"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."