New Math? Fuzzy Math? Fluffy Math!

[CountDeMoney]

Nontraditional math decried as 'fluff' in Frederick

Changes in math curriculum don't usually stir parents to political action, but in Frederick County, the introduction of a new math textbook has caused a minor revolt by county residents, 600 of whom have signed a petition to persuade the county school board to return to a more traditional approach.

"Our primary objection is the lack of solid math content. It has been replaced with fluff," said Tom Neumark, a Frederick resident whose daughter will be a kindergartener next year.

The battle these parents are having with the county is just one part of a larger discussion taking place around the nation and among federal education officials over how math should be taught. Some education experts are calling for states to adopt national standards that would narrow the focus of the math curriculum and make American students more competitive internationally. Maryland is on the verge of reviewing its standards and deciding whether to make changes.

On the grass-roots level, Stacey McGiffin is part of the parents group that would like to see changes at the state level. She says her second-grade son is bored with the math he is doing in a new curriculum called TERC. She and other parents say the program requires teachers to spend too much time explaining why math works and not enough time having students practice how to do problems. "TERC takes discovery learning to an extreme. In trying to make math 'fun,' TERC fails to teach the fundamentals that will ultimately prepare kids for algebra and beyond," McGiffin said.

TERC uses an approach called reform math that has grown popular in the past decade. Used in thousands of classrooms, according to its creators, TERC stresses conceptual thinking rather than a more traditional approach of solving problems. Frederick County schools adopted TERC in its elementary grades this school year after having used it at Lincoln Elementary School for several years.

A number of school systems in the Baltimore area have chosen to use textbooks that attempt to blend both methods and have not chosen to use TERC exclusively as Frederick has. In Baltimore County, teachers supplement a more traditional textbook with TERC. Anne Arundel County uses a similar approach, approving TERC only as a supplement, but no schools there have chosen to use it.

Such programs have come under scrutiny as national leaders suggest that math programs in foreign countries where students are most proficient give students more practice at solving difficult equations, and they do them at an earlier age.

Those views were supported by a multiyear study done for the U.S. Department of Education that indicated that TERC was the least successful of four math programs studied. In fact, the results gave high marks to one reform math textbook series and to another traditional approach. TERC spokesman Ken Maher said the study used an early version of TERC and that the program may do better later.

The choice of textbooks, which is left up to local school districts, is likely to come under scrutiny if the state school board takes action this year to make the changes that are being pushed nationally.

Donna Watts, who is in charge of math curriculum for the state, said an outside group called Achieve was hired to review the math standards. Achieve's review was completed in February, but Watts had said the document would not be released to the school board until fall, after a full review. After The Baltimore Sun filed a Freedom of Information Act request asking the document to be released, the department announced it would make the Achieve report public after the state school board receives a copy at its May meeting.

Watts said the Achieve report is not calling for major revisions to the state standards but rather "polishing."

Frederick County Associate Superintendent Bonnie Ward, who was involved in the selection of TERC, said the county based its choice on the success it had at Lincoln, which saw scores move up significantly over several years. She said the county is holding meetings for parents to better understand their children's homework.

Nevertheless, some Frederick County parents say they will be teaching their children math from different textbooks so that they don't fall behind.

[Martinus]

What is the typical US math curriculum like?

They are trying to dumb it down in Poland, too, btw. Back in my day, we've got trigonometry introduced in the fourth grade, and the calculus in the fifth or the sixth grade, and we liked it. 

[DisturbedPervert]

Just outsource math to the Chinamen.

[Habbaku]

What is the typical US math curriculum like?

For me, it was the basics for 1st-5th grade.  Pre-algebra and geometry for 6th-8th with Algebra 1/2, Geometry, Algebra 3 and Trigonometry courses through high school.

[Syt]

I recently realized that though I was always between A and B in maths grades I have thankfully forgotten all about integral and infinitesimal calculus.

[CountDeMoney]

Major Characteristics of TERC's Program: Investigations in Number, Data, and Space

   1. TERC insists on the ongoing use of hands-on tools (manipulatives, "models," and calculators).
          * They say concrete tools must always be available and regularly used.
          * TERC strongly rejects the idea that children must eventually migrate from hands-on to abstract thinking.
   2. TERC rejects the need for memorization and practice.
          * They say that student's familiarity with single-digit number facts must "grow out of lots of experience with constructing these facts on their own."  BA, Page 72   (emphasis added)
                             [Please click on  References for the meaning of the BA code.] There's no additional gain in conceptual understanding associated with the task of trying to "construct" one more basic number fact.
              o o TERC doesn't think it's possible to understand memorized information. But knowledge must first be loaded into the brain before it can connected to other knowledge and "understood."  Explicit memorization is sometimes the most efficient way to get it there.
                o TERC fails to understand that it's often desirable to move to automatic use of knowledge. The mind must be free to think at higher levels of complexity, without consciously revisiting underlying details. For example, the key idea of the standard algorithms is that multi-digit calculations are reduced to multiple single-digit calculations.  If children don't have instant recall of the single-digit number facts, they aren't equipped with the essential pre-knowledge for easily carrying out multi-digit computations.
   3. TERC fails to clearly define terms.
          * They regularly state: "We don't ask students to learn definitions of new terms."
          * They offer some "definitions," typically using multiple undefined terms to "define" a new term.
          * They favor "natural language" and "personal language."
   4. TERC emphasizes "familiar numbers."
          * The  "landmark numbers" are 5, multiples of 10, and multiples of 25.
                o Landmark number are also known as anchors.
          * The "familiar fractions" are limited to proper fractions with denominator equal to 2, 3, 4, 5, 6, 8, 10 or 12.  Thus 7 and 11 are not familiar denominators.  Perhaps TERC is opposed to gambling.
                o Note: 12 is included because it's needed for TERC's clock face method for adding fractions.
          * TERC doesn't believe in defining terms, so you won't find the preceding definitions in TERC materials.  This is what they appear to mean by these phrases.  We welcome their clarification.
          * Although TERC rejects explicit memorization of basic single-digit number facts, they expect students to remember many non-basic facts about landmark numbers and familiar fractions.
   5. TERC omits standard formulas.
   6. TERC emphasizes estimation and many right answers.
          * They suppress the concepts of precision and accuracy.
   7. TERC proudly rejects standard computational methods.
          * No standard algorithms for multi-digit computation.
          * No standard methods for operations with fractions and decimals.
          * No general methods for calculating with numbers.
          * TERC emphasizes special case methods involving landmark numbers and familiar fractions.
   8. TERC attempts to directly teach their shallow and misleading content.
          * They claim to offer a "constructivist" approach where students discover math as they play games and carry out investigations.  But they provide thousands of pages of teaching instructions and recommended scripts that identify the content they expect kids to "discover."
                o Thousands of pages for the teacher, but no text for the student.

[CountDeMoney]

An example I found:

[CountDeMoney]

NYT article from 2000.  This argument's been going on longer than I thought.

The New, Flexible Math Meets Parental Rebellion

By ANEMONA HARTOCOLLIS

Three years ago, one of New York City's most adventurous school districts set out to tackle a nagging problem: the math phobia that afflicts many students, and the disparity between the test scores of white middle-class students and their poorer black and Hispanic counterparts.

The district, which stretches from the Upper East Side to Chinatown, embraced a new "constructivist" curriculum without textbooks. This approach preaches that it is more important for children to construct their own solutions to mathematical problems than to learn the standard rules -- from multiplication tables to the value of pi -- handed down through the centuries.

Long ranked near the top of the city in mathematics, the district has held its place, although there is still a disparity in test scores between the poorest schools and the more affluent ones. But the new curriculum has enraged many parents who find that their children cannot multiply easily or understand basic algebra.

One parent, Anna Huang, said her son, Mack, a fourth grader, "felt a lack of clarity" when his teacher insisted that he estimate answers, rather than compute them precisely. Another parent, Anne Cattaneo Santore, said she was troubled because her son, William, a second grader at P.S. 124 in Chinatown, spent months counting with coins and solving equations using "friendly numbers," for instance, converting 71 + 19 into the easier 70 + 20.

"Those strategies don't work when you get to larger numbers," Ms. Santore said, "and they have been doing those strategies all year."

Ms. Huang and Ms. Santore have joined other parents, mathematicians and many teachers in a rebellion that has shaken education from New York City to Plano, Tex., and Lincoln, Mass. As school districts from affluent enclaves in Greenwich Village to poor minority neighborhoods like East New York have embraced constructivist math, parents have formed e-mail networks and turned out in force at school meetings to protest what they say is "fuzzy math" and the systematic "dumbing down" of mathematics teaching.

"Parents are worried," said Elizabeth Carson, an actress, the mother of a seventh grader and a leader of the protest movement in New York City. "They're scared that their kids are not going to be competitive. The math is not in their bones.'

The new math has at its core a passionate belief shared by tens of thousands of teachers around the country that they can reach more children, especially low-achieving minority students, by dropping standard rules in favor of exercises that allow students to discover the principles of math on their own. Constructivist programs are being tried in more than half of New York City's 1,145 schools, Board of Education officials said.

Educators who support the new math say that old-fashioned teaching through memorization and rules produced generations of people who hated math and never deeply understood it. Indeed, the manifesto of constructivist mathematicians, the 1989 standards of the National Council of Teachers of Mathematics, urges teachers not to demand too much accuracy too early. Math should be "flexible," the standards say, and "reasonable" answers should be valued over a single right answer.

The constructivist movement has led to the widespread rejection of textbooks, in favor of exercises using blocks, beans and other materials. One popular program, MathLand, suggests that students count a million grains of birdseed to get a feeling for the size of a million. Another, Everyday Mathematics, teaches children an ancient Egyptian method of multiplication.

It also suggests that fourth graders measure angles with bent straws instead of protractors. Connected Mathematics, a popular middle-school program used widely in New York, teaches sixth graders to add fractions by folding paper strips into segments representing halves or fourths or thirds, instead of by converting to common denominators.

Lucy West, the director of mathematics at Manhattan's District 2, where the new math has been most aggressively adopted, said that old-fashioned math had been oversold. "There is a misconception that in the good old days everybody could add and subtract, multiply and divide really easily and efficiently," she said.

But professional mathematicians say the activists have set up a false dichotomy between conceptual understanding and basic skills. Parents chafing at constructivist math tell stories of their children coming home confused and dispirited by lessons in which getting the right answer to problems is devalued in favor of strategies that are often primitive, cumbersome and indirect. Used by inexperienced teachers who are weak in math, they say, the curriculum can be murky.

And tutoring services say that they are seeing an epidemic of children coming to them for basic math instruction.

THE MOVEMENT
Good Intentions, Unproven Theories

How schools got to this point is a saga of good intentions, unproven theories and a progressive education movement that has has had its most profound impact in reading and math. In many ways, the math wars echo the once ferocious disputes about reading between advocates of the intuitive "whole language" approach, which stresses acquiring skills through simple reading of books, and the phonics method, which stresses decoding of letters and words. Until "whole language" became a dirty word, constructivist math was known as "whole math." One obvious solution is to mix a bit of both. But while educators have called a truce in the reading wars, deciding that compromise is best, the math wars continue to rage.

The high point for new math advocates came last October, when a panel set up by the United States Education Department endorsed 10 constructivist math programs as "exemplary" or "promising." Within a few weeks, nearly 200 university mathematicians and scholars sent an open letter to Education Secretary Richard W. Riley warning that the 10 programs had "serious mathematical shortcomings" and would leave students ill-prepared for college-level courses.

R. James Milgram, a mathematics professor at Stanford University, analyzed three programs and found that they consistently neglected to teach basic rules of multiplication, division, addition and subtraction. The programs are typically one or two years behind grade level, he said, and aimed at what he considered underachieving students.

On April 12, the National Council of Teachers of Mathematics, the nation's most influential group of math teachers, made a gesture to the critics as it revised its 1989 standards for teaching mathematics, the closest thing this country has to a national curriculum. Though not abandoning its original constructivist agenda, the council put the arithmetic back in math, by adding language emphasizing accuracy, efficiency and basic skills like memorizing the multiplication tables. The chairwoman of the standards committee declared that the group's new message was "Get the right answer."

Still, there is ambivalence in the teaching field. When the national council of mathematics teachers endorsed more of a balance between basics and the constructivist approach at the group's annual meeting in Chicago, the president-elect, Lee V. Stiff, ardently defended constructivism. "If I only teach it the way I understand, then only students who understand it the way I do will be successful," he said.

The evidence to support the educational virtues of new math is inconclusive at best. Publishers have provided studies in which they compare the results in pilot schools that adopted the program to other schools that made no change.

But those studies have been challenged by critics who say improvements may be the result of better training of teachers, and the extra attention given to pilot programs.

District 2, which has a high percentage of affluent students, has long ranked near the top of the city school system. Last year, when a new math test was introduced, scores across the city declined, and District 2 was the only district in the city to remain stable.

Ms. West, the math coordinator, said that was evidence the program was working. And other officials noted that the new test was closely aligned with the District 2 curriculum, and many parents attributed the good scores to tutoring by professionals and parents. And they noted that the district had been spending $800,000 a year on training math teachers.

Even experience in the classroom can be ambiguous. Roberta Schorr, an education professor at Rutgers, is using a computer simulation of a frog and a clown walking back and forth across a plane to help teach the concept of velocity at Central High School in Newark. Ms. Schorr sees the simulation as an intuitive introduction to calculus that does not require what she called "formal symbol structure." But during a recent class, half the students seemed baffled, while Tieyon Hendry and Rahul Patel, the star students, told a visitor that they had arrived at the answer to one exercise by using a mathematical formula: y = mx+ b.

THE OPPOSITION
Unconvinced by Unconventional

As the new math -- a cousin of the "new math" popular in the 1960's -- entered the educational mainstream, first in California in 1992 and then around the country, it sparked waves of opposition. Hostility came first from conservative parents who opposed any change in education, then from university math professors who felt it was not rigorous, and finally from liberal, affluent parents who were worried their children were not getting enough math to succeed in school and in life.

In Plano, Tex., parents have sued the school district for refusing to provide an alternative to Connected Math. In New York, the opposition first emerged not in a failing school but in one of the city's best, Public School 234 in TriBeCa. Edgy schoolyard conversations grew into a parent Math Committee, which gathered members across the district, sending out surveys and making angry statements at school board meetings.

Parents said they were stunned as they talked to their friends and realized how many had hired tutors. Those who cannot afford tutoring tell of scouring educational bookstores for workbooks and textbooks to help them make sense of the new math.

Ms. Huang said she became alarmed when her son, Mack, a fourth grader at the Bridges School in Chelsea, came home complaining that he hated math. The emphasis on estimation, she said, was confusing him. She bought him workbooks consisting of straightforward calculations and he enjoyed the sense of mastery.

Constructivist teachers celebrate the unconventional exercises they use as a way of keeping weaker children engaged, especially those from groups that have historically lagged in mathematics performance, like girls and black and Hispanic students.

But some parents are insulted by them. Ms. Weinberg, a dentist, said she was appalled when her daughter, Kelly, a sixth grader at East Side Middle School, came home with assignments to write her math autobiography and to write about her favorite number. "She was being graded on grammar and spelling," Ms. Weinberg said.

Wilfried Schmid, a professor of mathematics at Harvard, became a critic of constructivist programs after his daughter, Sabina, began using one of them, TERC Investigations in Number, Data and Space, at her elementary school in Lincoln, Mass. When she started second grade last fall, Sabina knew how to carry tens and add two-digit numbers, Mr. Schmid said. Sabina's teacher, who is well-intentioned but too inexperienced to deviate from the program, Mr. Schmid said, told the child that she was not allowed to use this method; she had to demonstrate her work with blocks or by counting on her fingers.

"So Sabina is reduced to drawing 39 little men to solve problems like 39-14," her father said.

He worries that this rudimentary and tedious approach is quashing Sabina's spirit. "Last year, she would have complained that this is below her level," Mr. Schmid said, "but she doesn't rebel anymore."

"I'm a professional mathematician, and I myself very often use mathematical methods that I understand only imprecisely," he said. "It is while I use them that I begin to understand. After a while, the use and the understanding are mutually supporting."

In their worst nightmares, parents fear that schools are producing a lost generation of math illiterate children. Bruce Winokur, a math teacher at Stuyvesant High School, New York City's most selective public school, says he is seeing more and more students who are gifted in math but unable to keep up with high school work. They understand concepts, he said, but have not internalized the rules.

THE FUTURE
Easing the Rules to Allow the Old

There are some signs of change.

Andrew Lachman, a spokesman for District 2 in New York City, said the district was responding to parent concerns. "We are not purists," he said.

California recently adopted new standards with a stronger focus on computation, and many districts will be putting them into place this fall. Some teachers, often the most experienced, have instinctively combined the old and the new.

The Daniel Boone School, in West Ridge, a tidy working class part of Chicago brightened by magnolia trees and the babushkas of Russian grandmothers, has been a laboratory for the development of TIMS Math Trailblazers, a constructivist program created by the University of Illinois. Math scores have risen since the program was put into effect. The principal, Paul Zavitkovsky, credits the program, but does not rule out increased attention to math, teacher training and collaboration.

In fifth grade the other day, Mila Kell, a Russian immigrant, taught a crisp lesson in probability, improvising riffs on the probability that the sun would rise in the morning and that she would fly to the moon. The class was enchanted.

Mrs. Kell said she loved the freedom and creativity of the new math. But on her desk was a secret weapon: a stack of worksheets -- the antithesis of constructivist math -- pages of classic problems in long division, the addition of fractions and reducing the sum of fractions to its simplest terms.

[Martinus]

What is the typical US math curriculum like?

For me, it was the basics for 1st-5th grade.  Pre-algebra and geometry for 6th-8th with Algebra 1/2, Geometry, Algebra 3 and Trigonometry courses through high school.

Uhm, does it mean you didn't have geometry until high school? Wtf.

I remember we were calculating volumes of spheres and cones by the fourth grade. 

[CountDeMoney]

That NYT article made me sick to my stomach.

Math should be "flexible," the standards say, and "reasonable" answers should be valued over a single right answer.

W + T = F?

[CountDeMoney]

Uhm, does it mean you didn't have geometry until high school? Wtf.

I remember we were calculating volumes of spheres and cones by the fourth grade. 

And a lot of good it's done for Eastern European competitiveness.

[Josquius]

An example I found:

I think the second way is probally more how I would do it.
Much quicker and takes less paper.

[Martinus]

An example I found:

I think the second way is probally more how I would do it.
Much quicker and takes less paper.

Err, how does it take less paper? The examples there show that it takes more paper. Are you retarded?

[Josquius]

Err, how does it take less paper? The examples there show that it takes more paper. Are you retarded?

Don't be retarded.
They're giving an example, they have to show everything. In practice you could just remember a number in your head though. (not that its really significant, I wasn't being entirely serious with that bit)

[Habbaku]

Uhm, does it mean you didn't have geometry until high school? Wtf.

I remember we were calculating volumes of spheres and cones by the fourth grade. 

What does "Pre-algebra and geometry for 6th-8th" mean to you?