Calculus in Context


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Calculus in Context
The Five College Calculus Course
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Support

Support for the different aspects of Calculus in Context has come from several sources.

Primary funding for curriculum development and dissemination was provided by the National Science Foundation in grants DMS-14004 (1988-95) and DUE-9153301 (1991-97), awarded to Five Colleges, Inc. Other curriculum development funding has been provided by NECUSE (New England Consortium for Undergraduate Science Education, funded by the Pew Charitable Trusts) to Smith College (1989) and Mount Holyoke College (1990). Five Colleges, Inc. also provided start-up funds.

Equipment and software for computer classrooms has been funded by NSF grants in the ILI program: USE-8951485 to Smith College and DUE/EHR-9551919 to Mount Holyoke College. The Hewlett-Packard Corporation contributed equipment to Mount Holyoke and Smith Colleges, and other equipment was contributed to Mount Holyoke College by IBM and the Sloan Foundation.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.

Overview

Calculus in Context is the product of the Five College Calculus Project. Besides the introductory calculus text, the product includes computer software and a Handbook for Instructors described below.

In this overview, we tell our "creation story" and then describe how it led to the text, spelling out our starting points, our curricular goals, our functional goals, and our view of the impact of technology.

Origins

The story of the Five College Calculus Project began almost forty years ago, when the Five Colleges were only Four: Amherst, Mount Holyoke, Smith, and the large Amherst campus of the University of Massachusetts. These four resolved to create a new institution that would be a site for educational innovation at the undergraduate level; by 1970, Hampshire College was enrolling students and enlisting faculty.

Early in their academic careers, Hampshire students grapple with primary sources in all fields--in economics and ecology, as well as in history and literature. And journal articles don't shelter their readers from home truths: if a mathematical argument is needed, it is used. In this way, students in the life and social sciences found, sometimes to their surprise and dismay, that they needed to know calculus if they were to master their chosen fields. However, the calculus they needed was not, by and large, the calculus that was actually being taught. The journal articles dealt directly with the relation between quantities and their rates of change--in other words, with differential equations.

Confronted with a clear need, those students asked for help. By the mid-1970s, Michael Sutherland and Kenneth Hoffman were teaching a course for those students. The core of the course was calculus, but calculus as it is used in contemporary science. Mathematical ideas and techniques grew out of scientific questions. Given a process, students had to recast it as a model; most often, the model was a set of differential equations. To solve the differential equations, they used numerical methods implemented on a computer.

The course evolved and prospered quietly at Hampshire. More than a decade passed before several of us at the other four institutions paid some attention to it. We liked its fundamental premise, that differential equations belong at the center of calculus. What astounded us, though, was the revelation that differential equations could really be at the center--thanks to the use of computers.

This book is the result of our efforts to translate the Hampshire course for a wider audience. The typical student in calculus has not been driven to study calculus in order to come to grips with his or her own scientific questions--as those pioneering students had. If calculus is to emerge organically in the minds of the larger student population, a way must be found to involve that population in a spectrum of scientific and mathematical questions. Hence, calculus in context. Moreover, those contexts must be understandable to students with no special scientific training, and the mathematical issues they raise must lead to the central ideas of the calculus--to differential equations, in fact.

Coincidentally, the country turned its attention to the undergraduate science curriculum, and it focused on the calculus course. The National Science Foundation created a program to support calculus curriculum development. To carry out our plans we requested funds for a five-year project; we were fortunate to receive the only multi-year curriculum development grant awarded in the first year of the NSF program. The text and software is the outcome of our effort.

Designing the curriculum

We believe that calculus can be for students what it was for Euler and the Bernoullis: a language and a tool for exploring the whole fabric of science. We also believe that much of the mathematical depth and vitality of calculus lies in connections to other sciences. The mathematical questions that arise are compelling in part because the answers matter to other disciplines. We began our work with a "clean slate," not by asking what parts of the traditional course to include or discard. Our starting points are thus our summary of what calculus is really about. Our curricular goals are what we aim to convey about the subject in the course. Our functional goals describe the attitudes and behaviors we hope our students will adopt in using calculus to approach scientific and mathematical questions.

Starting Points

Curricular Goals

Functional Goals

Impact of Technology

The text illustrates how we have pursued the curricular goals. Each goal is addressed within the first chapter which begins with questions about describing and analyzing the spread of a contagious disease. A model is built: a model which is actually a system of coupled non-linear differential equations. We then begin a numerical exploration on those equations, and the door is opened to a solution by successive approximations. Our implementation of the functional goals is also evident. The text has many more words than the traditional calculus book--it is a book to be read. The exercises make unusual demands on students. Most are not just variants of examples that have been worked in the text. In fact, the text has rather few ``template'' examples.

Shifts in Emphasis

It will also become apparent to you that the text reflects substantial shifts in emphasis in comparison to the traditional course. Here are some of the most striking:

How the emphasis shifts:
increase decrease
concepts techniques
geometry algebra
graphs formulas
brute force elegance
numerical solutions         closed-form solutions

Since we all value elegance, let us explain what we mean by "brute force". Euler's method is a good example. It is a general method of wide applicability. Of course when we use it to solve a differential equation like y'(t) = t, we are using a sledgehammer to crack a peanut. But at least the sledgehammer works. Moreover, it works with coconuts (like y' = y(1 - y/10)), and it will even knock down a house (like y' = cos2(t)). Students also see the elegant special methods that can be invoked to solve y' = t and y' = y(1 - y/10) (separation of variables and partial fractions are discussed in chapter 11), but they understand that they are fortunate indeed when a real problem will succumb to such methods.

Audience

Our curriculum is not aimed at a special clientele. On the contrary, we think that calculus is one of the great bonds that unifies science. All students should have an opportunity to see how the language and tools of calculus help forge that bond. We emphasize that this is not a "service" course or calculus "with applications," but rather a course rich in mathematical ideas that will serve all students well, including mathematics majors.

The student population in the first semester course is especially diverse. In fact, since many students take only one semester, the first six chapters stand alone as a reasonably complete course. We have also tried to present the contexts of broadest interest first. The emphasis on the physical sciences increases in the second half of the book.

Handbook and Supplements

We have prepared a Handbook for Instructors (a PDF file) based on our experiences, and those of colleagues at other schools, with specific suggestions for use of the text. We urge prospective instructors to consult it, since this course differs substantially from the calculus courses most of us have learned from and taught in the past.

There are also software programs are available at no charge for use with this text. These are QuickBasic versions of the Basic programs that appear in the text; you can also get QuickBasic itself.

sarah-marie belcastro has produced a collection of notebooks to accompany the text. They are written in both Mathematica and Sage; the latter is a "free open source alternative to Magma, Maple, Mathematica and Matlab."