Calculus from 1988 to 1995
The faculty members in the department of mathematics were not interested in using calculators and/or computers in the teaching of mathematics simply as a computational and graphing tool. The main focus was on the best way for students to learn and for faculty to teach mathematics. It was the calculus sequence where these issues were first addressed by the faculty in the department.
One of the issues facing the Department of Mathematics and Computer Science in the 1980’s was the decreasing number of mathematics faculty, especially faculty who taught in the calculus sequence. In the 1986-87 academic years, there were 17 faculty members in the Department of Mathematics and Computer Science. Three faculty members who had originally been hired as mathematics faculty (Phil Pratt, Ken Johnson, and Georgianna Klein) had retrained in computer science and were now teaching mostly computer science courses. In addition, no Ph.D. mathematician had been hired at Grand Valley between 1973 and 1986. (Dr. Carl Arendsen and Dr. Ted Sundstrom started at Grand Valley in 1973.) Of the 17 faculty members, probably 10 were mathematics faculty, but of those 10, only 4 or 5 regularly taught courses in the calculus sequence.
Things began to change in 1987 when the department was able to increase the number of mathematics faculty. In 1987, the department hired Dr. Alverna Champion, and in 1988, Dr. Charlene Beckmann, Dr. Tom Gruszka, and Dr. Karen Novotny joined the mathematics faculty. (Dr. Beckmann and Dr. Novotny are still members of the department.) The department was also able to add new mathematics faculty in the early 1990’s; Dr. Gary Klingler in 1990, Dr. Steve Schlicker in 1991, Dr. Salim Haidar in 1992, and Dr. Paul Fishback in 1993.
Both the new faculty members and the existing faculty members were very interested in exploring possible ways to use technology to enhance the teaching of mathematics, and in particular, the teaching of calculus. In fact, shortly after she came to Grand Valley in 1988, Charlene Beckmann and Ted Sundstrom began working on ways to use graphing calculators in the calculus courses. Their approach, as well as that of many others in the department, was to help students build a better understanding of calculus by exploring familiar applications, which were modeled graphically and numerically. At that time, the graphing calculator was the ideal tool for this approach. The graphing calculators of choice for the department at that time were the Casio fx-7000 (and later the fx-7700) and the Texas Instruments TI-81 (and later the TI-82). With these calculators, it was possible to:
- Graph polynomial, rational, algebraic, trigonometric, logarithmic, and exponential functions.
- Change the domain and range to graph a function on any interval.
- Trace a graph of a function to determine function values.
- Zoom in on a particular region of a graph (similar to zooming in on a particular part of a photograph).
A result of the collaboration of Beckmann and Sundstrom was the publication of two books dealing with using graphing calculators in the teaching of calculus.
Graphing Calculator Laboratory Manual for Calculus. Reading, MA: Addison-Wesley, 1990.
Exploring Calculus with a Graphing Calculator. Reading, MA: Addison-Wesley, 1992.
In addition, it was possible to write programs for these calculators to further explore graphs and to do repetitive computations, and these two books contained several programs for graphing calculators. Some of the programs were designed to:
- Sketch secant lines along the graph of a function of the form y = f(x).
- Sketch tangent lines along the graph of a function of the form y = f(x).
- Use Newton’s method for approximating a root of a function, both graphically and numerically.
- Graph left and right endpoint approximations for Riemann sums.
- Approximate a definite integral using midpoint, trapezoid, and Simpson’s rules.
- Draw the graph of a set of parametric equations.
These books were used in the calculus sequence at Grand Valley for a few years and graphing calculators have been required for students in calculus courses since 1990. This link will open one of the explorations from Exploring Calculus with a Graphing Calculator.
Because most instructors of calculus were now using a more exploratory approach to the teaching of calculus using graphing calculators and some computer software, the department approved a significant change in the descriptions of the calculus courses. These new descriptions first appeared in the 1991 – 92 catalog.
MTH 201 Calculus and Analytic Geometry I. A development of the fundamental concepts calculus using graphical, numerical, and analytic methods with algebraic and trigonometric functions of a single variable. Limits and continuity, derivatives, indefinite integrals, definite integrals, and the Fundamental Theorem of Calculus; applications of derivatives and integrals. Prerequisites: 122 and 123. General education course CGE/A. Five credits. Offered fall and winter semesters.
MTH 202 Calculus and Analytic Geometry II. Continuation of MTH 201 using graphical, numerical, and analytic methods to study exponential, logarithmic, hyperbolic, and inverse trigonometric functions. Indeterminate forms, improper integrals, integration techniques sequences and series, Taylor polynomials and power series. Prerequisite: 201. Four credits. Offered fall and winter semesters.
MTH 203 Calculus and Analytic Geometry III. Continuation of MTH 202 using graph numerical, and analytic methods to study parametric equations, polar coordinates, vector algebra in two and three dimension, differentiation and integration of vector functions of a single variable and scalar functions of several variables. Prerequisite: 202. Four Offered fall and winter semesters.
The descriptions of the calculus courses have not changed since then and these are the current descriptions of the calculus courses.
Graphing Calculators and Computers
One of the major problems with the early graphing calculators such as the Casio fx-7000 and the Texas Instruments TI-81 was that even though they were programmable, all students had to enter (fairly lengthy) programs into their calculators. (Eventually, the manufacturers eliminated this problem by making it possible to transfer programs from one calculator to another by a cable that linked the two calculators.) During this time, there were several computer programs available that had most (if not all) of the capabilities of graphing calculators. Members of the department were actively investigating the use of these programs as well as the use of graphing calculators. This can be seen by the titles of some of the department seminars conducted by faculty at that time.
Using MicroCalc in Calculus Courses. Tom Gruszka and Ted Sundstrom. (1988)
Using Graphing Calculators in Calculus Courses. Ted Sundstrom, Charlene Beckmann, Ralph Wiltse, and Pedro Rivera-Muniz. (1989)
Another Approach to Using Computers in the Calculus Classroom. Ralph Wiltse. (1990)
Teaching Mathematics with the TI-81 Graphing Calculators. Charlene Beckmann. (1990)
An Impromptu Introduction to the CASIO 7700 Graphic Calculator. Charlene Beckmann. (1992)
In addition, in 1991, Dr. Steve Schlicker left Luther College in Iowa to join the Department of Mathematics. In addition to having experience using graphing calculators, Dr. Schlicker had written his own software to help explore the concepts of calculus. This software was titled Calc and had all and more of the capabilities of the graphing calculators. Faculty used this software and MicroCalc in their calculus courses for several years. Students were able to use these computer programs since they were installed on the network in the microcomputer labs that were becoming more prevalent on campus.
Even though these computer programs were readily available in the computer labs, the department continued to require students to use graphing calculators, primarily because the students could bring these calculators to class and have them available whenever they studied calculus. Because students had these calculators in class, the faculty started to move away from the lecture-discussion format of teaching and began developing exploratory classroom activities for the students to complete. This is about the time when faculty seriously began using cooperative learning techniques in the classroom since many of these activities were designed to be completed in small groups.
Computer Algebra Systems
The programs MicroCalc and Calc were both excellent programs that helped instructors teach the concepts of calculus and more importantly, helped students learn the concepts of calculus. As was true of many other programs, these programs were numerically based. That is, they used the numerical processing capabilities of computers to plot graphs of functions, approximate solutions of equations, and perform many other tasks that helped in the teaching of calculus. However, in the early 1990’s, a different type of computer program called a computer algebra system was becoming more available for desktop computers. A computer algebra system (CAS) is a software program whose core functionality is the manipulation of mathematical expressions in symbolic form as opposed to manipulating the approximations of specific numerical quantities represented by those symbols. Members of the faculty began experimenting with two of these programs –Mathematica and Maple.
Note: Maple was first developed in 1980 by the Symbolic Computation Group at the University of Waterloo in Waterloo, Ontario, Canada. Since 1988, it has been developed and sold commercially by Waterloo Maple, Inc., also known as Maplesoft), also based in Waterloo, Ontario. On a Wikipedia web site, it is stated that “there is a common belief that the name, Maple, is an acronym standing for MAthematics with PLEasure. In fact, the name is a reference to Maple’s Canadian heritage.”
Since site licenses for both of these programs were quite expensive, the department needed to choose one of these for the campus networks. In all likelihood, the primary factor in the decision was cost and the department obtained a site license for Maple. By the 1994-95 academic year, Maple was available on the networks in the computer labs and faculty could have it installed on their desktop computers. The first documented mention of Maple was in course syllabi and other course materials for calculus courses from the 1994 – 95 academic year.