# Scholar Suggests 'Real-World' Approach to Mathematics Teaching

By Maureen Fleming Special to Education Week

Martha Matzke and Susan Walton contributed to this report.

Detroit--Building on the conclusions of nine recent national reports, a University of Chicago professor has proposed basic changes in the content and structure of junior- and senior-high-schoolmathematics curricula that would emphasize the use of computers and calculators, provide more work with applications of abstract principles, and introduce algebra and geometry earlier.

The revision--the work of Zalman Usiskin, a nationally known researcher and teacher educator--would also involve a shift away from the traditional division of students into "college bound" and "noncollege bound." Instead, according to Mr. Usiskin, the curriculum would be organized around the future needs of students, offering them new types of courses more closely tailored to their academic and career plans, while allowing them to keep their options for further study open.

The curriculum, Mr. Usiskin told mathematics educators at a recent meeting, is "in accord" with the findings of major mathematics-education studies conducted since 1975--including the National Council of Teachers of Mathematics' (nctm) 1980 Agenda for Action. His proposal, he said, is directly based on their consensus on the need for increased requirements; general use of calculators and computers; and greater attention to statistics, applications, and problem-solving in mathematics curricula.

Mr. Usiskin, who is the author of Applications in Algebra, is considered a leading proponent of using more varied "real-world" applications in the teaching of mathematics. His work has also included a study, issued last year, indicating that girls are as adept as boys at mastering complex abstract concepts. His presentation drew some 1,000 listeners at the 61st annual meeting of the nctm, held in Detroit last month.

Important First Step

The proposal is "a very important first step" in the process of deciding how to implement the recommendations of the various national groups that have studied mathematics education, said Mark A. Spikell, a professor of education at George Mason University. "What he's been able to do is to cull out the points of consensus of the very reputable commissions and come forth with the possible implementation of those suggestions to get to those goals," said Mr. Spikell, whose research has focused on the innovative implementation of school curricula.

The University of Chicago scholar's proposal coincided with the release of national reports--including that of the National Commission on Excellence in Education--whose conclusions suggest a need for modifications in the mathematics curricula, including a greater emphasis on applications and "higher-order" skills. (See Education Week, April 21 and April 27, 1983.)

The proposal is shaped to retain the successful parts of the current curriculum and to eliminate others that have become obsolete, Mr. Usiskin said. He estimated that the new structure could be put into effect by September of 1984.

Simply requiring students to take more mathematics, as states and school districts have begun to do, will not suffice, Mr. Usiskin warned, because both the substance and the arrangement of mathematics courses need refashioning.

The proposal would replace the current distinction between college-bound and noncollege-bound stu-dents with a three-track curriculum for: college-bound students with mathematical or scientific interests (Population I); college-bound students with nonscientific interests (Population II); and noncollege-bound students (Population III).

Computers and Statistics

All three groups would work more extensively with calculators and would be required to take a one-year course on computers and statistics, according to Mr. Usiskin.

Both college-bound groups would be introduced to "pre-algebra" and "pre-geometry" in the 7th grade, and would move on to Algebra I in the 8th grade. Reflecting his view that nonscience-oriented students require a somewhat different program, Mr. Usiskin called his 8th-grade course for Population II "World Algebra." It replaces word problems with "real-world" applications and eliminates some of the theory to which Population I would be exposed.

Populations I and II would go on to geometry in the 10th grade, with Population II taking the less theoretical "World Geometry." In the 11th grade, each would take the second year of algebra, with the scientific group receiving some trigonometry and the nonscientific group focusing again on applications, perhaps using computer programs to analyze advanced calculations.

In the senior year, Population I would take precalculus, while Population II would take a four-part course that includes trigonometry, elementary functions, discrete mathematics, and analytic geometry.

Population III, the noncollege-bound group, would start in the 7th and 8th grades with arithmetic courses utilizing calculators for complicated functions. This arrangement, Mr. Usiskin said, would give those students "the time and resources to cover ratio, percent, powers, and many of the other arithmetical ideas which they now often do not reach because they are immersed in paper-and-pencil work."

Grade 9 for these students would cover consumer mathematics, "a course so successful that many people think that college-bound students should take it," Mr. Usiskin said. In 10th grade, Population III would take a course covering "formulas, graphing, proportions, pattern description," and other elements of applied algebra and mathematics.

In grade 11, they would take a course in computers and statistics. Although some have argued that a three-year mathematics requirement is too stringent for noncollege-bound students, the Population III proposal has "obvious justification for future consumers and those who may attend vocational or technical schools," Mr. Usiskin said.

Major Weaknesses

The new curriculum is designed to remedy what Mr. Usiskin regards as major weaknesses in the current structure of the curriculum.

Under the present system, he said, all college-bound students are forced to take calculus-oriented courses. But these courses are later used by only 30 percent of college students--those who major in the physical and biological sciences and engineering.

A much larger percentage of college-bound students, however, subsequently major in fields--business and the social sciences, for example--that require knowledge of advanced algebra and statistics, but not of calculus or trigonometry. Ten percent of college students major in fields that require no mathematics at all.

"These students are better served by a broad mathematical background," Mr. Usiskin said.

In addition, he pointed out, raising requirements without making structural changes will not resolve the problems that students encounter in algebra, which he said has "the highest failure rate of any high-school course." And one consequence of that, he added, is that many students are frightened away from ge-ometry and further algebra.

Moreover, for the noncollege-bound students who take the current general mathematics course, "the material is almost always a review of the same material the students were unable to learn in previous years and ... though students do learn some arithmetic in this course, they do not become competent users of arithmetic, and they do not have facility with skills."

'A Transition Time'

In outlining his proposed changes, the mathematics educator acknowledged the view of some in the mathematics field that computers and calculators have already rendered obsolete much of current high-school mathematics. "Ultimately," he said, "no students will have to be proficient at paper and pencil arithmetic or algebra." But he added that "we are in a transition time."

"I don't know how long it will last," he said, "but for the next year at least, it seems that there are practical reasons--college-entrance tests, for example--for insisting on competence either way from our best students."

As for those who say that students' difficulties with algebra suggest that it is already introduced too early, Mr. Usiskin pointed to Japan and the Soviet Union, where algebraic concepts are introduced several years earlier. American teachers tend to greatly underestimate the potential of their students to grasp mathematical concepts, he said.

Mr. Usiskin told his colleagues, in effect, that the elementary curriculum raises another set of important and unresolved issues. "I consider that my proposal reflects about as far as we can go without making major changes in the elementary-school mathematics curriculum. The problems in elementary schools are quite a bit more serious than they are in high schools and seem more difficult to remove."

" ... Most of the time is spent teaching children robot skills, with the teaching done by individuals most of whom do not know enough to change what they are doing even if they wanted to."

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