Early in the morning, over half-full cups of cappuccino, groups of mathematics teachers here recently sat squabbling over how to equitably divide the piles of hard candies and strips of red licorice that were heaped on the tables before them.
While some chose to divide the cache by type of candy, others, unwilling to compromise, argued for division by sheer numbers.
Meanwhile, Rick Jennings, a math teacher at Eisenhower High School in Yakima, Wash., watched patiently before breaking in to explain that the concept being stressed here was “fair division’'--the process by which such decisions can be reached--and the importance of showing students that many problems have a variety of viable solutions.
His presentation was entitled “Connect With the Social Sciences--A Look at the Mathematics of Decisionmaking,’' but the concepts being taught readily fall under the rubric of “discrete math,’' a hot topic at the annual meeting of the National Council of Teachers of Mathematics, held here March 31 to April 3.
Unlike traditional math courses, discrete mathematics essentially encourages students to think about ways to solve problems, rather than focusing on reaching a predetermined solution.
Many presentations here on discrete math, like one given by Kathy Blackwood, who teaches the subject at Venice High School in Los Angeles, were popular events with admission by ticket only.
Interest in discrete math as an adjunct to traditional courses is so high, teachers here said, because its concepts often have direct application to problems faced in fields that run the gamut from politics to the airline industry.
Discrete math, they noted, can, in many cases, be described as the intersection between mathematics and the social sciences, and makes use of relatively recent discoveries in the field.
“We’re not really teaching anything very modern in the schools,’' said Ms. Blackwood. “Yet most of the math that we use [in business] is fairly new. Today’s revolution in mathematics has to do with the human aspects of math.”
Roots in Computer Science
Discrete mathematics comprises a host of subjects, from counting techniques to graph theory, some of which date from the late 18th century but are only now seeing practical application.
The rise of discrete math as a discipline in large measure can be traced to a recent explosion in the application of computers to problem-solving and the related need to develop formal algorithms, or procedures, for solving intricate, though concrete, problems.
Discrete math came into its own as a college-level subject only in the 1980’s, as computers became more prevalent both on campus and in the world of work. At the time, it was considered a specialized topic, primarily aimed at computer-science majors.
But with K-12 educators increasingly looking for connections between the classroom and the world outside, discrete-math concepts have begun working their way into the high school curriculum.
Margaret Kenney, who teaches math educators at the Boston College Mathematics Institute, noted that the attention paid to discrete math in the curriculum and evaluation standards published by the N.C.T.M. in 1989 has also prompted math teachers to take a closer look at the field.
“Computers are essentially finite, discrete machines, and thus topics from discrete mathematics are essential to solving problems using computer methods,’' a separate section devoted to discrete math in the standards notes. “In light of these facts, it is crucial that all students have experiences with the concepts and methods of discrete math.’'
Ms. Kenney, who discussed ways to integrate discrete math into the geometry curriculum during a session at the N.C.T.M. conference, also is the principal investigator of a multiyear grant from the National Science Foundation to help secondary school teachers integrate discrete-math topics into the curriculum.
“It is kind of a fast-growing concept,,” said Ron L. Millard, one of her students, who teaches discrete math at Shawnee Mission (Kan.) South High School. “When they devoted a whole standard to it, that made a lot of people aware of it.”
Solving ‘Real World’ Problems
Another significant development that may also hasten the widespread teaching of discrete math is the inclusion of its concepts in the nationally known and respected California math-curriculum framework, Ms. Kenney added.
A number of states considering revisions in their math curricula are using the California framework as a model for reform.
And a compilation of recommendations for infusing discrete math into the curriculum was published in the N.C.T.M.'s 1991 yearbook, which Ms. Kenney edited.
In an introduction to the yearbook called “Discrete Mathematics: The Math for Our Time,’' John Dossey, the council’s past president, stresses that “the [underlying] theory does not require learning a large number of definitions and theorems, but does require a sharp and inquisitive mind.’'
Discrete math, he continues, is an excellent tool for teaching students the value of mathematical modelling to solve real-world problems.
As an adjunct to the traditional curriculum, supporters say, discrete math fits perfectly with the standards’ call for exposing students to real-world applications of math.
“We’re not focusing just on computation any more,’' said Beverly J. Ferrucci, a professor of math education at Keene State College in New Hampshire. “We want students to look at the beauty of mathematics and how it relates to their world.’'
Mr. Jennings, who demonstrated discrete math with his candy problem, added that this new emphasis, and its implications for teachers, “kept me in teaching when I was thinking of getting out.’'
Even so, Ms. Blackwood noted, she previously taught discrete-math concepts “under a different name’’ because administrators were resistant to change.
Some math educators, however, sound a note of caution on the use of discrete math in classrooms.
Writing in the N.C.T.M. yearbook, Anthony D. Gardiner, a researcher at the University of Birmingham in Britain, observes that teachers should not be so lulled by the attractiveness of discrete-math problems that emphasize thinking over computation that they attempt very complex problems. Such problems, he argues, may have to be broken down into repetitive and predictable routines that could bore students.
“School mathematics all too easily degenerates into a succession of meaningless routines,’' he writes. “If we wish to exploit the educational advantages of easily understood problems that force students to think, then we must produce a revised curriculum that does not try to do too much.”
Calculator Use Important
One classic problem that is often used by teachers of discrete math, and one that Mr. Jennings allowed math teachers here to try, presents students with the names of three heirs to a sizable estate and the list of bids that each has made on three pieces of property contained in the estate.
Students then must devise a means of equitably dividing the property, which leads to discussions of the meaning of “fairness’’ and “equity,’' Mr. Jennings pointed out.
Another popular discrete-math problem, said Mr. Millard, the Kansas high school teacher, involves “recursion,’' a concept that he illustrates for students by discussing a hypothetical automobile-loan payment.
“If you tell yourself that your car payment is $120 a month, then ask how much is left on the balance after you’ve made the payment, you’re really doing a recursion process,’' he said.
Similarly, Ms. Blackwood noted during her presentation here that various electoral systems are mathematically based.
“You’re bringing mathematics into situations that most don’t think of mathematically,’' Mr. Millard said.
Also, the N.C.T.M. standards point out, the mathematical search for patterns can often lead from math into different, apparently unrelated fields of learning.
For example, an illustration of a recurring number sequence called the Fibonacci sequence can be found as close as the nearest window, where the arrangement of leaves on certain trees and the scales on pine cones conform to the pattern.
Another strength of the discrete-math topics is that they lend themselves to the use of electronic calculators, particularly graphing calculators--handheld devices with the computing power of early microcomputers. (See Education Week, April 10, 1991.)
A Markoff chain, for example, allows students to study the probability of a certain event through describing it either as the branch of a probability tree or as a portion of a matrix.
“But I couldn’t do the number-crunching of a Markoff chain if I didn’t have a [graphing] calculator,’' Mr. Millard said.
Math for Everyone
Some disagreement exists over whether discrete math should be taught as a separate subject or incorporated into existing math courses.
Mr. Millard has chosen a compromise position: He teaches one version of discrete math as a “problem-solving course’’ for advanced-math students, but he also incorporates discrete-math lessons into algebra and other courses.
“Even as a full-semester course, you don’t have a problem with coming up with the materials,’' he said. “I’ve taught it for three years and I’ve never taught it the same way twice.’'
However, Ms. Ferrucci of Keene State College said that some discrete-math topics, such as systems of counting, are applicable as early as kindergarten.
For example, she gave a kindergarten class pictures of pizzas and then asked the students to figure out how many ways different toppings could be arranged.
“They exhausted all of the possibilities,” she recalled with a laugh.
Seeing their students ably tackling a rather sophisticated concept often gives elementary teachers more confidence in their own ability to learn math beyond arithmetic, she added.
“Many of these people were never successful in their math,” Ms. Ferrucci said. “But anyone can learn discrete math.”
