A presentation of the following article was given at the Joint Mathematics Meetings, January 10-13, 1996, Orlando, Florida.

Mth 390, Mathematical Modeling, was conceived with the intention of having the students learn mathematics as they worked through extended mathematical modeling problems in a group setting. The mathematical modeling projects were chosen based on the area of mathematics that the instructor wanted the students to explore and use. A chronological list of the projects, their mathematical content, and some remarks on the process and outcomes follow. Note that the statement of the project is short which requires the students to spend a considerable amount of time talking among themselves and with their instructor(s), who sometimes assume the role of a "client", in deciding just what the problem is. This problem construction facet of the modeling process proves to be indispensable in all areas involving problem solving, not just in mathematics.

Project #1:

"Build a mathematical model of how you plan to read the textbook."

First build an individual model and then get together with your other two group members to build the group's model. This activity was to introduce the students to the ideas involved in math modeling and to disrupt their usual notions of what went on in a "math class". This activity also demonstrated to students that mathematical models could be developed in "non- mathematical" contexts. Finally, students were introduced to the interaction and negotiation that must go on in constructing a group consensus.

Project #2:

"Develop and explain a method to measure the height of an inaccessible object. This includes the construction of a device from readily available materials in order to measure distances and/or angles. Use that method to measure the height of two assigned objects, one of which was the roller coaster under construction at Elitch's. Your written report should enable a person, unexperienced in trigonometry, to obtain the height by following your instructions."

The groups quickly realized that this was "just a trig problem" that they had seen before. The purpose of this project was to force the students to notice that just knowing the mathematics involved does not mean that the problem is solved. As a matter of fact, the groups' self-assessment comments often expressed amazement as to how much time the problem took after they had solved the mathematics. Each group learned just how difficult it was to actually use the mathematics, to obtain accurate measurements, and to account for errors in measurement that gave erroneous results. In fact, this was a good exercise in spotting when a result was reasonable or not. The writing of clear instructions for a "novice user" to implement their methods was also a difficult task. When running this model in future classes I plan to have the students spend more time on the error analysis and the ways in which errors can be adjusted for and ultimately controlled.

Project #3:

"You are a retailer of coffee. Determine the best container for serving your product."

The mathematical intent of this project was to have the students experimentally determine Newton's Law of Cooling. Along the way the groups learned about defining the problem so that they knew just what they had to model. Initially the groups were upset that the problem was so "vague", but were mollified somewhat when told that part of the purpose of doing this course was to disrupt traditional notions of math courses. Each group developed their own definition of "best" through interviews with local businesses who served coffee. They discovered that many factors were considered (and not considered) by the companies who made and used the various coffee containers in the Denver area. They also learned about data collection and curve fitting as well as customer surveys and what constituted a good question on a survey. In addition, at this point in the course students were starting to become more comfortable with the structure of the course. We had 9 students, 6 males(including 1 minority ) and 3 females, and one significant choice I made as instructor was to not have a team with only 1 female student, which more often than not leads to the isolation of the single female student. With this in mind, the groups for the projects were composed of three students and one group was all female. This "3 groups of 3" structure held throughout the course until the last project where we had two groups of four with a female "consultant" (who was in fact a practicing high school math teacher).

Project #4:

"The conic sections. Each group was assigned a different conic and required to present a definition, at least one method of construction, development of the equations for their group's conic, and a demonstration of a physical manifestation of the conic section."

This project differed from the others in that we started with the mathematics and went in search of the application. The intent here was to have the students take a topic that they were accustomed to dealing with in terms of equations and graphing, but had lost touch with them on a concrete level. They were required to go back to the definition and applications of the conics so that they could see beyond the abstractions. Each team did a very good job of showing that they understood the definition through their hands-on construction of the particular conic, complete with detailed instructions that a person who was just learning about conics could follow. The teams also did physical demonstrations which showed that they had a grasp of the reflection properties of each of the conics.

Project #5:

"For a game of chance (craps, roulette, coin toss onto a grid), do an analysis of your chances of winning if you are the "house" . What if you are the "customer?" Your analysis should include both physical trials (i.e. actual plays of the game) and a computer simulation. The presentation will be a "Casino Day" where the other teams will try to beat the game which you have "rigged" as the house."

This was our foray into probability. Students learned about analyzing games of chance through analysis, through experiment and through computer simulation. Simulation was a new experience for most of the students and getting a computer or programmable calculator to simulate playing a game helped their writing skills. The different ways in which the students could approach the problem helped them see that mathematics gives one flexibility in solving problems.They also gained an appreciation for the gaming industry and what is involved in setting up a "saleable" game. Having the presentations in the context of a "Casino Day" resulted in a very lively class. No reports of students visiting Las Vegas and winning big followed this project.

Project #6:

"What relationships exist between mathematics and music? Use a monochord to find out what the harmonic sequence sounds like. What do musical sounds look like mathematically?" How do you decide where to place frets on a guitar and why is a grand piano have the shape it does? Along the way you will find out about the Fibonacci numbers and obtain physical objects that are natural occurrences of Fibonacci numbers. What is the Golden Ratio - how do you construct it with numbers and does it occur in nature?"

Starting with simple notes each group proceeded to explore the relationship between mathematics and music. This led them through discussions of rational and irrational numbers, exponentials and logarithms, Fibonacci numbers, and the Golden ratio. Each group had the use of a monochord with movable frets and a ruler to see what fractions "sounded" better. Their project was to complete a series of worksheets taken from the book , Algebra in Everyday Life. This set-up was different in that the main "product" was not a group report, but rather a collection of worksheets. In the future I plan to use the worksheets to construct a project which would hang together better than it did this time. Many of the student comments mentioned that the material, though interesting, seemed to be disjointed.

Project #7:

"A company wants to design an automatic cab following system for a new monorail. They are trying to decide whether a "Safe Distance Algorithm" or a "Speed of Approach Algorithm" is the best approach. Use a computer simulation and analysis to determine which is better."

Students are put into two groups of 4 (instead of the usual 3 groups of 3) with one student (who is actually an high school teacher) serving as a technical expert for both groups (How do you divide 9 in half?). This problem was chosen so that the groups learned about the cost benefits of simulation over building scale models. This project took the students' calculus knowledge and extended it while trying to solve a real problem (Denver was just putting in it's light rail line and th enew airport was having significant problems with it's automatic baggage system. They saw an application of the third derivative of the position function, namely, jerk. They did not solve DIA's baggage problems.

For Comments and Questions please contact:

Thomas Kelley
Department of Mathematical Sciences
Campus Box 038
P. O. Box 173362
Denver, CO 80217-3362
kelleyt@mscd.edu