Formulas for Math Problems

Scientists, doctors, engineers--you leave the room. While we’re at it, mathematicians and MBAs, you join them.

It’s math time. And for once, the rest of us--not merely the numbers whizzes--have to pay attention.

Today’s problem is a two-parter: 1) How many times will we redesign math classes before U.S. students measure up to the rest of the world? 2) Will doing so, again, finally get Americans over their aversion to math more complicated than balancing a checkbook?

Stumped? Don’t feel bad. So are many experts.

In November, Americans brought home a report card from the Third International Math and Science Study, the most comprehensive comparative test ever. Graded on a curve, eighth-graders in this country averaged about a C-minus, far back of such honor roll nations as Singapore, Japan and Belgium.

The result hardly surprised American educators--it merely repeated what had long been known.

Back in 1957, though, when the Russians blasted Sputnik into orbit, Americans were shocked. They realized their children were being taught math less suited for space travel than figuring how much seed to buy for the spring planting. As a result, the nation was dangerously short of top scientists and mathematicians.

Thus came the revolution known as New Math. Only to be toppled later by Back to Basics. Which was supplanted by Reform which, even now, is giving rise to. . .

Backlash.

For four decades, the United States has skittered from one math fad to another--each bringing rewritten textbooks, new training courses for teachers and new homework assignments to befuddle parents schooled under an earlier orthodoxy. Today’s parents, who were forced to memorize multiplication tables in their school days, may find their own kids being handed calculators in the first grade.

Our math instruction oscillates between the same poles that shape and reshape our culture, politics and even our morality. We are torn between discipline and liberation, between demanding performance and promoting self-esteem--a two-step that, in education, causes us to fixate on facts and formulas one moment, then complain the next that such “rote learning” fails to produce “true understanding.”

The mood of the moment places greater value on getting kids to feel good about math than on improving their test scores. But we’re reevaluating, of course, worried that things have gone too far. Here in California, a state panel is about to take yet another look at guidelines for teaching math.

Meanwhile, what you see as we begin 1997 is:

* Fantasy Lunch. Trying to make numbers fun, the Mathland lesson series has second-graders spend “math time” creating paper versions of favorite foods. All the cutting and pasting makes it more art lesson than charting exercise, some teachers complain.

* The same philosophy has university consultants telling high school algebra teachers that students must work in groups on problems that will “reveal” concepts. One class in Ventura County grows so frustrated trying to “discover 1,000 years of math” that the students beg the teacher to explain the material. So she lectures--but keeps the desks in groups so the principal won’t find out.

* Districts, including Los Angeles Unified, ban purchases of certain math books though teachers say they lift test scores. The problem? Reform philosophers find them too packed with formulas.

Such conflict among professional educators would be unthinkable in a country like Japan, where instruction changes slowly, guided by classroom successes.

In the United States, no central education authority, national curriculum or performance standards show the way. Philosophies of “teaching go back and forth, but there’s no sense of progress,” said James Stigler, a UCLA psychology professor who has compared math teaching around the world.

Reacting to November’s report card, U.S. Secretary of Education Richard W. Riley urged that we not simply abandon the current orthodoxy--the Reform agenda that spawned Fantasy Lunch--without having given it a real try.

But Riley is enough of a historian to know that competition is more American than consensus.

So a San Gabriel Valley school may be a microcosm of how the nation might sort it all out.

At Rosemead High, the math department proudly refuses to decide between the “reformers” and “traditionalists.” In some classrooms, you find the teacher conspicuously in the back while students up front present the nightly homework problem on an overhead projector. But down the hall, a teacher is at the chalkboard, firmly in control, leading the class through geometry exercises.

Students and parents decide which class works for them.

The key is, teachers at Rosemead do both methods well--with adequate training and with passion.

That has not always been the case in America’s math classrooms.

Launching of Sputnik --and New Math

Though Sputnik was the turning point, the seeds of New Math were planted years earlier during World War II. New technologies such as radar exposed the weakness of U.S. soldiers’ math skills.

After the war, mathematicians at the University of Illinois and Yale, helped by the National Science Foundation, looked for ways that children still riding bicycles with training wheels could be taught math concepts such as sets--the idea that numbers can be grouped by characteristics, such as whether they are even or odd.

If students understood, for example, that our numbering system is based on 10s (there are ten 10s in 100, ten 100s in 1,000, etc.), they would appreciate the brilliance of math and be able to use it in advanced classes--or on the job.

The approach sputtered along until Sputnik elevated math and science instruction to a national crisis. Overnight, thousands of teachers--many inexperienced--were asked to introduce new topics to classes packed with baby boomers.

New Math landed Philinda Denson in the Los Angeles Times in 1964. She was a newly minted math major from the University of Redlands who not only understood the approach but “loved it.”

“I thought it was very rigorous and true and honest,” recalled Denson, now one of the teachers at Rosemead High.

Thirty-three years ago, she was recruited to help ease parents’ anxiety over not understanding what their children were learning.

“The old math is not being discarded,” Denson was quoted as saying. “We want to teach children why as well as how. Instead of starting out by ‘borrowing’ and ‘carrying,’ we want children to understand what they’re doing.”

But even teachers, especially in elementary grades, were lost. Lynn Steen, a math professor at St. Olaf College in Northfield, Minn., said that led to absurd lessons in which students were drilled on how to spell terms such as “commutative,” as in the “commutative property”--which simply means that 2 + 3 equals 3 + 2.

By 1966, the movement was unraveling amid concerns that students were not learning basic skills. Over the next decade, New Math was satirized as “hopelessly abstract, elitist, confusing and impractical,” said San Francisco writer Jeffrey W. Miller, who studied the history of New Math after deciding he was one of its victims.

Back to Basics was the reaction.

A National Hero Emerges

Opinion polls showed that Americans wanted discipline and a familiar curriculum to offset the social disruption shaking the country. So even as dress codes were being liberalized and requirements eased in college--meaning students did not have to take much science, for instance--elementary schools returned to old-style arithmetic drills.

The approach eventually produced a national hero, East Los Angeles’ Jaime Escalante, who showed that hard work mastering formulas could lift low-income students into the math elite. His Garfield High students passed Advanced Placement calculus exams at astounding rates.

But many math educators were not convinced. They agreed that Escalante was a brilliant motivator, but how many were like him? They simply were not willing to concede that math instruction had to be top-down--or that calculating was the same as understanding.

In 1983, this group got the ammunition it needed. The National Commission on Excellence in Education released its “Nation at Risk” report, showing that too few high school students were taking math. Each year after the ninth grade, the number fell by half.

Another problem: women and non-Asian minorities were overwhelmingly being filtered out. Why couldn’t math be a sponge, soaking everyone up?

That was the language of Reform.

In a sense, it turned New Math upside down. Where New Math presented the students with theories (“sets”), Reform started with games, designed so students would discover such concepts.

And why did math have to be so abstract? Why ask students to divide the fraction 1/2 by 1/4? Why not ask: How many quarters are there in a half-dollar? Easier, right?

Under the Reform philosophy, students were given calculators, freeing up time previously spent number-crunching for “higher order thinking.” They were to work in groups to get a feel for how math was used on the job.

Students might figure out what products a bakery should have to maximize profits. The solution traditionally required complex equations. But the students here can draw diagrams and the answer is less important than getting them to “think about strategies, talk with other kids and then pull the math out of that,” said Judy Anderson, who directs a National Science Foundation project helping Southern California teachers develop nontraditional lessons.

The reformers won a big victory in 1985, when California adopted a framework for math instruction that promised to make students “mathematically powerful.”

The Reform movement remains on top: Education journals highlight ways to teach “African American math.” Conferences attracting 5,000 teachers suggest downplaying the difficulty of classwork by basing problems on fairy tales.

One missionary in the Reform cause is consultant Ruth Parker, who rejects long division and multiplication tables as nonsensical leftovers from a pre-calculator age. She urges audiences to “let kids play with numbers,” and they will figure out most any math concept.

Parker has spoken before 20,000 people over the last six months at the behest of school districts. But there’s an ominous reason for that: The districts are worried.

About backlash.

Reform now is facing the same sort of scrutiny--and ridicule--that killed New Math.

Why? The feel-good language presents an easy target. And the test score gap with other industrial nations is not closing.

This fall, the National Assessment of Educational Progress said 17-year-olds are no stronger in math than 20 years ago. Only six of 10 high school seniors can compute with decimals, fractions and percentages. Fewer than one in 10 can use beginning algebra.

Math professors shake their heads at the skills of freshmen--54% in the CSU system have to take remedial math. “Things the average students would know backward and forward 12 years ago, these students don’t know at all,” said Jerry Rosen of Cal State Northridge, lamenting how students now use calculators to add single-digit numbers.

Performance in elementary grades is shaky as well. Last year, after many California schools began using Reform lesson plans, test scores immediately plunged in Santa Barbara, San Francisco and elsewhere--stirring parent revolts.

“I don’t think parents would be skeptical if they thought the new ideas were firmly anchored in their kids being able to balance a checkbook when they’re older,” said Miller, the San Francisco writer. He put his daughter in a Catholic school where she is expected to memorize multiplication tables by the end of the third grade.

Just as California led the way to Reform, so is it experiencing backlash first. Critics compare the state’s math curriculum to its disastrous experiment in reading instruction. Officials embraced the “whole language” approach, downplaying fundamental phonics skills in favor of trusting that students would learn them through exposure to interesting stories.

In math, those leading the backlash say it’s a difficult subject, whether reformers admit it or not. And it is practice adding and subtracting--with a pencil--that prepares the mind for complex work such as calculus.

The fight gets ugly at times.

At San Fernando High, Dan Hart is following the example of Jaime Escalante. He touts “real academic standards” and uses the same texts and cram sessions to teach low-income Latino students Advanced Placement calculus.

Of 19 who took the AP test last spring, eight passed. Francisco Garcia also scored a perfect 800 on the college entrance SAT test.

But Hart is an outlaw in the Los Angeles district because he uses structured Saxon Publishing books, which reformers have stricken from approved lists. His students have them only because the publisher donated them.

“It’s astounding to me that these books are so vilified, because kids learn so much better,” Hart said.

Hart is optimistic, though, because the state now is rewriting its guidelines for the teaching of math and reading. In fact, the appointment of outspoken backlashers to the math panel enraged reformers, with 3,000 teachers signing protest petitions.

So what to do? Were we to repeat the patterns of the past, policymakers would order a retreat to traditional practices and declare the war won . . . until the next counterrevolution.

But no one--neither reformers nor their critics--believes that would improve our international standing.

Voices as prominent as Albert Shanker, the president of the American Federation of Teachers, say we need to decide exactly what math students should know at each level. And we should not flee from testing performance because failure may hurt some.

That still leaves room for different approaches.

The nations high on the international report card do not use one method. Japanese teachers use many Reform-type lessons, but students also attend private programs for extra drilling.

What’s more, Japanese lessons are better crafted and more likely to include challenging math ideas. That was the conclusion of Stigler, the UCLA professor, who supervised videotaping of eighth-grade classes in various nations.

American lessons, in contrast, were unfocused and often interrupted. Stigler said 95% of the teachers espoused Reform ideas, but the vast majority offered lessons not unlike those of the 1950s.

That finding was one reason that Education Secretary Riley urged Americans not to give up on Reform philosophy. Parents, he said, should demand classes that help kids really “understand.”

But Riley, a former governor of South Carolina, noted that unlike other countries, authorities here have limited power.

Americans “don’t like the federal government to come in and tell them how to teach,” he said.

The last four decades back him up. Ordering all teachers to teach a certain way--or taking away textbooks they like--seems futile. There are too many teachers to indoctrinate them all. There’s too much room for misunderstanding. And a disgruntled few can scuttle any method.

“The nature of our people, their diversity, the freedom that Americans enjoy has made this country great,” Riley said. “But another thing that’s made the country great is competition.”

That’s what you see at Rosemead High.

Competition Among Approaches

Even before the students settle in, it’s clear that Melody Martinez is in control of her math class for freshmen and sophomores, but not dictating to it. “The bell’s going to ring, have your calculators ready,” she says from the back of the room. “Presenters, get ready.”

A student named Claudia is ready to talk about the homework--devising a strategy for guessing what will be on the back of three cards: one with an X on both sides; another with an O on both sides; the third with an X and an O.

Students flip the cards 100 times as a trial, then work up a probability formula: Two-thirds of the time, the back will match the front.

Martinez is in the Reform vanguard. She and a few other teachers at Rosemead use the Interactive Mathematics Program (IMP), which replaces the usual high school sequence (algebra to geometry to advanced algebra) with a series of problems that each can take eight weeks to solve. Developed with grants from the National Science Foundation, it is used in 178 schools nationwide--and also widely in Japan.

“You don’t feel you are doing math most of the time but . . . when you put it all together it’s the same,” said senior Rene Cardona.

This is not “feel-good” math. If “you miss your homework, you’re busted,” Cardona said. “She’ll call your parents.”

In trials across the country, IMP students have done no worse--but no better--on college entrance exams than students taught traditionally. Still, Martinez believes weaker students stick with math longer because they enjoy the unconventional approach.

Linda Boyd teaches geometry down the hall. Her classroom brings back memories.

She starts by handing back the previous night’s homework and then going over the problems. She’s at the front of the room. The desks are in rows, not pushed together for group work.

Then she calls students’ attention to a lesson on how to tell whether two geometric shapes, such as triangles, are the same.

“Suppose that polygons QRPNL and ZYWXS are congruent. List all pairs of congruent angles and all pairs of corresponding sides.”

Boyd acknowledges that the lesson is abstract. But students “are learning to develop their minds.” And that is the way that two shapes would be compared in, say, construction jobs.

Rosemead’s teachers have reached a truce: Respect each other--but compete for students.

This school year, only a quarter enrolled in Reform classes.

“It’s been a struggle because people have very strong feelings on both sides,” Martinez said. “Some of them you can understand. They’ve been through so many new programs and . . . they find it hard to see that this one is going to be any different.”

Denson, who was featured 33 years ago as a New Math pioneer, now is among the traditionalists. But she borrows Reform methods.

Her biggest worry? That tomorrow’s math gurus will “want to make a big change and pretty much throw out what went before.”

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One lesson--and 3 Ways to Teach It

The Pythagorean Theorem, A + B = C , is one of the best known concepts of mathematics. Attributed to the Greek philosopher Pythagoras, who died in 495 B.C., it shows how the two shorter sides of a right triangle compare to the longest side--known as the hypotenuse. Part of a high school geometry, it has practical uses in many fields--carpentry or industrial design, for example. Here’s how it is taught by:

TRADITIONALISTS

* In a book commonly used in the mid ‘70s, the teacher would describe the theorem, talk about its history and state it: “In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.”

Sample problem: A man walks 2 miles north, 3 miles east, and then 2 more miles north. How far is he from where he started?

Step 1: Diagram the man’s walk then determine length of dotted line.

Step 2: To do this, create a right triangle.

Step 3: Apply the formula.

Answer: 5 miles

****

REFORMERS

* The theorem is the first geometry lesson in the College Preparatory Math curriculum used in about 750 schools. Students plot points on graph paper to create different size triangles. Then they experiment to understand what “squared” means: drawing squares off the three sides of a triangle. Then they can see the truth of the statement that A-squared plus B-squared equals C-squared.

Create squares to see how the size of A (squared) + B (squared) in fact equals the size of C (squared)

****

RADICAL REFORMERS

The Interactive Math Program introduces the theorem with a game. Two students arrange square “rugs” of various sizes to create a right triangle. One player gets points equal to the area of the largest rug. The second gets points equal to the combined area of the two smaller rugs.

Step 1: Students get rugs.

Step 2: They arrange the rugs three at a time, to form triangles.

Step 3: They discover that, when the triangle is a right triangle, both players receive an equal number of points--because the two smaller rugs equal the size of the larger one.

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Changing Math Fashions in Math Education

Since the World War II, math instruction in the United States has changed course time and again--with little improvement in test results.

Early 1950s: Students are grouped by ability and memorize multiplication tables in early grades, using timed drills. But a majority drop math after the ninth grade.

1952: Committee on School Mathematics is formed at the University of Illinois and begins developing one version of what would become known as “New Math.” Idea is that students should learn the laws governing math as well as how to calculate.

Mid-1950s: Small number of schools test committee’s curriculum, weaving together lessons in algebra and geometry. Students work with sets--groups of numbers having common characteristics.

October 1957: Sputnik is launched by Soviet Union.

Summer 1958: National Science Foundation begins funding four-week summer institutes on college campuses to train high school teachers in New Math. They are told that instead of giving students a rule--for instance that a series of multiplications can be combined if they have a common element--the youngsters should figure out for themselves that (6 X 4) + (7 X 4) is the same as 13 X 4.

1958: National Science Foundation funds School Mathematics Study Group at Yale University to write another version of New Math. Sale of Yale books, completed in 1959, jump from 23,000 the first year to 1.8 million after three years.

1962: Articles opposing New Math begin appearing in scholarly journals.

1964: Max Beberman, one of the founders of New Math, warns that because teachers had not been adequately trained in it, the nation is “in danger of raising a generation of kids who can’t do computational arithmetic.”

Mid-1960s: Many schools sponsor classes to explain the new teaching methods to parents. Though there are concerns about whether students are learning basic skills, California downplays drills in favor of encouraging students to “discover” math.

1967: Five-year study of 12 Western nations finds U.S. 13-year-olds and high school seniors far behind those in other countries. New Math is blamed.

Early 1970s: Gallup Polls show public concern about lack of basic skills and discipline in schools. Schools begin rejecting New Math materials in favor of a back-to-basics approach.

1974: “Why Johnny Can’t Add,” an indictment of New Math, is published.

Mid- to late-1970s: Back-to-basics approach spreads, but it has its critics as well, planting seeds of the Reform movement: Math conferences begin featuring sessions on how to help students gain understanding by solving problems on their own.

1983: “Nation at Risk” report warns that America is in danger because of the weakness of its schools.

1985: California’s issues a “framework” for math instruction that is the most advanced statement of the Reform agenda. Emphasis is on problem-solving, applications and student understanding.

1986: U.S. Dept. of Education releases a 17-nation comparison showing that America’s best students--those in the top 5%--last in algebra and calculus when matched against top students elsewhere. Back-to-basics movement is blamed.

1989: “Curriculum and Evaluation Standards for School Mathematics” the bible of Reform math, is published.

1992: Revised California math framework is published, further de-emphasizing teaching of basic skills in favor of greater thinking and understanding.

1994: Reform textbooks are adopted by state’s Board of Education. Critics say approach is filled with “fuzzy crap,” signaling start of backlash.

1995: State Supt. of Public Instruction Delaine Eastin appoints a panel to examine math instruction. It concludes that changes are necessary to restore emphasis on basic skills as part of a balanced approach that also includes conceptual understanding.

November 1996: Another state panel is created to rewrite the state’s math guidelines to carry out that vision.

November 1996: The Third International Mathematics and Science Study, comparing students in 41 countries, finds U.S. eighth-graders below average.