ICTCM - November 16-19, 1995 (Houston): Contributed Paper C45 Using the TI-82 and Spreadsheets in a 1st-Year Math Course for Business and Social Science Bruce Pollack-Johnson and Audrey Borchardt Department of Mathematical Sciences Villanova University Villanova, PA 19085 Phone: (610) 519-6926 Fax: (610) 519-6928 email: brucepj@ucis.vill.edu Abstract This course uses the TI-82 for calculus and spreadsheet software for multiple regression, matrices and linear programming. The authors have a FIPSE grant for further problem-driven interdisciplinary development focusing on modeling, concepts, realistic and interesting data and problems, and student-generated projects, plus videos, and are looking for test sites starting in the Fall of 1996. Introduction Several years ago, motivated by dissatisfaction among math and business faculty (not to mention students!) with our first-year math sequence designed mainly for business students, we began to look for ways to better connect the course to other business courses and the real world. We wanted an approach where realistic problems, developed collaboratively with our business and social science colleagues, would motivate and help teach math concepts. Around this time, we discovered the Clemson FIPSE project. We felt the emphasis on models, concepts, technology, and data-based real problems was on target, but the project did not include such major topics as matrix methods and linear programming that are essential to our sequence. We also wanted to add several other innovations we had been developing. The idea of our project is to build upon the work of the Clemson project, largely to make the approach available for a wider audience of students. In addition to our own materials on matrices, multivariable regresion, and linear programming (using spreadsheets), we will also be creating simple videos of business and social science faculty and people from the non-academic work world describing how the math topics of the course relate to their courses and jobs. These people also serve as an advisory committee to give us continual feedback and suggestions. Some of our other innovatons include the use of student-generated projects and student journals, encouraging group work, focusing more on translating between the real world (or words) and symbols (including formulation and clear definitions of variables with units specified), and the idea of incorporating "Instructor's Manual"-type comments within the text for students. The schedule of our project is to continue to develop the materials and recruit test sites in the first year, then train faculty from the test sites in the summer of 1996. We will be using the new materials in all of our classes at Villanova in 1996- 97 and working on the videos. In the summer of 1997 we will have another workshop to gather feedback and suggestions from everyone who has used the materials. Then in 1997-98 we will be making revisions in response to the feedback and recruiting a new group of test sites, to be trained in the summer of 1998 (with the original test sites involved to share experiences). Philosophy Our philosophy includes the following: - Connectedness of the topics covered to students' concurrent and following courses, to their personal lives, and to their eventual careers, - A problem-solving approach, in which we discuss the process of problem-solving from beginning to end and look at the entire course from that perspective, - Problem-driven development, in which problems motivate the concepts being introduced and serve as concrete examples throughout the develpment of the concepts and techniques, - Realistic, interesting, practical problems using real-world data whenever possible, - Emphasis on mathematical modeling and functions as they relate to the real world, - Interdisciplinary/inter-college development of problems and topics, as well as course structure, evaluation, and choice of technology (client-focused), - A focus on translating between the real world (or words) and symbols, which includes formulation and precise definitions of variables (including units), as well as interpretation of answers, - Emphasis on understanding and applying concepts over detailed mathematical theory and rote/symbol manipulation, - Technology as a tool: - All students should be able to solve simplistic problems by hand for a working understanding of the concepts, - Technology makes it possible to solve real-world problems that would otherwise be too messy and tedious, - Having technology do the calculations shifts the emphasis from algebraic algorithms to the processes of formulating problems, examining assumptions, modeling, and interpreting output results from the technology. - Multiple representations of concepts: numerical, graphical, symbolic, and verbal (the Rule of 4) for the deepest possible understanding. Methodology Our means of achieving the above goals include the following: - Short videos of non-academics and faculty from business and social science that show how they use mathematics in their everyday work and in the courses the students take concurrently and subsequently, - Incorporating many Instructor's Manual-type comments in the text for students, - Student-generated projects as a way to learn about the entire problem-solving process from start to end, in which students: - generate problem ideas (whether about business or their personal lives, the goal being problems about which the students are passionate/care deeply and which are both interesting and useful to them); - write proposals (including a description of the problem, how they plan to collect data, and how they plan to solve the problem); - collect and communicate data; - formulate, model, and use technology to solve the problem; - verify (double-check) their calculations by multiple solution methods (e.g., by different modes within one technology, by different technologies, independent solutions by different individuals, or by hand as well as by technology where possible), fully resolving any discrepancies; - validate their solution to make sure it squares with reality (e.g., trying their optimal solution to see if the result it yields is close to what their model predicts, or trying to solve the problem with a different model, or just using common sense and ballpark estimates); - perform sensitivity analysis (think about the reliability of their data, search for biases, and study the effects on the solution of changing data values to different conceivable values); - write up a well-written full description of the problem and the process (including the concepts and techniques involved, with communicative graphs wherever possible), ending with meaningful conclusions for the real-world problem, showing a good understanding of the degree of uncertainty (precision and accuracy) of the solution. - Group work (collaborative learning), both in the projects, in class for discovery activities, and in working on homework problems. Examples of Student-Generated Projects Some of the best student-generated projects to date have included the following: - Determining the optimal price and quantity of T-shirts for sale by an organization (similar projects have involved the sale of M&M's and hair scrunchies on campus); - Determining the optimal number of hours of studying per week to maximize happiness (finding the right balance between academics and other aspects of college life) for an individual student; - Determining the optimal number of hours of sleep to get each night; - Optimal allocation of study time between 2 courses at finals time; - Finding the optimal angle for an individual to shoot a 3- pointer in basketball; - Finding optimal (minimum cost) breakfast food combinations to meet nutritional goals; - Finding an optimal combination of forms of exercise; - Determining optimal prices and quantities for multiple product sales; and - Determining the best combination of gas octane and speed to minimize the cost per mile of operating a given car. As can be seen, these topics are clearly of interest to college students, and they find working on them very interesting and practical. Furthermore, although some of the topics do not sound much like business, they all involve optimal allocation of resources, such as time and money, which is the most fundamental category of business problem. Because students are working on projects about which they care deeply, they really learn the process and concepts. This makes it possible to later transfer the knowledge and skills to business and career problems about which they will have a similar passion, since their salaries and careers are likely to be at stake. Topics The topics covered in this course include the following: - The Process of Problem-Solving, - Mathematical Modeling and Functions, - Translating between the Real World (or Words) and Symbols (including Formulation), - Compound Interest / Present Value / Future Value, - Rates of Change / Derivatives, - Single-Variable Optimization, - Post-Optimality Analysis (Verification, Validation, and Sensitivity Testing; Accuracy/Precision), - Integration (Motivated by and Applied to Concepts in Probability and Statistics), - Solving Simultaneous Equations (including Breakeven and Equilibrium Analysis) / Matrices - Multivariate Functions (Partial Derivatives and Optimization Optional), - Least-Squares Regression, including Multivariate Regression, - Linear Programming, and - Selecting Appropriate Concepts/Techniques/Models. Sequence The sequence we recommend for this material is that outlined above. This way, matrices are covered before multivariate optimization, so that matrices can be used to solve the partial derivative equations from multivariate quadratic functions. Then partials can be used to derive least-squares formulas and explain some of the concepts of linear programming and the Simplex Method. Depth of Calculus Coverage We focus primarily on smooth continuous functions, and intuitive notions of limits rather than more explicit mathematical details, although we try to make students aware that complications can occur in the real world, and that they need to understand more theory than we will cover in this course to fully deal with them. With both derivatives and integrals, we do not emphasize rules and techniques beyond the basics, since numerical calculations done with technology are quite accurate, and the majority of these students will not need (or use) anything more in their later courses or careers. We show examples with parameters (e.g. the Economic Order Quantity inventory model, or finding the mean of a linear density function) to explain why symbolic solutions are useful and important, and explain that symbolic algebra systems exist to perform such operations if necessary. Advanced Mathematical Topics Our course is designed so that the single-variable calculus material is all covered in one semester (so-called "brief calculus"), with the other material (roughly what is often called "finite math") in another semester. Part of the reason for this is that students who have AP credit or knowledge of calculus can place out of the calculus semester and take the other material, which they are not likely to have seen in high school. At Villanova, we are looking to put into place a third semester math course for business students that would cover more advanced calculus and linear algebra concepts (discontinuous and non-differentiable functions, Lagrange multipliers, determinants, Taylor series, sequences and series, etc.) required for some majors and for admission to some graduate programs. Probability and Statistics This course does not aim to teach probability or statistics per se. It is designed to lay the foundation for a business statistics course, and to use some fairly intuitive concepts of probability and statistics as applications of, and motivation for, topics in calculus. For example, the mode of a continuous probability distribution is an example of function maximization (optimization): one can take the derivative of the density function, set it equal to 0, and solve for the mode. It can be shown intuitively that the probability of an interval with a uniform distribution corresponds to the area under the density function over that interval. The question of how to calculate the same probability when the density is not uniform is then wonderful motivation for integration, since it underlies most of the statistics these students will be learning and using. After we then explain the idea of limits of sums (e.g., of rectangles) to find area, we show that the process of recapturing an original function from its derivative (antidifferentiation) also corresponds to the area under the derivative. Thus, finding the area under any curve can be obtained by antidifferentiation, which we mathematicians call the Fundamental Theorem of Calculus. At this moment in the course, we point out that the median is defined in terms of the area under the density function. In fact, to find the median, one creates an integral equation with the variable to be found as one of the limits of integration. This is exactly the kind of integral used in the Fundamental Theorem of Calculus, and a perfect way to build intuition to understand it better. Once students understand integration, they can also be shown how means and variances can be calculated as integrals, as well as expected values of any kind. Final Comments The idea of this course is to give students in business and social science the mathematical foundations and tools they need for courses in their field and for their eventual careers, as determined in collaboration with faculty and non-academics in those fields. We emphasize the connectedness of the topics of this course to the students' current and future lives with videotapes of faculty and non-academics at work using the mathematical concepts and techniques. Our focus is on the process of problem-solving, which they experience in full through student-generated projects, developing teamwork as well as writing and presentation skills at the same time. We teach explicitly about problem-solving, as well as about the process of translating between the real world (or words) and symbols and the process of mathematical modeling. We use realistic, interesting problems to motivate and develop the mathematical concepts inductively through a discovery approach. We use technology as a tool to do tedious calculations, making real- world problems solvable. This shifts the emphasis to formulation and interpreting output, but we also require understanding of concepts (in as many cognitive modes as possible), demonstrated by being able to solve simplistic problems by hand, so that the technology will be used appropriately and intelligently. If you are interested in serving as a test site for these materials starting in the Fall of 1996, or if you know someone else who might be, please contact the authors.