Mathematical Connections, Modeling in Business Calculus

Stephen Gendler, Clarion Univeristy

This talk concerns my experiences working as a test site for the Villanova project. Audrey Borchardt and Bruce Pollock Johnson instituted this project at Villanova. The purpose of the project was to redesign the Business Calculus to better reach the students. They got input from the Business Faculty on what they thought was necessary, and then redesigned the course to make the material more student-friendly by involving the students and using technology to remove the drudgery of difficult algebraic computations. After they wrote a new text Mathematical Connections, Modeling in Business Calculus, they ran workshops for interested faculty and took their results to national conferences.

Dr. Gendler has taught the Business Calculus since 1971. He has extensive training in Business Administration with a major in Securities analysis. He has been using the same kind of text for years and was becoming bored with his presentation. The course he teaches includes Theory of Functions, Differentiation and Integration of Functions of One Variable and Applications. The students have gotten weaker and weaker over the years, and not only would do nothing beyond the minimum, but could not carry any ideas to their business courses. They were totally dissatisfied with having to take a course that was a tedious requirement.

The Faculty in the School of Business could not use any calculus in their courses without meeting student resistance. They complained that the students couldn’t even set up a simple problem that related to their courses at any level, let alone calculus. They also viewed the course as a necessary evil to meet accrediting requirements.

Dr. Gendler attended the workshop at Villanova University in the summer of 1998.It was a two day intensive workshop covering the basic ideas of the text and how to teach it. They demonstrated how to integrate graphing calculators or excel into the course. They emphasized the real-life experience and active problem solving component of the course and especially the use of modeling techniques to make the course more relevant. After the workshop they were there for follow-up and to give support where necessary.

Several people who attended the workshop tried out the materials during the 1998-1999 academic year. Dr. Gendler taught two sections using the text in Spring 1999. In the course modeling12 modeling principles including:

Definition of Variables

Bounds

Constraints

Assumptions

Data gathering

Formulating a model

Solving the problem mathematically

Double checking

Validating the solution

Checking the sensitivity of the model

The text and the course included problems that emphasized real life applications that required the use of modeling techniques to set up problems. They also took time to interpret the answers and investigate their implications. The process involved curve fitting from "real" data using either the TI82 calculator or the shareware Mac-Curve-Fit. There was a personal project required that involved collecting real data and interpreting it in a written paper that followed closely the steps in a modeling project and interpreted the results. There as a very difficult pre and post -test administered beyond the Final Exam that examined understanding of calculus concepts rather than algebraic manipulation of calculus terms. The final also included a choice from several reasonably difficult applications in addition to the standard problems.

The course was not memorization of formulas. The course was not complicated algebraic manipulations. The course was more fun to teach. The course did not seem to be as boring and meaningless.

The results:

Because the students were engaged , they worked harder and paid better attention. In the Villanova study they did significantly better both in their testing and in subsequent courses. They are presenting these results nationally. Dr. Gendler’s students seemed more motivated, and became involved in the data collection and interpretation phases of their projects. Several of them predicted the stock market, one did a time study of the family retail store, One did a study of how many deer visited their feeding station. Projects ran the gamut from ingenious and very good to mediocre but the were much better overall then one would expect. Even though EXCEL is not available i the Mathematics Laboratory, numerous students either owned it themselves and used it on their personal computers, or used laboratories in the School of Business or elsewhere on campus to help analyze and present their data. It was the tool of choice for about half of the projects.

Student performance in the course was significantly better also. In the past more than half of the students in these classes, which do not meet at a popular time failed, with even more in the later class which fills with two and three time repeaters who add during add-drop. The fail rate dropped to 1./3 in the earlier class and less than 1/2 in the later one.

Students for the first time in many years seemed to want to do more work on their own. They seemed to have a better idea of the meaning of composition of functions, slopes, concavity, points of inflection, all the major topics. But in addition they were able to choose the right formulas for standard problems in differentiation and integration and also for some that are not normally emphasized like x**x. They saw problems from Business, not as dead examples but as living real problems that needed to be solved. They had a much better grasp of the demand curve, where it comes from and how it is used to determine profit and the meaning of elasticity, They were much more comfortable with how to use these concepts in business applications.
Dr. Gendler would like to use Mathematical Connections again. But all courses in Business Calculus need not necessarily use this text. Our traditional faculty object to it as not as encyclopedic not having enough problem choice, and being too heuristic. But the real-life problems with the modeling approach is a breath of fresh air, and the hands on project where the students choose a personally meaningful project and gather and interpret their own data is clearly a wonderful motivator. An attempt to involve the faculty from the School of Business in supporting the approach seems positive also. Art Villanova they actually produced a film to show the students featuring Business faculty using calculus in their courses. So it needs to be mastered before you take their course.

Indeed the Villanova project has many innovative ideas that make teaching Business Calculus easier and more effective. Many of these ideas seem to carry over naturally to other courses.