What
is a mathematical model?
A new
emphasis in the calculus reform movement has been an attempt to balance
students' abilities to use and interpret information in numeric, graphic and
symbolic form with similar facility.[1] A traditional math curriculum studies and
develops algorithms that facilitate the relationships indicated in the pathways
below.
Much drill is provided as -- given a symbolic
formula, evaluate to numeric data; or -- generate a graph associated with the
symbolic formula. Exercises are provided
also for reading numeric data off of a graph, or graphing numeric data. Indeed, the richness of this bidirectional
pathway has been enhanced with new technologies (e.g., graphing calculators)
and early introduction of graphing techniques and strategies in descriptive
statistics.[2]
However,
exercises that develop students' abilities to move from the numeric or graphic
form to the symbolic form are newer territory in the curriculum. They can be introduced naturally in a context
of mathematical modelling. In
traditional calculus courses, these models are generally from physics, and some
special geometric applications such as volumes and arclength; they are
formulated theoretically, and students then use the models in exercises. And many of the specialty calculus courses
using the traditional format also provide a symbolic model for the special
applications in the discipline; at best there is theoretical justification or
development of the model in the text.
In
developing a richer understanding of function relationships and symbolic
models, it is helpful to introduce a classification of functions in a
trichotomy of real, observed, and model.[3] The characteristics of these three types of
functions are summarized in the following table.
Type Characteristics
Real Conceptual
Not
completely knowable
Not
expressible by formulas
Observed Sampled from real
relations
Finite
in extent or number
Not
expressible by formulas
Model Approximate fit to real or observed
functions
Expressible
by (some type of) formula
These
classifications add another facet to the original triangle drawn on the
previous page:
In traditional textbooks, the symbolic model,
sometimes theoretically developed, sometimes appearing from nowhere, often
masquerades as the real function.
Recent
access to technology, especially easy to use software that fits data to the
model curve of your choice, has opened up the paths from graphic or numeric to
symbolic. But how is an appropriate
model chosen? This can be approached
from theoretical considerations of the real function or from tests of the
observed numeric data or graphs.