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.


[i]Hughes-Hallett, Gleason, et al. (Harvard Calculus Consortium) Calculus

[ii]as in Tukey, Exploratory Data Analysis