Ithaca College Calculus: Unifying Threads

Unifying Threads

In our work on calculus, we have identified a number of unifying
threads that typically run through a successful first-year
course. We have noted with interest that a number of other
groups of mathematicians working on revitalizing calculus have
identified similar themes. We label our threads graphical calculus,
distance and velocity, multiple representation of functions, modeling,
top-down analysis, and approximation and estimation. We give
a brief description of each thread below.

We describe these themes as "threads" because they are
woven throughout the course, and serve to bind it together into
a unified whole. Most of the projects, and many of the activities
contain elements from more than one of the threads. In an
appendix to the Instructor's Guide to Calculus, An Active
Approach with Projects we include guides to the relationships
between the projects and activities in the volume and the corresponding
threads.

Threads in first-year calculus

Graphical calculus. We use graphs as important examples of
functions from the start of the first semester of
calculus. Many of the important concepts of calculus are
presented using graphs of functions (with no formula given)
during the first two weeks of the course. These concepts include
the slope of a function at a point, increasing and decreasing
functions, concavity, continuity, extrema, and the fundamental
theorem of calculus, as well as functions paired with their
derivatives in the form of graph/rate-graph pairs. We continue
to use many examples of functions represented by graphs
throughout the year.
Distance and velocity. While graphing (slopes and areas)
provides one convenient example of derivatives and
integrals, distance and velocity provides another. Velocity is
one of the few examples of a rate that students have experienced,
so students are able to understand it more intuitively than other
examples. It is easy for the instructor to model velocity by
simply moving around the classroom. Over the course of two or
three days, we are able to introduce the important concepts of
calculus by discussing average velocity, instantaneous velocity,
and approximations to distance traveled. It is also easy to
motivate the first derivative test, for example, by noting that
the distance is maximized (locally) "when you turn around;"
i.e., when velocity changes from positive to negative. Of
course, we don't necessarily refer to derivatives the first day,
but when we get there, this result looks familiar.
Multiple representations of functions. We emphasize the notion of
a function stressing that there are several
representations for a function: graphical, numerical (a table of
data), algebraic (a formula or expression), physical,
theoretical, and written or verbal. This is important, because students
need to be able to apply techniques they learn to functions
presented in any of these formats. It is not enough for students
to simply learn to manipulate symbols.

This idea also helps students see the unity of calculus. For
instance finding the area under a graph, finding the
antiderivative of a function, and computing a Riemann sum from a
table of values are all examples of the concept of integration.
We emphasize the interplay of all representations. If we have a
result for a function some typical questions from the instructor
are "What if you were given a graph? Then what does this result
tell you? What if you were given a table of values? What does
this result mean? Describe this situation in English."

Modeling. Modeling begins early in the course and
continues throughout the first year. Projects and classroom
examples typically begin with some "real" application, which
must be translated into a mathematical model.

Modeling is introduced early by way of classroom activities.
Graphical relationships are the first instance of modeling. For
example, in class students are asked to create and analyze a
graphical model of the height of a flag from the ground as a
function of time as the flag is being raised.

The students use modeling techniques on the projects. For
example, in a Calculus 1 project, students are asked to model the
motion for a detector that is guarding a hallway against
intruders.

This emphasis on modeling in class and in the projects seems to
have strengthened the students' belief that it is possible to
construct functions that model even complex situations and that
the concepts presented in calculus are valuable tools in this
process. Furthermore, it seems to have significantly reduced the
students' fear of word problems.

Top-down methodology. The top-down method is a very
effective tool in problem solving. This approach involves
breaking a problem into smaller problems and breaking those
problems into smaller problems until the resulting problems can
all be solved easily. This scheme includes a method for
assembling the solutions of the smaller problems into a solution
of the original problem.

We spend some class time introducing the top-down approach to
problem solving, and examples in class are approached top down
whenever appropriate. ("What things do I need to know? ...
Now, what do I need to do in order to know these things? ...'')
We emphasize using this approach on any large problem. Most
projects are designed to be attacked in a top-down fashion.
Students don't need to discover "big" solutions if they can
combine small solutions to solve big problems.

Approximation and estimation. Approximation and
estimation are stressed as fundamental and unifying concepts in
calculus in a theoretical sense (they appear in the concepts of
the limit of a function, the derivative, the integral, and the
convergence of sequences and series). Moreover, these notions
are stressed as appropriate solutions to specific problems. For
example, students are asked in class in a group setting to
estimate the slope of a graph of a function at a point using only
the graph of the function on graph paper and a ruler. They then
compare answers and discuss the question of who has the
"correct" answer. Similarly, they experience approximation by
applying at least one numerical method to the problem of root
finding to a given degree of accuracy and they study at least one
method of numerical integration, complete with an error estimate.

Students are also asked to compute Riemann sums and finite
differences from tables of values and solve certain problems
using these numerical estimates. Indeed, the students view some
modeling experiences as estimation or approximation of a "real"
situation. They also see Taylor polynomials and/or other
functions obtained from curve fitting serving as approximations
to the original function, which may be used in place of the
original function under certain conditions. Finally, where
appropriate, estimation is stressed in connection with common
sense. That is, students are asked if their answers to problems
make sense. In this way, "mathematical" common sense is both
developed and reinforced.

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This page maintained by: D. Schwartz, Ithaca College
schwartz@ithaca.edu

Last Modified: January 8, 2000