Calculus in Context
Calculus in Context
The Five College Calculus Course
- James Callahan, Smith College
- David Cox, Amherst College
- Kenneth Hoffman, Hampshire College
- Donal O'Shea, Mount Holyoke College
- Harriet Pollatsek, Mount Holyoke College
- Lester Senechal, Mount Holyoke College
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Support for the different aspects of Calculus in Context has come from
Primary funding for curriculum development and dissemination was
provided by the
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.
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
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.
The story of the Five College Calculus Project began almost forty years
ago, when the Five Colleges were only Four:
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,
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,
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.
- Calculus is fundamentally a way of dealing with functional
relationships that occur in scientific and mathematical contexts. The
techniques of calculus must be subordinate to an overall view of the
questions that give rise to these relationships.
- Technology radically enlarges the range of questions we can
explore and the ways we can answer them. Computers and graphing
calculators are much more than tools for teaching the traditional
- The concept of a dynamical system is central to science.
Therefore, differential equations belong at the center of calculus,
and technology makes this possible at the introductory level.
- The process of successive approximation is a key tool of calculus,
even when the outcome of the process--the limit--cannot be
explicitly given in closed form.
- Develop calculus in the context of scientific and mathematical
- Treat systems of differential equations as fundamental objects of
- Construct and analyze mathematical models.
- Use the method of successive approximations to define and solve
- Develop geometric visualization with hand-drawn and computer
- Give numerical methods a more central role.
- Encourage collaborative work.
- Enable students to use calculus as a language and a tool.
- Make students comfortable tackling large, messy, ill-defined
- Foster an experimental attitude towards mathematics.
- Help students appreciate the value of approximate solutions.
- Teach students that understanding grows out of working on problems.
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.
- Differential equations can now be solved numerically, so they can
take their rightful place in the introductory calculus course.
- The ability to handle data and perform many computations makes
exploring messy, real-world problems possible.
- Since we can now deal with credible models, the role of modelling
becomes much more central to the subject.
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 , we are using a sledgehammer to crack a peanut. But
at least the sledgehammer works. Moreover, it works with
coconuts (like , and it will even knock down a house (like . Students also see the
elegant special methods that can be invoked to solve and (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.
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
There are also
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
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."