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Opening a course to both mathematics and business faculty teaching as a team creates public dialogue about problems that straddle two departments.
Background and Purpose
Networking with client disciplines is a role that mathematics departments must be prepared to take seriously. Business disciplines have large enrollments of students in mathematics, and many mathematics departments offer one or two courses for business majors, generally including such topics as word problems in the mathematics of finance, functions and graphing, solution of systems of equations, some matrix methods, basic linear programming, introduction to calculus, and elementary statistics. This is a dazzling array of ideas, even for the mathematically mature. As a result, the required courses often have high drop and failure rates, much to the frustration of students and faculty. Nevertheless, at joint conferences, business faculty consistently urge the mathematics department to keep the plethora of topics, assuring the mathematicians that business has a strong interest in their students' development of a working knowledge of mathematics.
This article discusses one attempt to incorporate economic concepts with mathematics. What if students encountered mathematical topics in the context of economic reasoning? Would understanding of both disciplines be increased? Would student satisfaction improve? Would students stop asking, "What is this good for?" An integrated mathematics and economics course ensued: TEAM — Technology, Economics, Active learning, and Mathematics. The plan was to obtain a dual perspective on the problems of first year instruction in both economics and mathematics, and to use these assessments to feed back to both disciplines for further course development and more cooperation among the departments.
Jacksonville University is a small, private, liberal arts college which draws students from the Northeast and from Florida. Students tend to be career-oriented, fairly well-prepared for college, but lacking self-motivation. They view education as an accumulation of facts, and synthesis of these facts is largely a foreign notion. But if instructors make no synthesis of course content, it is unrealistic to expect freshmen to do so. The following is an account of success and failure of a totally integrated program* in which mathematical concepts are developed as needed, within the framework of a two-semester course in the principles of economics.
Method
We decided to develop mathematical concepts as needed in the study of the principles of economics. For example, slope would be couched within the topic of demand curves, and derivatives would emerge in an investigation of marginal cost and marginal revenue. The year-long syllabus included the standard topics of business calculus and elementary statistics, both required by the College of Business. The classes were taught in a two-hour block with little distinction between topics in economics and mathematics. Both the economics and the mathematics instructors were present for all class meetings, and each pair of students had a computer equipped with appropriate software. The subject matter involved a series of problem-solving exercises which enabled the students to construct their knowledge via active, cooperative learning techniques. The instructors served as facilitators and coaches, and lecturing was kept at a minimum. Some of the exercises were done by the student pairs, but most were completed by two pairs working together so that four students could have the experience of setting priorities and assigning tasks. Reports were group efforts.
Apart from a desire to increase conceptual understanding in both disciplines, the instructors' intentions, we found later, were initially too vaguely formulated. But as the course evolved over the two-year period, more definite goals emerged. Ultimately, we assessed a number of fundamentally important criteria: (1) student achievement, (2) student retention, (3) development of reasoning skills, (4) attitude of students, and (5) attitude of other faculty toward the required courses.
Findings
Student achievement. The Test of Understanding in College Economics (TUCE) was administered at the beginning and the end of each semester both to TEAM members and to students in the standard two-semester Principles of Economics course. Slightly greater increases were indicated among TEAM members, but the difference was not statistically significant. Student achievement in mathematics was measured by professor-devised examinations. Similar questions were included in the TEAM final examination and in examinations in the separate, traditional mathematics courses. No significant differences were noted, but students in the TEAM course performed on the traditional items at least as well as the others. Students in the TEAM course performed better on non-traditional questions; for example, open-ended items on marginal analysis within the context of a test on derivatives. Since confidence and a sense of overview are ingredients of successful achievement, this anecdotal evidence supports the hypothesis that integrated understanding of the subjects took place.
Student retention. In the TEAM course, the dropout rate was substantially reduced. In comparable mathematics courses it is not unusual for ten percent to fail to finish the course, but nearly every one of the TEAM members completed the courses. Numbers were small, just twenty each semester for four semesters, so it is possible that this improvement in retention would not be replicated in a larger setting. The students evidently enjoyed their work and expressed pride in their accomplishments.
Reasoning skills. TEAM students were more proficient than others in using mathematics to reason about economics. For example, class discussions about maximizing profit showed TEAM students had an easy familiarity with derivatives. Students on their own seldom make such connections, and this may be the most important contribution of the TEAM approach. When students link mathematics to applications, they become more sure about the mathematics.
Student attitudes. TEAM members demonstrated increased confidence in attacking "what-if" questions, and they freely used a computer to test their conjectures. For example, in a project about changes in the prices of coffee, tea, sugar, and lemons, the students first reasoned that increasing the price of lemons would have no effect on the demand for coffee, but after some arguing and much plotting of graphs the teams concluded that the price of lemons could affect the demand for coffee since coffee is what economists call "a substitute good" for tea.
TEAM students uniformly reported satisfaction with their improved computer skills. On the other hand, they showed some resentment, especially at the beginning of the course, that the material was not spoon-fed as they had come to expect. Even though some economic topics are highly mathematical, the resentment was more pronounced in what the students perceived to be "just math." Indeed, mathematical concepts often became more acceptable when discussed by the economist.
The emphasis on writing added some frustration. One student asked, "How can I do all the computations and get the right answers but still get a C?" Not every student was entirely mollified by the explanation that a future employer will want, not only the correct answers, but a clear report of the results.
Faculty attitudes. The authors were surprised that they did not meet the kind of opposition that has sometimes attended calculus reform efforts, but there was some reluctance among colleagues to consider expanding the program to a larger audience. Even people who are favorably inclined to participate in an integrated course are made nervous by the fact that some topics must of necessity be curtailed or eliminated. In hindsight, it would have been well to involve more faculty from both departments at the early stages.
Use of Findings
Although the TEAM course is no longer offered, this does not mean that the experiment failed, for much of what was learned is being incorporated into the existing courses. The lab assignments have been revised and expanded for use with a larger audience both in principles of economics classes and in the business calculus and elementary statistics sessions. Teachers are finding, often based on discussions with the two experimenters, the many advantages of cooperative learning together with an interdisciplinary approach to mathematical content. More importantly, plans are being laid to offer a new integrated course in the 1999-2000 academic year.
Success Factors
A small institution with computerized classrooms will find the experiment worthwhile, but the labor-intensive delivery system is probably too expensive for widespread application. Desirable as it is to have both economics and math ematics instructors present for all class meetings, it may not be feasible on a routine basis. A revision of the economics curriculum is under discussion. It has become desirable to offer an introduction to economics aimed at students with a strong mathematics background with a syllabus which would easily incorporate the experimental materials. Finally, and even more importantly, the dialogue between mathematicians and members of their client disciplines will continue, not just with business. Our work has set a model for networking and assessing across the curriculum, with physics, engineering, and others. We must decide what concepts students should carry with them and we must work for the development of those concepts in all related disciplines. A team approach teaches some lessons about how this can be done.
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* Supported in part by DUE grant 9551340.
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