Math & Ticktacktoe

The more he observed the goings-on in the Illinois high-school math class, the more uncomfortable the young mathematician became. At one point the teacher took up a "story" problem which involved two equations with two unknowns (10x+50y =320 and x+2=y). One boy ingeniously found a way to solve the problem with only one unknown, x+5 (x+2) = 32. But the teacher did not congratulate him: she told him he was "wrong" and sent him back to his seat. To Mathematician David Page, 31, of the University of Illinois' College of Education, this was just one more example of why high-school math is in the plight it is. "Mathematics," says he, "is normally regarded by teachers as a subject with cut and dried rules of procedure. The theory is that the teacher simply passes on the rules, and the kids absorb them without question." The result: math has become the subject most likely to be shunned by today's high-school student (TIME. June 18).

Since 1951 the University of Illinois has had a team working on a way to change all that. The project was launched when university authorities found engineering applicants were consistently ill-equipped mathematically, wasted months in learning basic math skills they should have learned in high school. It assigned young (then 26) Mathematician Max Beberman of the University's College of Education to show the high schools how to step up their math instruction. Beberman, later joined by Page, decided the trouble lay in the whole approach to math teaching. He junked old methods, drew up a new curriculum, now has five Midwest high schools trying his theories on college-bound students. Many mathematicians regard Beberman's new method as the most important reform in nearly a century, and the Carnegie Corporation has voted a $277,000 grant to expand and test it.

How Old Is John? As Beberman and Page see it, high-school math has sunk to its present state because students learn their theorems and formulas for an array of algebra problems ("If John is twice as old as Jane was four years ago . . ."), but never find out what makes the mathematician's brain work. In the hope of making arithmetic lively, some teachers insist on making each problem functional, as if there were nothing more to the subject than how to add up a grocery bill or compute compound interest. Such teachers completely misunderstand the adolescent, says Beberman. "The adolescent is the purest intellectual in our society. He doesn't have to be concerned with practical problems." The question that concerns him most is "Why?"

To a large extent, Beberman and Page have cast aside the traditional tags (algebra, geometry, trigonometry, etc.) that tend to make math seem a series of separate and unattached compartments. "Frequently," says Beberman, "our students do not know whether they are doing geometry or algebra at any given point.'' But the basic intent is to reveal math as a "creative process in which we want our students to participate." Instead of telling students how to solve equations, "we just explain to them what the root of an equation is and then give them 30 pages of problems and tell them to go ahead and solve them any way they can--until they get in too far over their heads."

"For Every y and x . . ," In freshman year high-school students are exposed to the philosophy behind variables, answer such questions as: For every y and x, if the cost of a book is 2y -3 dollars, then the total cost of 7x such books is --dollars. The next year students find themselves confronted with:

"If two sets are disjoint, and one set has three elements and another set has five elements, then how many elements does the intersection of the two sets have? A) 0, B) 2, C) 3, D) 5, E) 15."

(Answer: o.)

In the third year students take up complex numbers, polynomial calculus and more analytic geometry. Teachers are expected to lead their classes as far as they will explore. "If you dare," says a first-year teachers' manual at one point, "you can ask for a formula for finding the area of a triangle none of whose sides is necessarily parallel to either coordinate axis, that is, for every a, b, c, d, e, and f, the area of the triangle whose vertices are the graphs of (a.b), (c.d), and (e.f)."

What Is Zero? To emphasize the creative nature of math, students are encouraged to make up laws, theorems and symbols of their own. Thus a class might find itself with a Johnny Jones Law or using the symbol a, which one student invented to mean "approximately equal to." Students argue over abstractions ("talk about zero--is it a number or isn't it?"), and one class was even asked to write a mathematical description of ticktacktoe.*

The real obstacle to progress, admit Beberman and Page, is not so much the nation's students as the shortage of teachers who know enough to answer the question "Why?" Nevertheless, says Beberman optimistically, "if we can get even half the schools inthe nation into the program, that'll be a pretty valuable life's work."

* The most impressive description: "The game is won by getting your symbols in one of eight groups of three. These must be chosen so that one space is part of four such groups, four spaces are part of three such groups, and each of four belongs to two such groups. I will call the first one 4, the next set 3, and the next set 2. The eight winning combinations must include four 3-2-3s, two 3-4-3s and two 2-4-2s. In the 3-2-3s no 2 occurs twice and each 3 occurs twice and only twice. In the 2-4-2s no 2 may occur twice, and in the 3-2-3s with the 2s on the 2-4-2 in it no 3 may occur twice. And in the 3-4-3s no 3 may occur twice, and in the 3-2-3s with 3s in the 3-4-3 no one of these can contain both 3s."

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