# The New "New Math"?

Kalispell, Mont.

Like most tourist towns in the off-season, this gateway to Glacier National Park is largely populated by the locals on a chilly April evening.

So, if you've taught the children of a town of 20,000 for as long as Larry Kabor has, the chances of running into someone you know on the main street are fairly high.

It's no surprise, then, that Kabor, who heads the mathematics department at Flathead High School, encounters Tom Johnson, the father of one of his students while walking to dinner.

Moving within an arm's length of one another, the men nod noncommittally. But it isn't difficult to sense the underlying tension as they pass.

It's not a political or neighborhood or family dispute that has left the two men uncomfortable with one another. It's mathematics.

Kabor, who has taught at Flathead High for 34 years, supports the Systemic Initiative for Montana Mathematics and Science, or SIMMS project, an effort that aims to make math more meaningful to the vast majority of students who otherwise would drop the subject at the first opportunity.

Teachers and curriculum developers across Montana are working together to create and field-test a cross-disciplinary curriculum based on standards for exemplary math teaching developed by the National Council of Teachers of Mathematics.

Many of the precepts of the standards, particularly a greater reliance on calculators and other technologies to help transform instruction, are undergoing a trial by fire in Kabor's classroom and in others at Flathead High, one of Montana's largest.

Johnson, on the other hand, is what Kabor and his colleagues scornfully call a traditionalist.

Johnson and his wife, Anita, whose children both attend Kalispell schools, have met with both Kabor and Superintendent Bill Cooper to express their concerns that elements of the SIMMS approach, particularly the use of calculators, are inappropriately making their way into the elementary schools. As a result, the Johnsons contend, children aren't developing the mental discipline that would come from, say, rote memorization of the multiplication tables.

Tom Johnson is no stranger to math. He owns a local business that assembles sophisticated computers, bringing an international clientele to this town roughly 80 miles south of the Canadian border. The work is both technically and mathematically demanding. And Johnson values highly his ability, and that of his employees, to perform mental calculations without electronic aids.

As do most parents, the Johnsons, who decline to be quoted, favor a rigorous and demanding math curriculum for their children. They simply aren't sure that the new approach fits the bill.

Kabor also espouses high expectations for his students, particularly those who will not go on to study math in college.

"High school," he says over dinner, "is often the last chance that many of these kids will have to experience a liberal-arts education."

So, in a way, the chance encounter between the two men is symbolic of the clashes that inevitably occur when national academic standards get translated into a local curriculum--when idealism meets the reality of classrooms, teachers, and parents. Yet, if the math standards are to have any positive effect on teaching at all, it will be because their innovative approaches have taken root in Kalispell and thousands of other towns across the nation.

The movement to develop national standards for what students should know and be able to do in various disciplines has had its own encounter with reality in recent months. The most public blowup came over U.S. history standards, which political conservatives have slammed as ideologically tainted and presenting a largely negative picture of the American experience.

No such controversy accompanied the release of the N.C.T.M. standards in 1989. Indeed, they were scarcely noticed outside the relatively small community of mathematicians and math educators. And as other standards-setting groups got to work in the early 1990's, they looked to the math standards, and a companion document describing exemplary teaching practices, as a benchmark and a model.

The standards argue that most math teaching places too much emphasis on rote memorization and includes too many exercises divorced from real-world applications of math. Instead, the curriculum should include: an emphasis on math as a problem-solving activity with broader applications for the world outside the classroom; the idea of math as a means of communicating information and ideas, in equations or in prose; an emphasis on reasoning to find the appropriate solution to a problem, rather than depending on teachers to provide answers; and making explicit the "mathematical connections" between apparently unrelated subjects.

To hear some education reformers talk, you'd think the document has become the de facto national standard in that discipline. That schools all over the nation have dropped their multiplication drills for sessions with math "manipulables" and other tools of the trade.

Yet, it's difficult to point to a single state or school district that can truly be said to have adopted a standards-based approach to teaching math. A study by the Exxon Foundation in 1990, for example, found little evidence that an "integrated math curriculum," of the sort envisioned by the standards, existed in any substantial way in the nation's school districts.

"What they found is probably no surprise," says James Gates, the N.C.T.M.'s executive director. "One classroom is moving toward the standards and the classroom down the hall is not. It's uneven."

Moreover, the developers of the standards never expected to change overnight the way that math has been taught for decades.

"It was an orchestrated movement that we realized would take a long time," says Jack Price, the math group's president. "It's going to take us 10 or 12 years to get this done. We're talking about the turn of the century at least."

But it's also fair to say that the standards have already had an impact on the profession. Says Zalman Usiskin, the director of the University of Chicago School Mathematics Project and a frequent, if sympathetic, critic of the math standards: "The standards, one could argue, have been successful beyond anybody's dreams."

In part, Usiskin points out, that's because the standards were deliberately not written as a prescriptive document that could lend itself to wholesale "adoption" by a state or district.

"It's not as if you've got one thing that counts as adopting the standards," he notes. "You would have to look at each element of the standards and ask, 'Has this been adopted? Has that been adopted?'"

While Montana hasn't adopted the standards in whole, it has a strong record of grassroots efforts to use standards-based teaching that makes it a model for other states.

Even before the N.C.T.M. standards came into being, the state had a home-grown movement to develop an integrated approach to math teaching that would incorporate elements of language arts, science, and other disciplines.

In 1991, the National Science Foundation gave the movement a boost with a $10 million grant to the Montana Council of Teachers of Mathematics under the N.S.F.'s State Systemic Initiative program. The federal agency gives states funds for statewide efforts to overhaul science and math teaching.

The SIMMS approach, while standards based, is not simply a reiteration of the N.C.T.M. standards. Instead, the reform effort is In 1991, the National Science Foundation gave that movement a boost with a $9 million grant to the Montana Council of Teachers of Mathematics under the N.S.F.'s State Systemic Initiative program. The federal agency gives states funds for statewide efforts to overhaul science and math teaching.

The SIMMS approach, while standards based, is not simply a reiteration of the N.C.T.M. standards. Instead, the reform effort is based on the theory that children develop their individual world views by interpreting information they have largely derived for themselves. The SIMMS curriculum is being pilot-tested at 106 of Montana's 186 high schools.

Patricia Haffey, Gov. Mark Racicot's senior education adviser, concedes that her conservative Western state seems an unlikely place for a leading-edge curriculum overhaul to be taking root. But she also notes that Montanans are a pragmatic lot.

"I think they're 'conservative-slash-practical,'" she says. "And when you can demonstrate the importance of the science and mathematics applications, then Montanans see the value of it from a practical point of view."

To appreciate how difficult it is to build such a consensus around an untried reform, you need look only a few hundred miles to the west to the nation's most populous state.

In California, the state's math framework, which is similar, though by no means identical, to the N.C.T.M. standards, has become a lightning rod for parental criticism, particularly in affluent districts.

In the wake of disappointing student performance on the California Learning Assessment System tests last month, Maureen DiMarco, Gov. Pete Wilson's chief education adviser, went so far as to denounce the math outlined in the framework as "fuzzy" learning espoused by "radicals [who] are convinced that basic skills are not important."

Price, the N.C.T.M.'s president, has watched the fray at close range: He's the co-director of the Center for Science and Mathematics Education at California State Polytechnic University at Pomona. He believes that much of the backlash comes from parents who--even though they may have suffered through traditional math classes themselves--demand the same for their children.

"You have a very highly educated, high socioeconomic group who feel their kids are not going to get into Harvard and Stanford and the cream schools because of what [DiMarco] has characterized as 'fuzzy math,'" he explains.

He concedes that while the central premise of the standards is that all students should study meaningful math, the document "did not say enough about the mathematically promising child."

Gates, the N.C.T.M.'s executive director, acknowledges that because the California framework is so similar to his group's standards the implications of the parent revolt cannot be ignored.

Some angry parents in Palo Alto, for example, have dubbed the framework "the new 'new math,'" referring to the curriculum-reform effort of the 1960's that is widely viewed by the public as a failed experiment. Although the teaching of "new math" concepts varied widely, it essentially embodied more abstract ways of thinking about mathematics.

The Palo Alto parents formed a group to oppose the curriculum and persuaded school officials to postpone buying textbooks that agree with the teaching strategies in the framework.

Gates, who was a classroom teacher when the new math was introduced, is quick to draw distinctions between it and the standards.

"It was a curriculum prepared by mathematicians," he says of the new math. "It was not intended for every child. Basically, new math was aimed for the most part at college-bound students. It was an effort primarily aimed at catching up with the rest of the world. The new math failed to a great extent largely due to our faults in selling to the public and in teacher preparation."

Usiskin of the University of Chicago agrees that many in the general public remember the new math with horror. But he notes that the reform "changed the secondary school curriculum forever" by eliminating such courses as solid geometry.

One day, he argues, the precepts contained in the standards may similarly change the field.

Some of those precepts are on display daily in Kabor's SIMMS classroom in Kalispell. But those who come looking for a radically different math class will be disappointed.

The teaching in his SIMMS course, which is geared toward 10th and 11th graders, is all but indistinguishable on any given day from a traditional calculus course. Kabor stands at an overhead projector, posing questions about how changing the number in a given math function would affect the shape of a graph. The students sit in traditional rows. The only difference comes when they bang out their solutions on their powerful graphing calculators.

In other SIMMS classes, meanwhile, students sit in groups of four, discussing how to solve problems and giving each other tips on how best to use the graphing calculator to derive the appropriate answers.

"Some days, there's no difference," says Karen Longhart, who teaches both SIMMS and traditional math. "A lot of times you'll walk into a SIMMS class and not notice any difference. You can't always jump directly into 'applications.'"

The curriculum does vary from traditional instruction in the types of problems that students are expected to solve and in that they often are expected to work cooperatively to solve them.

In a SIMMS module on probability called "I'm Not So Sure Anymore," for example, students learn that determining the probability of an event requires a large number of trials. Working in teams, they use the calculator to generate random numbers to simulate a lottery. They then must determine the probability of a given number appearing at a given time.

Three years into the field-test, students, parents, and teachers remain divided on the merits of the SIMMS approach. In their written evaluations, some students praised the curriculum as making math more meaningful for them. But one parent offered this simple advice for improving the math curriculum: "Boycott the whole damned SIMMS project."

Even among SIMMS teachers, there is no consensus about what should and should not be included in the curriculum. Longhart, who serves on the statewide SIMMS professional-development committee, says that like many parents and teachers involved in the pilot-test, including her husband, Fred, who also teaches at Flathead High, she is constantly struggling with what should be taught.

"I think what we're trying to do is to make some decisions about what are 'the basics,'" she says. "It's tough and it's scary. We talk a lot, Fred and I, and say, 'Is this the right way?'"

Eventually, she adds, it is likely that the reform effort will produce some blend of the traditional coursework and the more open-ended SIMMS curriculum. "Not all the kids like the learning style of SIMMS. Some of the kids will say, 'Tell me how to do it and I'll do it,'" she says. "I think they're going to have to meet in the middle and come to some sort of happy marriage."

Perhaps more surprising, some critics don't believe that SIMMS goes far enough toward meaningful reform.

"What disturbs me about SIMMS is that the beauty that attracted me to mathematics isn't there," says Nancy Lindsey, a school board member from Polson.

Lindsey, a graduate of the Massachusetts Institute of Technology who works for a local software developer, adds that many of the "real-world applications" of math in SIMMS are too vocational in nature.

"We're not developing a solid concept of what the appropriate direction for reform is," she says. "Instead, we settle for a very short-term fix which is reductionist. In education, you can prepare people to solve the problems of today and yesterday. But you really should be preparing them to solve the problems of tomorrow."

Pete Stabio, an author of the SIMMS curriculum, counters that the traditional curriculum is far from desirable. "It's not a situation where we're taking a curriculum that's done a superfantastic job with the majority of kids and dumping this on top of it," he says. "A majority of kids still think that math has no application in the real world."

Most critics focus on the project's emphasis on the use of technology to replace much of the drudgery of the traditional math curriculum. "There're a lot of myths that we have to break down," argues Kabor. "And one of them is that calculators are going to destroy thinking."

At Flathead High, each math classroom contains a plastic rack full of graphing calculators. The graphing calculator is an integral technology in the reform effort and is used in both the SIMMS pilot courses and in traditional math classes.

"I spent too many years in high school getting one problem done in a day because we had to graph so meticulously," says Merrie Rampy, a SIMMS teacher at Loyola High School, a parochial school in Missoula. "With a calculator, students can do several problems a day. What I like about the calculator and the computer is the immediate feedback the students get."

But even SIMMS supporters continue to have some reservations about whether the emphasis on technology will have long-term detrimental effects.

Marge Simpson, a school board member in Kalispell, is one of them. Although she is impressed with the SIMMS curriculum at the high school, she worries about a similar standards-based reform that is getting under way at the elementary level.

Elementary teachers who favor standards-based reform have assured SimpsonSimpson that the use of technology in conjunction with other reforms will allow students to progress toward the same goal at their own pace.

"I have a problem with that," she adds. "I don't think they're going to learn to do the multiplication tables unless we do some drill and practice."

Simpson says that she plans to examine the scores of elementary school students on the Iowa Test of Basic Skills and compare them with 10-year-old data to see how students are faring.

Longhart herself admits that she depends more now on a calculator than she once did.

"My ability to do mental math is not what it was 10 years ago," she concedes. "But I also have the ability to estimate and see if the answer is reasonable."

In the long run, Usiskin of the University of Chicago believes, Longhart may be right in predicting that SIMMS will combine the best of traditional approaches and the N.C.T.M. standards.

In a speech at the N.C.T.M. annual meeting last month in Boston, he noted that reform movements, including the new math and the standards, are characterized by predictable cycles. First, those unhappy with the status quo issue a call to arms; then those who are satisfied with their teaching are pressured to join the reformers.

This tension, he says, leads to both overapplication and oversimplification of basic reform tenets. In the case of the new math, the overapplication, he says, was "we can use this approach with all students." With the standards, a common overapplication is "we're going to put all of the kids in the same class and teach them all of the math in the same time."

An oversimplification of the standards, he says, is that "students should create their own knowledge. That they can't learn any other way."

Usiskin has long argued that a second edition of the N.C.T.M. standards is needed both to account for improvements in technology, such as the graphing calculator, and to refine many of the tenets.

Despite the setbacks, Kabor believes that this time math reform is here to stay. In contrast to the new math, which was imposed on teachers from above, the standards were developed by educators, for educators.

"I think we'll fight to preserve these," he says.

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