Math wars: old vs. new: modern day Hatfield vs. McCoy: when traditionalists debate constructivists about math education.

When today's teachers attended school, they likely sat in desks arranged in neat rows, silently computing multiplication problems on a ditto sheet, practicing so often the answers became automatic. At home, their parents flashed cards in front of them--two times two is four, three times seven is 21--over and over again, until the right responses came quick and easy.

But, in classrooms across the country in savvy Manhattan and sleepy Wisconsin towns, rote computation has taken a back seat to problem solving and reasoning, reflecting a decade-old shift in mathematics instruction that has ignited academic debate and has befuddled parents. One camp says students should master the tried-and-true algorithms of traditional mathematics before they attempt higher-order problem solving. The other, often labeled "constructivist," believes that through group work and manipulatives, students can make sense of mathematics, uncovering ancient formulas and rules on their own.

In the constructivist program Connected Mathematics, for example, a teacher doesn't stand at the front of the classroom telling middle-schoolers that the area of a circle equals pi times the radius squared. Instead, students use scissors, glue and graph paper to determine that it takes just over three radius squares to fill the circle. The figure, more precisely, is 3.14, or pi. After the "guided discovery" session a teacher reconvenes the class and summarizes the lesson, making sure students have grasped the objective, says Jim Wolgehagen, secondary mathematics coordinator for the Plano, Texas, Independent School District, which adopted Connected Mathematics district wide in 1999.

The New-New Math

In Pittsburgh, Penn., students in an Everyday Mathematics program learn to add and multiply the "new-new" way: using manipulatives, fast-paced card games, "alternative algorithms" and other methods foreign to parents. One frustrated father told mathematics director Diane Briars: "Every night, I try to help my son with his homework. Every night, we argue. I tell him, you have to add on the right. He tells me there are other ways to do it."

Everyday Mathematics has led to a sharp increase in math competency among white and African-American students in the district, according to Briars, and, in 1999, the U.S. Department of Education named it one of the country's 10 exemplary and promising mathematics programs. But soon after, in a counterattack typical of the "math wars," 200 professionals, most of them university mathematicians, signed an open letter to then Education Secretary Richard Riley, warning that Every day Math, Connected Mathematics and the other selected programs had "serious mathematical shortcomings." Fearing the widespread acceptance of so-called "fuzzy math," they urged Riley to withdraw the recommendations.

The "math wars" have visited Palo Alto, Calif., Lincoln, Mass., and other well-educated communities across the country, often around the time of new textbook adoptions. The first round was fired in 1995, when California parent Michael McKeown, now a professor of medical science at Brown University, helped found a Web site called Mathematically Correct. The cyberspace consortium--made up mostly of parents trained as scientists, engineers and statisticians--worried that the team work and discovery learning encouraged in their kids' mathematics classes were pushing basic skills out the door. "We saw mathematics courses coming into our schools that were highly touted, that they said were going to teach our kids concepts and creative thinking," McKeown says. "But when we looked at the content, it was watered down."

In recent years, California has rewritten its state standards, adopted new textbooks, poured significant dollars into staff development, and stepped back from the contested curriculum it adopted in the early 1990s, according to Tom Lester, a mathematics consultant for the California State Department of Education. The curriculum, considered cutting-edge at the time, was based on a set of standards issued by the National Council of Teachers of Mathematics in 1989. Critics have blamed NCTM standards for opening the door to a math that considered precise answers less important than the thought processes students use to "construct" and discover mathematical concepts.

One critic, Arizona-based author/educator Elaine McEwan, says the NCTM standards threw "the baby out with the bathwater" by expecting students to make sense of abstract concepts without an adequate foundation of skills. "What you have is the equivalent of whole reading, where you don't teach kids decoding or grammar or phonics, you just throw them into a book," McEwan says. "I think constructivist learning is fabulous when you have some knowledge, some concepts, some understanding, but when you turn kids loose to constructivist math and you haven't taught them any of the basics, you have chaos. I'm afraid that in classrooms that are doing the NCTM math, there is more activity than achievement."

Briars says McEwan and others have missed the point. "While we talked in the standards about what needed to be different, we also meant that skills are still valued and still important," she says. In response to its critics, NCTM revised the standards in 2000 to include more explicit language about accuracy and efficiency. But the document still calls for a radical--and to some, unsettling--departure from the mathematics instruction of yesteryear.

Traditionalist Wayne Bishop, a California State University mathematics professor, scoffs at the idea that "kids are going to invent their own algorithms," or "get ideas well enough from playing with manipulatives." But NCTM President Lee V. Stiff argues that traditional math instruction leaves students unenthused and in the dark. "We need to uncover math to show how it works," Stiff says. "If there were nothing wrong with the traditional math of the '70s and '80s, our kids would be at the top of the pack, and adults wouldn't be lamenting the fact that they hated fractions or don't know algebra."

Stiff says he gets frustrated when his organization's quest for sound mathematics is clouded by misunderstanding. "Reducing the vision of Principles and Standards to one method of teaching or learning is a distortion of the facts and thwarts our progress toward providing a high-quality mathematics education for every student," he wrote to members in his August President's Message. The issue has become so stratified, in fact, that those in the thick of the math wars seem reluctant to consider the merits of either side.

Fuzzy Math

McKeown charges that school administrators who are far removed from the classroom and know little about mathematics instruction are wooed to trendy constructivist programs by the promise of grant dollars from major foundations. Other traditionalists accuse constructivist mathematicians of peddling unproven programs that disregard basic skills and precise answers--a characterization the latter strongly refutes. "There is a false dichotomy being perpetuated about reform mathematics that says right answers don't matter, that skills don't matter," says Lucy West, director of K-12 mathematics for New York City's District 2. "Reform mathematics has been reduced to sound bites like `fuzzy math' that raise parent anxiety."

Four years ago, West's district implemented a controversial math program called Investigations in Number, Data and Space, more commonly referred to as TERC for its founding institution, a nonprofit research and development organization based in Cambridge, Mass. According to West, standardized test scores rose the first three years, dipping slightly last year in conjunction with a citywide decline. District 2 is the city's second highest performing district in math despite a mixed population of Chinese immigrants, African-American, Hispanic and Caucasian students, she says. "This program pays attention to what kids are thinking about and allows us to plug in to their ideas so we can help them understand complex mathematics," West says. "It's real-world mathematics."

Some, however, including a number of university professors, charge that the TERC program is watered-down and misguided, turning out confused kids who need outside tutoring to bolster basic math skills. While they can appreciate the engaging games, the writing exercises and the discovery methods the programs emphasizes, says parent Elizabeth Carson, parents are deeply concerned "with the absence of practice and drill, the level to which the children are challenged, homework with questionable purpose and rigor, the encouraged use of calculators for simple computation and the lack of coherent and regular assessment of understanding and skills," she says.

Former District 2 parent Denise Matava Haffenden says her son's math workbook was filled with graphs, written instructions and just "a smattering of actual work." "He can't do algorithms, long division and carrying," says Haffenden, who spent the summer preparing her son for a traditional math program in parochial school this fall. "What he is good at is drawing pictures to work out things he cannot understand." Haffenden helped organize forums to protest New York City's adoption of NCTM-influenced programs.

Mathematical Litigation 

In Plano, Texas, a small group of parents took their frustrations even further, suing the school district in 1999 for refusing to allow the distribution of materials critical of the Connected Mathematics program. Parent Susan Sarhady, a Connected Mathematics opponent who is not part of the lawsuit, describes a sixth-grade assignment in which students are asked to make a poster reflecting their favorite number, and are then graded for their creativity. "That's not mathematics," Sarhady says. "What they're doing is unreal math, unconnected math, where kids have to figure it out for themselves."

Wolgehagen defends the program, which was piloted at four middle schools from 1996 to 1999. "We used the Texas Assessment of Academic Skills tests all three years of the pilot," he says. "We pulled the scores apart every possible way we could think of--by ethnicity, gifted students, socioeconomic strata--and the pilot schools still had better growth."

Briars reports similar success in Pittsburgh, an urban school district where more than half of the 40,000 students are African-American. In 1994 and 1995, the district began a major implementation of the National Science Foundation-funded program Everyday Mathematics. "When you walk into our classrooms, it's the kids, not the teachers, doing the mathematical thinking," Briars says. Every year, the district dedicates $1 million to teacher training, much of it to fund resource teachers who travel to different schools, helping to implement the program.

Seventy-nine percent of fourth-graders in "high-implementation" schools met the skills standard on Harcourt's New Standards Mathematics Reference Examination in 1998, according to district data. By contrast, 38 percent of students met the standard in schools where traditional instruction continued. "Kids need an opportunity to struggle with ideas, instead of us just telling them the answers," Briars says. "If the procedures are simply rote, you have no clue what to do if you forget a step."

A Seven is Still a Seven

Since 1991, University of North Carolina professor Jack Piel has taken his Comprehensive Applied Manipulative Mathematics Program to 30 area elementary schools. Kids are first introduced to dice, abacuses, blocks and other manipulatives, then given "transitional" assignments to connect what they've learned to pencil and paper activities. Standard algorithms come last. Charlotte's Thomasboro Elementary School has seen tremendous results, according to Piel. Last year, for example, 96 percent of fourth-graders performed at or above grade level on a state standardized math test compared to 50 percent two years prior.

"In every school, parents have been concerned," Piel says. "I tell them, `If we dug your basement with a shovel, should we still use the shovel today when we have much better tools and more research into how a child learns and where they are developmentally? This is not a different math. A seven is still a seven, it's just a matter of how we present it."

If only it were that simple. In Fairfax County, Va., where a new textbook adoption looms, two school board members recently called a "Citizens for Better Math" forum to talk about back-to-basics math. "We don't want `drill-and-kill,' but students should know their basic math skills before they move on to higher-order mathematics," says at-large member Mychele Brickner. Superintendent Daniel Domenech is unfazed by the hullabaloo. "This debate is more political than pedagogical, the math version of the phonics wars," he says. "It's a conservative versus liberal labeling of the curriculum."

Prominent mathematics educator Marilyn Burns just wishes the name-calling would stop. "Labels are damaging because they oversimplify complex concepts," she says. "The lack of civility and the passion that goes with these arguments is really unsettling." Burns leads Math Solutions courses and workshops, and her material has been used to train teachers in New York City.

Staff development and teacher buy-in is what ultimately decides the success of any new mathematics program, as administrators in California, New York City and Pittsburgh have found. While experienced teachers may be adept at blending the best of both techniques, balancing automaticity with group and hands-on learning, Burns says it's tougher for new teachers who may have grown up disliking math and received little graduate school training about how to teach it.

Amidst the squabbling and uncertainty, school administrators are left wrestling with a fundamental question: How do students acquire mathematical learning? In The Principal's Guide to Mathematics Reform (Corwin Press, 2000), McEwan writes: "Must mathematical understanding be `constructed' by the student through meaningful situated experiences and activities? Or, can some key learnings be mastered as a result of more direct instruction?"

Mathematics may have a foundation as rock-solid as the Egyptian pyramids, but in our ever-evolving world of knowledge, discovery and technology, these are questions educators may ponder for years to come.

SUGGESTED READING

Sensible Mathematics: A Guide for School Leaders, Steven Leinwand, Heinemann, $17.00 This book helps school leaders shape and sell a vision of learner-centered mathematics and is closely aligned with the NOTM standards.

Garbage Pizza, Patchwork Quilts, and Math Magic: Stories About Teachers Who Love to Teach and Children Who Love to Learn, Susan Ohanian, W.H. Freeman & Co., $12.05

This award-winning book is full of ideas for dynamic classroom math activities along with a thoughtful justification for progressive mathematics education.

How Big Is the Moon? Whole Maths in Action, Dave Baker, Cheryl Semple Tony Stead, Heinemann, $15.95

This book uses kids' own wondering about the world as a basis for serious problem solving. Mathematics becomes the focus of curriculum integration. A clearly delineated framework for connecting kids' own interests with the curriculum is presented for teachers to follow. Numerous suggestions for exploration topics are included.

Young Children Reinvent Arithmetic : Implications of Piaget's Theory (Early Childhood Education Series, Constance Kamii, Teachers College Press, $23.95

Kamii's research on how children really learn arithmetic (it's not through drill & practice) will transform your thinking. Teachers College Press also sells videos of young children solving complex problems in personal ways.

Math: Facing An American Phobia, Marilyn Burns, Marilyn Burns Education Associates, $14.05

This book explores the causes, consequences and cures of America's fear of mathematics. Excellent ideas for improving classroom practice are included.

Jennifer Covino, jencovino@hotmail. com, is a contributing editor and an education writer based in New Canaan, Conn.