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.