# Add Understanding, Subtract Drills

Thirty years ago, a movement that became the "new math" led to major revisions in the mathematics curriculum. Ten years ago, the back-to-basics movement brought a renewed emphasis on "basic skills."

Today, fueled by evidence of declining student performance and growing remediation efforts at all levels, the wheels of reform are turning again. Learning from the mistakes of the earlier reforms will be critically important, math educators suggest, if the process of reform is to succeed.

Working in a climate favorable to change, math specialists are trying to steer a course between the extreme of the new math, which focused on abstract, formal ideas, and that of the back-to-basics movement, which many see as erring in the opposite direction by overemphasizing low-level computation skills.

The advent of new technologies and the current concern for educational improvement, coupled with reports in recent years from various professional associations, have laid the groundwork for reform. One key document was the National Council of

Teachers of Mathematics' "Agenda for Action," issued in 1980.

MORE AND DIFFERENT

The consensus that is beginning to emerge suggests that the next stage will involve both more and different mathematics. But the proponents of change, still mindful of the sad fate of the new math, are equally vehement in saying that any "reform" must be considered with great care.

The result this time is likely to be a shift toward a mathematics curriculum that blends the most useful ideas from both earlier revisions, jettisons some obsolete material, and incorporates the use of computers and calculators.

Among the major points of agreement among math educators:

Students at all levels should receive more instruction in mathematics;

Emphasis on "problem solving" should increase, but there should be a parallel shift away from excessive drill and practice. That modification would most affect the elementary-school curriculum, but it would filter through to secondary-school math;3Schools should make more--and more effective--use of computers and calculators in the math curriculum at all grade levels;

Greater emphasis should be placed on teaching students how to estimate; and

The standard secondary-school mathematics sequence should give way to a greater diversity of courses that better meet the needs of all students, not only those who plan to attend college.

RAISING REQUIREMENTS

Math educators are generally in sympathy with moves by state and local school officials to raise graduation requirements as a signal to the public that they are concerned about improving the performance of students. But most argue that such actions are only one component of needed reform.

"There are two schools of thought," says Zalman Usiskin, professor of mathematics education at the University of Chicago. "One involves increasing requirements. It's very easy to implement those kinds of changes because they can do it by legal means. The second--changing the curriculum--is a lot harder to implement because it requires new materials. My hope is that we get the best of both."

The growing pressure to revise the mathematics curriculum arises mainly from widely publicized declines in student achievement on standardized tests. Scores for most students on the National Assessment of Educational Progress had declined until last year, when students improved in their ability to perform basic calculations.

But mathematics educators who analyzed the assessment scores were quick to point out that although students had improved their ability to perform relatively routine computations, they still were no better at solving more complex problems or at applying low-level skills to problems.

Observers conclude from these phenomena that the back-to-basics movement produced students who can perform routine calculations but who are considerably less skilled at using mathematical principles to solve more complex problems.

Scores on the Scholastic Aptitude Test in mathematics have shown similar declines for many students. Among students who plan to major in science or mathematics in college, however, the scores have held steady, according to an analysis by Lyle V. Jones of the University of North Carolina.

Those trends suggest, in part, how the curriculum might be modified to produce a

better balance in achievement. Students who do poorly on the math sat would probably benefit from more math courses. Students who know their multiplication tables cold, but are uncertain just what those tables mean, could benefit from more instruction in applications.

CURRICULAR TRADEOFFS

But adding more of one thing probably means dropping something else, given the limitations of the school day and the school year. To obtain Mr. Usiskin's "best of both," educators will have to decide what aspects of the curriculum to keep.

One report, prepared for the National Science Board's Commission on Precollege Education in Mathematics, Science and Technology by the Conference Board of the Mathematical Sciences, summarizes much of the current thinking on what is "fundamental" in the math curriculum and should remain so.

For the elementary grades, the report suggests reversing the trends of the 1970's and focusing more on higher-level skills.

Basic knowledge of one-digit number facts, place value, decimals, and fractions will all remain extremely important, but teachers should devote less time to these skills once pupils have mastered the basics, according to the report.

Teachers should spend more time teaching students how to use the basic skills as means to understanding real-life situations, rather than as ends in themselves, the report suggests. In that context, estimation, statistics, probability, and other more sophisticated topics would gain greater importance.

Central to teaching these topics, say mathematics educators, is a renewed emphasis on problem-solving--teaching students how to apply, and to enjoy applying, math skills to real-world problems that involve numbers.

"To me, the thing that we ought to be pushing very hard is to make sure the children understand the mathematics and enjoy it," says Stephen S. Willoughby, director of mathematics education at New York University and president of the National Council of Teachers of Mathematics.

"That's really at the crux of the matter," agrees Robert B. Davis, associate director of the curriculum laboratory and of the computer-based education laboratory at the University of Illinois. "If you look at school mathematics, it is heavily based on telling people explicitly what to do. If math is simply an exercise in following instructions, it isn't much of a winner. It's not a normal thing for human beings to do, and it's not the way mathematicians and engineers think about it."

COMPUTERS AND CALCULATORS

Computers and calculators, too, should be used to enhance students' abilities to work with and understand numbers and as useful tools in problem solving, mathematics educators agree. The conference-board report, drawing on other mathematics-curriculum reports, urges a "heavy and continuing emphasis on problem solving, including the use of calculators or a computer."

"Every child should learn to use a calculator effectively and accurately," the board recommends. "Calculators and computers should be introduced into the classroom at the earliest grade practicable."

"It's very clearly the time to begin a careful transition from the concentration on the traditional computation to a curriculum which takes advantage of the calculating tools," says Shirley Hill, professor of mathematics and education at the University of Missouri-Kansas City and a former nctm president.

Mathematics educators emphasize that a child who can use a calculator will still have to know the fundamental principles of the discipline. But the capabilities of calculators and computers, according to Ms. Hill, will "downplay the two major goals of elementary mathematics: to learn computational skills and to store important factual information in the brain."

"That isn't to say [computing skills] are no longer important," she continued. "But I don't think we need to spend one or two years of the curriculum on multi-digit long division. The better part of grades 5 or 6 is spent on that. That precious instructional time ought to be spent on how to apply the skill. Learning some division is necessary--one-digit divisors, but that's it."

But the key question--when and how much should children replace their pencils with calculators?--remains to be answered.

Currently, calculators are not widely used in the schools, according to Mr. Crosswhite, who is writing the U.S. section of a study of math education in 19 countries. Teachers' apparent reluctance to use calculators, he and others suggest, may be due partly to their uncertainty about how they fit in with the standard sequence.

"That's certainly an issue," says Usiskin. "When you juxtapose it with people saying kids are not skillful enough, you see the dilemma. It isn't clear that they need the skills with paper and pencil, even though they need the skills in some way."

Now, the elementary math curriculum remains "driven by paper and pencil," Mr. Usiskin points out. It is also "very carefully sequenced"; stus learn to multiply two-digit numbers one year, decimals the next, and so on.

"The calculator plays havoc with that, because it's just as easy to multiply by a three-digit decimal as by a two-digit number," Mr. Usiskin says. "It's just as easy to multiply by negative numbers," a concept generally taught five years after studentsarn two-digit multiplication.

"So if one puts calculators in the classroom," Mr. Usiskin says, "one must realize these difficulties with standard sequence. Once you bring the calculator in, no kid will use it on only two-digit numbers.'' No one, he notes, has yet devised a remodeled elementary-level curriculum adapted to the use of the calculator.

Educators already agree that as calculators come into wider use in the schools, it will become more important for children to learn to estimate answers mentally. "Informal mental arithmetic should be emphasized at all levels, first aimed at exact answers and later at approximate ones," the conference-board report says.

"Such activity is necessary if students are to be able to decide whether computer or calculator printouts or displays are reasonable and/or make sense," the board notes. Estimation also helps children to establish the "sense of numbers" that mathematics educators view as critical to gaining both understanding and enjoyment of math.

"It really has not been systematically done, and there is more and more work showing how it can be," says Ms. Hill, referring to the systematic teaching of estimation. "The trick is to teach people to think of a reasonable answer without knowing the precise answer in advance. That becomes a very crucial skill, and involves the analysis of when, where, what it means. You still have to understand what division is and where it applies."

TOO MUCH DRILL?

The amount of time spent on re is also an issue in elementary math. Although some review is clearly necessary, many math educators agree that the current practice--involving about one-third of the total time spent on arithmetic--is excessive.

"A significant amount of time is wasted in elementary school for a student who learned what he was supposed to learn," says Mr. Usiskin. "They're forced in 5th grade to sit through half a year, or at least a third, of review. That happens in grades 4, 5, 6, and a little in grade 7. If you're not ability-grouped, it happens in 8th grade as well."

Review, drill, and other much-used practices, however, have one thing in common--they produce students who tend to do well on standardized tests of "minimum competency." The success of any revisions in current programs, educators point out, will hinge to a degree on whether they produce children who achieve high test scores.

"The resistance will occur only if we continue to have as our criterion for success a high score on tests that continue to employ the mechanistic skills," Ms. Hill says. "We really have a public that believes that success on those tests is synonymous with achievement. And it is, if the tests reflect what the curriculum should be. But you can have a totally anachronistic curriculum."

Many in the mathematics-education community already regard testing as an inadequate measure of the success of a curriculum and of students' achievement. In its 1980 "Agenda for Action," the nctm recommended that both programs and students be evaluated by other means. "The evaluation of probleming performance will demand new approaches to measuring," the agenda states. "Certainly present tests are not adequate.''

"Where minimal-competency tests are mandated," the council states in its recommendations, "they should be implemented with extreme caution to assure that adverse effects on the program do not result."

THE 'WASTELAND' AND BEYOND

At the secondary level, the curriculum issues become more complex, math educators say, primarily because questions of content must be considered in relation to diverse student populations.

Some of the key issues--how to use computers and calculators, for instance--are closely linked to what occurs at the elementary level and will require reformers to focus on the "articulation" between the various curriculum levels. Another issue is whether junior-high math, which now consists mostly of review, should cover more algebra, or other topics.

The mathematics conference board report offers some recommendations on what mathematics students planning to go to college should learn. The "basic thrust" of algebra, in both the first- and second-year courses, is still worthwhile. Even if technology produces calculators that can do algebra much more quickly and efficiently, students will still need a good understanding of the principles.

However, the board notes, emphasizing the field axioms of algebra and "rigorously proving obvious facts from them" is both "uninteresting and unproductive for most stus." Similarly, in geometry, the board advises that "not everything has to be proved rigorously, certainly not obvious things, and certainly not in column proofs, which mathematicians do not use. It suffices to work through short sequences of rigorously developed material, say a unit on angles inside circles."

The power of computers and calculators to change the focus of the secondary math curriculum is as strong as it is for elementary courses, but in traditional high-school courses, it is not yet clear how the technology can best be used, according to the conference board's evaluation of traditional secondary-school mathematics. "Powerful, yet low-cost, machines now perform many of the arithmetic, algebraic, and graphic operations that have been taught to generations of students," the report states. "With those operations now mechanized, it is urgent to re-examine the content, emphases, and approaches in traditional courses."

However, the report notes, "Since the significant breakthroughs in technology and software are so recent and conceivable implications so revolutionary, it is not yet clear what specific changes in current programs are appropriate."

As is true for elementary math, the report notes, technology offers an opportunity to give less time to traditional skills and more to complex problem solving. To date, however, there is little research on how feasible this change would be, or how best to go about doing it.

One approach would be to infuse the technology into the curriculum as it stands; another would be to reorganize--radically--the sequencecontent of the courses.

The traditional sequence of mathematics in high school, math educators suggest, has been best suited for those students who plan to attend college. In most schools, the sequence begins in 9th grade with algebra and proceeds through geometry, a second year of algebra, and, for serious students of math or science, precalculus. The new math, with its emphasis on rigorous, formal problems, was especially tailored to the best students. When other students did not succeed in mastering that material, the most common response on the part of curriculum planners was to revert more to a pre-1950's curriculum, heavily influenced by the focus on "skills."

"These materials were designed for the best students, and the poorest students don't know what's going on," Mr. Usiskin says of the new-math curriclum. "Having gotten a curriculum designed for college-bound students and having tried to use it for everybody, they then got a curriculum for the noncollege-bound and tried to use that for almost everybody. Now we see the reaction to that setting in. This is the wave of education."

A more diverse curriculum would retain the classical mathematics courses needed by students who plan to major in math or science. But it would also offer courses in consumer math, statistics, and the like, for those who plan nonnce careers or who do not expect to continue their education after high school.

"I think that consumer math or statistics might be particularly suitable for those who do not aim to go on to college, or do not aim to go into social and mathematical science," notes Henry Alder, professor of mathematics at the University of California at Davis and a member of the California State Board of Education.

"There is a strong push to give a little more emphasis to those abilities that will finally result in learning about statistical information--handling data and information intelligently, how we use it, read it, draw inferences from it," says Ms. Hill.

"By the time you get to secondary school, that strand will be called statistics," she continues. "But its foundations have to be developed in the elementary school, with finding and using information. That, in a way, is a content change. The current texts have a few pages devoted to it, but it's very much a lip-service approach."

Both consumer math and statistics, as well as computer science, are among the choices offered in California's new model curriculum. Under that model, students must take one year each of algebra and geometry. Then, if they opt for a third year of math, they may branch out.

SEQUENCE CHANGE

From an analysis of nine reports on high-school math curricula, Mr. Usiskin developed a model that would significantly change the substance and sequence of secondaryinued on next page

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mathematics.

One of the major revisions that he proposes would be a break with the traditional division of students into college-bound and noncollege-bound populations. Instead, Mr. Usiskin suggests, students would fall into three categories: college-bound with a special interest in math and science (population I); college-bound with nonscientific interests (population II); and noncollege-bound (popion III).

The curriculum would be structured in a fashion that allowed students to cross from one group to another if their interests changed. All three groups would work extensively with calculators and would be required to take a one-year course in computers and statistics.

Both college-bound groups would begin their study with "pre-geometry" and "pre-algebra" in the 7th grade, followed by algebra I in the 8th grade. Those students whose insts lay outside science, however, would take "World Algebra" that year, a course that would include heavy emphasis on practical problems and less stress on the theory that their scientifically inclined peers would learn. In the 9th grade, both groups would take a course in computers and statistics.

The same general structure would continue throughout high school. Science-oriented students would take more theoretical geometry in 10th grade; those without a special interest in science would take "World Geometry." In the 11th grade, both groups would take a second year of algebra, with the population II students spending more time on applications and population I learning some trigonometry. In the 12th grade, population I would take precalculus, while population II would take a four-part course that included trigonometry, elementary functions, discrete mathematics, and analytic geometry.

The curriculum would be somewhat different for students who did not plan to attend college. That group, population III, would begin with arithmetic courses that used calculators for complicated problems in the 7th and 8th grades. In grade 9, these students would proceed to consumer math, and in grade 10 they would take a course that covered formulas, graphing, proportions, pattern description, and other elements of applied algebra and mathematics. In the 11th grade, the noncollege-bound students would study computers and statistics.

Not everyone agrees that students who do not plan to go to college need three years of math. Mr. Usiskin, however, argues that the curriculum he proposes has "obvious justification for future consumers and those who may attend vocational or technical schools."

The response to the proposal has been positive, according to Mr. Usiskin. "It's clear that there are a lot of people thinking in the same direction," he says. "It's natural that there should be that kind of reaction because I was following all those reports. I wasn't trying to think of a totally new picture of mathematics education."

"The thing that is questioned than anything else is whether we can teach algebra as an 8th-grade course," he notes. "It has not been questioned that 8th grade is a wasteland, and we could teach some algebra. The question is whether we can do a full course."

Ms. Hill offers another proposal for that age group. "I would not suggest that curriculum be accelerated. I think it's a mistake to take content and simply keep pushing it down." Middle school, however, would be "a pretty good place to make a push for computer literacy."

Her view is widely shared. In a statement issued in May, the nctm voiced its opposition to "vertical acceleration"--offering more advanced material in earlier grades--as a general practice, and its support for "horizontal acceleration," or enrichment. "In almost all cases, a student would benefit more from a program of [enrichment] than from one of vertical acceleration," the statement said.

Mathematics educators are hopeful that the growing visibility of problems in the mathematical training of American students will catalyze action on the various reform proposals. But their past experiences with the forces of change temper the optimism.

Five years ago, an nctm group asked to react to the National Science Foundation's studies on the status of math and science education wrote: "The most discouraging feature of the studies is the consistent pattern of great differences between the apparent reality of mathematics education in most schools and the recommendations or practices of many prominent teachers, supervisors, and professional organizations."

"As an institution, I think education is extraordinarily resistant to change," notes Ms. Hill. "But society changes without it."

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