way through some problems with minimal guidance.  As a
result on the final exam I often see surprising things like students cross
applying methods successfully.  I have also seen students invent some ideas
from introductory calculus even though they are taking only pre-cal.  When I
was in the position of teaching standard algorithms and noodling the
students, I never saw that sort of inventiveness.  Remember noodling a goose
produces spoiled livers.  In students it produces spoiled minds.


John M. Clement
Houston, TX