Date: Apr 9, 1993 5:07 PM
Author: Evelyn Sander
Subject: Some creative teaching techniques
"The world comes to you in a mess, and out of the mess you
have to discover something mathematical." This is the
opinion of Arnie Cutler, Education Consultant at the
Minnesota Geometry Center. These words seem to serve as
a guide to him; in addition to his position at the center,
he has instructed in the math education department at
the University of Minnesota, in three weeks will begin
his term as President of the Minnesota Council of Teachers
of Mathematics, and a math teacher at a local
high school. In this article, rather than talk about
organizations and committees, I will describe
Cutler's actions and opinions as a teacher.
"I make a big distinction between problems and exercises,"
he tells me. "Exercises are canned versions of what has
already been done in the book. They have an easy answer,
and the student doesn't learn much by doing them. Problems
are more difficult. They are often taken from real
situations. The students have to figure out what method
to use to start them.
"In my classes I try to have a lot of problems and very few
exercises. There are plenty of exercises in the books if
they want to work them. In class we need to do something
more challenging. Sometimes students complain, saying that
they don't know what to do to solve a problem. I tell them
that there would be no point giving them a problem that
they already knew how to solve."
In order to solve a problem, Cutler breaks the class
into small groups and let them work on it for a while.
Solving may take days, or even weeks. He encourages them to
think for themselves, always saying, "The real world doesn't
have an answer book."
After the groups have worked for some time, Cutler
chooses someone in the class to go to the
board and present what their group has done. The students
quickly learn that there is more than one method to do a
problem. Even if someone gives a correct answer to a problem,
he asks other to show their solutions.
In past years, Cutler has assigned a variety of
problems to his classes. For example, last year his
calculus class considered the path and length of path
taken by a man standing on a ladder as the ladder slips
down a wall. The work groups each designed a model for
the problem, the most successful being a lego construction
with a pen attached; the pen traced the path on a wall
beside the ladder. Another problem from earlier this year
was to calculate center of mass of the textbook example
of a meter stick with weights at the ends. The students
realized that they were going to have to use
integration now that the meter stick itself had nonzero mass.
Most recently, Cutler's calculus class spent
several weeks working on the problem of designing a food
drop for Bosnia. They spent the first week doing background
research. For example, they had to read the paper and
decide which cities most needed the supplies. Then they
had to find the coordinates of these cities and the coordinates
of the air bases. They considered which airplane to use
for the drop, evaluating the specifications on each plane.
After some work, the class decided that the Defense
Department was right; the C-130 is the best plane for the
job. It turned out to be quite difficult to find a weather
service with information on the current prevailing winds
near the cities chosen for the drop.
The next step was to come up with a mathematical model.
At first the class got bogged down in small details.
They worried about whether there would be a small hill
or someone standing right where they were trying to make
the drop. They soon realized the need for simplifying
assumptions in mathematical modelling; they assumed the
wind would have the predicted average for direction and
Finally the students were ready to write down some
equations, using a variety of mathematical ideas. They had
to find the angles between two planes when decribing the
airplane's bearing. They needed to coordinatize the earth
working with spherical and rectangular coordinates. They used
the idea of a great circle when calculating the airplane's
path. In order to avoid having the boxes break, they had to
figure out at what height the parachutes on the boxes needed
to open. This meant they had a two stage drop, so after they
were all done with the equations, they had to calculate
derivatives to make sure the pre and post parachute stages
The students really enjoyed the experience. It was problem
of current news interest. However, they mainly were happy
that they could pick up their text books and turn thirty
pages, on each page seeing at least one concept they had
used in their solution.
I asked Cutler to tell me how he thinks of these
problems: "Keep your eyes open. You can see math everywhere.
I always look for a problem that is interesting and deals
with real life. It also needs to apply topics that the
students have learned." He points out the window at a bridge
spanning the Mississippi river. "Look at that bridge. I
wonder what curve the underside of it traces. I
can't tell off the top of my head, but with a few
measurements and a bit of work, I bet we could make a good