The Rule of Four

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. The original dictate of this
idea was "The Rule of Three: Every topic should be presented
geometrically, numerically and algebraically."[i] A traditional math curriculum would study and
develop 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.[ii] Regression analysis available in recent
generation calculators has enabled students to also practice the moves from the
numeric or graphic form to the symbolic form.

As calculus reform came into its next phase, and in
concert with the NCTM Standards of 1989, a new focus on mathematical writing
became evident. Writing to learn
mathematics, math class journals, and written reports became common usage in
mathematics courses. It was evident that
the “rule of three” had in fact become a “Rule of Four: every topic should be
presented numerically, graphically, symbolically, and *verbally*”.

This new vision has a three dimensional diagram to
show six pathways:

In traditional mathematics instruction, students
have been expected to interpret the verbal to symbolic pathway (the greatly
beloved story problems); now there is comparable emphasis on verbal
interpretation and explanation, both to specify problem response (answer the
problem question with a sentence) and problem solving methodology used by the
student.

Facility with all six of the pathways gives
opportunity for fuller understanding of the concept studied. Exercises should be balanced to develop and
strengthen skill in moving from any node to another. Students should be aware of this structure
and able to determine which pathway a question or exercise is using.

There are twelve pathways; some examples of how each
pathway can be used are given.

symbolic®numeric evaluate
the function or expression for

particular
values of the variable(s);

find
solutions for given equations.

numeric®symbolic calculate
the equation of a line from

two points; calculate a
regression equation from data.

graphic®numeric identify
coordinates of points on a graph.

numeric®graphic plot
points to form a graph.

symbolic®graphic graph
a function or inequality.

graphic®symbolic determine
the equation for a function from its graph.

verbal®numeric sort
data information into a table.

numeric®verbal describe
what a table of data is measuring;
interpret the results of solving an equation.

verbal®graphic use
verbal descriptions to draw the shape of a graph.

graphic®verbal give
verbal descriptions that adequately identify the shape of a graph; provide a
descriptive relationship that is represented by the graph.

verbal®symbolic translate
a description to mathematical symbols; formulate an equation from a descriptive
relationship.

symbolic®verbal provide
a descriptive relationship that is represented by the symbolic representation;
interpret the symbol equation into natural language.