New Angle on Algebra

By Richard Stengel

One teacher takes on the math Establishment

His features are as austere and regular as a mathematical formula, his carriage as straight as the vertical axis in a Cartesian graph. Teacher John Saxon's eyes are the one variable in the equation: they burn with a visionary gleam. His vision is simple: a future in which the basics of algebra, the building blocks of all higher mathematics, become understandable to all American students.

Saxon, 58, is an unlikely mathematical messiah. He still resembles what he once was: a professional soldier. A graduate of West Point, he is a World War II veteran, a decorated Korean War combat pilot and a former engineering instructor at the U.S. Air Force Academy. After retiring in 1970, Saxon began teaching math and algebra at Oscar Rose Junior College in Midwest City, a suburb of Oklahoma City.

He soon discovered that his students could not remember what he taught them from one week to the next, and as the year progressed their accumulated comprehension of algebra did not. Unlike many teachers, he did not blame the students or himself. The culprit, he finally decided, was the textbook.

Saxon became convinced that most introductory algebra texts used in high schools are unclear and confusing, often hindering students from learning the rudiments. He attributed the frightening decline in mathematics test scores across the country to abstruse textbooks written in the name of the New Math by "arrogantly inept" mathematicians who do not teach beginners. Like many other classroom algebra teachers, he found that such textbooks emphasize mathematical theory at the expense of practice and are usually written in baffling jargon. Emphasis is placed on rapid exposure to many "topics," or procedures. Before students can master one topic, explains Saxon, they must move on to a new one. The great sin, he insists, is that the books teach abstract theory first and skills second--the reverse of the order in which children normally learn.

Saxon decided to do something about the teaching of algebra. He decided, in fact, to write his own textbook. His notion was simple enough, a readable text requiring the students to do continuous review. Algebra, he points out, is a skill, and like any other, it must be learned thoroughly. In 1979 Saxon came to New York City and tried to sell his idea for an Algebra I text. The conservative world of high school textbook publishers, where new books often tend to be virtual clones of the most successful standard text, abruptly turned him away.

Undeterred, he returned to Oklahoma to work on his book and began knocking on the doors of school principals. He managed to convince 20 of them to participate in an experiment that would test his method of teaching algebra against the standard texts. He also persuaded the Oklahoma Federation of Teachers to oversee the experiment and certify the results. In all, 1,360 students participated The control group consisted of 841 who used the regular textbooks, while an experimental group of 519 used Saxon's text--Algebra I, an Incremental Development. In each school the same teacher taught one section using Saxon and one or more classes using the standard text. The 16 short tests taken during 1980-81 were drawn from questions submitted by the teachers themselves.

The results were astonishing. Students using Saxon's book showed an overall gain of 159% as compared with the control group. The tests also revealed that Saxon students in the lowest-ability group (classified on the basis of their scores on the California Achievement Tests, given in August 1980) outscored their control counterparts by a staggering 246%. Perhaps even more surprising is the fact that the "low-ability" Saxons outscored the "high-medium" controls on every test.

Saxon soon enlisted others in his crusade. New York Publisher Bob Worth provided guidance and advice in bringing out the book under the name Grassdale Publishers. Saxon himself paid the $60,000 cost of publishing 10,000 volumes.

He also sent off a barrage of letters to the media. William Buckley, editor of the conservative National Review, responded and became a believer. With Buckley's help, Saxon got a small foundation grant and wrote two pugnacious and polemical articles in the magazine about textbooks and the teaching of algebra. They stirred considerable controversy, bringing Saxon a national audience and, eventually, hundreds of letters from teachers and parents curious about the book. Because of the articles and the Oklahoma test results, schools all over the country are now requesting Saxon's book. Fifty schools in Oklahoma have adopted it, plus scattered schools in California, New Jersey, New York and Maryland.

Just what is it about Saxon's book that is different? The initial contrast is that the book is written in simple, straightforward, clear prose. In many standard algebra texts, students are immediately confronted by mystifying postulates, like "The reflexive property of equality states that any number is equal to itself."

They are often expected to memorize a good deal of difficult abstract theory about the basic properties of real numbers such as "associative," "distributive" and "commutative." The last term means simply that three and four always make seven in whatever order they happen to be added. Saxon, by contrast, uses diagrams and examples to make the same point clear and only afterward says:

"We may exchange the order of the numbers in an addition problem without changing the answer to the problem." According to Gerry Murphy, head of the math department at Hackley School in Tarrytown, N.Y., Saxon also tries to confront two fundamental weaknesses that afflict most Algebra I texts: the lack of a sense of continuity and connection among topics, and student failure to remember the material already covered. Saxon presents the material in small linked units, without the traditional division into chapters. Saxon treats the problem of retention by the obvious and old-fashioned device of a large number of daily cumulative review questions, examples and problems. One result, says Lionel Garrison, head of the math department at Dwight Englewood School in New Jersey, is that the book reinforces the "point that math is a reasonable approach to reasonable problems."

The report card on Saxon includes some low grades. Some teachers who have examined the book are put off by the absence of traditional chapters. Others find the book "mechanistic" and too repetitive, and think it might be boring to use in class. Bruce Vogeli, professor of mathematics at Columbia University's Teachers College, sees Saxon's innovations as insignificant and ineffectual: "One can't teach algebra only as a skill. Drill and practice are only part of the problem." He likens drilling to the lowest common denominator of algebra. Mathematical literacy, assert Saxon's critics, is not simply the ability to calculate, but the ability to reason quantitatively.

Criticism of Saxon tends to divide along the same lines as the debate about the future of American mathematics teaching. There are those who advocate a return to basics through practice and drill, and those who insist that practice without abstract theory is ultimately limiting. Both sides are in a sense right. Yet Saxon's main point contradicts neither. He simply affirms that Algebra I is not the place for obscure theory, which can be introduced later, when students know how to use algebra well enough to profit from it. "Algebra is the basic language of all mathematics beyond arithmetic," he says. In his view, today's puzzled students do not really master the introductory grammar. They must first become fluent with fundamentals.

The root of the word algebra is the Arabic al-jabr, which means "bringing together." Saxon's synthesis of traditional practice and drill with the fundamentals of modern algebraic theory taught clearly may provide an alternative to the present dismal state of mathematics teaching. Alfred North Whitehead, the English mathematician and philosopher, once noted that "the study of mathematics is apt to commence in disappointment." If John Saxon is right, the study of algebra may not end so. --By Richard Stengel. Reported by Jeanne-Marie North/New York

With reporting by Jeanne-Marie North/New York

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