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

magnitude.

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

matched up.

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

guess."