Purpose

Although the last two decades have seen numerous efforts to reform mainstream calculus, until recently nonstandard or "business" calculus-which is increasingly being required of students in the social sciences as well as business, economics and management-had been left largely untouched.

Business calculus texts, mostly watered-down versions of traditional calculus, reflected this neglect. They offered irrelevant content, failed to incorporate realistic applications, and did not require students to use technology. Although firms in business and industry need employees who can think logically, write well, use technology proficiently, and solve problems as part of a team, the available textbooks in nonstandard calculus only addressed the first of these skills.

Furthermore, the high level of algebraic manipulation demanded by these texts overwhelmed non-technical students, who often suffered from math anxiety, had poor algebra skills, and could not see how the course would help their careers. The last mathematics course these students would ever take, traditional business calculus failed to reflect the uses of mathematics in an information society, and to provide students with the applications they would need in their future work.

Innovative Features

The project originated in the wake of a FIPSE grant to introduce graphing calculators into undergraduate mathematics courses taken by science and engineering students, when faculty realized the parallel need to integrate technology into the mathematics courses for non-science majors. Soon after they began to address this problem, it became apparent that a much more fundamental transformation would be necessary to make these courses responsive to students' needs. The new project led to the complete reform of the two-semester business calculus sequence.

Project staff believed that what non-science students need is not mastery of algebraic technique, but rather conceptual understanding and the ability to interpret correctly the mathematics in real-life situations. Therefore, they focused on the relevance of key concepts to the world of change, eliminated topics that stifle the mathematical development of non-technical students, and designed courses with less lecturing and more student involvement in interpretation, mathematical decision making, and team projects. Seeking to create a classroom environment where students would be led to construct their own knowledge and to conduct their own inquiries, they encouraged dialogue between faculty and students as well as among the students themselves, and they included in the course concepts and topics of relevance to non-scientific careers. Because the business calculus sequence is taken by students majoring in fields from philosophy to animal science, along with a majority of business students, the courses featured in similar proportions business topics and topics related to other fields.

The focus of the revised courses-on rates of change and their interpretation in non-technical settings, the derivative as a rate of change, and the integral as the accumulation of change-is much narrower than that of the traditional approach. The courses feature the interplay between the discrete and the continuous, because many real-life situations involving change are discrete processes which can often be represented by continuous mathematical models allowing the application of the concepts and methods of calculus. When real-life discrete situations are represented by continuous mathematical models, the role of algebra becomes to describe, and to enable the student to reason about, the quantities that are undergoing change. This departs markedly from algebra's traditional function as a collection of organized manipulations used to obtain a numerical answer to a well-formulated problem.

Because most students are likely to work in manufacturing and marketing, the courses introduce early on linear, quadratic, cubic, exponential, and logistic models and go on to provide modeling experiences using real data. The focus, however, is not on the subtleties of modeling, but on using elementary models to obtain functional relationships between variables. Students conduct their calculus investigations on the functions represented by the models.

The new courses require the use of graphing calculators to work with real data and to develop insight through graphical and numerical representations of problem situations. Faculty use the Texas Instruments 82, but students may use the TI-82, the TI-85 or the Hewlett-Packard 48G/GX calculators.

Evaluation and Project Impact

The external evaluator examined student questionnaires, projects, journals and tests, and interviewed faculty at 13 institutions which had tested the materials. He also compared the performance of students at Clemson and at six other institutions (two groups of 273 and 247, respectively) who were taught according to the new approach.

Overall, the courses resulted in lower failure rates and improved student attitudes (85 percent of students believed they were learning as much as or more than students in traditional courses), although not in improved retention.

Because available tests concentrate primarily on procedural knowledge, they are not appropriate for assessing the range of student performance elicited by the new courses. Nevertheless, a small set of test items appropriate to both traditional and project classes was developed. The performance of students being taught by the new approach was deemed, on the whole, acceptable. The student projects and presentations, on the other hand, exhibited a large range of quality.

Lessons Learned

As part of the project, faculty and teaching assistants were trained to teach the new approach by means of summer workshops and weekly seminars. An unanticipated need for additional instruction arose as soon as the first pilot courses were carried out-the tutors whom students were accustomed to hiring to get them through the course had to be trained in the use of graphics calculators and in the reformed methodology.

Both students and faculty were sometimes hampered in the use of the new materials by their preconceptions about calculus: students who had been exposed to innovative teaching in prior courses reacted better to the reformed approach than those accustomed to traditional lecturing.

Project Continuation

Currently all 44 business calculus classes at Clemson-more than 1,500 students per semester-are being taught according to the new approach.

Dissemination and Recognition

The project has been the topic of over 150 workshops and presentations across the nation, as well as ten articles, an Internet discussion group, and a Web site. The textbook, published by D.C. Heath and Company in 1995, and the brief and complete first editions, published by Houghton Mifflin, have been or are currently being used in over 75 colleges, universities and high schools.

The project won the grand prize in the INPUT (Innovative Programs Using Technology) international competition funded by the Annenberg/CPB Project and the National Science Foundation. The competition focused on programs that take advantage of modern technology-calculators, computers, video, multimedia and telecommunications-to improve student learning in mathematics service courses. The project also received a grant to spread reform business calculus to ten institutions under FIPSE's Disseminating Proven Reforms competition.

Available Information

Further information may be requested from:

Iris B. Fetta
Department of Mathematical Sciences
Clemson University
Clemson, SC 28634-1907
Telephone: 864-656-5212

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