Los Angeles Times
Sunday, February 11, 2001

MATH LESSONS: BEYOND RHETORIC, STUDIES IN HIGH ACHIEVEMENT

By David Klein

Schools with low-income students tend to have low test scores. Low academic achievement, especially in mathematics, is often one of the consequences of poverty. Nevertheless, some schools beat the odds.

Bennett-Kew Elementary School in Inglewood is an example. At Bennett-Kew 51% of the students are African American, 48% are Latino, 29% are not fluent in English and 77% of all students qualify for free or reduced-price lunch, a standard measure of poverty in schools. Yet test scores at Bennett-Kew require no excuses. The average third-grader at Bennett-Kew scored at the 83rd percentile in mathematics on the most recent Stanford Achievement Test, double the score for Los Angeles Unified School District.

In the summer of 2000, the Brookings Institution, a Washington, D.C., think tank, commissioned me to find three high-achieving, low-income schools in the Los Angeles area, and to write a report about how they teach math. That report is available from www.mathematicallycorrect.com. In addition to Bennett-Kew, the report describes William H. Kelso Elementary School, also in Inglewood, and Robert Hill Lane Elementary School in Monterey Park, part of LAUSD. Students at these outstanding schools also exhibit unusually high achievement in mathematics despite modest resources.

What accounts for the high academic achievement of these schools? Can their successes be replicated?

For starters, consider how they are alike. All three closely follow the California mathematics content standards. Direct instruction, as opposed to “student discovery,” is the primary mode of instruction. All three schools focus on basic skills as prerequisites to problem solving and understanding of concepts. Calculator use is rare or nonexistent. Faculty at all three schools are well-coordinated and work together. Principals at these schools are strong leaders, and they are careful to hire dedicated teachers. The principals have found that noncredentialed teachers are sometimes better than credentialed teachers. All three schools have programs that provide remediation, and the principals closely monitor student achievement. But the most important characteristic of all three schools is that students are held to high expectations. The principals were adamant about high expectations and dismissive of excuses.

These days almost everyone uses buzzwords like “high expectations.” But Nancy Ichinaga, the former principal of Bennett-Kew and now a member of the California State Board of Education, took her students beyond the rhetoric of these words to their actual substance. The same may be said for retired principal Marjorie Thompson of Kelso and principal Sue Wong of Lane Elementary.

What prevents hundreds of L.A. schools from following suit? Part of the answer is that ideology trumps common sense in LAUSD. School administrators have long believed that “learning styles” are strongly correlated with race and gender, and that “dead white male math” is just not appropriate for minority students. As a consequence, the LAUSD board decided last year to prevent its elementary schools from buying the successful but traditional math program used at Bennett-Kew, called Saxon Math. This California state-approved curriculum is also a component of the math program at Melvin Elementary School in Reseda. Melvin, an LAUSD campus, was highlighted in Gov. Gray Davis’ State of the State speech for its dramatic improvement in test scores during the last two years.

So, what has LAUSD deemed appropriate for minority students? Following recommendations of the Los Angeles-based Achievement Council, LAUSD last year left hundreds of schools saddled with vacuous calculator-based, anti-arithmetic programs like MathLand, which is not even remotely aligned to the state standards upon which students are tested.

Perhaps the worst blunder is yet to come. Instead of focusing on California’s standards, written by world-renowned mathematicians at Stanford University, LAUSD Supt. Roy Romer is now promoting standards from the National Center on Education and the Economy, or NCEE. These standards are inconsistent with the California standards. They are faddish, low level and incoherent. Judy Codding, a vice president of the NCEE, made no secret of her organization’s hostility to California’s rigorous standards when she announced at an NCEE conference, “I will fight to see that California math standards are not implemented in the classroom.”

She might succeed. If teachers are forced to serve two contradictory masters, the high-caliber California standards and the dubious NCEE standards, the result will be more confusion and misdirection. Although LAUSD deserves some praise for recent steps to purchase state-approved textbooks, school board members should put an end to the continual bombardment of students and teachers with the latest education fads. It is far more constructive to maintain clarity of purpose, and to join successful schools that follow the state standards.

David Klein (david.klein@csun.edu) is a Professor of Mathematics at Cal State Northridge

Copyright 2001 Los Angeles Times

Stand and Deliver Revisited       July 2002

The untold story behind the famous rise — and shameful fall — of Jaime Escalante, America’s master math teacher.

By Jerry Jesness

 

Thanks to the popular 1988 movie Stand and Deliver, many Americans know of the success that Jaime Escalante and his students enjoyed at Garfield High School in East Los Angeles. During the 1980s, that exceptional teacher at a poor public school built a calculus program rivaled by only a handful of exclusive academies.

It is less well-known that Escalante left Garfield after problems with colleagues and administrators, and that his calculus program withered in his absence. That untold story highlights much that is wrong with public schooling in the United States and offers some valuable insights into the workings — and failings — of our education system.

Escalante’s students surprised the nation in 1982, when 18 of them passed the Advanced Placement calculus exam. The Educational Testing Service found the scores suspect and asked 14 of the passing students to take the test again. Twelve agreed to do so (the other two decided they didn’t need the credit for college), and all 12 did well enough to have their scores reinstated.

In the ensuing years, Escalante’s calculus program grew phenomenally. In 1983 both enrollment in his class and the number of students passing the A.P. calculus test more than doubled, with 33 taking the exam and 30 passing it. In 1987, 73 passed the test, and another 12 passed a more advanced version (“BC”) usually given after the second year of calculus.

By 1990, Escalante’s math enrichment program involved over 400 students in classes ranging from beginning algebra to advanced calculus. Escalante and his fellow teachers referred to their program as “the dynasty,” boasting that it would someday involve more than 1,000 students.

That goal was never met. In 1991 Escalante decided to leave Garfield. All his fellow math enrichment teachers soon left as well. By 1996, the dynasty was not even a minor fiefdom. Only seven students passed the regular (“AB”) test that year, with four passing the BC exam — 11 students total, down from a high of 85.

In any field but education, the combination of such a dramatic rise and such a precipitous fall would have invited analysis. If a team begins losing after a coach is replaced, sports fans are outraged. The decline of Garfield’s math program, however, went largely unnoticed.

Movie Magic

Most of us, educators included, learned what we know of Escalante’s experience from Stand and Deliver. For more than a decade it has been a staple in high school classes, college education classes, and faculty workshops. Unfortunately, too many students and teachers learned the wrong lesson from the movie.

Escalante tells me the film was 90 percent truth and 10 percent drama — but what a difference 10 percent can make. Stand and Deliver shows a group of poorly prepared, undisciplined young people who were initially struggling with fractions yet managed to move from basic math to calculus in just a year. The reality was far different. It took 10 years to bring Escalante’s program to peak success. He didn’t even teach his first calculus course until he had been at Garfield for several years. His basic math students from his early years were not the same students who later passed the A.P. calculus test.

Escalante says he was so discouraged by his students’ poor preparation that after only two hours in class he called his former employer, the Burroughs Corporation, and asked for his old job back. He decided not to return to the computer factory after he found a dozen basic math students who were willing to take algebra and was able to make arrangements with the principal and counselors to accommodate them.

Escalante’s situation improved as time went by, but it was not until his fifth year at Garfield that he tried to teach calculus. Although he felt his students were not adequately prepared, he decided to teach the class anyway in the hope that the existence of an A.P. calculus course would create the leverage necessary to improve lower-level math classes.

His plan worked. He and a handpicked teacher, Ben Jimenez, taught the feeder courses. In 1979 he had only five calculus students, two of whom passed the A.P. test. (Escalante had to do some bureaucratic sleight of hand to be allowed to teach such a tiny class.) The second year, he had nine calculus students, seven of whom passed the test. A year later, 15 students took the class, and all but one passed. The year after that, 1982, was the year of the events depicted in Stand and Deliver.

The Stand and Deliver message, that the touch of a master could bring unmotivated students from arithmetic to calculus in a single year, was preached in schools throughout the nation. While the film did a great service to education by showing what students from disadvantaged backgrounds can achieve in demanding classes, the Hollywood fiction had at least one negative side effect. By showing students moving from fractions to calculus in a single year, it gave the false impression that students can neglect their studies for several years and then be redeemed by a few months of hard work.

This Hollywood message had a pernicious effect on teacher training. The lessons of Escalante’s patience and hard work in building his program, especially his attention to the classes that fed into calculus, were largely ignored in the faculty workshops and college education classes that routinely showed Stand and Deliver to their students. To the pedagogues, how Escalante succeeded mattered less than the mere fact that he succeeded. They were happy to cheer Escalante the icon; they were less interested in learning from Escalante the teacher. They were like physicians getting excited about a colleague who can cure cancer without wanting to know how to replicate the cure.

 

The Secrets to His Success

How did Escalante attain such success at Garfield? One key factor was the support of his principal, Henry Gradillas.

Escalante’s program was already in place when Gradillas came to Garfield, but the new principal’s support allowed it to run smoothly. In the early years, Escalante had met with some resistance from the school administration. One assistant principal threatened to have him dismissed, on the grounds that he was coming in too early (a janitor had complained), keeping students too late, and raising funds without permission. Gradillas, on the other hand, handed Escalante the keys to the school and gave him full control of his program.

Gradillas also worked to create a more serious academic environment at Garfield. He reduced the number of basic math classes and eventually came up with a requirement that those who take basic math must concurrently take algebra. He even braved the wrath of the community by denying extracurricular activities to entering students who failed basic skills tests and to current students who failed to maintain a C average.

In the process of raising academic standards at Garfield, Gradillas made more than a few enemies. He took a sabbatical leave to finish his doctorate in 1987, hoping that upon his return he would either be reinstated as principal of Garfield or be given a position from which he could help other schools foster programs like Escalante’s. He was instead assigned to supervise asbestos removal. It is probably no coincidence that A.P. calculus scores at Garfield peaked in 1987, Gradillas’ last year there.

Escalante remained at Garfield for four years after Gradillas’ departure. Although he does not blame the ensuing administration for his own departure from the school, Escalante observes that Gradillas was an academic principal, while his replacement was more interested in other things, such as football and the marching band.

Gradillas was not the only reason for Escalante’s success, of course. Other factors included:

The Pipeline. Unlike the students in the movie, the real Garfield students required years of solid preparation before they could take calculus. This created a problem for Escalante. Garfield was a three-year high school, and the junior high schools that fed it offered only basic math. Even if the entering sophomores took advanced math every year, there was not enough time in their schedules to take geometry, algebra II, math analysis, trigonometry, and calculus.

So Escalante established a program at East Los Angeles College where students could take these classes in intensive seven-week summer sessions. Escalante and Gradillas were also instrumental in getting the feeder schools to offer algebra in the eighth and ninth grades.

Inside Garfield, Escalante worked to ratchet up standards in the classes that fed into calculus. He taught some of the feeder classes himself, assigning others to handpicked teachers with whom he coordinated and reviewed lesson plans. By the time he left, there were nine Garfield teachers working in his math enrichment program and several teachers from other East L.A. high schools working in the summer program at the college.

Tutoring. Years ago, when asked if Garfield could ever catch up to Beverly Hills High School, Gradillas responded, “No, but we can get close.” The children of wealthy, well-educated parents do enjoy advantages in school. Escalante did whatever he could to bring some of those advantages to his students.

Among the parents of Garfield students, high school graduates were in the minority and college graduates were a rarity. To help make up for the lack of academic support available at home, Escalante established tutoring sessions before and after school. When funds became available, he arranged for paid student tutors to help those who fell behind.

Escalante’s field-leveling efforts worked. By 1987, Gradillas’ prediction proved to be partially wrong: In A.P. calculus, Garfield had outpaced Beverly High.

Open Enrollment. Escalante did not approve of programs for the gifted, academic tracking, or even qualifying examinations. If students wanted to take his classes, he let them.

His open-door policy bore fruit. Students who would never have been selected for honors classes or programs for the gifted chose to enroll in Escalante’s math enrichment classes and succeeded there.

Of course, not all of Escalante’s students earned fives (the highest score) on their A.P. calculus exams, and not all went on to receive scholarships from top universities. One argument that educrats make against programs like Escalante’s is that they are elitist and benefit only a select few.

Conventional pedagogical wisdom holds that the poor, the disadvantaged, and the “culturally different” are a fragile lot, and that the academic rigor usually found only in elite suburban or private schools would frustrate them, crushing their self-esteem. The teachers and administrators that I interviewed did not find this to be true of Garfield students.

Wayne Bishop, a professor of mathematics and computer science at California State University at Los Angeles, notes that Escalante’s top students generally did not attend Cal State. Those who scored fours and fives on the A.P. calculus tests were at schools like MIT, Harvard, Yale, Berkeley, USC, and UCLA. For the most part, Escalante grads who went to Cal State-L.A. were those who scored ones and twos, with an occasional three, or those who worked hard in algebra and geometry in the hope of getting into calculus class but fell short.

Bishop observes that these students usually required no remedial math, and that many of them became top students at the college. The moral is that it is better to lose in the Olympics than to win in Little League, even for those whose parents make less than $20,000 per year.

Death of a Dynasty

Escalante’s open admission policy, a major reason for his success, also paved the way for his departure. Calculus grew so popular at Garfield that classes grew beyond the 35-student limit set by the union contract. Some had more than 50 students. Escalante would have preferred to keep the classes below the limit had he been able to do so without either denying calculus to willing students or using teachers who were not up to his high standards. Neither was possible, and the teachers union complained about Garfield’s class sizes. Rather than compromise, Escalante moved on.

Other problems had been brewing as well. After Stand and Deliver was released, Escalante became an overnight celebrity. Teachers and other interested observers asked to sit in on his classes, and he received visits from political leaders and celebrities, including President George H.W. Bush and actor Arnold Schwarzenegger. This attention aroused feelings of jealousy. In his last few years at Garfield, Escalante even received threats and hate mail. In 1990 he lost the math department chairmanship, the position that had enabled him to direct the pipeline.

A number of people at Garfield still have unkind words for the school’s most famous instructor. One administrator tells me Escalante wanted too much power. Some teachers complained that he was creating two math departments, one for his students and another for everyone else. When Escalante quit his job at Garfield, John Perez, a vice president of the teachers union, said, “Jaime didn’t get along with some of the teachers at his school. He pretty much was a loner.”

In addition, Escalante’s relationship with his new principal, Maria Elena Tostado, was not as good as the one he had enjoyed with Gradillas. Tostado speaks harshly about her former calculus teachers, telling the Los Angeles Times they’re disgruntled former employees. Of their complaints, she said, “Such backbiting only hurts the kids.”

Escalante left the program in the charge of a handpicked successor, fellow Garfield teacher Angelo Villavicencio. Escalante had met Villavicencio six years previously through his students — he had been a math teacher at Griffith Junior High, a Garfield feeder. At Escalante’s request and with Gradillas’ assistance, Villavicencio came to Garfield in 1985. At first he taught the classes that fed into calculus; later, he joined Escalante and Ben Jimenez in teaching calculus itself.

When Escalante and Jimenez left in 1991, Villavicencio ascended to Garfield’s calculus throne. The following year he taught all of Garfield’s AB calculus students — 107 of them, in two sections. Although that year’s passing rate was not as high as it had been in previous years, it was still impressive, particularly considering that two-thirds of the calculus teachers had recently left and that Villavicencio was working with lecture-size classes. Seventy-six of his students went on to take the A.P. exam, and 47 passed.

That year was not easy for Villavicencio. The class-size problem that led to Escalante’s departure had not been resolved. Villavicencio asked the administration to add a third section of calculus so he could get his class sizes below 40, but his request was denied. The principal attempted to remove him from Music Hall 1, the only room in the school that could comfortably ac-commodate 55 students. Villavicencio asked himself, “Am I going to have a heart attack defending the program?” The following spring he followed Escalante out Garfield’s door.

Scattered Legacy

When Cal State’s Wayne Bishop called Garfield to ask about the status of the school’s post-Escalante A.P. calculus program, he was told, “We were doing fine before Mr. Escalante left, and we’re doing fine after.” Soon Garfield discovered how critical Escalante’s presence had been. Within a few years, Garfield experienced a sevenfold drop in the number of A.P. calculus students passing their exams. (That said, A.P. participation at Garfield is still much, much higher than at most similar schools. In May of 2000, 722 Garfield students took Advanced Placement tests, and 44 percent passed.)

Escalante moved north to Sacramento, where he taught math, including one section of calculus, at Hiram Johnson High School. He calls his experience there a partial success. In 1991, the year before he began, only six Johnson students took the A.P. calculus exam, all of whom passed. Three years later, the number passing was up to 18 — a respectable improvement, but no dynasty. It had taken Escalante over a decade to build Garfield’s program. Already in his 60s when he made his move, he did not have a decade to build another powerhouse in new territory.

Meanwhile, Villavicencio moved to Chino, a suburb east of Los Angeles. He had to take a pay cut of more than $7,000, since his new school would pay him for only six of his 13 years in teaching. (Like many districts, the Chino Valley Unified School District had a policy of paying for only a limited number of years of outside experience.) In Chino, Villavicencio again taught A.P. calculus, first in Ayala High School and later in Don Lugo High School.

In 1996 he contacted Garfield’s new principal, Tony Garcia, and offered to come back to help revive the moribund calculus program. He was politely refused, so he stayed at Don Lugo. Villavicencio worked with East Los Angeles College to establish a branch of the Escalante summer school program there. This program, along with more math offerings in the district’s middle schools, allowed Villavicencio to admit even some ninth-graders into his calculus class.

After Villavicencio got his program running smoothly, it was consistently producing A.P. calculus passing scores in the 60 percent to 70 percent range. Buoyed by his success, he requested that his salary be raised to reflect his experience. His request was denied, so he decided to move on to another school. Before he left, Don Lugo High was preparing to offer five sections of AB calculus and one section of BC. In his absence, there were only two sections of AB and no BC.

Meanwhile, after seeing its calculus passing rate drop into the single digits, Garfield is experiencing a partial recovery. In the spring of 2001, 17 Garfield students passed the AB calculus exam, and seven passed the BC. That is better than double the number of students passing a few years ago but less than one-third the number passing during the glory years of Escalante’s dynasty.

And after withering in the absence of its founder, the Escalante program at East Los Angeles College has revived. Program administrator Paul Powers reports that over 1,000 high school students took accelerated math classes through the college in the year 2000.

Although the program now accepts students from beyond the college’s vicinity, the target pupils are still those living in East L.A.

Nationally, there is no denying that the Escalante experience was a factor in the growth of Advanced Placement courses during the last decade and a half. The number of schools that offer A.P. classes has more than doubled since 1983, and the number of A.P. tests taken has increased almost sixfold. This is a far cry from the Zeitgeist of two decades ago, when A.P. was considered appropriate only for students in elite private and wealthy suburban public schools.

Still, there is no inner-city school anywhere in the United States with a calculus program anything like Escalante’s in the ’80s. A very successful program rapidly collapsed, leaving only fragments behind.

This leaves would-be school reformers with a set of uncomfortable questions. Why couldn’t Escalante run his classes in peace? Why were administrators allowed to get in his way? Why was the union imposing its “help” on someone who hadn’t requested it? Could Escalante’s program have been saved if, as Gradillas now muses, Garfield had become a charter school? What is wrong with a system that values working well with others more highly than effectiveness?

Barn Building

Lyndon Johnson said it takes a master carpenter to build a barn, but any jackass can kick one down. In retrospect, it’s fortunate that Escalante’s program survived as long as it did. Had Garfield’s counselors refused to let a handful of basic math students take algebra back in 1974, or had the janitor who objected to Escalante’s early-bird ways been more influential, America’s greatest math teacher might just now be retiring from Unisys.

Gradillas has an explanation for the decline of A.P. calculus at Garfield: Escalante and Villavicencio were not allowed to run the program they had created on their own terms. In his phrase, the teachers no longer “owned” their program. He’s speaking metaphorically, but there’s something to be said for taking him literally.

In the real world, those who provide a service can usually find a way to get it to those who want it, even if their current employer disapproves. If someone feels that he can build a better mousetrap than his employer wants to make, he can find a way to make it, market it, and perhaps put his former boss out of business. Public school teachers lack that option.

There are very few ways to compete for education dollars without being part of the government school system. If that system is inflexible, sooner or later even excellent programs will run into obstacles.

Escalante has retired to his native Bolivia. He is living in his wife’s hometown and teaching part time at the local university. He returns to the United States frequently to visit his children. When I spoke to him he was entertaining the possibility of acting as an adviser to the Bush administration. Given what he achieved, he clearly has valuable advice to give.

Whether the administration will take it is another question. We are being primed for another round of “education reform.” One-size-fits-all standardized tests are driving curricula, and top-down reforms are mandating lockstep procedures for classroom instructors. These steps might help make dismal teachers into mediocre ones, but what will they do to brilliant mavericks like Escalante?

Before passing another law or setting another policy, our reformers should take a close look at what Jaime Escalante did — and at what was done to him.

Jerry Jesness is a special education teacher in Texas’ Lower Rio Grande Valley.

 

Music-Math Analogy

By Nakonia (Niki) Hayes
Columnist EdNews.org

 

Mathematics is the heart of music, so shouldn’t we teach music as constructivist/ reformist mathematics educators insist that children learn that discipline? That is, shouldn’t music students be taught to play by ear?

 

Suppose your child had to learn to play a musical instrument by ear. There would be no focus on the symbols of music, sounds of specific notes, practicing of scales, learning classical pieces, or even learning some standard tunes (“Chop Sticks”) from which creative “extensions” could be made.

The small percentage of those students or teachers who could play an instrument by ear could not help you or your child. The intuitive players wouldn’t know and thus couldn’t translate their innate abilities into the internationally-known music symbols.

So the adopted method for all these “other” students would be called “discovery learning.” They would “manipulate” their instruments with teachers “facilitating” the their efforts in order to discover how to formulate a particular tune, which, of course, they had created themselves.

There would be no continuous practice—no “drill and kill” of repetition. All tunes would be considered acceptable because they were the original, personal creation of each student. Comparisons to respected or classical renditions might be possible, but that would be extremely time consuming, and it would not be considered “relevant” in today’s modern classroom.

Students who needed to learn by the old-fashioned methods, such as studying music symbols, their related sounds, and repetitive practice would need extra tutoring. Supplemental materials might be allowed that taught some “basic skills,” but the bigger picture to learning music, or the conceptual approach, must be maintained.

All of this supplementary material would cost extra money for the schools—and extra time for the students and teachers.

Schools of education that train teachers would insist this “discovery” method of learning music is progressive and provides social justice for girls and students of color in the music profession. They would base much of their beliefs on a few education researchers in the 1970s who had concluded that inductive and intuitive methods–those that focus on process rather than product–were needed by these two “subgroups.”

They assert while traditional music lessons that teach procedures and memorization without understanding may lead to a facility with technique, note reading and instrument mastery, those lessons do not lead to improvisation or playing music with feeling.

Further, with a glowing love for the advent of technology in music – such as computer sampling, electronic instruments, and digital recording technology that can improve the sound, including fixing pitch problems so that all singers sound like they’re on pitch no matter how flat (or sharp) they sing—education schools say music students no longer need to learn the basics of good vocal production, music composition, or even tuning their instruments.

Finally, music education tells teachers that white males and Asian students were the only ones who had benefited from the traditional methods of learning music for the past several thousand years. The progress made in music by the “ancients” and their methods are to be considered of no significance or relevance in the child-directed, “discovery” teaching classroom.

Many elementary school teachers liked the discovery method because it did not require their learning the music symbols and the many complicated relationships that could result from those symbols. High school music teachers hated the discovery method because they had difficulty finding enough qualified students to form a school band, symphony, or choir.

Many parents of elementary students accepted the discovery learning because the students seemed to “enjoy” it and they always had good grades in the subject. After all, the grading was based on subjective judgments about the student’s process of creating his or her own musical piece, and it was not a comparison to another’s work.

The consequence, however, is a growing lack of new musicians. This is impacting, among many music-related scenarios, high school bands, symphonies, and musical productions in theatres. Foreign students who had studied traditional music lessons are becoming the heart of America’s shrinking music scene.

How long before the public refuses to tolerate this destruction of music education and ultimately music’s contribution to society and the world? Will it take five years, 10 years, or 20 years? Will college music teachers stand by quietly as their incoming students’ proficiencies continually disintegrate? Will professional music companies and businesses ignore the shrinking pool of talent? Will business leaders believe the progressive philosophy that insists we must focus on “creativity thinking” and not worry about the significance of foundational work in the music discipline?

Now substitute “mathematics” for “music” and you have a picture of what has been happening in American mathematics education for the past 40 years.

“Whole math,” based on conceptual, intuitive, process-based thinking has replaced traditional mathematics education. (Yes, it is the parallel universe to the “whole language” fiasco that produced two generations of poor readers and writers in American education.)

Algorithms, symbolic manipulation, and basic skills are no longer mastered in elementary mathematics—and therefore in high school classes—because those represent the traditional, classical education formerly reserved only for white males, according to the leaders of “reform mathematics.” The traditional program represents “drill and kill,” they say. Traditionalists say the program offers “drill and skill,” as well as mastery of concepts.

This reform pedagogy was codified in 1989 by a private group called The National Council of Teachers of Mathematics (NCTM) when they published their Curriculum Standards for K-12 mathematics education. The National Science Foundation bought into their ideas, probably due to their emphasis on egalitarianism. From 1991 through 1999, the NSF pumped $83 million into universities and publishers that would create math curricula that supported the reformists’ social engineering agenda.

In 1999, more than 200 professional mathematicians sent a letter to Richard Riley, Education Secretary, asking him to withdraw support for the reform math products, due to their poor quality of mathematics instruction.

He ignored them.

In fact, even more multi-millions have been funneled into the programs from both government and private sources through today.

Educators have latched onto these cash cows as money is offered to “pilot” reform programs and students have become research subjects. Math wars have erupted among parent groups and districts in pockets across the country as parents (and a few teachers) try to change the direction of mathematics education in their schools. Parents are learning, however, that schools really don’t want parent involvement if it means they are going to question curriculum choices.

And test scores continue to show the disintegration of mathematics’ skills among American students.

When educators and businesses wonder why this is happening, they should think about students learning to play music by ear. That’s the real picture of mathematics education today. It’s been going on, officially, for almost two decades.

When will the people who can make a real difference—parents, colleges, and businesses who must look to foreign workers to bring in mathematical skills—conduct a reality check on the “whole math” philosophy?

When will they stop being schmoozed by an education establishment that’s protecting its turf and special interest groups? When will they demand a truthful answer to the question, “Whose interest is being served here?”

In essence, when will our children have advocates who understand proven mathematical logic and reasoning with regards to performance and product?

Or, will we continue to follow the false concept that equity and excellence can be achieved by everyone learning to play by ear.

Published July 19, 2007

 

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MATH PROBLEMS
Why the U.S. Department of Education’s
recommended math programs don’t add up

By David Klein

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What constitutes a good K-12 mathematics program? Opinions differ. In October 1999, the U.S. Department of Education released a report designating 10 math programs as “exemplary” or “promising.” The following month, I sent an open letter to Education Secretary Richard W. Riley urging him to withdraw the department’s recommendations. The letter was coauthored by Richard Askey of the University of Wisconsin at Madison, R. James Milgram of Stanford University, and Hung-Hsi Wu of the University of California at Berkeley, along with more than 200 other cosigners. With financial backing from the Packard Humanities Institute, we published the letter as a full-page ad in the Washington Post on Nov. 18, 1999, with as many of the endorsers’ names and affiliations as would fit on the page. Among them are many of the nation’s most accomplished scientists and mathematicians. Department heads at more than a dozen universities–including Caltech, Stanford, and Yale–along with two former presidents of the Mathematical Association of America also added their names in support. With new endorsements since publication, there are now seven Nobel laureates and winners of the Fields Medal, the highest award in mathematics. The open letter was covered by several newspapers and journals, including American School Board Journal (February, page 16).

Although a clear majority of cosigners are mathematicians and scientists, it is sometimes overlooked that experienced education administrators at the state and national level, as well as educational psychologists and education researchers, also endorsed the letter. (A complete list is posted at http://www.mathematicallycorrect.com.)

University professors and public education leaders are not the only ones who have reservations about these programs. Thousands of parents and teachers across the nation seek alternatives to them, often in opposition to local school boards and superintendents. Mathematically Correct, an influential Internet-based parents’ organization, came into existence several years ago because the children of the organization’s founders had no alternative to the now “exemplary” program, College Preparatory Mathematics, or CPM. In Plano, Texas, 600 parents are suing the school district because of its exclusive use of the Connected Mathematics Project, or CMP, another “exemplary” program. I have received hundreds of requests for help by parents and teachers because of these and other programs now promoted by the Education Department (ED). In fact, it was such pleas for help that motivated me and my three coauthors to write the open letter.

Common problems

The mathematics programs criticized by the open letter have common features. For example, they tend to overemphasize data analysis and statistics, which typically appear year after year, with redundant presentations. The far more important areas of arithmetic and algebra are radically de-emphasized. Many of the so-called higher-order thinking projects are just aimless activities, and genuine illumination of important mathematical ideas is rare. There is a near obsession with calculators, and basic skills are given short shrift and sometimes even disparaged. Overall, these curricula are watered-down math programs. The same educational philosophy that gave rise to the whole-language approach to reading is part of ED’s agenda for mathematics. Systematic development of skills and concepts is replaced by an unstructured “holism.” In fact, during the mid-’90s, supporters of programs like these referred to their approach as “whole math.”

Disagreements over math curricula are often portrayed as “basic skills versus conceptual understanding.” Scientists and mathematicians, including many who signed the open letter to Secretary Riley, are described as advocates of basic skills, while professional educators are counted as proponents of conceptual understanding. Ironically, such a portrayal ignores the deep conceptual understanding of mathematics held by so many mathematicians. But more important, the notion that conceptual understanding in mathematics can be separated from precision and fluency in the execution of basic skills is just plain wrong.

In other domains of human activity, such as athletics or music, the dependence of high levels of performance on requisite skills goes unchallenged. A novice cannot hope to achieve mastery in the martial arts without first learning basic katas or exercises in movement. A violinist who has not mastered elementary bowing techniques and vibrato has no hope of evoking the emotions of an audience through sonorous tones and elegant phrasing. Arguably the most hierarchical of human endeavors, mathematics also depends on sequential mastery of basic skills.

The standard algorithms

The standard algorithms for arithmetic (that is, the standard procedures for addition, subtraction, multiplication, and division of numbers) are missing or abridged in ED’s recommended elementary school curricula. These omissions are inconsistent with the mainstream views of mathematicians.

In our open letter to Secretary Riley, we included an excerpt from a committee report published in the February 1998 Notices of the American Mathematical Society. The committee was appointed by the American Mathematical Society to advise the National Council of Teachers of Mathematics (NCTM). Part of its report discusses the standard algorithms of arithmetic. “We would like to emphasize that the standard algorithms of arithmetic are more than just ‘ways to get the answer’–that is, they have theoretical as well as practical significance,” the report states. “For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not by accident, but by virtue of the construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials.”

This statement deserves elaboration. How could the standard algorithms of arithmetic be related to algebra? For concreteness, consider the meaning in terms of place value of 572:

572 = 5 (10²) + 7(10) + 2

Now compare the right side of this equation to the polynomial,

5x² + 7x + 2.

The two are identical when x = 10. This connection between whole numbers and polynomials is general and extends to arithmetic operations. Addition, subtraction, multiplication, and division of polynomials is fundamentally the same as for whole numbers. In arithmetic, extra steps such as “regrouping” are needed since x = 10 allows for simplifications. The standard algorithms incorporate both the polynomial operations and the extra steps to account for the specific value, x = 10. Facility with the standard operations of arithmetic, together with an understanding of why these algorithms work, is important preparation for algebra.

The standard long division algorithm is particularly shortchanged by the “promising” curricula. It is preparatory for division of polynomials and, at the college level, division of “power series,” a useful technique in calculus and differential equations. The standard long division algorithm is also needed for a middle school topic. It is fundamental to an understanding of the difference between rational and irrational numbers, an indisputable example of conceptual understanding. It is essential to understand that rational numbers (that is, ratios of whole numbers like 3/4) and their negatives have decimal representations that exhibit recurring patterns. For example: 1/3 = .333…, where the ellipses indicate that the numeral 3 repeats forever. Likewise, 1/2 = .500… and 611/4950 = .12343434….

In the last equation, the digits 34 are repeated without end, and the repeating block in the decimal for 1/2 consists only of the digit for zero. It is a general fact that all rational numbers have repeating blocks of numerals in their decimal representations, and this can be understood and deduced by students who have mastered the standard long division algorithm. However, this important result does not follow easily from other “nonstandard” division algorithms featured by some of ED’s model curricula.

A different but still elementary argument is required to show the converse–that any decimal with a repeating block is equal to a fraction. Once this is understood, students are prepared to understand the meaning of the term “irrational number.” Irrational numbers are the numbers represented by infinite decimals without repeating blocks. In California, seventh-grade students are expected to understand this.

It is worth emphasizing that calculators are utterly useless in this context, not only in establishing the general principles, but even in logically verifying the equations. This is partly because calculator screens cannot display infinite decimals, but more important, calculators cannot reason. The “exemplary” middle school curriculum CMP nevertheless ignores the conceptual issues, bypassing the long division algorithm and substituting calculators and faulty inductive reasoning instead.

Steven Leinwand of the Connecticut Department of Education was a member of the expert panel that made final decisions on ED’s “exemplary” and “promising” math curricula. He was also a member of the advisory boards for two programs found to be “exemplary” by the panel: CMP and the Interactive Mathematics Program. In a Feb. 9, 1994, article in Education Week, he wrote: “It’s time to recognize that, for many students, real mathematical power, on the one hand, and facility with multidigit, pencil-and-paper computational algorithms, on the other, are mutually exclusive. In fact, it’s time to acknowledge that continuing to teach these skills to our students is not only unnecessary, but counterproductive and downright dangerous.”

Mr. Leinwand’s influential opinions are diametrically opposed to the mainstream views of practicing scientists and mathematicians, as well as the general public, but they have found fertile soil in the government’s “promising” and “exemplary” curricula.

Calculators

According to the Third International Mathematics and Science Study, or TIMSS, the use of calculators in U.S. fourth-grade mathematics classes is about twice the international average. Teachers of 39 percent of U.S. students report that students use calculators at least once or twice a week. In six of the seven top-scoring nations, on the other hand, teachers of 85 percent or more of the students report that students never use calculators in class.

Even at the eighth-grade level, the majority of students from three of the top five scoring nations in the TIMSS study (Belgium, Korea, and Japan) never or rarely use calculators in math classes. In Singapore, which is also among the top five scoring countries, students do not use calculators until the seventh grade. Among the lower achieving nations, however, the majority of students from 10 of the 11 nations with scores below the international average–including the United States–use calculators almost every day or several times a week.

Of course, this negative correlation of calculator usage with achievement in mathematics does not imply a causal relationship. There are many variables that contribute to achievement in mathematics. On the other hand, it is foolhardy to ignore the problems caused by calculators in schools. In a Sept. 17, 1999, Los Angeles Times editorial titled “L.A.’s Math Program Just Doesn’t Add Up,” Milgram and I recommended that calculators not be used at all in grades K-5 and only sparingly in higher grades. Certainly there are isolated, beneficial uses for calculators, such as calculating compound interest, a seventh-grade topic in California. Science classes benefit from the use of calculators because it is necessary to deal with whatever numbers nature gives us, but conceptual understanding in mathematics is often best facilitated through the use of simple numbers. Moreover, fraction arithmetic, an important prerequisite for algebra, is easily undermined by the use of calculators.

Specific shortcomings

A number of the programs on ED’s list have specific shortcomings–many involving use of calculators. For example, a “promising” curriculum called Everyday Mathematics says calculators are “an integral part of Kindergarten Everyday Mathematics” and urges the use of calculators to teach kindergarten students how to count. There are no textbooks in this K-6 curriculum, and even if the program were otherwise sound, this is a serious shortcoming. The standard algorithm for multiplying two numbers has no more status or prominence than an Ancient Egyptian algorithm presented in one of the teacher’s manuals. Students are never required to use the standard long division algorithm in this curriculum, or even the standard algorithm for multiplication.

Calculator use is also ubiquitous in the “exemplary” middle school program CMP. A unit devoted to discovering algorithms to add, subtract, and multiply fractions (“Bits and Pieces II”) gives the inappropriate instruction, “Use your calculator whenever you need it.” These topics are poorly developed, and division of fractions is not covered at all. A quiz for seventh-grade CMP students asks them to find the “slope” and “y-intercept” of the equation 10 = x – 2.5, and the teacher’s manual explains that this equation is a special case of the linear equation y = x – 2.5, when y = 10, and concludes that the slope is therefore 1 and the y-intercept is -2.5. This is not only false, but is so mathematically unsound as to undermine the authority of classroom teachers who know better.

College Preparatory Math (CPM), a high school program, also requires students to use calculators almost daily. The principal technique in this series is the so-called guess-and-check method, which encourages repeated guessing of answers over the systematic development of standard mathematical techniques. Because of the availability of calculators that can solve equations, the introduction to the series explains that CPM puts low emphasis on symbol manipulation and that CPM differs from traditional mathematics courses both in the mathematics that is taught and how it is taught. In one section, students watch a candle burn down for an hour while measuring its length versus the time and then plotting the results. In a related activity, students spend a whole class period on the athletic field making human coordinate graphs. These activities are typical of the time sacrificed to simple ideas that can be understood more efficiently through direct explanation. But in CPM, direct instruction is systematically discouraged in favor of group work. Teachers are told that as “rules of thumb,” they should “never carry or grab a writing implement” and they should “usually respond with a question.” Algebra tiles are used frequently, and the important distributive property is poorly presented and underemphasized.

Another program, Number Power–a “promising” curriculum for grades K-6–was submitted to the California State Board of Education for adoption in California. Two Stanford University mathematics professors serving on the state’s Content Review Panel wrote a report on the program that is now a public document. Number Power, they wrote, “is meant as a partial program to supplement a regular basic program. There is a strong emphasis on group projects–almost the entire program. Heavy use of calculators. Even as a supplementary program, it provides such insufficient coverage of the [California] Standards that it is unacceptable. This holds for all grade levels and all strands, including Number Sense, which is the only strand that is even partially covered.”

The report goes on to note, “It is explicitly stated that the standard algorithms for addition, subtraction, and multiplication are not taught.” Like CMP and Everyday Math, Number Power was rejected for adoption by the state of California.

Interactive Mathematics Program, or IMP, an “exemplary” high school curriculum, has such a weak treatment of algebra that the quadratic formula, normally an eighth- or ninth-grade topic, is postponed until the 12th grade. Even though probability and statistics receive greater emphasis in this program, the development of these topics is poor. “Expected value,” a concept of fundamental importance in probability and statistics, is never even correctly defined. The Teacher’s Guide for “The Game of Pig,” where expected value is treated, informs teachers that “expected value is one of the unit’s primary concepts,” yet teachers are instructed to tell their students that “the concept of expected value is nothing new … [but] the use of such complex terminology makes it easier to state complex ideas.” (For a correlation of lowered SAT scores with the use of IMP, see Milgram’s paper at ftp://math.stanford.edu/pub/papers/milgram.)

Core-Plus Mathematics Project is another “exemplary” high school program that radically de-emphasizes algebra, with unfortunate results. Even Hyman Bass–a well-known supporter of NCTM-aligned programs and a harsh critic of the open letter to Secretary Riley–has conceded the program has problems. “I have some reservations about Core Plus, for what I consider too shallow a coverage of traditional algebra, and a focus on highly contextualized work that goes beyond my personal inclinations,” he wrote in a nationally circulated e-mail message. “These are only my personal views, and I do not know about its success with students.”

Milgram analyzed the program’s effect on students in a top-performing high school in “Outcomes Analysis for Core Plus Students at Andover High School: One Year Later,” based on a statistical study by G. Bachelis of Wayne State University. According to Milgram, “…there was no measure represented in the survey, such as ACT scores, SAT Math scores, grades in college math courses, level of college math courses attempted, where the Andover Core Plus students even met, let alone surpassed the comparison group [which used a more traditional program].”

And then there is MathLand, a K-6 curriculum that ED calls “promising” but that is perhaps the most heavily criticized elementary school program in the nation. Like Everyday Math, it has no textbooks for students in any of the grades. The teacher’s manual urges teachers not to teach the standard algorithms of arithmetic for addition, subtraction, multiplication, and division. Rather, students are expected to invent their own algorithms. Numerous and detailed criticisms, including data on lowered test scores, appear at http://www.mathematicallycorrect.com.

How could they be so wrong?

Perhaps Galileo wondered similarly how the church of Pope Urban VIII could be so wrong. The U.S. Department of Education is not alone in endorsing watered-down, and even defective, math programs. The NCTM has also formally endorsed each of the U.S. Department of Education’s model programs (http://www.nctm.org/rileystatement.htm), and the National Science Foundation (Education and Human Resources Division) funded several of them. How could such powerful organizations be wrong?

These organizations represent surprisingly narrow interests, and there is a revolving door between them. Expert panel member Steven Leinwand, whose personal connections with “exemplary” curricula have already been noted, is also a member of the NCTM board of directors. Luther Williams, who as assistant director of the NSF approved the funding of several of the recommended curricula, also served on the expert panel that evaluated these same curricula. Jack Price, a member of the expert panel is a former president of NCTM, and Glenda Lappan, the association’s current president, is a coauthor of the “exemplary” program CMP.

Aside from institutional interconnections, there is a unifying ideology behind “whole math.” It is advertised as math for all students, as opposed to only white males. But the word all is a code for minority students and women (though presumably not Asians). In 1996, while he was president of NCTM, Jack Price articulated this view in direct terms on a radio show in San Diego: “What we have now is nostalgia math. It is the mathematics that we have always had, that is good for the most part for the relatively high socioeconomic anglo male, and that we have a great deal of research that has been done showing that women, for example, and minority groups do not learn the same way. They have the capability, certainly, of learning, but they don’t. The teaching strategies that you use with them are different from those that we have been able to use in the past when … we weren’t expected to graduate a lot of people, and most of those who did graduate and go on to college were the anglo males.”

Price went on to say: “All of the research that has been done with gender differences or ethnic differences has been–males for example learn better deductively in a competitive environment, when–the kind of thing that we have done in the past. Where we have found with gender differences, for example, that women have a tendency to learn better in a collaborative effort when they are doing inductive reasoning.” (A transcript of the show is online at (http://mathematicallycorrect.com/roger.htm.)

I reject the notion that skin color or gender determines whether students learn inductively as opposed to deductively and whether they should be taught the standard operations of arithmetic and essential components of algebra. Arithmetic is not only essential for everyday life, it is the foundation for study of higher level mathematics. Secretary Riley–and educators who select mathematics curricula–would do well to heed the advice of the open letter.

David Klein is a professor of mathematics at California State University at Northridge.

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Marks of a good mathematics program

It is impossible to specify all of the characteristics of a sound mathematics program in only a few paragraphs, but a few highlights may be identified. The most important criterion is strong mathematical content that conforms to a set of explicit, high, grade-by-grade standards such as the California or Japanese mathematics standards. A strong mathematics program recognizes the hierarchical nature of mathematics and builds coherently from one grade to the next. It is not merely a sequence of interesting but unrelated student projects.

In the earlier grades, arithmetic should be the primary focus. The standard algorithms of arithmetic for integers, decimals, fractions, and percents are of central importance. The curriculum should promote facility in calculation, an understanding of what makes the algorithms work in terms of the base 10 structure of our number system, and an understanding of the associative, commutative, and distributive properties of numbers. These properties can be illustrated by area and volume models. Students need to develop an intuitive understanding for fractions. Manipulatives or pictures can help in the beginning stages, but it is essential that students eventually be able to compute easily using mathematical notation. Word problems should be abundant. A sound program should move students toward abstraction and the eventual use of symbols to represent unknown quantities.

In the upper grades, algebra courses should emphasize powerful symbolic techniques and not exploratory guessing and calculator-based graphical solutions.

There should be a minimum of diversions in textbooks. Children have enough trouble concentrating without distracting pictures and irrelevant stories and projects. A mathematics program should explicitly teach skills and concepts with appropriately designed practice sets. Such programs have the best chance of success with the largest number of students. The high-performing Japanese students spend 80 percent of class time in teacher-directed whole-class instruction. Japanese math books contain clear explanations, examples with practice problems, and summaries of key points. Singapore’s elementary school math books also provide good models. Among U.S. books for elementary school, Sadlier-Oxford’s Progress in Mathematics and the Saxon series through Math 87 (adopted for grade six in California), though not without defects, have many positive features.–D.K.

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For more information

Askey, Richard. “Knowing and Teaching Elementary Mathematics.” American Educator, Fall 1999, pp. 6-13; 49.

Ma, Liping. Knowing and Teaching Elementary Mathematics. Mahwah, N.J.: Lawrence Erlbaum, 1999.

Milgram, R. James. “A Preliminary Analysis of SAT-I Mathematics Data for IMP Schools in California.” ftp://math.stanford.edu/pub/papers/milgram

Milgram, R. James. “Outcomes Analysis for Core Plus Students at Andover High School: One Year Later.” ftp://math.stanford.edu/pub/papers/milgram/andover-report.htm

Wu, Hung-Hsi. “Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education.” American Educator, Fall 1999, pp. 14-19; 50-52.

Why Math Always Counts

Respect   Responsibility   Readiness

 

The following article was published in the Los Angeles Times   12/20/2004

Why Math Always Counts

By Arthur Michelson

American middle school students don’t care that they’re worse at math than their counterparts in Hong Kong or Finland. “I don’t need it,” my students say. “I’m gonna be a basketball star.” Or a beautician, or a car mechanic, or a singer.
It’s hard to get much of a rise out of adults over the fact, released earlier this year, that the United States ranked 28th out of 41 countries whose middle school students’ math skills were tested by the Organization for Economic Cooperation and Development. So what if we tied with Latvia, while nations like Japan and South Korea leave us in the dust? After all, when was the last time you used algebra?
But math is not just about computing quadratic equations, knowing geometric proofs or balancing a check book. And it’s not just about training Americans to become scientists.
It has implicit value. It is about discipline, precision, thorough-
ness and meticulous analysis. It helps you see patterns, develops your logic skills, teaches you to concentrate and to separate truth from falsehood. These are abilities that distinguish successful people.
Math helps you make wise financial decisions, but also informs you so you can avoid false claims for advertisers, politicians and others. It helps you determine risk. Some examples.

It can open our minds to logic and beauty

* If a fair coin is tossed and eight heads come up in a row, most adults would gamble that the next toss would come up tails. But a coin has no memory. There is always a 50-50 chance. See you at the casino?
* If you have no sense of big numbers, you evaluate the consequences of how government spends your money. Why should we worry? Let our kids deal with it…
* Enormous amounts of money are spent on quack medicine. Many people will reject sound scientific studies on drugs or nutrition if the results don’t fit their preconceived notions, yet they might leap to action after reading news stories on the results of small, inconclusive or poorly run studies.
* After an airplane crash, studies show that people are more likely to drive than take a plane despite the fact that they are much more likely to be killed or injured while driving. Planes are not more likely to crash because another recently did. In fact, the most dangerous time to drive is probably right after a plane crash because so many more people are on the road.
The precision of math, like poetry, gets to the heart of things. It can increase our awareness.
Consider the Fibonacci series, in which each number is the sum of the preceding two.(0,1,1,3,5,3,13……..). Comparing each successive pair

yields a relationship known as the Golden Ratio, which often shows up in nature and art. It’s the mathematical underpinning of what we consider beautiful. You’ll find it in the design of the Parthenon and the Mona Lisa, as well as in human proportion; for instance, in the size of the hand compared to the forearm and the forearm to the entire arm. Stephen Hawking’s editor warned him that for every mathematical formula he wrote in a book he would lose a big part of his audience. Yet more than a little is lost by dumbing things down.
It is not possible to really understand science and the scientific method without understanding math. A rainbow is even more beautiful and amazing when we understand it. So is a lightning bolt, an ant, or ourselves.
Math gives us a powerful tool to understand our universe. I don’t wish to overstate. Poetry, music, literature and the fine and performing arts are always gateways to beauty. Nothing we study is a waste. But the precision of math helps refine how we think in a very special way.
How do we revitalize the learning of math? I don’t have the big answer. I teach middle school and try to find an answer one child at a time. When I can get one to say, ”Wow, that’s tight.” I feel the joy of a small victory.

Arthur Michelson teaches at the Beechwood School in Menlo Park, California.

 

Why Education Experts Resist Effective Practices (And What It Would Take to Make Education More Like Medicine)

by Douglas Carnine
04/01/2000

 

 

In perhaps no other profession is there as much disputation as in education. Phonics or whole language? Calculators or no calculators? Tracked or mixed-ability classrooms? Should teachers lecture or “facilitate”? Ought education be content-centered or child-centered? Do high-stakes exams produce real gains or merely promote “teaching to the test”? Which is the most effective reform: Reducing class size? Expanding pre-school? Inducing competition through vouchers? Paying teachers for performance?

And on and on and on. Within each debate, moreover, we regularly hear each faction citing boatloads of “studies” that supposedly support its position. Just think how often “research shows” is used to introduce a statement that winds up being chiefly about ideology, hunch or preference.

In other professions, such as medicine, scientific research is taken seriously, because it usually brings clarity and progress. We come close to resolving vast disputes, and answering complex questions, with the aid of rigorous, controlled studies of cause and effect. Yet so much of what passes for education research serves to confuse at least as much as it clarifies. The education field tends to rely heavily on qualitative studies, sometimes proclaiming open hostility towards modern statistical research methods. Even when the research is clear on a subject—such as how to teach first-graders to read—educators often willfully ignore the results when they don’t fit their ideological preferences.

To Professor Douglas Carnine of the University of Oregon, this is symptomatic of a field that has not yet matured into a true profession. In education, research standards have yet to be standardized, peer reviews are porous, and practitioners tend to be influenced more by philosophy than evidence. In this insightful paper, Doug examines several instances where educators either have introduced reforms without testing them first, or ignored (or deprecated) research when it did not yield the results they wanted.

After describing assorted hijinks in math and reading instruction, Doug devotes considerable space to examining what educators did with the results of Project Follow Through, one of the largest education experiments ever undertaken. This study compared constructivist education models with those based on direct instruction. One might have expected that, when the results showed that direct instruction models produced better outcomes, these models would have been embraced by the profession. Instead, many education experts discouraged their use.

Carnine compares the current state of the education field with medicine and other professions in the early part of the 20th century, and suggests that education will undergo its transformation to a full profession only when outside pressures force it to.

He knows the field well, as Director of the National Center to Improve the Tools of Educators, which works with publishers to incorporate research-based practices into education materials and with legislative, business, community and union groups to understand the importance of research-based tools. Doug can be phoned at 541-683-7543, e-mailed at dcarnine@oregon.uoregon.edu, and written the old fashioned way at 85 Lincoln St., Eugene, OR 97401.

The Thomas B. Fordham Foundation is a private foundation that supports research, publications, and action projects in elementary/secondary education reform at the national level and in the Dayton area. Further information can be obtained at our web site (www.edexcellence.net) or by writing us at 1627 K Street, NW, Suite 600, Washington, DC 20006. (We can also be e-mailed through our web site.) This report is available in full on the Foundation’s web site, and hard copies can be obtained by calling 1-888-TBF-7474 (single copies are free). The Foundation is neither connected with nor sponsored by Fordham University.

 

Introduction

Education school professors in general and curriculum and instruction experts in particular are major forces in dictating the “what” and “how” of American education. They typically control pre-service teacher preparation, the continued professional development of experienced teachers, the curricular content and pedagogy used in schools, the instructional philosophy and methods employed in classrooms, and the policies espoused by state and national curriculum organizations.

Although they wield immense power over what actually happens in U.S. classrooms, these professors are senior members of a field that lacks many crucial features of a fully developed profession. In education, the judgments of “experts” frequently appear to be unconstrained and sometimes altogether unaffected by objective research. Many of these experts are so captivated by romantic ideas about learning or so blinded by ideology that they have closed their minds to the results of rigorous experiments. Until education becomes the kind of profession that reveres evidence, we should not be surprised to find its experts dispensing unproven methods, endlessly flitting from one fad to another. The greatest victims of these fads are the very students who are most at risk.

The first section of this essay provides examples from reading and math curricula. The middle section describes how experts have, for ideological reasons, shunned some solutions that do display robust evidence of efficacy. The following sections briefly examine how public impatience has forced other professions to “grow up” and accept accountability and scientific evidence. The paper concludes with a plea to hasten education’s metamorphosis into a mature profession.

 

Embracing Teaching Methods that Don’t Work

The reaction of a large number of education experts to converging scientific evidence about how children learn to read illustrates the basic problem. Data strongly support the explicit teaching of phonemic awareness, the alphabetic principle, and phonics, which is often combined with extensive practice with phonic readers. These are the cornerstones of successful beginning reading for young children, particularly at-risk youngsters. The findings of the National Reading Panel, established by Congress and jointly convened by the Department of Education and the Department of Health and Human Services, confirm the importance of these practices. Congress asked the panel to evaluate existing research on the most effective approaches for teaching children how to read. In its February 1999 Progress Report, the panel wrote,

[A]dvances in research are beginning to provide hope that educators may soon be guided by scientifically sound information. A growing number of works, for example, are now suggesting that students need to master phonics skills in order to read well. Among them are Learning to Read by Jeanne Chall and Beginning to Read: Thinking and Learning about Print by Marilyn Adams. As Adams, a senior scientist at Bolt Beranek and Newman, Inc., writes, “[It] has been proven beyond any shade of doubt that skillful readers process virtually each and every word and letter of text as they read. This is extremely counter-intuitive. For sure, skillful readers neither look nor feel as if that’s what they do. But that’s because they do it so quickly and effortlessly.1

Even the popular media have recognized this converging body of research. As James Collins wrote in Time magazine in October 1997: “After reviewing the arguments mustered by the phonics and whole-language proponents, can we make a judgment as to who is right? Yes. The value of explicit, systematic phonics instruction has been well established. Hundreds of studies from a variety of fields support this conclusion. Indeed, the evidence is so strong that if the subject under discussion were, say, the treatment of the mumps, there would be no discussion.”2 Yet in the face of such overwhelming evidence, the whole-language approach, rather than the phonics approach, dominated American primary classrooms during the 1990s. Who supports whole language? As Nicholas Lemann wrote in the Atlantic Monthly in 1997, “Support for it is limited to an enclosed community of devotees, including teachers, education school professors, textbook publishers, bilingual educators, and teacher trainers. Virtually no one in the wider public seems to be actively promoting whole language. No politicians are crusading for it. Of the major teachers’ unions, the American Federation of Teachers (AFT) is a wholehearted opponent and the National Education Association (NEA) is neutral. No independent scientific researchers trumpet whole language’s virtues. The balance of parental pressure is not in favor of whole language.”3

This phenomenon is not just the story of reading. Math education experts also live in an enclosed community. In 1989, the National Council of Teachers of Mathematics (NCTM) developed academic content standards that have since been adopted by most states and today drive classroom practice in thousands of schools. The standards not only specified what children were to learn, but how teachers were to teach. According to the NCTM, these standards were designed to “ensure that the public is protected from shoddy products,” yet no effort was made by the NCTM to determine whether the standards themselves were based on evidence. Indeed, the document setting them forth also urged that the standards be tested, recommending “the establishment of some pilot school mathematics program based on these standards to demonstrate that all students—including women and underserved minorities—can reach a satisfactory level of mathematics achievement.”4 There’s nothing wrong with testing the NCTM approach to math education. But should NCTM’s standards become the coin of the realm before they have proven their efficacy in rigorous experimental settings?

What is striking about the math episode is the NCTM’s inconsistent stance toward evidence. At one point there seems to be a reverence for evidence. “It seems reasonable that anyone developing products for use in mathematics classrooms should document how the materials are related to current conceptions of what content is important to teach and should present evidence about their effectiveness,” wrote the NCTM experts.5 The NCTM pointed to the Food and Drug Administration (FDA) as a model for what it was doing in creating content standards.

Yet it is impossible to imagine the FDA approving a drug—indeed, urging its widespread use—and later proposing “the establishment of some pilot … program” to see whether the drug helps or harms those to whom it is given. The FDA uses the most reliable kind of research to identify what works: dividing a population into two identical groups and randomly assigning treatment to one group, with the other group serving as a control. Properly done, the “patients” don’t know which group they’re in and neither do the scientists dispensing the medications and placebos. (This is known as a “double blind” experiment.) Such research is virtually unknown in education.

The resistance of education experts to evidence is so puzzling that it is worth closely investigating what educators say about research. In 1995, the Research Advisory Committee of the NCTM expressed its disdain for the kind of research that the FDA routinely conducts: “The question ‘Is Curriculum A better than Curriculum B?’ is not a good research question, because it is not readily answerable.” In fact, that is exactly the kind of research question that teachers, parents, and the broader public want to see answered. This kind of research is not impossible, though it is more complicated to undertake than other kinds of research—particularly the qualitative research that most education experts seem to prefer. (The role of qualitative research is discussed later in this essay.)

For some education professors, the problem with experimental research runs deeper. One prominent member of the field, Gene Glass, a former president of the American Educational Research Association, introduced an electronic discussion forum on research priorities with the following remarks: “Some people expect educational research to be like a group of engineers working on the fastest, cheapest, and safest way of traveling to Chicago, when in fact it is a bunch of people arguing about whether to go to Chicago or St. Louis.”6

With research understood in this way, it should not be surprising to find that the education profession has little by way of a solid knowledge base on which to rest its practices. But if we don’t know what works, how are teachers to know how to respond in a sure and confident way to the challenges they face? Hospitalized some months ago with a pulmonary embolism, Diane Ravitch, former assistant secretary of the U.S. Department of Education, looked up at the doctors treating her in the intensive care unit and imagined for an instant that she was being treated by education experts rather than physicians. As she recounts:

My new specialists began to argue over whether anything was actually wrong with me. A few thought that I had a problem, but others scoffed and said that such an analysis was tantamount to “blaming the victim.” . . .

Among the raucous crowd of education experts, there was no agreement, no common set of standards for diagnosing my problem. They could not agree on what was wrong with me, perhaps because they did not agree on standards for good health. Some maintained that it was wrong to stigmatize people who were short of breath and had a really sore leg; perhaps it was a challenge for me to breathe and to walk, but who was to say that the behaviors I exhibited were inappropriate or inferior compared to what most people did?

A few researchers continued to insist that something was wrong with me; one even pulled out the results of my CAT-scan and sonogram. But the rest ridiculed the tests, pointing out that they represented only a snapshot of my actual condition and were therefore completely unreliable, as compared to longitudinal data (which of course was unavailable).

. . . The assembled authorities could not agree on what to do to make me better. Each had his own favorite cure, and each pulled out a tall stack of research studies to support his proposals. One group urged a regimen of bed rest, but another said I needed vigorous exercise. . . . One recommended Drug X, but another recommended Drug Not-X. Another said that it was up to me to decide how to cure myself, based on my own priorities about what was important to me.

Just when I thought I had heard everything, a group of newly minted doctors of education told me that my body would heal itself by its own natural mechanisms, and that I did not need any treatment at all.7

This may read like caricature, yet it is clear that many education experts have not embraced the use of rigorous scientific research to identify effective methods. But this is not the only thing that affects their judgments. In other cases, what prevents them from being guided by scientific findings is a misunderstanding of the inherent limits of descriptive or qualitative research. Such research has its place. It can aid, for example, in the understanding of a complex problem and can be used to formulate hypotheses that can be formally evaluated (in an experiment with control groups, for instance). But such research cannot provide reliable information about the relative effectiveness of a treatment, of “Drug X” vs. “Drug Not-X.”

Despite this simple fact of logic, many education experts assume that descriptive research will determine the relative effectiveness of various practices. Claims made by two national organizations of mathematics educators illustrate the problem. In a letter to the president of the California State Board of Education, the American Educational Research Association’s Special Interest Group for Research in Mathematics Education wrote, “[D]ata from the large-scale NAEP tests tell us that children in the middle grades do well in solving one-step story problems but are unable to solve two-step story problems. A qualitative study, involving observations and interviews with children, can provide us with information about why this is the case and how instructional programs can be changed to improve this situation”8 (emphasis added). In another letter to the same board, Judith T. Sowder, editor of the NCTM’s Journal for Research in Mathematics Education, wrote that “by in-depth study of children’s thinking we have been able to overcome some of our past instructional mistakes and design curricula that allows (sic) students to form robust mathematical concepts”9 (emphasis added).

Both statements illustrate a serious reasoning fallacy, one that is pandemic in education: deriving an ‘ought’ from an ‘is.’ A richly evocative description of what a problem is does not logically imply what the solution to that problem ought to be. The viability of a solution depends on its being compared to other options.
What is clear from these examples is that lack of evidence does not deter widespread acceptance of untested innovations in education; indeed, a pedagogical method can even be embraced in the face of contradictory evidence. Conversely, the evidence for an instructional approach may be overwhelmingly positive, yet there is no guarantee that it will be adopted. The case of Direct Instruction is a prime example.

 

A Large-scale Education Experiment

In the annals of education research, one project stands out above all others. Project Follow Through was probably the largest education experiment ever conducted in the United States. It was a longitudinal study of more than twenty different approaches to teaching economically disadvantaged K-3 students. The experiment lasted from 1967 to 1976, although Follow Through continued as a federal program until 1995. Project Follow Through included more than 70,000 students in more than 180 schools, and yearly data on 10,000 children were used for the study. The project evaluated education models falling into two broad categories: those based on child-directed construction of meaning and knowledge, and those based on direct teaching of academic and cognitive skills.

The battle between these two basic approaches to teaching has divided educators for generations. Each is rooted in its own distinctive philosophy of how children learn. Schools that have implemented the child-centered approach (sometimes called “constructivist”) have a very different look and feel from schools that have opted for the more traditional, teacher-directed approach (often called “direct instruction” in its most structured form).

First graders in a constructivist reading classroom might be found scattered around the room; some children are walking around, some are talking, some painting, others watching a video, some looking through a book, and one or two reading with the teacher. The teacher uses a book that is not specifically designed to be read using phonics skills, and, when a child misses a word, the teacher will let the mistake go by so long as the meaning is preserved to some degree (for instance, if a child reads “horse” instead of “pony”). If a child is stuck on a word, the teacher encourages her to guess, to read to the end of the sentence and then return to the word, to look at the picture on the page, and, possibly, to look at the first letter of the word.

In a direct instruction classroom, some children are at their desks writing or reading phonics-based books. The rest of the youngsters are sitting with the teacher. The teacher asks them to sound out challenging words before reading the story. When the children read the story, the teacher has them sound out the words if they make mistakes.

In the category of child-directed education, four major models were analyzed in Project Follow Through:

•Constructivism/Discovery Learning: The Responsive Education Model, sponsored by the Far West Laboratory and originated by Glenn Nimnict. The child’s own interests determine where and when he works. The goal is to build an environment that is responsive to the child so that he can learn from it.
•Whole Language: The Tucson Early Education Model (TEEM), developed by Marie Hughes and sponsored by the University of Arizona. Teachers elaborate on the child’s present experiences and interests to teach intellectual processes such as comparing, recalling, looking, and relationships. Child-directed choices are important to this model; the content is less important.
•Developmentally Appropriate Practices. Cognitively Oriented Curriculum, sponsored by the High/Scope Educational Research Foundation and developed by David Weikart. The model builds on Piaget’s concern with the underlying cognitive processes that allow one to learn on one’s own. Children are encouraged to schedule their own activities, develop plans, choose whom to work with, etc. The teacher provides choices in ways that foster development of positive self-concept. The teacher demonstrates language by labeling what is going on, providing interpretations, and explaining causes.
•Open Education Model. The Education Development Center (EDC) sponsored a model derived from the British Infant School and focused on building the child’s responsibility for his own learning. Reading and writing are not taught directly, but through stimulating the desire to communicate. Flexible schedules, child-directed choices, and a focus on intense personal involvement characterize this model.

The major skills-oriented, teacher-directed model tested in Project Follow Through was Direct Instruction, sponsored by the University of Oregon and developed by Siegfried Engelmann and Wes Becker. It emphasizes the use of small group, face-to-face instruction by teachers and aides using carefully sequenced lessons in reading, mathematics, and language in kindergarten and first grade. (Lessons in later grades are more complicated.) A variety of manuals, observation tools, and child assessment measures have been developed to provide quality control for training procedures, teaching processes, and children’s academic progress. Key assumptions of the model are: (1) that all children can be taught (and that this is the teacher’s responsibility); (2) that low-performing students must be taught more, not less, in order to catch up; and (3) that the task of teaching more requires careful use of educational technology and time. (The author of this report was involved with the Direct Instruction Follow Through Project at the University of Oregon.)

Data for the big Follow Through evaluation were gathered and analyzed by two independent organizations—Stanford Research Institute and Abt Associates.10 Students taught according to the different models were compared with a control group (and, implicitly, with each other) on three types of measures: basic, cognitive, and affective.

Mean percentile scores on the four Metropolitan Achievement Test categories—Total Reading, Math, Spelling, and Language—appear in Figure 1. Figure 1 also shows the average achievement of disadvantaged children without any special help, which at that time was at about the 20th percentile.

In only one approach, the Direct Instruction (DI) model, were participating students near or at national norms in math and language and close to national norms in reading. Students in all four of the other Follow Through approaches—discovery learning, language experience, developmentally appropriate practices, and open education—often performed worse than the control group. This poor performance came in spite of tens of thousands of additional dollars provided for each classroom each year.

Researchers noted that DI students performed well not only on measures of basic skills but also in more advanced skills such as reading comprehension and math problem solving. Furthermore, DI students’ scores were quite high in the affective domain, suggesting that building academic competence promotes self-esteem, not vice versa.11 This last result especially surprised the Abt researchers, who wrote:

The performance of Follow Through children in Direct Instruction sites on the affective measures is an unexpected result. The Direct Instruction model does not explicitly emphasize affective outcomes of instruction, but the sponsor has asserted that they will be consequences of effective teaching. Critics of the model have predicted that the emphasis on tightly controlled instruction might discourage children from freely expressing themselves, and thus inhibit the development of self-esteem and other affective skills. In fact, this is not the case.12

An analysis of the Follow Through parent data found moderate to high parental involvement in all the DI school districts.13 Compared to the parents of students from schools being served by other Follow Through models, parents of DI students more frequently felt that their schools had appreciably improved their children’s academic achievement. This parental perception corresponded with the actual standardized test scores of the Direct Instruction students.

These data were collected and analyzed by impartial organizations. The developers of the DI model conducted a number of supplementary studies, which had similarly promising results.

Significant IQ gains were found in students who participated in the program. Those entering kindergarten with low IQs (below 71) gained 17 points, while students entering first grade with low IQs gained 9.4 points. Children with entering IQs in the 71-90 range gained 15.6 points in kindergarten and 9.2 points in first grade.
Longitudinal studies were undertaken using the high school records of students who had received Direct Instruction through the end of third grade as well as the records of a comparison group of students who did not receive Direct Instruction. Researchers looked at test scores, attendance, college acceptances, and retention. When academic performance was the measure, the Direct Instruction students outperformed the control group in the five comparisons whose results were statistically significant. The comparisons favored Direct Instruction students on the other measures as well (attendance, college acceptances, and retention) in all studies with statistically significant results.14

Additional research showed that the DI model worked in a wide range of communities. Direct Instruction Follow Through sites were located in large cities (New York, San Diego, Washington, D.C.); mid-sized cities (Flint, Michigan; Dayton, Ohio; East St. Louis, Illinois); rural white communities (Flippin, Arkansas; Smithville, Tennessee); a rural black community (Williamsburg, South Carolina); Latino communities (Uvalde, Texas; E. Las Vegas, New Mexico); and a Native American community (Cherokee, North Carolina).

More than two decades later, a 1999 report funded by some of the nation’s leading education organizations confirmed the efficacy of Direct Instruction. Researchers at the American Institutes of Research who performed the analysis for the Educators’ Guide to Schoolwide Reform found that only three of the 24 schoolwide reform models they examined could present solid evidence of positive effects on student achievement. Direct Instruction was one of the three.15

 

Direct Instruction after Project Follow Through

Before Project Follow Through, constructivist approaches to teaching and learning were extremely popular. One might have expected that the news from Project Follow Through would have caused educators to set aside such methods and embrace Direct Instruction instead. But this did not happen. To the contrary.
Even before the findings from Project Follow Through were officially released, the Ford Foundation commissioned a critique of it. One of the authors of that study, the aforementioned Gene Glass, wrote an additional critique of Follow Through that was published by the federal government’s National Institute of Education. This report suggested that the NIE conduct an evaluation emphasizing an ethnographic or descriptive case-study approach because “the audience for Follow Through evaluations is an audience of teachers that doesn’t need statistical finding of experiments to decide how best to teach children. They decide such matters on the basis of complicated public and private understandings, beliefs, motives, and wishes.”16

After the results of the Follow Through study were in, the sponsors of the different programs submitted their models to the Department of Education’s Joint Dissemination Review Panel. Evidently the Panel did not value the differences in effectiveness found by the big national study of Follow Through; all of the programs—both successful and failed—were recommended for dissemination to school districts. According to Cathy Watkins, a professor of education at Cal State-Stanislaus, “A program could be judged effective if it had a positive impact on individuals other than students. As a result, programs that had failed to improve academic achievement in Follow Through were rated as ‘exemplary and effective.’ ”17 The Direct Instruction model was not specially promoted or encouraged in any way. In fact, extra federal dollars were directed toward the less effective models in an effort to improve their results.

During the 1980s and early 1990s, schools that attempted to use Direct Instruction (originally known as DISTAR)—particularly in the early grades, when DI is especially effective—were often discouraged by members of education organizations. Many experts were convinced that the program’s heavy academic emphasis was “developmentally inappropriate” for young children and might “hinder children’s development of interpersonal understanding and their broader socio-cognitive and moral development.”18 “DI is the answer only if we want our children to swallow whole whatever they are told and focus more on consumption than citizenship,” argued Lawrence Schweinhart of the High/Scope Educational Research Foundation.19 (High/Scope had developed one of the constructivist models.)

Faced with the evidence of Direct Instruction’s effectiveness, some experts still advocated methods that had not proved effective in Project Follow Through. “The kind of learning DISTAR tries to promote can be more solidly elicited by the child doing things,” argued Harriet Egertson, an early childhood specialist at the Nebraska Department of Education. “The adult’s responsibility is to engage the child in what he or she is doing, to take every opportunity to make their experience meaningful. DISTAR isn’t connected to anything. If you use mathematics in context, such as measuring out spoons of sugar in a cooking class, the notion of addition comes alive for the child. The concept becomes embedded in the action and it sticks.”20

Tufts University professor of child development David Elkind argued that, while Direct Instruction is harmful for all children, it

is even worse for young disadvantaged children, because it imprints them with a rote-learning style that could be damaging later on. As Piaget pointed out, children learn by manipulating their environment, and a healthy early education program structures the child’s environment to make the most of that fact. DISTAR, on the other hand, structures the child and constrains his learning style.21

The natural-learning view that underlies the other four Follow Through models described above is enormously appealing to educators and to many psychologists. The dominance of this view can be traced back to Jean-Jacques Rousseau, who glorified the natural at the expense of the man-made, and argued that education should not be structured but should emerge from the natural inclinations of the child. German educators developed kindergartens based on the notion of natural learning. This romantic notion of learning has become doctrinal in many schools of education and child-development centers, and has closed the minds of many experts to actual research findings about effective approaches to educating children.22 This is a classic case of an immature profession, one that lacks a solid scientific base and has less respect for evidence than for opinion and ideology.

 

Learning from Other Professions

Education could benefit from examining the history of some other professions. Medicine, pharmacology, accounting, actuarial sciences, and seafaring have all evolved into mature professions. According to Theodore M. Porter, a history professor at the University of California at Los Angeles, an immature profession is characterized by expertise based on the subjective judgments of the individual professional, trust based on personal contact rather than quantification, and autonomy allowed by expertise and trust, which staves off standardized procedures based on research findings that use control groups. 23

A mature profession, by contrast, is characterized by a shift from judgments of individual experts to judgments constrained by quantified data that can be inspected by a broad audience, less emphasis on personal trust and more on objectivity, and a greater role for standardized measures and procedures informed by scientific investigations that use control groups.

For the most part, education has yet to attain a mature state. Education experts routinely make decisions in subjective fashion, eschewing quantitative measures and ignoring research findings. The influence of these experts affects all the players in the education world.

Below is a description that could very well describe the field of education:

It is hard to conceive of a less scientific enterprise among human endeavors. Virtually anything that could be thought up for treatment was tried out at one time or another, and, once tried, lasted decades or even centuries before being given up. It was, in retrospect, the most frivolous and irresponsible kind of human experimentation, based on nothing but trial and error, and usually resulting in precisely that sequence.24

Yet this quote does not describe American education today. Rather, it was written about pre-modern medicine by the late Dr. Lewis Thomas (1979), former president of the Memorial Sloan-Kettering Cancer Center. Medicine has matured. Education has not. The excerpt continues:

Bleeding, purging, cupping, the administration of infusions of every known plant, solutions of every known metal, most of these based on the weirdest imaginings about the cause of disease, concocted out of nothing but thin air—this was the heritage of medicine up until a little over a century ago. It is astounding that the profession survived so long, and got away with so much with so little outcry. Almost everyone seems to have been taken in.25

Education has not yet developed into a mature profession. What might cause it to? Based on the experience of other fields, it seems likely that intense and sustained outside pressure will be needed. Dogma does not destroy itself, nor does an immature profession drive out dogma.

The metamorphosis is often triggered by a catalyst, such as pressure from groups that are adversely affected by the poor quality of service provided by a profession. The public’s revulsion at the Titanic’s sinking, for example, served as catalyst for the metamorphosis of seafaring. In the early 1900s, sea captains could sail pretty much where they pleased, and safety was not a priority. The 1913 International Convention for Safety of Life at Sea, convened after the sinking of the Titanic, quickly made rules that are still models for good practice in seafaring.

The metamorphosis of medicine took more than a century. As the historian Theodore Porter explains:

In its pre-metamorphosis stage, medicine was practiced by members of an elite who refused . . . to place the superior claims of character and breeding on an equal footing with those of scientific merit. . . . These gentlemen practitioners opposed specialization, and even resisted the use of instruments. The stethoscope was acceptable, because is was audible only to them, but devices that could be read out in numbers or, still worse, left a written trace, were a threat to the intimate knowledge of the attending physician.26

External pressure on medicine came from life insurance companies that demanded quantitative measures of the health of applicants and from workers who did not trust “company doctors.” The Food and Drug Administration, founded in 1938 as part of the New Deal, initially accepted both opinions from clinical specialists and findings from experimental research when determining whether drugs did more good than harm. However, the Thalidomide disaster led to the Kefauver Bill of 1962, which required drugs thereafter to be proven to be effective and safe before they could be prescribed, with little attention paid to the opinions of clinical specialists. (Medical interventions and intervention devices, such as coronary stents, are subject to similar reviews of safety and efficacy.)

The catalyst that transformed accounting in the United States was the Great Depression. To restore investor confidence, the government promulgated reporting rules to guard against fraud, creating the Securities and Exchange Commission.

In general, it appears that a profession is not apt to mature without external pressure and the attendant conflict. Metamorpho-sis begins when the profession determines that this is its likeliest path to survival, respect, and prosperity. Porter writes that the American Institute of Accountants established its own standards to fend off an imminent bureaucratic intervention.27 External pressures had become so great that outsiders threatened to take over and control the profession via legislation and regulation. There are signs today that this is beginning to happen in education.

 

Making Education a Mature Profession

The best way for a profession to ensure its continued autonomy is to adopt methods that ensure the safety and efficacy of its practices. The profession can thereby deter extensive meddling by outsiders. The public trusts quantified data because procedures for coming up with numbers reduce subjective decision-making. Standardized procedures also are more open to public inspection and legal review.

American education is under intense pressure to produce better results. The increasing importance of education to the economic well-being of individuals and nations will continue feeding this pressure. In the past—and still today—the profession has tended to respond to such pressures by offering untested but appealing nostrums and innovations that do not improve academic achievement. At one time or another, such practices have typified every profession, from medicine to accounting to seafaring. In each case, groups adversely affected by the poor quality of service have exerted pressures on the profession to incorporate a more scientific methodology.

These pressures to mature are inevitable in education as well. Its experts should hasten the process by abandoning ideology and embracing evidence. Findings from carefully controlled experimental evaluations must trump dogma. Expert judgments should be built on objective data that can be inspected by a broad audience rather than wishful thinking. Only when the profession embraces scientific methods for determining efficacy and accepts accountability for results will education acquire the status—and the rewards—of a mature profession.

 

Notes

1 National Reading Panel Progress Report, 22 February 1999. <www.nationalreadingpanel.org>
2 James Collins, “How Johnny Should Read,” Time, 27 October 1997, 81.
3 Nicholas Lemann, “The Reading Wars,” Atlantic Monthly (November 1997), 133-134.
4 National Council of Teachers of Mathematics, Curriculum and evaluation standards for school mathematics (Reston, VA: Author, 1989), 253.
5 Ibid, 2.
6 Gene Glass, “Research news and comment-a conversation about educational research priorities: A message to Riley,” Educational Researcher 22, no. 6 (August-September 1993), 17-21.
7 Diane Ravitch, “What if Research Really Mattered?” Education Week, 16 December 1998.
8 Personal communication with California State Board of Education.
9 Personal communication with California State Board of Education.
10 L. Stebbins, ed., “Education experimentation: A planned variation model” in An Evaluation of Follow Through III, A (Cambridge, MA: Abt Associates, 1976), and L. Stebbins, et al., “Education as experimentation: A planned variation model,” in An Evaluation of Follow Through IV, A-D (Cambridge, MA: Abt Associates, 1977).
11 Stebbins et al., 1977.
12 Abt Associates, “Education as experimentation: A planned variation model,” An Evaluation of Follow Through IV, B (Cambridge, MA: Author, 1977), 73.
13 Walter Haney, A Technical History of the National Follow Through Evaluation (Cambridge, MA: Huron Institute, August 1977).
14 R. Gersten and T. Keating, “Improving high school performance of ‘at risk’ students: A study of long-term benefits of direct instruction,” Educational Leadership 44, no. 6 (1987), 28-31.
15 Although the data supporting Direct Instruction are quite strong, it is important to note that the model is demanding to implement and results from a poor implementation may be poor.
16 Gene Glass and G. Camilli, “FT Evaluation” (National Institution of Education, ERIC document ED244738), as cited in Nina H. Shokraii, “Why Congress Should Overhaul the Federal Regional Education Laboratories,” Heritage Foundation Backgrounder, no. 1200 (Washington, DC: The Heritage Foundation, 1998).
17 Cathy L. Watkins, “Follow Through: Why Didn’t We,” Effective School Practices, 15, no. 1 (Winter 1995-96), 5.
18 Rheta DeVries, Halcyon Reese-Learned, and Pamela Morgan, “Sociomoral development in direct instruction, eclectic, and constructivist kindergartens: a study of children’s enacted interpersonal understanding,” Early Childhood Research Quarterly 6, no. 4, 473-517, as cited in Denny Taylor, Beginning to Read and the Spin Doctors of Science (Urbana, IL: National Council of Teachers of English, 1998), 231.
19 Lawrence Schweinhart, “Back to School,” letter appearing in National Review, 20 July 1998.
20 As quoted in Ellen Ruppel Shell, “Now, which kind of preschool,” Psychology Today (December 1989).
21 Ibid.
22 E.D. Hirsch, Jr., “Reality’s revenge: Research and ideology,” American Educator (Fall 1996), excerpted from E.D. Hirsch, Jr., The Schools We Need and Why We Don’t Have Them (New York, NY: Doubleday, 1996). An interesting perspective on this topic can be found in an unpublished paper by Thomas D. Cook of Northwestern University called “Considering the Major Arguments against Random Assignment: An Analysis of the Intellectual Culture Surrounding Evaluation in American Schools of Education.”
23 Theodore M. Porter, Trust in Numbers: The Pursuit of Objectivity in Science and Public Life (Princeton, NJ: Princeton University Press, 1996).
24 Lewis Thomas, “Medical Lessons from History,” in The Medusa and the Snail: More Notes of a Biology Watcher (New York: Viking Press, 1979), 159.
25 Ibid, 159-160.
26 Porter, 202.
27 Ibid, 93.

 

 

 

 

 

 

 

Chester E. Finn, Jr., President
Thomas B. Fordham Foundation
Washington, DC
April 2000

Why Do Supporters of the National Council of Teachers of Mathematics Insist Only They Are Right?

 

By Nakonia (Niki) Hayes
Guest Columnist EdNews.org
October, 2006

 

The history of mathematics education in the United States is a complex one, with long-running philosophical conflicts among various groups. But there’s something to be said about listening to those with credentials, practical experience, and “seasons” on them in the mathematics education field. Since the elders in my culture are respected for their insight and ability to bring clarity to conflicting issues, and I am now an elder, it is not surprising for me to think this way.

Criticism of the undeniable impact, and surely an unintentional disastrous one, of the National Council of Teachers of Mathematics on math education since 1989 in the United States is such an issue. A subsequent “math war” began openly in California in 1996 with professional mathematicians and angry parents who opposed the NCTM pedagogy of “whole (discovery) math,” with its insistence on teaching “process” (concepts) over mathematic principles (basic skills). That conflict has grown among individual school districts in other states, as more pro-fessional mathematicians are joining organized parent groups in their desire to return a balance of both basic skills and concepts to math classrooms in the U.S.

At this time, California, Michigan, and Massachusetts have changed their mathematics standards, moving away from the NCTM program, to reflect such a balance. California, in particular, has continued to serve as a model for such change.

My own strong resistance to NCTM’s reform pedagogy of constructivism is based on training and professional experience in five disciplines: journalism, counseling, special education, mathematics, and administration.

With 17 years in journalism and another 30 in public education, I maintain there is mounting evidence in the NCTM “reformed” mathematics curriculum of the following: inaccurate views based on poor research, reverse discrimination (against white males), stereo-typed learning styles that have helped increase achievement gaps for minorities, opaque and convoluted lessons about mathematical procedures, and a disrespect for the historical importance of texts that represent the rich concepts and principles of mathematics.

Yet, the struggle by mathematicians and mostly middle class parents to stop the financial and human costs of this entrenched curriculum, passionately promoted by those who support its philosophy, including the National Science Foundation, is little recognized or understood by disenfranchised parents and, worse, by legislators. A “critical mass” has yet to understand the consequences of the NCTM domination, with its constructivist programs in math education.

And even though NCTM issued a new document in August of this year called Focal Points, the group strongly resists any suggestion this publication indicates they may have been wrong in their pedagogical stance. They explain the document is simply the “next step” of telling educators which mathematics topics should be the focus at each grade level from kindergarten through eighth grade. They sidestep the critical issue of teaching methodology.

My work in this paper is designed to help people understand the passion and the motives behind the NCTM philosophy, as codified in their discipline-rattling 1989 publication, Curriculum and Evaluation Standards for School Mathematics.

I first use a summarized point, which is based on direct quotes from the 1989 NCTM manual or NCTM officials, except for number 8, which discusses the funding relationship between NCTM and the National Science Foundation. There are some related quotes from A Nation at Risk, another public-rattling publication, from 1983, and clearly a basis for much of the NCTM philosophy. Additional sources are indicated as “boxed” information. My final conclusion regarding each numbered item is then offered.

Bold, italicized, and underlined words indicate my own emphases.

An explanation of how my professional background impacted my thinking on this issue is included as an appendix.

1) NCTM designed their Standards with a primary goal of socially promoting egalitarianism via mathematics education.

1) The opportunity for all students is at the heart of our vision of a quality mathematics program. (p. 5) From A NATION AT RISK: “At the heart of such a [learning] society is the commitment to a set of values and to a system of education that affords all members the opportunity to stretch their minds to full capacity…”

2) The social injustices of past schooling practices can no longer be tolerated. Creating a just society in which women and various ethnic groups enjoy equal opportunities and equitable treatment… Equity has become an economic necessity. (p.4) From A NATION AT RISK: “…public commitment to excellence and educational reform must be made [for] equitable treatment of our diverse population. The twin goals of equity and high-quality schooling have profound and practical meaning for our economy and society.

3) If all students do not have the opportunity to learn this mathematics, we face the danger of creating an intellectual elite and a polarized society. (p. 9) From A NATION AT RISK: “To deny young people a chance to learn…would lead to a generalized accommodation to mediocrity in our society on the one hand or the creation of an undemocratic elitism on the other…John Slaughter, a former Director of the NSF, warned of ‘a growing chasm between a small scientific and technological elite and a citizenry ill-inform-ed, indeed uninformed…”

4) Becoming confident in one’s own ability… doing mathematics is a common human activity. (p.6) From A NATION AT RISK: A high level of shared education is essential to a free democratic society and to the fostering of a common culture…

5) Curriculum Standards for Grades 5-8…No student should be denied access to the study of one topic because he or she has yet to master another. (p.69)

6) Curriculum Standards for Grades 9-12…In view of existing disparities in educational oppor-tunity in mathematics…each standard identifies content or processes [and] activities for all students. (p.123)

The core curriculum provides equal access and opportunity to all students… By… recognizing that there frequently is not a strict hierarchy among the proposed mathematics topics at this [9-12] level, we are able to afford all students more opportunities to fulfill their mathematical potential. (p.130)

In choosing not to trap students in one of the two conventional linear patterns, we ensure that doors to college programs and vocational training are kept open for all students. (p.130)

…no student will be denied access to the study of mathematics because of a lack of computa-tional facility. (p.124)

7) Goals are broad statements of social intent. (p.2) New social goals for education include 1) mathematically literate workers, 2) lifelong learning, 3) opportunity for all, and 4) an informed elector-ate. (p.3)

Historically, societies have established schools to…transmit aspects of the culture to the young…[to] provide them opportunity for self-fulfillment. (p.2)

…[the goal is to] focus attention on the need for student awareness of the… impact [their]… interaction has on our culture and our lives. (p.6)

In “An Overview of the Curriculum and Evaluation Standards,” … the curriculum and evalu-ation standards that reflect our vision of … societal and student goals. (p.7)

…standards are value judgments based on…societal goals… research on teaching and learning, and professional experience. (p.7)

8) Research findings from psychology indicate that learning does not occur by passive absorption alone (Resnick, 1987). (p.10)

9) Implications for the K-4 Curriculum: Overall goals must do the following (p.16):

vAddress the relationship between young children and mathematics.

vRecognize the importance of the qualitative dimensions of children’s learning.

vBuild beliefs …about children’s view of themselves as mathematics learners.

10) [grades 9-12] We believe the opportunity to study mathematics that is more interesting and useful and not characterized as remedial will enhance students’ self-concepts as well as their attitudes …students no longer will be confronted with the demeaning prospect of studying…the same content topics as their twelve-year-old siblings. (p.130) …for each individual, mathematical power involves the development of personal self-confidence. (p.5)

In summary, the [9-12] core curriculum seeks to provide a fresh approach to mathematics for all students—one that builds on what students can do rather than on what they cannot do. (p.131)

Deborah Loewenberg Ball, Imani Masters Goffney, Hyman Bass, “Guest Editorial… The Role of Mathematics Instruction in Building a Socially Just and Diverse Democracy,” The Mathematics Educator, 2005, Vol. 15, No. 1, 2-6:  Instead of seeing mathematics as culturally neutral, politically irrelevant, and mainly a matter of innate ability, we see it as a critical lever for social and educational progress if taught in ways that make use of its special resources.

…the disparities in mathematics achievement are tightly coupled with social class and race… learning to examine who and what is being valued and developed in math class is essential.”Mathematics instruction, we claim, can offer a special kind of shared experience with understanding, respecting, and using difference for productive collective work.

David Klein, writes in “A quarter century of U.S. ‘math wars’ and political partisanship,” British Society for the History of Mathematics, 2006, http://www.tandf.co.uk/journals/titles/17498430.asp:

The utilitarian justification of mathematics was so strong that both basic skills and general mathematical principles were to be learned almost invariably through ‘real world’ problems. Mathematics for its own sake was not encouraged.

The arguments in support of these changes took two major themes: social justice in the form of challenging racial and class barriers on the one hand, and the needs of business and industry on the other.

The Standards, buttressed by NCTM’s call for ‘mathematics for all’ and the equity agenda in Everybody Counts, clearly sat in the education-for-democratic-equality [camp]…These powerful indict-ments [of elitism] demanded radical solutions. Mathematics reform was social reform…

The NCTM reform was an attempt to redefine mathematics in order to correct social inequities.

CONCLUSION: Students who master both the concepts of a discipline and its principles, which are used to transcend those concepts across all of life’s domains, have the best opportunity of creating equitable opportunities for themselves, regardless of their backgrounds. And mastering the full body of any discipline formed by many peoples across thousands of years of study, such as mathematics, also helps bring its new learners together with a common respect and under-standing—including a true integration—of their thoughts and actions. Learning the content is a priority of any discipline to be mastered. Methodology is the teacher’s magic for getting it taught successfully, as long as it’s legal, ethical, and moral!

2) The secondary goal of NCTM is development of “problem-solving” skills among all learners, as they prepare for a technological world.

1) The first recommendation of An Agenda for Action (NCTM, 1980): “Problem solving must be the focus of school mathematics.” (p.6)

,…problem solving is much more than applying specific techniques to the solution of classes of word problems. It is a process by which the fabric of mathematics as identified in later standards is both constructed and reinforced. (p. 137)

The standards specify that instruction should be developed from problem situations. (p. 11)

2) Traditional teaching emphases on practice in manipulating expressions and practicing algorithms as a precursor to solving problems ignore the fact that knowledge often emerges from the problems. This suggests that instead of the expectation that skill in computation should precede word problems, experience with problems helps develop the ability to compute. (p.9)

3) A strong emphasis on mathematical concepts and understandings also supports the development of problem solving. (p.17)

4) …present strategies for teaching may need to be reversed; knowledge often should emerge from experience with problems. (p.9)

As reported in “An oral history of New Math and New-New Math – Based on a Series of Postings to the NYC Hold Math Reform Web Group, Nov. 18-19, 2003,” compiled by William Hook, Jan. 15, 2004, the following comments of W. Stephen Wilson, professor of mathematics at Johns Hopkins University, are offered:

I think we [Jim Milgram and Wilson] agree that there is “low level” problem solving, meaning “routine” problem solving, which does not require having an “idea,” but requires mastery of material and the use of a “problem solving algorithm (loosely).” We agree there is “high level” problem solving…for real problem solving, we only have vague ideas of what goes on, and none of how to teach it…[but] at that point the NCTM jumps in and says, “but we know how to teach it; that’s what our reforms are all about.” …Basically, the reformers claim to be able to teach “high level” problem solving without bothering with “low level” problem solving or basic mastery of material…I would claim that mathematicians know enough about “high level” problem solving to know you can’t do it if you can’t do “low level” and you haven’t mastered the material. Consequently, we think it is important to teach mastery of material and “low level” problem solving, since it is certainly a prerequisite to “high level” problem solving. Now it turns out that this is already pretty hard to do…it isn’t hard to teach it, but for some reason it seems to be pretty hard to learn it.

Jim Milgram, professor of mathematics at Stanford University, responded: …we should make a real point of the fact that we don’t know much about problem solving and neither do NCTM members. We should, loudly, defend our ignorance.

But what we shouldn’t do is directly attack NCTM ideology…[from his experience with writing the new California standards, he says…] people in NCTM actually want to learn more mathematics than they know, and if given a chance, will do so…it is probably best to challenge them to learn by making a large point of the fact that what NCTM actually does in “problem solving” is to develop systematic ways to do routine problems, something even the traditional programs always did—though they did them using sequenced series of exercises and problems. What NCTM did was to make a list of standard methods, such as “look for a simpler problem,” and say that this is problem solving.

The confusion comes [when] they talk about problem solving, but what they mean is “routine problem solving.” What we hear is “problem solving”…that means “real problem solving.”…[We] explain there are two kinds of problem solving—what we want, that nobody knows how to teach in a ver-bal way, and the routine part, that does scale up and has been generally understood for many, many years.

CONCLUSION: From the affective side of learning, problem-solving could be compared to developing a new habit, which, according to author Stephen Covey in the Seven Habits of Highly Effective People, requires knowledge (content), skills (process), and the desire for successful completion. A major focus for NCTM, it seems, is creating the desire among students to solve problems. NCTM has chosen to do this by using psychological methods based on questionable research about the learning styles of race and gender. In other settings, this practice would be labeled discriminatory and would not be tolerated.

3)NCTM believes that, historically, mathematics instruction had focused on deductive, analytical, and linear thinking skills in teacher-dominated classrooms, with

a competitive environment that met only the learning styles/needs of white (Anglo) males.

1)These four years [9-12]…will revolve around a broadened curriculum that includes extensions of the core topics and for which calculus is no longer viewed as the capstone experience. (p.125)

2)…a demonstration of good reasoning should be rewarded even more than students’ ability to find correct answers. (p.6)

3)Change has been particularly great in the social and life sciences…the fundamental mathematical ideas needed in these areas are not necessarily those studied in the traditional algebra-geometry-precalculus-calculus sequence, a sequence designed with engineering and physical science applications in mind. (p.7)

Jack Price, president of the National Council of Teachers of Mathematics, said in 1996 during a radio interview, “. . .Women and minority groups do not learn the same way as Anglo males . . . males learn better deductively in a competitive environment . . .” (Reported by Sandra Stotsky in Chapter 13, What’s at Stake in the K-12 Standards War, 2000.)

David Klein, “A quarter century of U.S. ‘math wars’ and political partisanship,” British Society for the History of Mathematics, 2006, http://www.tandf.co.uk/journals/titles/17498430.asp.

In The Math Wars, [Alan] Schoenfeld…describes the traditional curriculum as elitist and portrays the math wars as a battle between equality and elitism… …the traditional curriculum was a vehicle for… the perpetuation of privilege…Thus the Standards could be seen as a threat to the current social order.

“…the traditional curriculum, with its filtering mechanisms and high dropout and failure rates (especially for certain minority groups) has had the effect of putting and keeping certain groups ‘in their place.’

CONCLUSION:Although admittedly limited in number because of historical social constraints, and certainly not mental ones, women and “non-Asian minorities” (see #4 below) indeed have esta-blished themselves with respected places in mathematics. And while the design of mathematics instruction has been the same for 2,000 years—analytical, deductive, and linear—we know its history is rich and powerful because of contributions from many cultures and races—not just white males. To eliminate this historical type of teaching/ learning means 50% to 75% of all students, who learn concretely rather than intuitively, are ignored. (See http://www.virtualschool.edu/mon/Academia/ KierseyLearningStyles.html.) The consequence can be seen with higher failure rates now among boys

in all subjects, as those subjects have become more intuitive, inductive, verbal, and “cooperative” in scope.

4)NCTM used post-1960s research that defined “learning styles” of “non-Asian minorities and girls” as being inductive, intuitive, holistic, group-oriented, cooperative, and non-verbal. This meant such students determined the direction of their learning, needed to work in groups, and should focus on the “bigger picture” of “conceptual understanding” rather than the principles (rules) of mathematics.

Sandra Stotsky, in Chapter 13 of What’s at Stake in the K-12 Standards War, points out that mathematics education has been built upon stereotyped “learning styles” of “non-Asian minorities and girls.” She said there is a strain of thought [among mathematics reformists] that suggests non-Asian minorities and women need to be taught with less emphasis on deductive and analytical methods and more emphasis on inductive, intuitive methods because of gender and racial/ethnic differences in learning.

She wrote that two researchers in math education also suggested that African-American students‘ learning may be characterized as having a social and affective emphasis, harmony in their communities, holistic perspectives, expressive creativity and nonverbal communication. They are flexible and open-minded, rather than structured in their perceptions of ideas.

Stotsky asks, “Does this imply that African-Americans cannot engage in rigorous analytical thinking and articulate their ideas in academic prose?”

In addition, Dr. Stotsky explained how researchers said American Indians are “right brained.” This implies they cannot engage in structured forms of learning because . . . the functions of the left brain are characterized by sequence and order, while the right brain functions are holistic and diffused.

David Klein, “A quarter century of U.S. ‘math wars’ and political partisanship,” British Society for the History of Mathematics, 2006, http://www.tandf.co.uk/journals/titles/17498430.asp,

writes, “Ironically, progressivists’ advocacy of programs [that eliminated basic skills and the intellectual content that depends on those skills] for the supposed benefit of disenfranchised groups contributed to racial stereotying, in contradiction to core progressive values.

CONCLUSION: Rather than racial differences being addressed, the cultural deprivation of stu-dents due to a poverty of parenting and/or home values, for whatever reason, should be the educa-tional community’s mountain to climb. To suggest sweeping, stereotyped patterns of cognitive or mental behavior among a general group of people is bigotry. The use of such stereotyping has balkanized math education within America’s already diverse student body. It has increased achievement gaps among students of color, English language learners, and white students. It has, in fact, caused the loss of mathematical achievement across all subgroups, as the educational focus shifted from the proven discipline of mathematics to the perceived affective-based learning styles of learners.

5)NCTM thus adopted the “constructivist” form of pedagogy, which says children should “construct” their own learning for true cognitive understand-ing. (The terms “constructivist,” ” reform,” and now “progressive” are all used to denote the NCTM agenda.) This resulted in the NCTM curriculum develop-ment around holistic presentations, discovery learning (student-dominated), personal relationships encouraged between activities and students, and literary-based (verbal and written) exchanges among students and teachers about the “processes” of their learning within the activities.

1)Our premise is that what a student learns depends to a great degree on how he or she has learned it. (p.5)

…”instruction should persistently emphasize “doing” rather than “knowing that.” (p.7)

2)…in many situations individuals approach a new task with prior knowledge, assimilate new infor-mation, and construct their own meanings…As instruction proceeds, children often continue to use these [self-constructed] routines in spite of being taught more formal problem-solving procedures…This con-structive, active view of the learning process must be reflected in the way much of mathematics is taught. (p.10

3)…knowledge often should emerge from experience with problems. In this way, students may recognize the need to apply a particular concept or procedure and have a strong conceptual basis for reconstructing their knowledge at a later time. (p.9)

4)Programs that provide limited developmental work, that emphasize symbol manipulation and computational rules, and that rely heavily on paper-and-pencil worksheets do not fit the natural learning patterns of children. (p.16)

5)A conceptual approach enables children to acquire clear and stable concepts by constructing meanings in the context of physical situations and allows mathematical abstractions to emerge from empirical experience.(p.17)

6)Curriculum Standards for Grades 5-8…Instructional approaches should engage students in the process of learning rather than transmit information for them to receive.

7)Learning to communicate mathematically…This is best accomplished in problem situations in which students have an opportunity to read, write, and discuss ideas in which the use of the language of mathematics becomes natural…

8)…mathematics must be approached as a whole. Concepts, procedures, and intellectual processes are interrelated. (p.11)

Howard Gardner, father of “multiple intelligences,” stated in his 1981 book, The Unschooled Mind, that it is higher performing children from motivated families who do well with discovery learn-ing methods. This is due to their “readiness” skills in organization, focus, and learned behavior for school settings. By inference, lower-performing students do not use discovery learning methods successfully.

In his 2006 book, Concept-Rich Mathematics Instruction, Meir Ben-Hur also points out that discovery learning is not effective with children from disadvantaged backgrounds. Ben-Hur is a colleague of Reuven Feuerstein, an Israeli cognitive psychologist who studied with Piaget and is touted by constructivists as “one of their own.” Feuerstein has also stated that discovery learning does not work with at-risk students.

CONCLUSION: Research can be declared on all sides of any pedagogy. Research is not the basis for curriculum choices, however. It is the prioritized values a community wants to find in its classrooms: college prep, applied math for consumer use, self-esteem toward mathematics, equity outcomes, etc. We know that research is then found to support those values.(See “Relationships Between Research and the NCTM Standards,” by James Hiebert, University of Delaware, Journal for Research in Mathematics Education 1999, Vol. 30, No. 1, 3-19.)

Of special significance are the constructivists who declare that “discovery learning” does not work with disadvantaged students. In fact, I studied with both Feuerstein and Ben-Hur in Israel, so I’m confident in quoting them on their views of discovery learning. In addition, the NCTM’s paper by Dr. James Hieber states that standards are chosen because of a community’s values—and then research is found to support those choices.

6) NCTM significantly reduced the need for learning basic skills because, they believed, calculators and computers would replace “tedious” paper-and-pencil practice and the need to know algorithms.

1)Scientific calculators with graphing capabilities will be available to all students at all times… A computer will be available at all times… technology in our society further argues for a curriculum that moves all students beyond computation… By assigning computational algorithms to calculator or computer processing, this curriculum seeks not only to move students forward but to capture their interest. (p.130)

2) Calculators do not replace the need to learn basic facts, to compute mentally, or to do reasonable paper-and-pencil computation…young children take a common-sense view… and recognize the importance of not relying on them when it is more appropriate to compute in other ways. (p.19)

3) Some proficiency with paper-and-pencil computational algorithms is important, but such knowledge should grow out of the problem situations that have given rise to the need for such algorithms. (p.8)

4) By removing the “computational gate” to the study of high school mathematics and recognizing that there frequently is not a strict hierarchy among the proposed mathematics topics at this [9-12] level, we are able to afford all students more opportunities to fulfill their mathematical potential. (p.130)

5) [9-12] Standard 5: Algebra…The proposed algebra curriculum will move away from a tight focus on manipulative facility to include a greater emphasis on conceptual understanding. (p.150)

6) The new technology not only has made calculations and graphing easier, it has changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them. (p.7)

7) For the core program [9-12], this represents a trade-off in instructional time as well as in emphasis…available and projected technology forces a rethinking of the level of skill expectations. (p.150)

8) …although students should spend less time simplifying radicals and manipulating rational exponents, they should devote more time to exploring examples of exponential growth and decay that can be modeled using algebra. (p.150)

9) No student will be denied access to the study of mathematics in grades 9-12 because of a lack of computational facility. (p.124)

10) Although quantitative considerations have frequently dominated discussions in recent years, qualitative considerations have greater significance. Thus, how well children come to understand mathematical ideas is far more important than how many skills they acquire.(p.16)

11) A strong conceptual framework also provides anchoring for skill acquisition. (p.17)

CONCLUSION: By not recognizing the importance of long division and fractions/ratios as algorithms required in algebra, for example, or the many geometric theorems used in advanced mathematics, NCTM’s conceptual standards have crippled high school and college students in mathematics. This can be seen in data reflecting poor mathematics scores among a wide variety of state, national, and international tests. The $4 billion paid by parents annually for private tutoring, the $80 million tutoring industry on the Internet from India, the 50% of community college students requiring remedial math, and the 25% of university students requiring the same remediation further reflect disturbing data about the conceptually-based NCTM pedagogy.

Pg 2: Why Do Supporters of the National Council of Teachers of Mathematics Insist Only They Are Righ

7) NCTM maintains that basic skills are included in their pedagogy.

1) Although arithmetic computation will not be a direct object of study [grades 9-12]… number and operation sense, estimation skills, and judging reasonableness of results will be strengthened in the context of applications and problem solving…and scientific computation. (p.124)

2) The availability of calculators means, however, that educators must develop a broader view of the various ways computation can be carried out and must place less emphasis on complex paper-and-pencil computation. (p. 19)

3) Basic skills today and in the future mean far more than computation proficiency. Moreover, the calculator renders obsolete much of the complex paper-and-pencil proficiency traditionally emphasized in mathematics courses. (p. 66)

4) Some proficiency with paper-and-pencil computational algorithms is important, but such knowledge should grow out of the problem situations that have given rise to the need for such algorithms. (p.8)

CONCLUSION:The question must be asked of NCTM about when, and how, and how much basic skills instruction is included in the many curriculum materials that have been created by publishers to support the NCTM Standards since 1989. Their assurances that basic skills are not ignored in the Standards must be justified with specifics from NCTM.

8) The alliance forged between NCTM and the National Science Founda-tion, with millions of dollars given to universities, school districts, educators, and private entities to write and support the NCTM Standards, has created a huge power base of money, politics, and ideology.

Ralph Raimi, October, 1995, “Whatever Happened to the New Math?” writes, “The year 1958 kicked off the largest and best financed single reform effort ever seen in mathematics education, the School Mathematics Study Group (SMSG), upon which the National Science Foundation (NSF) spent mllions of dollars over its twelve-year lifetime…the teachers’ colleges, the National Council of Teachers of Mathematics, and all the state and federal departments of education and nurture, who though loosely organized did still govern all teaching below the college level, were compelled for the time being to follow our [mathematicians] lead.

“Experimental scientists like [Oliver Wendell] Holmes understand that reality is not to be pushed round, neither by nine old men nor by a prestigious bunch of mathematical geniuses with a pipeline to the U.S. Treasury…The cadre of teachers already out there had preexisting interests and capabilities, the public patience was shorter than experiments that could lose a generation of children, and the educational experts, the professional education bureaucracy (PEB), was gathering its strength for the political battle that finally turned the pipeline back in their direction.

“The books for grades 1-8 come packaged for teachers with mountainous ‘Teachers’ Guides,’ in which the mathematics is swamped into insignificance by the instructions on engaging the attention and improving the self-esteem of students…”

David Klein, in “A quarter century of U.S. ‘math wars’ and political partisanship,” British Society for the History of Mathematics, 2006, http://www.tandf.co.uk/journals/titles/17498430.asp:

“The conflict [development of new California standards] was more than just a theoretical disagreement. At stake was the use of NCTM aligned textbooks in California, the biggest market in the nation.”

Michael McKeown, What’s at Stake in the K-12 Standards War, Chapter 13, “National Science Foundation Systemic Initiatives: How a small amount of federal money promotes ill-designed mathematics and science programs in K-12 and undermines local control of education.”

A Primer for Educational Policy Makers, edited by Sandra Stotsky (Peter Lang, New York, 2000). Also, see nychold.org.

“Many states and districts have accepted NSF Systemic Initiatives grants to make”fundamental, comprehensive, and coordinated changes in science, mathematics, and technologyeducation through attendant changes in policy, resource allocation, governance, management, content and conduct.” This article shows how it is all for the worse, and explains the dynamics behind acceptance of these grants…

“The NSF has provided many grants for the development and dissemination of fuzzy math programs. For example, here is a listing of some of the NSF grants that supported the Connected Mathematics Project (CMP): #9986372 Connected Mathematics Phase II # #9980760 Adapting and Implementing Conceptually-Based Mathematics Instructional Materials for Developmental-Level Students # #9950679 Preparing Elementary Mathematics Teachers for Success: Implementing a Research-Based Mathematics Curricula # #9911849 Teaching Reflectively: Extending and Sustaining Use of Reforms in the Mathematics Classroom # #9714999 Show-Me Project: A National Center for Standards-based Middle School Mathematics Curriculum Dissemination and Implementation #9619033 The Austin Collaborative for Mathematics Education # #9150217 Connected Mathematics Project.”

In 1977, the president of the National Council of Supervisors of Mathematics (NCSM), created by NCTM, stated, “The most important issue NCSM faced during the seventies was countering the ‘back to basics’ movement. At a 1975 meeting, members called on NCSM to quickly develop a clear and concise position that they could use as ammunition in the back-to-basics battle. NCSM obtained $4,500 of funding from the National Institute of Education to write and publish this influential statement. A defining point for NCSM was the publication of the NCSM Position Paper on Basic Mathematics Skills in 1976.”

The recognized disregard toward “basic skills” in the mathematics education community has been around for at least 50 years.

But it hit is zenith in 1989 when the federal government began massive financial support to the privately-run NCTM in its self-appointed status to determine the national trends in U.S. mathematics education. Parents, educators, and administrators assumed such federal support meant the programs, with their attached dollars, were worthwhile. As the millions of dollars have flowed to NCTM supporters in universities, school districts, consultants, and private businesses, their power, reputations and ideology have become profoundly and rigidly entrenched.

The Math and Science Partnership (MSP) with NSF has granted $600 million in 48 partnerships and more than 30 other “tool-development and evaluation projects, plus $400 million in 10,000 new fund-ing awards in professional and service contracts.

See nychold.org or http://www.nsf.gov/news/news_summ.jsp?cntn_id=105812&org=NSF.

Barry Garelick, “An A-Maze-ing Approach To Math,” Education Next magazine, No. 2, 2005. The National Science Foundation (NSF) promoted the NCTM standards beginning in 1990 and awarded millions of dollars in grant money for the writing of math texts that embraced them and to state boards of education whose math standards aligned with them.

CONCLUSION:When one reads, “A Nation at Risk,” the startling 1983 federal report about America’s decline in public education, it becomes clear why NCTM selected “equity” as its focus. Instead of helping teachers learn how to adapt lessons and activities to the increasingly diverse student populations—much of which can be related to economic classes, rather than racial groups—NCTM converted the “discipline” of mathematics into a manual of “processes” to produce egaliatarianism among American students.

Unless the U.S. Department of Education increases its own influence within mathematics education, as it has been doing under the unpopular No Child Left Behind and through the What Works Clearinghouse (which has established rigorous research standards), mathematics education cannot be rescued from the powerful social engineering of NCTM supporters. (In addition, it appears that better oversight of how NSF is spending tax dollars is needed.)

Schools and districts, one at a time, are being led by parents and professional mathematicians in a battle to bring the balance of basic skills and conceptual understanding back into mathematics education. Only in California, Michigan, and Massachusetts has there been a critical mass of concerned adults that brought such change to those states’ standards, curriculum, and assessments. American school children deserve better than having to wait one state at a time for the greatness of mathematics to make sense to them.

APPENDIX

I have measured my solutions in any field of study against the “ABCs of (good) journalism,” which are accuracy, brevity, and clarity. These were “drilled” into me, I’m glad to say, at a small, rural-based teachers’ college called East Texas State University, which is now part of the Texas A&M University system, where I earned my bachelor’s degree in journalism and art.

As a journalist for 17 years, I learned, and taught students in high school and junior college, not to use clichés, loaded words, and always to verify information for accuracy. Most important: Never put your own beliefs, political views, or values in your work for the general public.

Second most important: Think of a girl’s bikini when you write a story: short enough to be interesting but long enough to cover the subject. (Of course, this was in the 1960s and 1970s.)

With a master’s degree in counseling, I spent most of my time as a high school and middle school counselor unraveling emotionally-laden perceptions of events and bringing clarity to situations.

Getting certified in special education, and subsequently being a teacher for grades 6-12, I learned to resist the subconscious, and even open, stereotyping of learning disabled students and the “sorting” of deficient learners into categories.

Earning my certification in mathematics at age 45, I continued to choose to work with inner-city, high-risk minority kids in both middle and high schools, primarily in central Texas, as I had for 15 years.

Following that were the years in Washington state, where I confronted inappropriate images of American Indians while I served as their K-12 principal and teacher and earned a principal’s certification at Gonzaga University. Administrators, I learned, spend a large proportion of their time unraveling problems among different groups of people,—especially on an Indian reservation that must abide by tribal, federal, and state regulations—finding solutions, and offering clarity on the issues to all stakeholders.

Then, as a K-5 principal in an upper middle class school in Seattle, I worked with teachers to build a written, vertically aligned, and cohesive course of study in mathematics, since individual teachers and administrators—often with varying abilities—do not always offer a consistent learning environment for students. Good books and written materials, however, have historically provided solid references in the lives of all learners—and teachers. Those assets had not been present or utilized in the school, so that became a clear priority.

Finally, before retiring as a principal, I decided to work half-time as a high school math teacher in Seattle. My frustration became unbearable as I saw seniors who couldn’t work with fractions and therefore couldn’t work algebra problems, regular education freshmen who didn’t know their multiplication tables, English language learners who were lost in literary-based mathematics lessons, and special education students in my mainstream remedial classes who were just lost in the whole “processing” of mathematics. After working for 47 years, and seeing the monstrous problems we had allowed to develop in mathematics education, I said, “I’m done.”

Oh, yeah. I even earned 15 semester hours in the doctoral program of mathematics education at the University of Texas. Once I learned my training in mathematics education would be under the control of NCTM supporters with their constructivist philosophy, I had to choose not to continue in that field of study.

 

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Where’s the Math?
– DEBRA J. SAUNDERS
Sunday, October 17, 1999

 

THIS MONTH, the U.S. Department of Education came out with a list of 10 “exemplary” or “promising” math education programs. Kings County fourth-grade teacher Doug Swords was shocked at the department’s bad choices.

Some three years ago, his school district adopted MathLand, a math curriculum that prefers not to give lessons with “predetermined numerical results.” The department of Educrats, oops, I mean, Education, rated MathLand as “promising.” Today, he said 14 out of 18 teachers use MathLand only as a supplement. “I stashed away my Addison-Wesley textbooks, as did a few other teachers,” he explained.

Do you teach your students how to multiply? I asked him. (You wouldn’t think that would be something I’d have to ask, but these days, it is.) Yes, he said. Is MathLand helpful in teaching kids to multiply? “No, quite frankly,” Swords answered.

UC Berkeley math professor Hung-Hsi Wu couldn’t believe the department described MathLand as “promising.” He’d describe MathLand as “execrable.”

Or how about: “I can’t believe it’s math class.” A second-grade MathLand exercise called Fantasy Lunch instructs students to think up their fantasy lunch, draw it on paper, then cut out the “food” and place their drawings into a bag.

A frantic teacher wrote to me two years ago, furious that she had spent 75 minutes on that exercise and there was no math in it. It was “like therapy,” she said. On more than one occasion, her students asked her, “Can we do some real math now?”

Wu had problems with the other nine picks as well. While there were things he liked about the high school programs, they lacked what he called “mathematical closure. You start something, you ought to finish it.”

He said almost all of his students took more traditional math classes not cited as “exemplary” or “promising” by the Department of Education. That wouldn’t surprise Melissa Lynn, who got As in high-school math, then placed in the bottom 1 percent in the University of Michigan math placement test. She blames the Core-Plus program which the department rated as “exemplary.” “It had very good intentions, and wanted you to apply real principles to real life scenarios,” she explained this spring, “but it was missing the crucial element of algebra.“

Wayne Bishop, a math professor at Cal State L.A. who is the Ralph Nader of math curricula, sees the department’s move as a reaction against California’s return to math sanity — after a mad fling when state educrats embraced “there is no right answer” new-new math curricula.

He’s right. The selection panel appoint ed by the department had as a main criterion that the math series ascribe to trendy standards put out by the National Council for Teachers of Mathematics (NCTM).

Don’t ask me why. Last year Bishop looked at the scores of some of the students subjected to the brilliance of new- new math wizards. In 1995, NCTM Chairman Jack Price boasted about a program on which he worked. Turns out, Price’s star school ranked in the bottom quartile nationally in the STAR test last year. Only 12 percent of the school’s eighth graders scored above the national average. Price called that a successful program.

The department cited data that show schools whose test scores improved with MathLand. Bishop isn’t impressed. “They appear to have excluded data where MathLand scores dropped,” he noted.

An administrator from an urban district that stopped using MathLand had just visited a school that had seen a 27 percent increase in its math scores after buying a traditional math series that didn’t rate in the department’s Top 10. Under ideal circumstances, he said, MathLand could work, but urban districts don’t have too many ideal circumstances.

Bill Evers of Stanford’s Hoover Institution called the department’s Top-10 picks “unconditional surrender to fuzziness.”

Fuzziness? The department praised one K-6 math program because, “Features include problem solving; linking past experience to new concepts; sharing ideas; developing concept readiness through hands-on explorations; cooperative learning through small-group activities; and home-school partnerships.”

Sounds more like marriage counseling than math class.

The problem: It’s not the kids who need counseling here. It’s the adults who care so little about children’s success that they would assert that Fantasy Lunch makes for a “promising” math program.

 

You can reach Debra J. Saunders on The Gate at sfgate.com.

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©2005 San Francisco Chronicle

Was the Texas State Board of Education correct to reject the 3rd Grade curriculum of Everyday Math?

By Nakonia (Niki) Hayes
Columnist EdNews.org

Part One: The facts on Texas Mathematics Standard 3.4(a) and Everyday Math
Part Two: The facts on the physical construction of Everyday Math, 3^rd Grade, 3^rd Edition
Part Three: An introduction to the author of this report

Faulty construction was the complaint of the Texas State Board of Education (SBOE) of Everyday Math‘s third grade materials up for adoption in November of this year. It wasn’t about the 109, 263 proofreading mistakes in mathematics materials submitted by all the publishers this year in Texas. It was about an issue that speaks to the heart among many in the mathematics education debate today: the multiplication tables.

With a 7-6 vote (and one abstention), the SBOE rejected the third grade program of the third edition of Everyday Math. They said it did not meet the Texas mathematics standards that required third graders to learn their multiplication tables to “automaticity” (or “by heart”) through the 12s:

3.4(a) The student is expected to learn and apply multiplication facts through 12 by 12 using concrete models and objects.

The SBOE’s members who voted to reject the book have been blasted by Everyday Math representatives who maintain the book does meet the standard. They say with proofreading corrections the problem can be corrected. There’s also a fringe group who declares the board is “censoring” materials by their voting to reject the 3^rd grade curriculum. Lawsuits are being threatened.

Let’s look at the facts. In their Math Masters and Home Link Masters, which is composed of worksheets to be sent home to the family, Everyday Math states in the “Introduction of Unit 7, pp. 202-232, on “Multiplication and Division”:

“The goal is for children to demonstrate automaticity with x0, x1, x2, x5, and x10 multiplication facts and to use strategies to compute remaining facts up to 10 x 10 by the end of the year.”

Conclusion: “Automaticity” is not expected for x3, x4, x6, x7, x8, x9, and certainly not for x11 and x12.

Yet, in their Correlation to the Texas Knowledge and Skills (TEKS) manual, EM lists 39 pages that say the publisher has lessons/activities to support the Texas standardrequirement of learning and applying multiplication facts through 12 by 12.

All of those pages do support multiplication topics. Only one page, however, refers to working with 12 by 12 multiplication facts: P. 280 in the Teacher’s Lesson Guide (TLG) Vol. 1 says in a framed graphic at the bottom of the page:

ADJUSTING THE ACTIVITY (of “Playing Baseball Multiplication”): The basic game uses facts through 6 x 6. The advanced version of Baseball Multiplication, described on pages 276-277 in the Student Reference Book, uses products up to 12 x 12. Children use Math Masters, page 444 to keep score.

Since it is a Teacher’s Guide that determines the “objective for the lesson” each day, and with which supplemental workbooks or resources are correlated, it is important to review the objectives cited as “supporting” Texas Math Standard 3.4(a):

1) Vol. 1, P. 248-253: To provide opportunities to use arrays, multiplication/division diagrams, and number models to represent and solve multiplication number stories. (Lesson 4.3, “Multiplication Arrays”)

2) Vol. 1, P. 272-277: To review fact families and the Multiplication/Division Facts Table; and to guide children as they practice multiplication and division facts. (Lesson 4.6, “Multiplication and Division Fact Families”)

3) Vol. 1, P. 278-280: To practice multiplication facts. (Lesson 4.7, “Baseball Multiplication.”)

4) Vol. 2, P. 576-579: To review square-number facts, multiplication, and division patterns. (Lesson 7.1, “Patterns in Products”)

5) Vol. 2, P. 582-587: To guide children as they determine which multiplication facts they still need to learn. (Lesson 7.2, “Multiplication Facts Survey”)

Other sources cited by EM that support Texas Math Standard 3.4(a) include four worksheet activities in the Student Math Journal, four pages of review in the Student Reference Journal, and three activities in Math Masters (not an assessment source).

Two conclusions can be drawn from this factual information. First, Everyday Math does not meet the Texas Mathematics Standards for 3^rd Grade. Second, the claim by representatives that it does is, at best, a misinformed one among themselves. At worst, the claim is a deliberate attempt to mislead the board.

The academic value placed by some Texas state board members on the role of automaticity with multiplication facts [i.e., cognitive, not calculator], with 3^rd grade students learning and applying required mathematical knowledge and skills for future success, was made clear with their votes.

This mistake by Everyday Math is not a “proofreading” error. It is a clear indicator of the publisher’s philosophy regarding multiplication facts for 3^rd graders. It will also require more than a “supplement” to correct or improve the program.

Part Two: The facts on the physical construction of Everyday Math, 3^rd Grade, 3^rd Edition

As a preface, it is necessary to remind every adult involved (or not) with elementary education the daily schedule of those teachers. They are teaching the four core subjects of mathematics, language arts (reading/writing, which includes phonemic awareness, spelling, and grammar), social studies, and science. Most are required to cover other areas such as art, music, character education, keyboarding skills, etc.

When a 2003 survey was done in Seattle Public Schools’ to determine actual “seat time” for learning purposes, we discovered that elementary students received about 4.5 hours of actual academic learning time per day out of the 6.5 hours we had them for 176 student days. (This excludes holidays or days off for teacher in-service, etc, during the year). The limited availability of learning time had become apparent after we removed the following times of “non-academics” that had to take place each day:

• “Passing time”: Bringing children in from their lineup areas when the bell rang to start the school day and settling them into class routines.
• Recess twice daily (once in the morning and once in the afternoon).
• “Passing time x2”: Bringing children in from each recess and settling them into class routines.
• “Passing time: Taking children to the cafeteria for lunch (and going to restrooms before lunch

and after recess to wash hands, etc.)

• “Passing time”: Bringing children in from lunch recess and settling them into class routines.
• “Passing time”: Taking children to and retrieving them from the gym for P.E.
• “Passing time”: Taking children to and retrieving them from the library.
• Setting aside time for foreign language instruction, or, at our school, other electives such as journalism or photography to improve writing skills (which included the 3^rd grade).
• Preparing children for the end of the school day with assignments, coats, etc., and ready for buses.

Then, we subtracted times for early dismissals (teacher professional development, holidays, parent-teacher conference weeks), assemblies, field trips, fire drills, earthquake drills, lockdown drills, two class parties allowed each year, and at least two weeks for TESTING for the Direct Reading Assessment, Direct Writing Assessment, and the state-mandated WASL (Washington Assessment of Student Learning).

There are, of course, other events that cause children to be out of their “academic setting.” One of the biggest is the absence of the regular teacher from the classroom. Substitutes are generally warm bodies who babysit the students. (Add a curriculum with unique and/or unknown procedures, such as lattice multiplication and substitutes will definitely not be able to teach math.) High teacher absenteeism means lost learning time. Another major source of time lost is caused by student misbehaviors that take away from a classroom’s learning environment.

We realized we were lucky to get 4.5 hours per day for “real learning.”

It is therefore crucial to remember the class schedules, hours, and academic requirements that are expected of elementary teachers as the following information is reviewed from Everyday Math‘s 3^rd grade curriculum materials.

The number of pages in each published “component” (manual, workbook, journal, etc.) is shown in red font. Worksheets, or “paper-and-pencil” tasks, are shown in blue font. For those who claim they avoid the “traditional” use of paper-and-pencil tasks and thus prefer Everyday Math’s “activity-based learning,” this should be enlightening.

A total of 2,997 pages are in the 3^rd grade “components” of Everyday Math for teacher review and use. Of those, 925 are worksheets for students.

In addition, when the following materials are stacked, they measure 8 inches in height and weigh 18 pounds. The following body of information, without the page count, is from http://www.wrightgroup.com/index.php/componentfeatures?isbn=007608972X.

Grade 3 Core Classroom Resource Package includes:

• Teacher’s Lesson Guides (Volume 1 & 2) – The core of the Everyday Mathematics program, the Teacher’s Lesson Guide provides teachers with easy-to-follow lessons organized by instructional unit, as well as built-in mathematical content support. Lessons include planning and assessment tips and multi-level differentiation strategies to support all learners. 848 pages + 152 pages of “reference” information (glossary, charts, etc.) = 1000 pages.
• Teacher’s Reference Manual (Grades 1-3) – Contains comprehensive background information about mathematical content and program management. 290 pages
• Assessment Handbook – Grade-specific handbook provides explanations of key features of assessment in the Everyday Mathematics program. Includes Assessment Masters. 144 pages of examples for teacher information + 83 forms for teacher use = 227 pages + 64pages of blank assessment worksheets
• Differentiation Handbook – Grade-specific handbook provides that helps teachers plan strategically in order to reach the needs of diverse learners. 145pages
• Home Connection Handbook (Grades 1-3) – Enhances home-school communication for teachers and administrators. Includes masters for easy planning. 102 pages
• Minute Math (Grades 1-3) – Contains brief activities for transition times and for spare moments throughout the day. 112 pages
• Math Masters – Blackline masters for routines, activities, projects, Home Links/Study Links, and games. 468pages
• Number Grid Poster
• Sunrise/Sunset Chart
• Content by Strand Poster
• One set of Student Materials
  • Student Math Journals (Volumes 1 & 2) – These consumable books provide daily support for classroom instruction. They provide a long-term record of each student’s mathematical development. 281 pages
  • Student Reference Book (Grades 3) – This book contains explanations of key mathematical content, along with directions to the Everyday Mathematics games. 308 pages
  • Pattern Block Template – A clear, green, plastic tracing template contains a variety of geometric shapes with six of the shapes exactly matching the sizes of the pattern blocks.

To cover the Texas Mathematics Standards, the two volumes of the Teacher’s Lesson Guide would have to be completed; i.e., 848 pages of teaching directions and content in 176 days, supported by assignments from other EM “component” materials. Even though each day’s lesson/objective in the Guide covers an average of four pages, that still requires 212 days to get through all of the recommended lessons—without assessments. And that’s assuming each day’s lesson is accomplished in one day, with no “extensions” or reteaching required of the lesson.

A buzz phrase now being circulated by Everyday Math‘s publisher, the University of Chicago Center for Elementary Mathematics and Science Education, is “fidelity of implementation” (FOI), explained at http://cemse.uchicago.edu/node/3. This leads one to believe that only EM materials may be used to assure promised outcomes by the publisher, which also suggests a district should buy all of the above materials or EM should not be held accountable for negative learning results.

Questions come to mind

The first question has to center on the costs, in millions of dollars, for these materials. That includes the published materials and the professional development required to train teachers how to use those “effectively”—or at least with “fidelity.” And, in today’s “eco-friendly world,” the costs to the environment should be considered with how many trees it takes to print these materials.

Second, since such massive teaching resources seem to be written for every conceivable situation a teacher might face in a 3^rd grade mathematics class, what is the underlying message? Is this an effort to “teacher-proof” the materials?

Third, has any data been collected to see how much of EM material is actually covered each year?

Lastly, has anyone surveyed third grade teachers to see if they need or want 3,000 pages of materials to cull for only ONE of the four core subjects they must teach in 176 days?

Part Three: An introduction to the author, Nakonia (Niki) Hayes

Even though I retired from teaching in 2006, my interest in mathematics education has continued. For that reason, I decided to study the issues around the recent rejection of Everyday Math‘s third grade curriculum by the Texas State Board of Education. In order to understand my frame of reference in reporting on this situation, I offer a brief introduction to my background as an educator and journalist.

First, my bachelor’s degree is in journalism; my master’s, in counseling. I began my doctoral work in mathematics education at the University of Texas-Austin but decided its philosophy did not square with mine and I left the program. My work in journalism fields for 17 years included being a newspaper reporter, public information officer, and in public relations positions for two state senators. I’ve taught journalism in three high schools, at a community college, and I established a journalism program at a K-5 elementary school in 2001 that is still being used.

As an educator for 28 years, I became certified and experienced in special education, counseling, mathematics, and administration. While working as a special education teacher, I found that teaching mathematics to my middle and high school learning disabled students was a valuable way for them to learn structure, cause-and-effect, and linear thinking, all traits they needed to incorporate in their episodic learning and living. This led me to earn a certification in mathematics. Subsequently, I taught grades 6-12 for 15 years in high, at-risk populations in Central Texas and Washington state. My students were in special, regular, and gifted education classes. Because of my training, I was usually asked to take the English language learners.

I became acutely aware of the growing deficiencies of math skills among all of my students during 1987-1991. My question was, “What is happening at the elementary level that our students are coming to us with so many deficiencies in mathematics?”

As a middle and high school guidance counselor, I saw failure rates and “remedial courses” becoming the norm for students, both in public education and colleges.

I figured being a principal would allow me to affect curriculum and the teaching of mathematics in elementary schools. So, I became one. That experience includes my being a P-12 principal/ teacher on an American Indian reservation and principal at a K-5 school in Seattle, WA, with an 80%, upper-middle-class white, student population.

Lastly, my training in Jerusalem, Israel in 1998 and 1999 with Reuven Feuerstein introduced me to a pioneer in constructivism who knows how to use it effectively. Prof. Feuerstein’s International Center for the Enhancement of Learning Potential is dedicated to teaching cognitive remediation strategies. His work has been used throughout Europe with business leaders, in South Africa for children from apartheid policies, and for a half million children in Brazil. The American “home office” is with IRI Skylight Publishers in Chicago, IL, since there are some U.S. districts that use his programs.

In essence, my approach to mathematics education is not that of a mathematician. It is one of a “diversified” educator who happens to appreciate the reality and potential of mathematics and what it can mean for learners who master its power.

And in summary, this report isn’t about pedagogy, the primary focus of the math wars across this nation. It is about the accuracy of EM‘s content as compared to the Texas state standards and an accountability of simple quantitative facts. As a teacher and an administrator, I would not accept any program that clearly maintains the mile-wide-inch-deep approach in mathematics education, whether it’s in teacher materials or in content for students, and that is what Everyday Math offers.

Published November 27, 2007

Understanding the Revised NCTM Standards

Arithmetic is Still Missing!

by Bill Quirk  ( wgquirk@wgquirk.com)

• Contrary to Recent Reports, the NCTM Has Not Changed Its Philosophy
• They Still Call It “Arithmetic”
  • Mastery of Basic Facts or Derive Them When Needed?
  • Mastery of Standard Algorithms or Student-Invented Algorithms?
  • Mastery of Fractions or Simple-Case Methods for “Familiar Fractions”?
• Liping Ma’s Book: U.S. Elementary School  Teachers Don’t Understand Arithmetic
• Not Subtle in Connecticut:  Arithmetic is Obsolete
• Role of Mathematicians: Advice Solicited,  Advice Received,  Advice Ignored
• Published in American Educator (AFT), But Also Ignored by The NCTM

Contrary to Recent Reports, the NCTM Has Not Changed Its Philosophy

On April 12, 2000, The National Council of Teachers of Mathematics (NCTM)  released Principles and Standards for School Mathematics (PSSM), a 402 page revision of  the NCTM Standards. The next day The New York Times reported: “In an important about-face, the nation’s most influential group of mathematics teachers announced yesterday that it was recommending, in essence, that arithmetic be put back into mathematics, urging teachers to emphasize the fundamentals of computation rather than focus on concepts and reasoning.”  It was further reported that “the council added strong language to its groundbreaking 1989 standards, emphasizing accuracy, efficiency and basic skills like memorizing the multiplication tables.”

Compare the preceding New York Times quotes to the following contradictory quote, published by the NCTM (in the third PDF file, Commonsense Facts to Clear the Air) under “News and Hot Topics” at  NCTM Speaks Out: Setting the Record Straight about changes in Mathematics Education.

When calculators can do multidigit long division in a microsecond, graph complicated functions at the push of a button, and instantaneously calculate derivatives and integrals, serious questions arise about what is important in the mathematics curriculum and what it means to learn mathematics. More than ever, mathematics must include the mastery of concepts instead of mere memorization and the following of procedures. More than ever, school mathematics must include an understanding of how to use technology to arrive meaningfully at solutions to problems instead of endless attention to increasingly outdated computational tedium.   -NCTM,  Commonsense Facts to Clear the Air

It’s clear that The New York Times was fed misleading NCTM propaganda, perhaps designed to placate “math wars” opponents.  Not surprisingly, we will show here that the NCTM has not rediscovered arithmetic.  Similar to the original NCTM Standards, PSSM is vague about the major components of arithmetic mastery:

1. Memorization of of basic number facts
2. Mastery of the standard algorithms of multidigit computation.
3. Mastery of fractions

The NCTM has toned down the constructivist language, but they still stress content-independent “process skills” and student-centered “discovery learning”.  Similar to the NCTM Standards, PSSM emphasizes manipulatives, calculator skills, student-invented methods, and simple-case methods.

Although PSSM contains five “Connections” sections,  there continues to be no acknowledgement of the vertically-structured nature of  mathematics.  Mastery of math requires a step-by-step build up (in the brain) of specific content knowledge.  PSSM omits this aspect of the “connections” within mathematics.  The idea conveyed by the following example is not found in PSSM.

Example: Migrating up the math learning curve

Each of the following skills serves as a preskill for acquiring all higher skills. To move up to the next skill level, the student must remember all preskills.

1. The ability to instantly recall basic multiplication facts
2. The ability to factor integers
3. The ability to reduce a fraction to lowest terms.

The NCTM says they want to maximize “understanding”, but they still fail to recognize that specific math content must first be stored in the brain as a necessary precondition for understanding to occur.  Although rarely the preferred method, intentional memorization is sometimes the most efficient approach.  The first objective is to get it into the brain!  Then newly remembered math knowledge can be connected to previously remembered math knowledge and understanding becomes possible. You have to “know math” before you can “understand math”, “do math”, or “solve math problems.”

We conclude this introductory section by noting that there is evidence of a battle within the NCTM, with some voices crying out for genuine arithmetic. These voices were heard in the Principles and Standards for School Mathematics: Discussion Draft (PSSM Draft), published in October, 1998. At later points in this document you will find quotes from both PSSM and PSSM Draft. The quotes from PSSM Draft do not appear in PSSM, the final version published in April, 2000.  The voices of reason have been largely silenced!  Here is an example of the silencing.

Technology: PSSM Draft vs. PSSM

However, access to calculators does not replace the need for students to learn and become fluent with basic arithmetic facts, to develop efficient and accurate ways to solve multidigit arithmetic problems, and to perform algebraic manipulations such as solving linear equations and simplifying expressions.  -PSSM Draft, Page 43

Technology should not be used as a replacement for basic understandings and intuitions; rather it should be used to foster those understandings and intuitions.  -PSSM, page 24

They Still Call It “Arithmetic”

Mastery of Basic Facts or Derive Them When Needed?

Similar to the original NCTM Standards, PSSM fails to clearly acknowledge that the ability to instantly recall basic number facts is an essential preskill, necessary to free up the mind, first for mastery of the standard algorithms of multidigit computation, and next for mastery of fractions. Then, once this knowledge is also instantly available in memory, the mind is again free to focus on the next level, algebra.

As the quotes below show, the PSSM Draft emphasized quick recall, but it appears that PSSM has reverted to the NCTM Standards idea that “the ability to efficiently derive” is preferable to “the ability to instantly recall.”

Quotes From The PSSM Draft:  Recall Number Facts Quickly

Most students should be able to recall addition and subtraction facts quickly by the end of grade 2 and recall multiplication and division facts with ease and facility by the end of grade 4.  -PSSM Draft, Page 51

A certain amount of practice is necessary to develop fluency with both basic fact recall and computation strategies for multi-digit numbers.  Anderson, Reder, and Simon (1996) point out that practice is clearly essential for acquiring cognitive skills of almost any kind. -PSSM Draft, Page 114

Quotes From PSSM:  Derive Number Facts Quickly

Fluency with basic addition and subtraction number combinations is a goal for pre-K-2 years.  By fluency we mean that students are able to compute efficiently and accurately with single-digit numbers.   -PSSM, Page 84

When students leave grade 5, they should be able to . . . efficiently recall or derive the basic number combinations for each operation.  -PSSM, Page 149

Fluency with the basic number combinations develops from well understood meaning for the four operations and from a focus on thinking strategies. -PSSM, Page 152-153

If by the end of the fourth grade, students are not able to use multiplication and division strategies efficiently, then they must either develop strategies so that they are fluent with these combinations or memorize the remaining “harder” combinations. -PSSM, Page 153

The changes may be subtle, but notice that fluency with basic number facts is defined as the ability to “compute efficiently and accurately.”   It does not mean the ability to instantly recall.  Also, in PSSM it’s “recall or derive” by the end of grade 5, not “recall” by the end of grade 4 as recommended in the PSSM Draft.  Basic number fact “thinking strategies” appear to be preferred by the writers of PSSM.  They give grudging admission that memorization may be necessary.  There is some discussion of activities to teach fact relationships, but there is no discussion of mastery activities to facilitate fact memorization.

Mastery of Standard Algorithms or Student-Invented Algorithms?

Considering that the NCTM appears to prefer basic number fact derivation strategies, it’s not surprising that they also appear to prefer student-invented algorithms for multidigit computation. Here are some  relevant quotes from PSSM:

In the past, common school practice has been to present a single algorithm for each operation.  However, more than one efficient and accurate computational algorithm exists for each arithmetic operation.   In addition, if given the opportunity, students naturally invent methods to compute that make sense to them.  -PSSM, Page 153

Many students are likely to develop and use methods that are not the same as the conventional algorithms (those widely taught in the United States).  For example, many students and adults use multiplication to solve division problems or add starting with the largest place rather than with the smallest.  The conventional algorithms for multiplication and division should be investigated in grades 3 – 5 as one efficient way to calculate. -PSSM, Page 155   (bold emphasis added)

Students’ understanding of computation can be enhanced by developing their own methods and sharing them, explaining why their methods work and are reasonable to use, and then comparing their methods with the algorithms traditionally taught in school.  In this way, students can appreciate the power and efficiency of the traditional algorithms and also connect them to student-invented methods that may sometimes be less powerful or efficient but are often easier to understand.  -PSSM, Page 220

This last quote contains a hint of truth.  If a student is given a problem and allowed to struggle for a while, trying to solve the problem, then the student becomes motivated to listen and learn about the most efficient, general solution to the problem.  This is the essence of the lesson method used in Japan.  The mistake is to elevate the value of “easier to understand” student-invented methods, while not stressing the power, mathematical importance, and universal acceptance of the efficient, general “algorithms traditionally taught in school.”

Efficient, Accurate, and (Possibly) General Methods

On page 32 of PSSM,  the term “computational fluency” is defined as “having and using efficient and accurate methods for computing”.  Later on the same page, we are told that students should “see the usefulness of methods that are efficient, accurate, and general.”  On page 87  we are told  “Teachers also must decide what new tasks will challenge students and encourage them to construct strategies that are efficient and accurate and that can be generalized.”

Has the definition of computational fluency been (appropriately) expanded to include “general”?   No, the original definition of computational fluency, including only efficient and accurate,  is restated on pages 79 and 153.   It appears that some PSSM writers recognized that all three characteristics contribute to the power of the standard algorithms of arithmetic.  But the standard algorithms are not mentioned in this context (page 87).  Instead, on this page we are advised:  “As students encounter problem situations in which computations are more cumbersome or tedious, they should be encouraged to use calculators to aid in problem solving.”

The Truth is in The Examples

Within the five “Number and Operations” sections, PSSM includes (only) three illustrations of  multidigit computation. All are “student-invented” strategies.  On page 85 we learn “In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different.”  There is no discussion of the accuracy, efficiency or generality of any method found in these three illustrations. Apparently those who wrote about “computational fluency” failed to communicate with those who developed the examples.

1. Figure 4.3 on page 85 presents six student solutions for computing 25 + 37. Student 2’s method utilized 12 “tallies” (four vertical marks crossed by a horizontal mark) followed by two additional vertical marks, with the 12 tallies identified by  5 written above the first tally, 10 written above the second tally, 15 written above the third tally, and so on until 62 is written above the concluding pair of vertical lines.  The NCTM is pleased with the “completeness” of Student 2’s thinking.  Student 4 correctly used the standard algorithm for addition, but the NCTM appears not to notice, even remarking that student 4’s thinking is “not as apparent.”
2. Figures 4.4 and 4.5 on page 86 describe student strategies for computing 153 + 273. Randy’s method is described first.  He used beans, bean sticks (10 beans), and rafts of bean sticks (100 beans).  The “conventional algorithm” is used successfully by some nameless students,  but unsuccessfully by other nameless students.  “Becky finds the answer using mental computation and writes nothing down except her answer.”    Subtle, but effective.  Randy and Becky are worth recognizing by name.
3. Page 153 presents two student solutions for dividing 728 by 34.  Henry used the method of repeated subtraction of multiples of 10, which he apparently invented. Michaela used long division, which she apparently invented. Mrs. Sparks “saw the relationship between the two methods described by the students, but she doubted that any of her students would initially see these relationships”.  This is a surprising lack of confidence, considering the remarkable discovery abilities demonstrated by Henry and Michaela.(We used “long division” to briefly describe the method used by Michaela, but the phrase long division in not found in PSSM.)

In this last illustration, we again we have a hint of the lesson method used in Japan.  But we learn that Mrs. Sparks objective is to “help the students understand, explain, and justify their computational strategies,”  rather than working to achieve closure by  connecting the two methods and teaching long division, emphasizing the efficiency and generality of the long division algorithm.

Mastery of Fractions or Simple-Case Methods for “Familiar Fractions”?

Consistent with the lukewarm treatment of number fact recall, PSSM fails to emphasize the importance of the ability to factor integers, and PSSM never discusses any of the details related to the addition, subtraction, multiplication, division, and simplification of fractions.  The phrase “common denominator” is not found in PSSM.

The NCTM says they want students to “develop and analyze algorithms for computing with fractions” and “develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.”  – PSSM, Page 214.

PSSM’s treatment of fractions offers just two illustrations, one for comparing fractions,  and the other for dividing fractions.  Both are simple-case methods, and neither is efficient or general.

• For comparing 7/8 to 2/3,  PSSM recommends the use of physical “fraction strips”, never mentioning the concept of converting to a common denominator.  -PSSM, Page 216
• For dividing 5 by 3/4 they recommend the method of “repeated subtraction,” after first suggesting (see the following quote) that  “invert and multiply” is too difficult for today’s kids.
  • How about 3/4 divided by 5 using repeated subtraction?  Do they expect that  kids will find it easy to use repeated subtraction to show that 9/11 divided by 3/121 equals 33?  No, they will tell you that these are unreasonable divisions, and 11 and 121 are unreasonable denominators (see Connecticut).  They say that students “need to see and explore a variety of models of fractions, focusing primarily on familiar fractions such as halves, thirds, fourths, fifths, sixths, eighths, and tenths.”  -PSSM, Page 150
  • A comment from Professor Richard Askey:  “Given that the authors had a very nice chapter on this topic in Liping Ma’s book with varied word problems and comments  from teachers about such things as objecting to using 1 3/4 divided by 1/2 to see if students understood division of fractions since this is so easy to do without understanding how to divide fractions, I find it shocking that successive subtraction is pushed as the way to do division of fractions, and the final step when successive subtraction does not work is just 1/4 divided by 1/2.  Even that is not adequately explained, it is just done.  As the Chinese teacher suggested, this is too
    easy to see if division of fractions is understood or not.”
    • Note: For the 5 divided by 3/4 problem discussed above in PSSM, the final step, 1/2 divided by 3/4, is not explained.  They just say that 2/3 is left after 6 subtractions of 3/4.
    • How does the student actually carry out the 6 subtractions of 3/4?  We are told “students can visualize repeatedly cutting off 3/4 yard of ribbon” from 5 yards of ribbon.  One wonders if they use scissors to help them “visualize”.

PSSM on “invert and multiply””

The division of fractions has traditionally been quite vexing for students.  Although “invert and multiply” has been a staple of conventional mathematics instruction and although it seems to be a simple way to remember how to divide fractions, students have for a long time had difficulty doing so.  Some students forgot which number is to be inverted, and others are confused about when it is appropriate to apply the procedure.  A common way of formally justifying the “invert and multiply” procedure is to use sophisticated arguments involving the manipulation of algebraic rational expressions—arguments beyond the reach of many middle-grade students.  This process can seem very remote and mysterious to many students. Lacking an understanding of the underlying rationale, many students are therefore unable to repair their errors and clear up their confusions about division of fractions on their own. An alternate approach involves helping students . . . understand the meaning of division as repeated subtraction.”  -PSSM, page 219   (underline added)

For the sophisticated arguments, see pages 2-3 in Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education, By H.Wu

PSSM recommends that probability be covered in every grade, offering five PSSM sections on “Data Analysis and Probability”.  The NCTM is not bothered by the fact that any meaningful discussion of elementary probability requires prior mastery of fractions.

Liping Ma’s Book: U.S. Elementary School Teachers Don’t Understand Arithmetic

U.S. elementary school teachers frequently don’t understand the underlying “whys” of arithmetic, but the same can’t be said of Chinese math teachers.  This is one message of the new book,  Knowing and Teaching Elementary Mathematics (KTEM) by Liping Ma.  (Please see Roger Howe’s Review of Liping Ma’s Book  and  Richard Askey’s Review of Liping Ma’s Book).

U.S. teachers fared poorly when asked questions related to the teaching of:

1. Subtraction with regrouping
2. Multidigit multiplication
3. Dividing fractions

Don’t blame the teachers!  None of these topics appear in PSSM or the PSSM Draft.  The term “subtraction with regrouping (or renaming)” is never used.  There is one example of multidigit multiplication, found on page 220 of PSSM. The text states: “the cost of 1.37 pounds of cheese at $2.95 a pound might be estimated, although a calculator would probably be the preferred tool.”  The failure to cover division by fractions was discussed above.

Liping Ma’s book is referenced in two ways in PSSM.  In each case her ideas have been misused:

1. Ma uses the phrase “profound understanding of fundamental mathematics” (PUFM).  A teacher who possesses PUFM has a comprehensive understanding of the “network of procedural and conceptual topics” that comprise elementary mathematics. Such a teacher “is able to reveal and represent connections among mathematical concepts and procedures to students.” -Ma, Page 124
   • Ma is referenced on page 17 of PSSM, where we are told that teachers who know “fractions can be understood as parts of a whole, the quotient of two integers, or a number on a line” have an understanding that may be characterized as ‘profound understanding of fundamental mathematics’ (Ma, 1999).”
     • This is not an illustration of PUFM. It is an example of a basic learning expectation for all students.
2. The terms “compose” and “decompose” appear frequently in PSSM and in KTEM (but not in the PSSM Draft, which preceded the publication of KTEM). In KTEM these words are only used relative to place value.  If “compose” appears alone in KTEM, it’s always shorthand for “compose a unit of higher value” (old term is carrying).  If “decompose” appears alone in KTEM, it’s always shorthand for “decompose a unit of higher value” (old term is borrowing).  PSSM never uses these terms this way. PSSM never discusses carrying or borrowing  (or any other equivalent terms).

The NCTM says in boldface  “The status quo of traditional mathematics isn’t working.”  (See the second fact sheet at NCTM Speaks Out: Setting the Record Straight about changes in Mathematics Education.)   Liping Ma’s book shows that the real problem is the failure to correctly teach “traditional mathematics.”

Not Subtle in Connecticut:  Arithmetic is Obsolete

It’s difficult to see what’s missing, but in Connecticut it’s in boldface.  The Connecticut Mastery Test (CMT), Third Generation, Mathematics Handbook states ( pages 5 – 6 ):

• 4th graders will continue not to be expected to demonstrate pencil-and-paper mastery of:
  • subtraction with regrouping.
• 6th graders will continue not to be expected to demonstrate pencil-and-paper mastery of:
  • addition and subtraction of numbers greater than 10,000 or money amounts greater than $100;
  • multiplication and division by 2-digit or larger factors or divisors;
  • addition and subtraction of fractions with unlike denominators; and
  • computation with non-money decimals.
• 8th graders will continue not to be expected to to demonstrate pencil-and-paper mastery of:
  • addition and subtraction of numbers greater than 10,000 or money amounts greater than $100;
  • addition and subtraction of fractions, except halves and thirds or when one denominator is a factor of the other; and
  • division with fractions or mixed numbers.

Connecticut teachers are told they have:

Permission to Omit. An amazing amount of time and energy is still expended by you and by your students on increasingly obsolete skills.  Teachers need to give each other permission to skip textbook pages that no longer serve a useful purpose.  So give yourself and your colleagues permission to omit such things as:

• pencil and paper multiplication problems with two-digit or larger factors (3 digits by 1 digit should be enough);
• paper and pencil division problems with two-digit or larger divisors (4 digits by 1 digit should be enough); and
• computation with fractions with unreasonable denominators like sevenths or 11ths (halves, fourths, eighths; thirds and sixths; fifths and tenths should be enough).

Update March 2002: A toned-down version of “Permission to Omit” is now found at the top of page 7 via this link.

Steven Leinwand, mathematics consultant for the Connecticut Department of Education,  wrote  “I believe that CT’s expectations are in fact aligned with the NCTM Standards – both old and new.  However, since these Standards cover grade bands and tend to be more general that our test specifications, it is often difficult to do a direct correlation.”
-email message to wgquirk@wgquirk.com, August 2, 2000

Don’t expect any change in the 13 NSF-sponsored “Standards based” math programs.  Their promoters will reason similarly.

Role of Mathematicians: Advice solicited,  Advice Received, Advice Ignored

The NCTM solicited advice from mathematicians:

In order to provide for this complex advisory function, the NCTM petitioned each of the professional organizations of the Conference Board of the Mathematical Sciences (CBMS) to form an Association Review Group (ARG) that would respond, in stages, to a series of substantial and focused questions framed by the Principles and Standards writing group in the course of its work.   -PSSM, Page xv

The NCTM received excellent input (see examples below), but ignored it. None of the several response reports (including the two quoted below) are referenced in either PSSM or the PSSM Draft.

One set of questions, via a letter from Joan Ferrini-Mundy and Mary Lindquist on April 1, 1997, asked:

1. What is meant by “algorithmic thinking”?
2. How should the Standards address the nature of algorithms in their more general mathematical context?
3. How should the Standards address the matter of invented and standard algorithms for arithmetic computation?
4. What is it about the nature of algorithms that might be important for children to learn?

Roger Howe, Professor of Mathematics at Yale University, responded in the American Mathematical Society NCTM2000 Association Resource Group Second Report in June, 1997 (See following first report via pdf file at Reports of AMS ARG ). Here are five excerpts from Professor Howe’s response:

An important feature of algorithms is that they are automatic and do not require thought once mastered. Thus learning algorithms frees up the brain to struggle with higher level tasks.  On the other hand, algorithms frequently embody significant ideas, and understanding of these ideas is a source of mathematical power.  -Howe, Page 273

 . . . we suspect it is impractical to ask all children personally to devise an accurate, efficient, and general method for dealing with addition of any numbers—even more so with the other operations. Therefore, we hope that experimental periods during which private algorithms may be developed would be brought to closure with the presentation of and practice with standard algorithms. Also we hope care would be taken to ensure that time spent developing and testing private algorithms will not significantly slow overall progress. -Howe, Page 274

Standard algorithms may be viewed analogously to spelling:  to some degree they constitute a convention, and it is not essential that students operate with them from day one or even in their private thinking; but eventually, as a matter of mutual communication and understanding, it is highly desirable that everyone (that is nearly everyone—we recognize that there are always exceptional cases) learn a standard way of doing the four basic arithmetic operations. -Howe, Page 275

We do not think it wise for students to be left with untested private algorithms for arithmetic operations—such algorithms may only be valid for some subclass of problems.  The virtue of standard algorithms—that they are guaranteed to work for allproblems of the type they deal with—deserves emphasis.  -Howe, Page 275   (bold added)

We would like to emphasize that the standard algorithms of arithmetic are more than just “ways to get the right answer”—that is, they have theoretical as well as practical significance.  For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not be accident, but by virtue of construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials. The division algorithms is also significant for later understanding of real numbers. -Howe, Page 275

Kenneth Ross, Professor of Mathematics, University of Oregon, also responded to these four questions in The Mathematical Association of America’s Second Report from the Task Force.  Here are three excerpts from Professor Ross’s June 17, 1997 report:

The NCTM Standards emphasize that children should be encouraged to create their own algorithms, since more learning results from “doing” rather than “listening” and children will “own” the material if they create it themselves. We feel that this point of view has been over-emphasized in reaction to “mindless drills.”  It should be pointed out that in other activities in which many children are willing to work hard and excel, such as sports and music, they do not need to create their own sports rules or write their own music in order to “own” the material or to learn it well. In all these areas, it is essential for there to be a common language and understanding. Standard mathematical definitions and algorithms serve as a vehicle of human communication. In constructivistic terms, individuals may well understand and visualize the concepts in their own private ways, but we all still have to learn to communicate our thoughts in a commonly acceptable language. -Ross, Page 1

The starting point for the development of children’s creativity and skills should be established concepts and algorithms. As part of the natural encouragement of exploration and curiosity, children should certainly be allowed to investigate alternative approaches to the task of an algorithm. However, such investigation should be viewed as motivating, enriching, and supplementing standard approaches. Success in mathematics needs to be grounded in well-learned algorithms as well as understanding of the concepts. None of us advocates “mindless drills.”  But drills of important algorithms that enable students to master a topic, while at the same time learning the mathematical reasoning behind them, can be used to great advantage by a knowledgeable teacher. Creative exercises that probe students’ understanding are difficult to develop but are essential.  -Ross, Page 1-2

The challenge, as always, is balance. “Mindless algorithms” are powerful tools that allow us to operate at a higher level. The genius of algebra and calculus is that they allow us to perform complex calculations in a mechanical way without having to do much thinking. One of the most important roles of a mathematics teacher is to help students develop the flexibility to move back and forth between the abstract and the mechanical. Students need to realize that, even though part of what they are doing is mechanical, much of mathematics is challenging and requires reasoning and thought. -Ross, Page 2

Published in American Educator (AFT), But Also Ignored by the NCTM

Below you will find links to two articles that were published in the Fall 1999 issue of American Educator/American Federation of Teachers.  Both of these were available well prior to the release of PSSM.  Neither is referenced in PSSM.

Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education, By H.Wu, Professor of Mathematics, University of California, Berkeley

Please see Professor Wu’s discussion of the division of fractions and the standard algorithms.  The following two quotes are from the beginning of Professor Wu’s article.

Education seems to be plagued by false dichotomies. Until recently, when research and common sense gained the upper hand, the debate over how to teach beginning reading was characterized by many as “phonics vs. meaning.”  It turns out that, rather than a dichotomy, there is an inseparable connection between decoding—what one might call the skills part of reading—and comprehension. Fluent decoding, which for most children is best ensured by the direct and systematic teaching of phonics and lots of practice reading, is an indispensable condition of comprehension. -Wu, Page 1

“Facts vs. higher order thinking” is another example of a false choice that we often encounter these days, as if thinking of any sort—high or low—could exist outside of content knowledge. In mathematics education, this debate takes the form of “basic skills or conceptual understanding.” This bogus dichotomy would seem to arise from a common misconception of mathematics held by a segment of the public and the education community: that the demand for precision and fluency in the execution of basic skills in school mathematics runs counter to the acquisition of conceptual understanding. The truth is that in mathematics, skills and understanding are completely intertwined. In most cases, the precision and fluency in the execution of the skills are the requisite vehicles to convey the conceptual understanding. There is not ‘conceptual understanding’ and ‘problem-solving skill’ on the one hand and ‘basic skills’ on the other. Nor can one acquire the former without the latter. – Wu, Page 1

 Knowing And Teaching Elementary Mathematics, By Richard Askey, John Boscom Professor of Mathematics, University of Wisconsin-Madison

The title of this article is also the title of the new book by Liping Ma. Please see Professor Askey’s discussion of the division of fractions.  The following quote is from Professor Askey’s article.

As the word ‘understanding’ continues to be bandied about loosely in the debates over math education, this book provides a much-needed grounding. It disabuses people of the notion that elementary school mathematics is simple—or easy to teach. It cautions us, as Ma says in her conclusion, that  ‘the key to reform…[is to] focus on substantive mathematics.‘  And at the book’s heart is the idea that student understanding is heavily dependent on teacher understanding.  -Askey, Page 2
(Bold emphasis added)

Next?

• How the NCEE Redefines K-12 Math
• TERC Hands-On Math: The Truth is in the Details
• Searching For The Truth About the TIMSS 4th Grade Math Test
• The Anti-Content Mindset: The Root Cause of the “Math Wars”
• Memorize Multiplication Facts?  Cheney, Yes.  Romberg, Abstain.
• Understanding the Original NCTM Standards
• Understanding the New Jersey Math Standards
• Home Page: Understanding the NCTM’s Math “Standards” and NCTM “Math Reform”

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Copyright 2000-2002 William G. Quirk, Ph.D.