B. The Development Plan

Mathematics faculty members currently are transforming all of the courses this proposal would affect so that they feature discovery-based learning in which computer programs and packages play an integral role. Our primary goal in doing so is to deepen students' understanding of the mathematical concepts presented in these courses, and to develop students' creativity and critical thinking skills more than we have in the past. We believe that presenting mathematical concepts in the context of real life applications whenever we can is consistent with this goal and also makes the concepts more interesting, relevant, and useful to the students who take the courses. Computers are essential to accomplishing these goals: students will use software to analyze mathematical models numerically, to visualize mathematical concepts, and to discover and explore mathematical concepts by generating examples from which to make and test conjectures. Our aim is for the computer to become a tool to which students routinely turn to explore or analyze mathematical ideas and models.

The students in our Calculus in Context courses already use computer packages and programs on a daily basis to discover, explore and analyze calculus concepts and models. The first-semester course (Mathematics 103-104 or 105) begins with a model for a measles epidemic. After setting up rate (differential) equations for the susceptible, infected, and recovered populations, the students use these equations to predict future sizes of the three populations. The computer is first introduced as a tool to facilitate calculation and plotting of these values. The students soon notice that their approximations of future population sizes seem to approach limiting values as they recompute them using smaller and smaller step sizes. In this way, students discover, and gain an intuitive understanding of, the limit process. Later in the first course, the derivative is introduced as the slope of the straight line the students see when they look at a tiny section of the graph of a (locally linear) function under a "computer microscope." Already it appears that our students have gained a better understanding of the derivative as slope than they ever have in the past.

The second-semester calculus course (Mathematics 106) will use the Calculus in Context curriculum during the 1993 spring semester; the third-semester course (Math 107) will use it beginning in Fall, 1993. All three calculus courses use the True BASIC programming language. The first-semester course uses the True BASIC Calculus package as a graphing tool; we intend for the second- and third-semester courses to use Mathematica as a graphing tool and for symbolic computation.

Our goal for Spring, 1994, is to conduct our differential equations and linear algebra courses using a laboratory format similar to that used in the calculus courses. In the meantime, during the 1993 spring semester, these courses will meet in a classroom equipped with a computer with an overhead projection system, which the instructors will use for in-class demonstrations and explorations. In addition, students will complete homework requiring computer use, and will participate in weekly or biweekly exploratory laboratory sessions in a computer laboratory in the University's Academic Computing Center. (We have been able to schedule a classroom in the computer center at least once each two weeks for these sessions.) Again, so that discovery-based learning can take place in these courses every day, our aim for the 1993-94 academic year is for students to have computers available during every class session.

Computer use in Mathematics 109, Differential Equations, will enable students to perform more numerical and more qualitative analysis of differential equations than they have in the past. Early in the course, students will use the MacMath software package to graph solutions to a series of carefully selected first-order differential equations. They will then use the software to explore the effects of changes in step size, in initial conditions, or in the parameters of the differential equation. We believe that this sort of exploration will lead students to a deeper understanding of differential equations and their solutions and will improve their critical thinking skills. Using as his primary sources materials from an NSF-funded "Teaching Differential Equations with Computer Experiments" workshop (see Section D), the instructor (Killpatrick) will develop the laboratory sessions for the course during the 1993 January term. Later sessions will utilize the Mathematica package.

Beginning during the 1993 spring semester, students in Mathematics 144, Linear Algebra, will use the computer package MATLAB as a learning tool. Although the course will continue to emphasize matrices, vector spaces, linear transformations, and eigenvectors and eigenvalues, computer use will allow students to spend less time row reducing matrices, computing inverses, and computing determinants and more time exploring concepts and applications. The course will feature a series of laboratory sessions, to include some that introduce concepts by having students explore examples, detect patterns, and make conjectures; some that introduce concepts in the context of an application; and some in which students apply their knowledge to a new application. For instance, students will use the computer to discover and explore relationships between such concepts as row rank and solutions of systems of equations. Eigenvalues will be introduced in a context such as Leslie population models. Such concepts as linear independence and linear transformation will be reinforced by having students apply them in the context of algebraic coding theory. The instructor (Beery) will write these laboratory sessions during the 1993 January term.

In Spring, 1994, the differential equations and linear algebra courses will shift from biweekly laboratory sessions and instructor-led explorations to the daily laboratory/discussion format used in the calculus courses. This will complete our transition to student-centered, discovery-based learning in our first- and second-year courses.

C. Equipment

(1) The Equipment Request

In Section B, we have described why and how we are using and plan to continue to use computing as an integral classroom activity. To facilitate this, we need two classrooms in which each pair of students has a computer available at all times. We plan to replace the 14 inadequate computers in our existing computer classroom with Macintosh Centris 650 computers with high resolution color monitors. One machine will be the system server, and another will be incorporated into an overhead projection system to be used for presentations by the instructor or students. The new, second laboratory will require exactly the same computer equipment as the first laboratory. Because of the mathematics faculty's commitment to the project described in this proposal, the University has agreed to establish the second laboratory with its half of the requested funds beginning in January, 1993. Anticipated costs are given in the Detailed Budget.

The Department offers ten sections of calculus, differential equations and linear algebra each semester, and two sections of calculus during its January term, to approximately 400 students. Each of the two computer classrooms will be used for five classes per semester, and for one class during the January term.

The software packages we use or wish to use in these courses are True BASIC (Math 103-107), True BASIC Calculus (103-105), Mathematica (106, 107, 109), MacMath (109), and MATLAB (144). While these packages can run on smaller computers, they run faster and more efficiently on the machines requested in this proposal. In particular, the relatively large, color machines and monitors we have requested optimize the packages' graphing capabilities.

(2) The Equipment on Hand for the Project

While the equipment that has been used to pilot the project has been useful in convincing us of the viability and merit of our approach to teaching the courses, it is inadequate for long-term use. In addition, the department's two portable computer/projection units, one of which is being used in our computer classroom, are needed in other classrooms. The department has site licenses for the True BASIC programming language and the MATLAB package, and has several copies of True BASIC Calculus and individual copies of MacMath and Mathematica.

We note that, in addition to providing matching funds for the requested computer purchases, the University of Redlands will spend approximately $10,000 for laboratory construction and furnishings.

(3) Equipment Maintenance

The university's Science Center employs one technician who is trained to perform Apple maintenance and who receives assistance from the technical staff of the Academic Computing Center.

D. Faculty Expertise

The mathematics faculty, and especially the four principal investigators, have the expertise necessary to complete the project successfully. Two of the four (Beery, Scherer) attended a Sloan Foundation-sponsored "Computers and the Teaching of Mathematics" workshop during Summer, 1990, which prompted all four investigators to begin incorporating computer use into their calculus courses in Fall, 1990. For comparison, one of them (Cornez) taught back-to-back sections of a computer and a non-computer course that semester; the results of this experiment will appear in [2]. During 1990, computers were used for classroom demonstrations and for some student homework. Since then, the faculty has gradually increased computer use in its calculus courses. (See [6].) During Spring, 1991, Beery began constructing a series of laboratory sessions for first-year calculus. She based her work on several sources, many of which were subsequently included in The Laboratory Approach to Teaching Calculus [4]. Her work is detailed in [1].

After gradually increasing computer use, geometric thinking, and conceptual understanding in our calculus courses, we decided to make substantial changes in the content of these courses by implementing one of the context-based NSF curriculum projects. This meant choosing between Duke University's Project CALC curriculum (see [8; 51-74]) and the Five Colleges' Calculus in Context curriculum. We chose the Calculus in Context curriculum because we felt it presented the mathematical ideas more coherently and because we already possessed the software necessary to implement it immediately in our first-semester course. Project CALC would have required software and equipment we did not possess. Beery, Killpatrick, and Scherer received NSF-funded training in using the Calculus in Context curriculum during March, 1992. Of course, all four investigators currently teach calculus courses utilizing this curriculum.

Beery, Cornez and Killpatrick participated in a "Calculus with Computer Algebra Systems" short course, led by Donald Small, during June, 1991. Killpatrick took part in an NSF-sponsored "Teaching Differential Equations with Computer Experiments" workshop during June, 1992; Beery participated in an NSF-sponsored "Augment the Teaching of Linear Algebra through the use of Software Tools" (ATLAST) workshop during June, 1992. In addition, Scherer was part of an NSF-sponsored "Algorithmic Number Theory with Mathematica" workshop during June, 1992, and Cornez regularly has students use programming and computer packages in courses in statistics, probability theory, and numerical analysis.

E. Dissemination Plan

The faculty has shared information about its innovations in calculus courses with mathematics colleagues from other universities through a poster session at the Spring, 1992 meeting of the Southern California Section of the Mathematical Association of America, and through two talks at the Conference on the Teaching of Calculus held at Harvard University in June, 1992. Three reports on the faculty's innovations in calculus courses are to appear shortly in two journals (see [1], [2], [6]). Beery has responded to requests for sets of laboratory instructions from approximately 40 calculus instructors. As noted above, Killpatrick is training teachers at our city's 3200-student high school. Two other large, local high schools and a local community college have discussed with us training for their mathematics faculty.

The faculty will continue to report the results of its curricular changes at conferences and in appropriate journals, and to respond to individual requests for information resulting from these presentations and articles. We believe that information about our computer classrooms will be of interest not only to those considering teaching calculus, differential equations and linear algebra as laboratory courses, but to all mathematics faculty planning to use discovery-based learning of any sort in their courses.

F. Bibliography

[1] Beery, Janet L. Calculus with Weekly Exploratory Laboratories. 1992. PRIMUS. To appear.

[2] Cornez, Richard, Janet Beery, and Mary Scherer. A Computer-Based Calculus Curriculum. 1993. College Teaching. To appear.

[3] Five College Calculus Project. Calculus in Context, I, II, III. 1992. Amherst, MA: Five Colleges, Inc.

[4] Leinbach, L. Carl, Joan R. Hundhausen, Arnold M. Ostebee, Lester J. Senechal, and Donald B. Small, eds. The Laboratory Approach to Teaching Calculus. 1991. MAA Notes 20. Washington, DC: Mathematical Association of America.

[5] Moving Beyond Myths: Revitalizing Undergraduate Mathematics. 1991. Washington, DC: National Academy Press.

[6] Scherer, Mary, Janet Beery, and Richard Cornez. Starting Small: Gradual Introduction of Computers into Calculus Courses. 1992. PRIMUS. To appear.

[7] Teaching with Technology. Chronicle of Higher Education. 28 October, 1992.

[8] Tucker, Thomas W., ed. Priming the Calculus Pump: Innovations and Resources. 1990. MAA Notes 17. Washington, DC: Mathematical Association of America.
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