MATHEMATICS “COUNCIL”
LOSES HARD-EARNED CREDIBILITY

The National Council of Teachers of Mathematics,
now led by theoreticians from our Schools of Education,
imposes policies that distort the teaching process
and heavily impair the learning of school mathematics.

By Frank B. Allen
Professor of Mathematics Emeritus, Elmhurst College,
National Advisor for Mathematically Correct,
and former President of NCTM

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When about 20,000 math “teachers” convene to attend a convention of the National Council of Teachers of Mathematics (NCTM), as they did in Chicago on April 12, some high ranking official is expected to welcome them. In earlier years, when the NCTM had a well-deserved reputation as a constructive force strongly focused on improving the teaching of school mathematics, this would have been a pleasant assignment. But, sadly, the publication in 1989 of the first of the NCTM’s three “Standards” reports (which are not standards because they do not set levels of student achievement) marked a drastic change in the Council’s status. Now, Its hard-won reputation squandered by its shrill advocacy of failed procedures, the NCTM stands before the nation as a rogue organization whose Standards-based policies are largely responsible for the undeniable fact that school mathematics in the USA is a disaster. Publication of the “Standards” also marked equally drastic changes in both the Council’s role and in the roles of its members. Any city’s welcome must be tempered by the following facts.

 

THIS IS NOT A COUNCIL. The NCTM is no longer a “Council”, i.e. “An assembly convened for consultation, advice or agreement”. In pre-Standards years it served that purpose beautifully. Its meetings provided classroom teachers with a place where they could assemble as peers to discuss, in a collegial atmosphere, ways to improve the teaching of mathematics. These free and open discussions were conducted without fear of censorship. No more. Standards-based policies dominate all NCTM meetings and the mounting evidence which discredits these policies is ignored. Most speakers are theoreticians from our Schools of Education where the false doctrines expressed in the “Standards” reports originated. In the eleven years since the publication of the first Standards Report triggered a controversy which is now so intense that it is aptly described as “the math wars,” NCTM publications have been closed to those who strongly oppose Standards-based policies. This is not a council.

THESE ARE NOT TEACHERS. Many of the procedures advocated by the “Standards” cannot be described as teaching in the accepted sense of this word. The constructivist-discovery theory, advocated by the NCTM, places heavy emphasis on cooperative or group learning and relegates the teacher to the role of “Facilitator”. As a result of the widespread application of this theory, math teachers who serve as directors of learning, and as expositors who impart knowledge and understanding by direct whole-class instruction, have largely disappeared from the nation’s classrooms. They have been replaced by “Facilitators” whose roles are hard to define. They move from group to group, sometimes answering a question with a question because facilitators are discouraged from giving help and from answering questions directly. The facilitator serves as “A guide on the side” and not as “A sage on the stage”. Many facilitators seem to believe that these bumper sticker slogans, provide ample justification for this drastic change in the teacher’s role. A more responsible view is that the effectiveness this profound change in the way the cultural heritage of the human race is transmitted from each generation to the next should be verified by replicable research BEFORE it is applied nationwide. No such verification exists. Nor is there any proof that teacher-directed instruction necessarily inhibits discovery or discourages student generated conjectures. There IS mounting evidence that facilitators are not effective teachers as measured by their student’s performance on objective tests.

THIS IS NOT INSTRUCTIVE MATHEMATICS. The standards-based subject (SBS) purveyed by the NCTM is so laden with major defects, so over-adjusted to alleged student learning deficiencies, that it no longer retains the properties of mathematics that make its study worthwhile. Mathematics is EXACT, ABSTRACT and LOGICALLY STRUCTURED. These are the ESSENTIAL and CHARACTERIZING properties of mathematics which enable it, WHEN PROPERLY TAUGHT to make unique and indispensable contributions to the education of all youth.

Students need the experience of working in a subject where answers are exact and can be checked for consistency with known facts. But in the SBS the importance of correct answers is minimized and student problems are often deliberately ambiguous. Hence the term “Fuzzy math”.

Students also need help in taking the crucial step from using manipulatives to illustrate various aspects of a general principle to understanding and formulating a general (abstract) statement of this principle. Without this step the extensive use of manipulatives is of little value. Heavy sales of manipulative materials and the scores of “Workshops” at NCTM meetings, suggest that many teachers are reluctant or unable to take this step. They want to stay with manipulatives (training wheels) AS LONG AS POSSIBLE.

 

Indeed, methods and gimmicks are a popular cop-out in teachers education programs. Universities seem to produce teachers who cannot understand the theory, research or principles underlying their subject, but rather want methods and techniques to satisfy and pacify their charges.
Gerald L. Peterson, Saginaw Valley State University in National Forum, Summer 1992, p. 48.

 

Cheating Our Children

The philosophy of moral relativism, which condones deviate behavior and insists that nothing is really wrong, now dominates the mathematics classroom. Students must not be told that they are wrong because this might impair their “self esteem” and the teacher might be seen as a judgmental despot. Math must be made easy and fun. In earlier years it was well recognized that math, properly taught, is a difficult subject whose mastery requires hard work and sustained concentration. Education was seen as the process of ADJUSTING STUDENTS to the subject. Now, NCTM policy seeks to ADJUST THE SUBJECT to students and to whatever learning deficiencies or “learning styles” they may have. THIS IS EDUCATION TURNED ON ITS HEAD.

Below are some examples of how the widespread use of this policy is cheating our school children. Note that the learning deficits are “adjusted to” rather than remedied as good educational policy would require.

 

* If the student is a poor reader or has a short attention span, don’t try to remedy these defects by demanding intensive study of elementary mathematical concepts. Instead, submerge him in a cooperative learning group where these weaknesses will not be noticeable but will remain to handicap him for the rest of his life.

* If the students do poorly on objective tests, avoid them. Resort to some form of highly subjective “authentic” assessment which conceals the student’s serious misconceptions. Better yet, use group testing which conceals them even more effectively,

* Adjust to his supposed learning difficulties by watering down or oversimplifying mathematics to insure that everybody passes. Failure must not be recognized, much less confronted and remedied.

* Eliminate competition from the mathematics classroom so that nobody loses. Let competition be confined to extracurricular activities such as athletics, where it is intense and to the real world, where it is all-pervasive.

The conjunction of these statements clearly implies that NCTM policies tend to produce students who have not learned how to read intensively for meaning, how to listen, how to concentrate, how to think or HOW TO LEARN. These children have not reaped any of the benefits that should be obtained from a properly taught course in school mathematics. Many of them graduate with self esteem, but totally unprepared to cope with the competitive world that confronts them. This adds up to a MIND-WASTING FORM OF CHILD ABUSE.

Still another of the destructive results of the “adjust the subject” process is based on the reformer’s strongly held conviction that certain minorities, such as African Americans and Hispanics cannot learn structured mathematics. This attitude deprives these minorities of the opportunity to learn. It is distressing to see this from people who profess concern for minorities under the banner of “Equity.”

 

Effect on Higher Education

Colleges must have students. They must be concerned with the bottom line. In the last ten years, they have been forced to, 1) lower the threshold for admission to accommodate hoards of less qualified applicants, and 2) devote an ever increasing portion of their time and resources to the remedial instruction that is necessary to bring these less qualified students up to speed. In some colleges more than 70 percent of the students enrolled in mathematics must take remedial courses covering material they should have learned in high school. The same situation exists in other subjects. Thus the lower academic standards in our schools are resulting in lowered standards in post-secondary education. To paraphrase an old adage “An ebbing tide lowers all ships”. Our system of higher education, once our pride and joy, is now in jeopardy.

 

The Lineage of “Reform”

Reformers often accuse their critics of wanting to go “Back to Basics”, implying that these critics cannot understand new theories of learning. Actually, there is nothing new about the theories promulgated by the Standards and it is the SBS reformers who are going back to old and discredited doctrines. This “adjust the subject to the student” theory is just another recycling of the “child-centered school” ideas that came out of Columbia University in the twenties. If the NCTM Board members regard them as new, it is because they do not know the history of American education since 1900. (For elaboration of this theme see the section on “Orthodoxy Masquerading as Reform,” page 48 in The Schools We Need and Why We Don’t Have Them by E.D. Hirsch, Jr.)

 

The Significance of Structure

The structured character of mathematics enables us to derive new facts (conclusions) from certain previously established facts (hypotheses) by building logical arguments (proofs). This proof process, which is the very essence of mathematics, establishes meaningful connections between existing facts and builds structure by adding to our fund of known facts. Proof, properly introduced, does not make mathematics more austere and difficult. On the contrary, it can be an exciting adventure which marks the student’s optimum path to understanding. A proof confers understanding on the student by showing how a formula or theorem can be derived from previously accepted facts, i.e. how it fits into a hierarchy of mathematical facts. What other kind of understanding exists?

In SBS this structured path to understanding is blocked in three devastatingly effective ways.

 

1. The neglect of the fundamental operations of arithmetic in the early grades. The early use of calculators, which detracts from the importance of learning the number facts, the algorithms for multiplication and division and the procedures for manipulating fractions, also destroys the foundation on which the student’s understanding of algebra is based.

2. Neglect of language skills. While the “Standards” speak of “Higher thinking skills”, they do not provide the student with the gradually formalized natural language which is needed to acquire and use such skills. This language which requires understanding such words as “and”, in conjunctive statements, “or” in disjunctive statements, and, “if-then” or “implies” in implicative statements, could be learned in grades 6-8. Introduced there, it could be used to construct simple essay and flow proofs in algebra where the study of formal proof should begin. The total lack of this vocabulary in SBS is a tremendous handicap to students in dealing with proof in Geometry. This may explain why The NCTM has watched, without protest, as proof has practically disappeared from the bloated, 900-page, expensive, multi-colored coffee table books that pass for geometry texts in America.

3. Neglect of clarifying, structure-building proof. The NCTM’s attitude toward proof is revealed by a key statement that appears in the CURRICULUM AND EVALUATION STANDARDS on page 150.

Although the hypothetical deductive nature of geometry first developed by the Greeks should not be overlooked, this standard proposes that the organization of geometric facts from a deductive perspective should receive less emphasis, whereas the interplay between inductive and deductive experiences should be strengthened.

Now it is precisely in this organization of facts from a deductive perspective that the student encounters proof. This standard is readily interpreted by teachers as “go easy on proof”. One wonders why the NCTM saw the need for this “Standard”. When it was published in 1989, the trend toward downgrading proof, without which deductive organization is impossible, was already far advanced. Students entering high school at that time had already had at least three years of “inductive experiences” where they had encountered many of the FACTS of plane geometry. Now it would seem to be time to use the proof process to forge connections between these facts, to organize them into logical structures and to consider extending the deductive organization involving theorems and proof to algebra. Instead of proposing this, the NCTM advocates a reduction of deductive organization in geometry! This is bizarre behavior by people who are fond of talking about “Structure” and “Connections.”

 

An Analysis of the Present Situation and How it Developed

The convention in Chicago is an assembly of facilitators and delegates from the Education Establishment whose irresponsible policies have caused the present crisis in school mathematics, and in many basic school subjects.

In many articles, written by reformers, a statement beginning “Research shows” is used to justify “Reform” policies. In most cases this research is entirely anecdotal or fatally flawed by the lack of control groups. Challenge to NCTM leaders: Cite supporting well-designed research with control groups that can be replicated by reputable researchers.

The advocates of “Reform” know that they cannot meet this challenge. If they had had confidence in their ability to do so, they should and perhaps would have proceeded much differently. The American public has always been receptive to new and better ways of doing things. The NCTM could have said “Look, we have research results which can be replicated, that prove that a constructivist-discovery approach involving cooperative learning will substantially raise the level of student achievement in school mathematics. This will be shown by standardized, objective tests that are externally set, externally graded and are comparable to world norms, such as those used in other industrialized nations.” Realizing that this statement could not be supported, the leadership of the NCTM, a small but well-connected group which has seized control of that once prestigious organization on behalf of the Education Establishment, went blundering ahead advocating new and untried programs that have no support in either research or experience and run counter to strong caveats expressed by their own Research Advisory Committee. In doing so they turned the nation’s school system into a giant laboratory for testing experimental, untried theories. This is CENSURABLY IRRESPONSIBLE.

 

Moving the Goal Posts

When, as the result of widespread use of Standards-based programs, test scores on objective tests, such as those used in the Third International Math and Science Study, came crashing down, it was belatedly evident to our “reformers ” that these tests or, for that matter, any standardized, objective tests, do not measure the subtle nuances of student understanding which are discernable only by using a complicated, highly subjective procedure called “authentic assessment”.

At this point the NCTM joined the Education Establishment (EE) in a nation-wide assault on standardized tests. Most parents see this for what it is, a determined and disgraceful effort by the EE to avoid accountability. These parents want their children to take these tests in order to qualify, on graduation, for a diploma that certifies that they have learned something. Other parents may agree with professional wailers, like Alfie Kohn, who say that “Our kids are being tested to death.” and “Preparation for high stakes tests has replaced any focus on real learning”. These parents may demand that their children be exempted from taking these examinations and thus qualify, on graduation, for a certificate of attendance. Each of these groups should be free to exercise its option without interfering with the other’s right to do the same.

 

 

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This paper, written before the NCTM’s national meeting in Chicago last April, formerly ended with a section entitled “The Path to Redemption”. This section has now been revised in order to reflect the significance of the concessions and reversals of policy expressed by NCTM speakers at the Chicago meeting. This revision entitled “A New Mission for NCTM“, begins with a review of these concessions and ends with the demand that NCTM endorse the ten points stated in the original paper, which now seem to be wholly consistent with NCTM’s revised position.

 

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Note: Frank B. Allen is the former Chairman of the NCTM’s “Secondary School Curriculum Committee,” whose report “The Secondary Mathematics Curriculum” was published in the May 1959 issue of the Mathematics Teacher. Please note the personnel of this prestigious committee and of its subcommittees. Note too, the number of mathematicians involved and how far the NCTM has strayed from the consensus of forty years ago.

 

 

Indictment of the Theoreticians “If the past has nothing to say to the present then the present has nothing to say to the future.” Frank B. Allen 1909-200? AD   Pity the NCTM today A worthy group that’s gone astray A group completely under the sway Of theoreticians, far away From schoolroom events of everyday Who conduct research in a curious way It matters little what they say This is the message their deeds convey: “Standardized tests are an awful bane, They reveal little or negative gain, And we regard them with disdain. A little logic might cause some pain, From proof that’s tough we will abstain. We’ll appeal to the hand instead of the brain. Subject teacher time to a terrible drain, With an assessment system that’s hard to explain. We’ll repeat sixth grade, like an old refrain, Recycling the facts all over again.“ “If you disagree with us at all You are a Neanderthal.” If we can’t stop them then let us pray For secondary math in the USA.

 

Seebach: Race-gap study launches 3-stage rant about readers

April 30, 2005

Everybody knows that American blacks and Hispanics are at a disadvantage to whites and Asians both in education and income. Three economists have written a paper demonstrating that the patterns of disadvantages for blacks and for Hispanics are very different, raising questions about the explanations often given for those disadvantages.

In the authors’ own words, here are some take- away points:

 “For black males, controlling for an early measure of ability cuts the black-white wage gap in 1990 by 76 percent. For Hispanic males, controlling for ability essentially eliminates the wage gap with whites. For women the results are even more striking. Wage gaps are actually reversed, and controlling for ability produces higher wages for minority females.”

• “When we control for the effects of home and family environments on test scores, the Hispanic-white test score gap either decreases or is constant over time while the black-white test score tends to widen with age.”

• “Hispanic children start with cognitive and noncognitive deficits similar to those of black children. They also grow up in similar disadvantaged environments, and are likely to attend schools of similar quality. Hispanics have substantially less schooling than blacks. Nevertheless, the ability growth by years of schooling is much higher for Hispanics than blacks. By the time they reach adulthood, Hispanics have significantly higher test scores than blacks.”

• “Our analysis of the Hispanic data illuminates the traditional study of black-white differences and casts doubt on many conventional explanations of these differences since they do not apply to Hispanics who also suffer from many of the same disadvantages.”

I know this is contrary to just about everything you’ve heard or read, so you’re asking, “Who are these people?” They’re Pedro Carneiro, University College London; James J. Heckman, University of Chicago, American Bar Foundation and University College London (and winner of the 2000 Nobel Prize in economics for developing the kind of technical statistical analysis that undergirds this paper) and Dimitriy V. Masterov. The paper was written for the Institute for Labor Market Policy Evaluation, a part of the Swedish Ministry of Industry, Employment and Communications, in Uppsala, Sweden.

The paper is “Labor market discrimination and racial differences in premarket factors” and it’s at www.ifau.se/swe/pdf2005/wp05-03.pdf on the Web.

They don’t argue against current policies on affirmative action – though they certainly could, based on their evidence – merely that policies addressing very early skill gaps are likely to do more good than additional affirmative action policies aimed at the workplace.

One possible explanation of persistent wage gaps is that there is “pervasive labor market discrimination against minorities.” Another, which they observe is equally plausible, is that “Minorities may bring less skill and ability to the market.” And of course both could be true in varying degrees, but I think this is the most important thing they say: “The two polar interpretations of market wage gaps have profoundly different policy implications.”

And how. So if you’re a policy-maker, Go Read The Whole Thing.

Now, since I have room for only a tiny bit of what’s significant in this paper anyway, I’m going to address a different issue that invariably comes up when I write about something so contrary to received opinion.

OK, (/turn rant on/) don’t waste your time writing me that I “haven’t considered” whatever particular bee is buzzing around your bonnet. You have no information about what I have considered; you know only what I have mentioned. And let me tell you, when I’m writing an 800-word summary of a highly technical 50-page paper bristling with statistical analysis, there’s a lot I’ve considered that I don’t mention. There’s even more data that the researchers have considered that they don’t mention – although the existence of Web appendices to scholarly papers has ameliorated that problem.

Next, don’t think you’re being erudite by citing some cliche about “lies, damned lies or statistics,” which I understand is properly credited to Benjamin Disraeli, but is often attributed to Mark Twain. Yes, it is possible to lie with statistics – there’s a charming and useful little book with that in the title – but it’s a lot harder to lie with statistics than without them.

Case in point, the current flap over the number of deaths statistically attributable to obesity. If you’re one of those people who fatuously asserts that “you can prove anything with statistics” I challenge you to find me a peer-reviewed journal article proving that smoking enhances longevity, or that women are taller than men.

Last, don’t talk about motives. You have no evidence about my motives aside from what I tell you – and I could be wrong about that; lots of people are. Even if we were both right about my motives, it would have no bearing on the cogency of my arguments, which do not adduce them; that’s the ad hominem fallacy. “Fallacy,” please note, which means that even if your premises are correct, your conclusion may be wrong. (/Turn rant off/)

Oh, I feel much better now. Excellent paper.

 

 

Linda Seebach is an editorial writer for the News. She can be reached by telephone at (303) 892-2519 or by e-mail at seebach@RockyMountainNews.com.

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

 

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

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DEBRA J. SAUNDERS — Man of Science Has a Problem With Real Math
– DEBRA J. SAUNDERS
Friday, December 19, 1997

THIS STORY demonstrates why you can’t trust Clinton’s education gurus to write national tests for America’s students. If there’s a sure thing in life, it’s that D.C. educrats will dumb down any subject, given half a chance and millions of dollars.

The tale begins this month as California’s state Board of Education was about to vote on math standards for public school students. A standards panel had written a document rich in trendy educratese. (“Show mathematical reasoning in solutions in a variety of ways.”) The board wanted — and ultimately approved — a meatier document with solid standards for computation and less fluff about writing about math. [an error occurred while processing this directive]

By injecting more math into math — actually expecting kids to memorize multiplication tables in the third grade and master long division in the fourth — the board invited the ire of state schools chief Delaine Eastin and the federal government. On December 11, the day before the final vote, Luther S. Williams, assistant director of the federally funded National Science Foundation, fired off a letter to board president Yvonne Larson. Basics wags call it “the blackmail letter.”

Williams, who didn’t call me back, criticized the new standards for not “elevating problem-solving and critical thinking.” His letter chided the board for preferring the “wistful or nostalgic `back-to-basics’ approach,” which he wrote, “has chronically and dismally failed.”

He then reminded Larsen that his bureaucracy gives grants totaling more than $50 million of taxpayer money to six California school districts, including Oakland. “You must surely understand,” he wrote, that his group “cannot support individual school systems that embark on a course that substitutes computational proficiencies for a commitment to deep, balanced, mathematical learning.”

On what planet does this man of science live?

First, Williams has a little jurisdictional problem. President Clinton says he doesn’t want the federal government to butt into local school business. Also, the guy works for a science — not math — agency. But he is so arrogant and power drunk that he feels free to sic his Science Foundation on California math dissidents.

Second, the state’s commitment to “deep, balanced mathematical learning” — aka new-new math — has resulted in computational deficiencies, as well as general arithmetical idiocy. For some years, trendy California educators have focused on students writing about math, repeatedly explaining how equations work and exploring their feelings about math. They’ve also taken to giving students credit for wrong answers. Thus, “critical thinking” has come to mean not being critical of students.

The result: In the last National Assessment of Educational Progress math test, California fourth-graders scored behind students from every state but Mississippi and Louisiana. Only 13 percent were rated proficient. Eastin has suggested that the state board should “get out of the dark ages.” She ought to get the schools out of the dark ages.

No wonder some parents are “nostalgic,” as Williams put it, for the days when basics were emphasized, and cash registers all had numbers on them instead of pictures of hamburgers. Back in the days of what Williams classified as failure, students scored an average of 22 points higher on math SATs.

Here’s a novel thought. Let the National Science Foundation give a grant to solve the great mystery of modern education: How is it that swells like Williams can look at the 1950s as years of math failure, but see no problem with high- school kids needing a calculator to compute 10 percent? How can you say you stand for problem solving without being able to recognize a problem?

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Los Angeles Times
Friday, September 17, 1999

 

L.A.’S MATH PROGRAM JUST DOESN’T ADD UP

Education: We’re starting with basics for reading; why not give students the basics of arithmetic, algebra and geometry, too?

By DAVID KLEIN and R. JAMES MILGRAM

The new Los Angeles Unified School District Board of Education deserves praise and encouragement for its efforts to improve student academic achievement. Unfortunately, in the case of mathematics education, the board is getting bad advice from district staff.

Phonics and other basic language skills have received well-deserved national attention in recent years. As a result, “whole language” is disappearing from the curriculum. By contrast, “whole math,” the philosophical sibling of whole language, is still entrenched in district schools.

The Los Angeles Systemic Initiative, or LASI, is a multiyear, federally funded district program with the worthy goal of improving mathematics and science education. The problem is that LASI has done more harm than good. The initiative’s recommendations have caused many district schools to abandon credible arithmetic, algebra and geometry instruction. LASI has implemented the worst mathematics curricula that we are aware of, and we are aware of many due in part to our service on the California Content Review Panel for K-8 mathematics books. In that capacity, we made recommendations to the state Board of Education for statewide adoption on a huge number of math textbooks submitted by publishers–finding only a small fraction of these worthy of use by California students.

LASI has promoted an experimental K-6 math curriculum, Mathland, which has no textbooks for students. Its manual for teachers tells them not to explain the standard algorithms of arithmetic to children. In other words, children are not taught the traditional procedures for addition, subtraction, multiplication and division. Nowhere in any of these K-6 materials is the usual way to multiply two numbers, like 35 times 76, ever explained.

For high school, LASI recommends so-called integrated math curricula such as Interactive Mathematics Program. Like other integrated math programs, IMP suppresses basic algebra at all grade levels. For example, it delays an important eighth-grade algebra topic, called the quadratic formula, until the 12th grade. This defect alone puts Los Angeles students at a serious disadvantage on the California standardized testing and reporting, or STAR, exam that tests this topic in the eighth grade.

Mathland and IMP are not the only questionable programs implemented by LASI in Los Angeles schools. All of LASI’s recommendations are problematic. The heavy emphasis on calculators is particularly damaging. This often results in students needing their calculators for even the most rudimentary figuring. It is our view that calculators should be used sparingly in grades 6-12 and not at all in grades K-5. The base 10 structure of our number system together with the standard arithmetic algorithms carry the seeds of algebra. Depriving children of mastery of arithmetic closes doors to more advanced mathematics courses in ways that district staff members do not seem to understand.

Statistics from the U.S. Department of Education show that success in secondary school algebra is the single greatest predictor of success in college–not just for engineering and science majors, but for majors in all fields.

Particularly troubling to us is the justification for LASI’s watered-down mathematics programs as reported in The Times in August. An LASI supporter is quoted as saying, “There’s a move to eliminate anything but old-style math. But it’s only striking against inner-city schools where kids need a different approach–they need to see, touch and feel what they are learning.”

We vigorously disagree. Independent of skin color and wealth, students need the same rigorous foundations, including the all important “old-style math” subjects of arithmetic, algebra and geometry. The legendary Jaime Escalante, depicted in the movie “Stand and Deliver,” catapulted his disadvantaged students to national prominence using “old-style math.” The high-achieving African American and Latino students at Bennett-Kew Elementary School in Inglewood provide another example. Sacramento Unified School District abandoned the faddish LASI-style curricula for its multiethnic students and increased its first and second grade SAT-9 test scores by more than 16 percentile points this year.

Data from the recent STAR exam show that students taking integrated math courses in California–such as those promoted by LASI–scored lower than their counterparts enrolled in traditional math courses.

All of the mathematicians who served on the Content Review Panel for the State Board agree about what constitutes a good mathematics curriculum. We urge the new Los Angeles school board to set aside the recommendations of LASI, and seek advice from the broader mathematics community instead.

 

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David Klein, a CSUN Mathematics Professor, was appointed by the State Board of Education to evaluate mathematics teacher professional development programs.

R. James Milgram is a Stanford University mathematics professor who regularly advises the state Board of Education on math issues.

 

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Integrated Mathematics in LAUSD

 

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Summary

In California, the integrated mathematics option refers specifically to an alternative to the algebra 1, geometry, algebra 2 secondary sequence wherein districts are allowed to provide the same content but in a different sequence over three years. All mathematics programs for K-7 are integrated in that topics from each strand of mathematics are included each year. Because the secondary integrated programs make heavy use of pedagogical approaches often called reform mathematics, these terms have unfortunately been used interchangeably. This confuses the issues.

With respect to the secondary integrated mathematics programs in use in LAUSD:

The content of these courses is not equivalent to the content required by the state standards.

 

No integrated 1 books were approved by the state under AB2519 – indeed most were not even submitted – as they are unsatisfactory relative to the state learning standards.

The integrated programs cannot be legitimately certified as aligned with the state standards.

Students in these programs learn less mathematics than those in traditional programs.

LAUSD students in integrated mathematics score lower than those in traditional mathematics in grades 8, 9, and 10 on the state standards-based tests according to state records.

This deficit is true for economically disadvantaged students as well as others.

This deficit is true for LEP students as well as others.

This deficit is true for male and female students.

These programs produce students who are less well prepared.

Integrated mathematics has been promoted through LA-SI (the Los Angeles Systemic Initiative, a federally funded project to implement integrated math programs) in schools around the district. Of the eleven schools associated with LA-SI the longest, all but one have experienced decline in SAT participation over the past two years. The average decline is 12% as reported by the IAU (the “Independent Analysis Unit” of LAUSD) to the Board in a report of May 12, 1999.

SAT administration across all LA-SI focal schools is down about 5% while it is up roughly 5% in non-focal schools. The SAT math average in focal schools is 445 while at the non-focal schools have an average of 462.

With respect to reform mathematics, the programs approved by the state for K-8 include a range of reform approaches and frequently note their alignment with NCTM. These approaches will still be available even when programs that fail to align with the standards are avoided.

 

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Integrated Mathematics in LAUSD

 

by Paul Clopton
Member, LAUSD Mathematics Curriculum Committee

 

Introduction

Historically, secondary mathematics in California has been taught using the courses algebra 1, geometry, and algebra 2. More recently, some schools have switched to courses that mix these topics across three courses. This is called integrated mathematics and is an optional sequence in the state Mathematics Framework. However, the integrated programs in use differ in many respects beyond the sequence of topic presentation, and thus integrated has taken on other meanings that have to do with pedagogy, presentation style, and other factors.

The new California Mathematics Standards and the Mathematics Framework require a mixture of topics from the strands of mathematics for all students in grades K-7. Districts may use either the traditional or the integrated sequence starting in grade 8. The standards stipulate exactly the same objectives for either sequence – that students learn the required mathematics.

To go along with this option, the standards-based portion of the state testing program (STAR) in mathematics has two choices for grades 8 to 10 – algebra 1, geometry, and algebra 2 or integrated 1, integrated 2, and integrated 3. The two sequences contain exactly the same items overall, but they are assigned in a different sequence across the three years.

Also, in grades 8 to 10, only students enrolled in the corresponding traditional or integrated sequence take the standards-based part of the exam. In grade 11, all students take a cumulative form of the standards-based exam covering all of these topics, regardless of what mathematics courses they have taken.

In general, the performance on these standards-based examinations has been poor. This is expected since students have not previously been expected to meet the new standards throughout their academic careers. Achievement in LA has been poor as well. However, certain comparisons are already possible given the baseline test results from the spring 1999 test administration.
Results for LAUSD

The California data file for these test results gives means for the standards-based mathematics tests in grades 8 to 10 only for those students who are “on-track” for meeting the standards, meaning that they are taking the first year in grade 8, or the second year in grade 9, or the third year in grade 10. Here are the average number of correct answers for all of LAUSD

Grade	Traditional	Integrated
8	21.4	19.2
9	23.0	21.6
10	22.5	20.0

On average, the traditional sequence scores are about 10% higher than the integrated sequence scores. We cannot be certain that the curriculum accounts for this difference, since we don’t know the characteristics of the students or teachers in each case. However, these results suggest that the integrated programs are less effective than the traditional ones in LAUSD.
Results for Economically Disadvantaged Students

The integrated programs also show weaker results for disadvantaged students. The STAR data file does not contain information on ethnic minorities, but it does summarize scores for economically disadvantaged students vs other students. Both groups achieved lower scores with integrated programs. This is not consistent with the idea that the integrated mathematics programs are better for the disadvantaged students. These results are consistent with the idea that the integrated programs lack equivalent mathematical content.

	Economically Disadvanted		All Others
Grade	Traditional	Integrated	Traditional	Integrated
8	20.0	18.3	23.8	21.0
9	20.9	20.3	25.1	23.6
10	20.3	19.2	24.0	21.8

 

Results for LEP Students

From the state data file, it is also possible to inspect the LAUSD results for limited English proficiency students (LEP) compared to other students. Again, both groups achieved lower scores with integrated programs than with traditional programs across the three grade levels.

	LEP Students		All Others
Grade	Traditional	Integrated	Traditional	Integrated
8	17.0	15.7	22.3	19.9
9	18.5	17.9	23.8	22.3
10	18.8	16.8	23.0	20.4

 

Results for Male and Female Students

The state data file breaks down scores by student gender. Again, both groups achieved lower scores with integrated programs than with traditional programs across the three grade levels.

	Female Students		Male Students
Grade	Traditional	Integrated	Traditional	Integrated
8	21.3	18.8	21.5	19.8
9	22.4	21.1	23.8	22.2
10	21.6	19.7	23.4	20.4

 

Integrated and Traditional High Schools

It is possible to characterize high schools as traditional or integrated based on the tests taken by the students (counts of tests taken are given even when the scores are not reported). In this example, schools were identified as traditional if at least 75% of these augmented tests were in the traditional sequence, and they were called integrated if at least 75% of the tests were in the integrated sequence. These high schools were then compared on the basis of their average scores for the 11th grade where all students take the same standards-based mathematics exam. The results were:

	Traditional	Integrated
Number of Schools	38	28
Average Number Correct	16.1	15.0

Again, we cannot be certain about the actual cause of this difference, but the result favors the traditional approach. What about the “middle” group, those with somewhere between 25% and 75% traditional score reports? There were 15 high schools in this group with an average of 15.1 correct.
The Stanford 9 scores for these same high schools give an indication of achievement on a less rigorous assessment. The results using the NPR for the average student at each school are:

	Economically Disadvantaged		All Others
Grade	Traditional	Integrated	Traditional	Integrated
9	38.6	35.9	40.1	33.5
10	35.3	33.3	36.8	31.3
11	41.9	37.5	41.9	34.2

The results for the percentage of students above the 50th percentile are:

	Economically Disadvantaged		All Others
Grade	Traditional	Integrated	Traditional	Integrated
9	30.9	28.3	32.7	24.0
10	29.3	27.6	31.2	24.6
11	36.0	31.5	36.0	25.9

In all cases, economically disadvantaged or not, the means for the traditional program schools are higher than the means for the integrated program schools.
LA-SI Schools

The influx of integrated mathematics programs in LAUSD high schools is related to involvement with LA-SI (the Los Angeles Systemic Initiative, a federally funded project to implement integrated math programs). Schools with the longest involvement are Phase I schools. Of the eleven Phase I schools, all but one have experienced decline in SAT participation over the past two years, some rather substantially; 12% is the average reported by the IAU (the “Independent Analysis Unit” of LAUSD) to the Board in a report of May 12, 1999.

According to the IAU numbers, SAT performance across all of the LA-SI Focal schools is down about 5% in the number of takers and has a math average of 445 while the number of takers is up roughly that same 5% at the non-Focal schools with an average of 462. Neither of these numbers is terribly impressive but they suggest a reason why several high schools have abandoned integrated mathematics and are returning to more traditional programs.

The IAU only looked at the two years 1996-1998 but a longer perspective on these LA-SI Phase I schools is informative. Palisades is not on the state’s data base because of its conversion to magnet status but data from the other ten is available on the Internet. That data starts with 1992, the year in which two of them, Roosevelt and Marshall, became pilots for an integrated program called IMP. From 1992 to 1998, the data year of the IAU report, these ten schools dropped an average of 13 points in their SAT math scores while experiencing a 30% overall drop in SAT participation. These numbers compare with the overall statewide math SAT average holding steady at 516 while participation increased by 24%.

Another useful measure of success is the Entry Level Mathematics Exam (ELM) required of students admitted to any CSU campus. The most recent data at that website is for those students admitted sometime during the 1997-8 year. A successful mathematics assessment is an SAT math score of 550 or a passing score on the ELM. For the LA-SI Phase I schools, the collectively failure rate was 78%. This compares with a statewide failure rate of 55%.
Alignment with Standards

The integrated secondary programs in use in LAUSD not only mix up the order of topic presentation, they also reduce the level of mathematics covered. The state has recently approved 5 algebra 1 programs under AB2519, while no integrated 1 programs were approved. In general, the integrated secondary programs were not even submitted because of their lack of alignment with the state standards. The district cannot legitimately certify these integrated programs as being aligned to the state standards.
Reform Methods without Integrated Secondary Programs

Even without these integrated secondary programs, LAUSD students will still have integrated content in grades K-7. Even without these integrated secondary programs, schools will be able to select books with varying degrees of reform mathematics methods at all grade levels. Indeed, the state-approved texts often note their inclusion of these new methods and make reference to the NCTM which is recognized for promoting these methods. Many include leading NCTM members as authors.

LAUSD can comply with state requirements and still encourage classroom teachers to use their own professional judgment in selecting the best methods for meeting the needs of their students. Indeed, this is exactly what is recommended in the state mathematics framework.

If math were a color . . .
By Marcia Tsicouris

In 1993/94 District 205 adopted the University of Chicago Everyday Math project as its K-5 curriculum and currently utilizes it to some degree in the middle schools and at the high school level. Since then, it has come up for review and the committee elected to re-instate it for another term. Why? I can only deduce the decision was based on economics and not the program’s effectiveness. Everyday Math does not require the purchase of textbooks or workbooks. Copies are made from masters or other copies. We purchased the program only a few years after it went on the market. Is it possible to include Everyday Math among those of best practices after such a short time? The U of C acknowledged some of its shortcoming and published an optional Skills Link supplement in July 1998.

Everyday Math. On which days exactly is this program effective? If you’re a 5th grader, maybe it works on the days you have art. Or, possibly, on the days you study nutrition, or on the days you discuss weather conditions. In lieu of practicing long division or mastering multiplication facts our 5th graders spend math time on exercises such as this:

A. If math were a color, it would be –, because –.
B. If it were a food, it would be –, because –.
C. If it were weather, it would be –, because –.

If this type of, so called, math activity takes you by surprise, I’ll allow time here for primal screams as did the author of the Wall Street Journal article where I first learned of this particular atrocity. (I verified its occurrence with my 5th grader!)

In 4th grade, my son’s fraction assignment was marked wrong when he identified 1/5 of the dogs pictured on his Home Link as being spotted. After checking it myself and talking with the teacher I found the copy quality was so poor it was nearly impossible to detect that a 2nd dog out of 5 was spotted.

1st through 3rd graders are encouraged to become dependent on calculators, peers, and parents to accomplish their goals. Calculators are introduced early and often. Internationally, U.S. students’ math scores ranked among the lowest in the world. Countries with the highest scores, Japan, China, and East Asian countries don’t permit the use of calculators until high school. They feel students must first master the concepts and operations necessary for mathematical problem solving.

Everyday Math lessons are based on a spiraling curriculum providing no room for mastery in any one area. New concepts are introduced one after another assuming children will pick up on the material as it is sporadically revisited throughout the year. It’s difficult to find two of the same type of math problem on any one Home Link. (If you do, its likely your child’s teacher has opted to supplement with worksheets from other programs such as Addison Wesley.)

In an attempt to promote problem solving skills, many exercises are done in groups. Unfortunately, it is not possible to develop mathematical problem solving skills without the basic tools necessary to arrive at a correct answer. However, Everyday Math is not concerned with correct answers. This program prefers to emphasize the creative processes used to arrive at any answer. I hope my financial advisor, banker, pharmacist, etc. don’t share the U of C’s position on this. A math problem isn’t solved until you’ve reached the correct answer. Like a carpenter, a students’ problem solving skills are useless with out proper tools!

Everyday Math places no importance on math facts, and no benchmarks are established in the program for mastering them. In place of math facts, students are required to learn a multitude of algorithms. Defenders of the program insist this clumsy process provides each student the opportunity to select the algorithm that works best for him/her. Yet, in 5th grade, instructions continue to specify which algorithm to use for the assignment.

Virtually every Home Link is prefaced with the words: Show someone at home; Have someone at home; With someone at home; Tell someone at home. The message sent to my 3rd grader is that she’s incapable of doing math independently. Thanks to this program, essentially, she is incapable. I have to re-teach each concept as it arises (in addition to teaching basic math facts) because the U of C sees no merit in mastery.

Some may argue they like the program. As with Whole Language, there is a small population of students that possess a natural aptitude for the subject. These students will excel regardless how effective or ineffective the program. Whole Language is a testament to that. For the majority, Everyday Math will create generations of math disabled students as Whole Language created generations of reading disabled students.

Back to the ridiculous 5th grade exercise: If math were a color, it would be black and white, for math is an exact science with concrete, absolute, correct solutions. If it were a food, it would be something high in nutrition like fruits and vegetables, as these would nourish and develop the brain. If it were weather, it would be clear, bright and crisp to keep skills sharp and the mind alert.

Given our students are burdened with Everyday Math; the color of math is gray and fuzzy with little importance placed on correctness and none placed on mastery. The food choice is junk food with little nutritional value serving only to clog arteries and provide immediate gratification. And, weather, no doubt, it’s a tornado whose spiraling winds leave our students strewn at the bottom of the scale.

By my calculations, Everyday Math equates to educational malpractice!

Check it out yourself. There are some great websites on the Internet. One of my favorites is Mathematically Correct at http://www.mathematicallycorrect.com.

Marcia Tsicouris
Elmhurst, Illinois

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Reform Mathematics Education
How to “Succeed” Without Really Trying

by Paul Clopton
Cofounder, Mathematically Correct

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Since the 1980’s, there have been substantial efforts nation wide to weaken mathematics education in America, and these efforts have largely been successful. This is not a communist conspiracy [Note 1]. It flows from an honest desire to help the less fortunate. This effort is based on the misguided notion that weaker mathematics will be helpful to the traditionally disadvantaged groups in our society. It is this effort, curiously known as reform, that is the root cause of what has come to be known as the math wars.

You won’t find many reformers who will openly admit that they favor “dumbed-down” mathematics. In fact, the reform movement is characterized by a plethora of rhetoric to the contrary. The diatribes are extensive and frequent and are laden with phrases like “higher order thinking” and “conceptual understanding” and “real-world problems” while shy on terms like “arithmetic” and “algebra.” Reformers have learned their scripts well, and the rhetoric comes gushing forth with little provocation.

The conditions that prompted this movement are obvious. Poor people, minorities, and women are under-represented among those who reach high levels of mathematical achievement. Those who cannot master arithmetic and algebra are unlikely to achieve a decent college education. There is no question that the educational system in this country is not successful for a great many students.

One way to deal with this problem is to make the mathematics easier. This means less rigor, less emphasis on arithmetic and algebra, more reading and art and creative projects, less emphasis on correct answers, more calculators, and a host of other reform-minded solutions. Stylish pedagogical methods combined with rhetoric about higher order thinking while downplaying or condemning outright both computation skills and mathematical proof complete the package. This is reform mathematics education.

Sometimes dubbed traditional or anti-reform, the second perspective has come in abreaction to the first and is mainly supported by parents and mathematicians. This perspective holds out that the mathematics must not be “dumbed-down.” The key in this perspective is to increase achievement rather than to decrease expectations. Central to this position is that the traditionally less fortunate are not well-served by weaker mathematics and, in fact, should be insulted by it. The real key to success is real mathematics achievement, and every effort should be made to foster this achievement.

Ironically, the struggle to promote real mathematics education is left up to those outside of the field – mostly parents. The perspective is traditional in the sense that it seeks to prevent learning expectations from being further eroded away by putative reform efforts. Mathematics education in America has not been very successful. However, do not look for relief in the reform notions. We would be better off if all the energy behind the reform was redirected toward clearly defined achievement goals and we measured progress toward those goals frequently and objectively.

Obliterating Distinctions between Success and Failure

The reform designs open the door to claims of successfully teaching mathematics without really doing so. The reform writings and methods are many and varied, but a common feature is that they end up obscuring the failure to teach mathematics. In reform mathematics education, the goal of success for all is not supported by achievement but rather by redefining success and, mostly, by obscuring failure. Here are but a few examples:

Group Learning and Group Tests – The story of Apollo 13 is used to promote group learning and group assessments with the argument that our students must learn to work together like people do in the real world. Never mind that people in the real world don’t sit in groups doing algebra problems. Group learning is plagued by inequities that most parents identify quickly – some do the work while others learn that they can “succeed” without learning the material and without effort. Group assessments effectively erase the ability to monitor individual achievement or to provide useful diagnostic information. Whether or not individuals are learning is obscured by these methods.

Calculators – Many argue that routine skills are out of date, and that technology has changed the mathematics that today’s students need to know. The position includes multiplication and division, obviously. However, today’s calculators can manipulate fractions and solve equations as well. Distancing students from these activities takes away the learning experiences that help form the foundation of mathematical understanding. By far, most American parents want their children to be able to solve problems without calculators. The reliance on calculators allows reformers to claim success even when children do not learn the fundamental operations of arithmetic. Soon they will claim success in algebra for students who have not learned how to solve equations.  Authentic Assessment – One of the greatest evils from the reform perspective is objective testing. It would have to be because these measures can identify failure. Many arguments are advanced for this perspective, but addressing them in detail is beyond the scope of this report [Note 2]. The proposed alternative is frequently called authentic assessment. Translating this bit of jargon into English isn’t easy. Basically, it refers to a variety of procedures that involve less mathematics, more writing or talking, and very subjective evaluation. In the worst instances, students suffer if they do not support the intended politically correct perspective in their response. But, politics aside, these methods are reliably unreliable. The subjective nature leaves little opportunity for valid information to be obtained. Sometimes, one cannot even tell who actually did the work. In the long run, many invalid assessments tend to average out (false equity) and, again, real differences in achievement go undetected.

Measuring Content With the educational bureaucracy in this country prone to jump on the bandwagon of pedagogical fads, assuring that children receive a decent education becomes the responsibility of their parents. Effective parenting now includes keeping a watchful eye on what happens in school and what the children are and are not learning. When deficiencies are found, parents can try to change the schools, to increase learning experiences at home, or to find outside resources to provide the needed learning experiences. The entire process of monitoring and remedying this situation is very demanding. The first stage of this process, examining the content of the school program, can be a little easier for parents who make use of existing resources that identify content by grade level. Coming on the heels of failed reform efforts in California, expectations for achievement that are roughly in line with those of the most successful countries of the world were developed. These documents identify achievement levels in terms that are sufficiently clear for parents to evaluate. Parents are encouraged to measure the school programs against these contents as a way of finding out whether or not important content is being covered. The California Mathematics Standards The San Diego Mathematics Standards The NCITE-LA Achievement Test Items Number Sense in California With the aid of these materials, parents can more easily find what is present and what is absent in the programs used in local schools. These documents enable parents to match local content to grade levels according to high-level standards.

Projects – The reform programs are loaded with projects and activities, often called investigations. Part of the argument for these methods relates to stimulating student interest. There are also claims of richer mathematics and the importance of context. Even a casual inspection of these activities will show that they tend to be very time consuming while involving very little mathematics. Time for mathematics, both in class and at home, is seriously limited and must be used as efficiently as possible. These activities are inefficient learning methods. But, beyond that limitation, they promote the evaluation of students on the basis of non-mathematical dimensions such as how artistic the display is or the writing style of the report or the social value of the application.

Standards – The reform movement claims to be based on standards, although most parents will be surprised by what they find – and what they don’t find – in reform standards documents. It is contrary to the goal of the reform to produce explicit statements about what students know and should be able to do – again, spotting failure would be too easy. Consequently, the reform movement produces standards that are so vague that one cannot tell whether they have been met or not. Any attempt to write tests for these standards, for example, will be unreliable because the required content is unclear. Reformers hate lists of clearly stated objectives and call them laundry lists. However, vague learning expectations are effectively the same as no learning expectations at all. Again, it becomes impossible to differentiate success from failure.

Strands – When attempts are made to subdivide mathematics into content areas, such as algebra and geometry, the subdivisions are often called strands. The reform movement uses this technique while simultaneously avoiding explicit content. Thus, all of the elementary school work with arithmetic falls into one strand which becomes just one of many topic areas students are supposed to address. The consequence is that students can still succeed while failing in arithmetic. The same thinking reduces algebra to just one component of mathematics in later grades with similar consequences.

Pedagogical Fads – The reform movement places great emphasis on classroom methods, such as those that involve groups, calculators, activities and projects, manipulatives, explorations, art work, and non-mathematical themes. Irrespective of any relationship between these methods and learning (or lack thereof), there are consequences of the fact that the emphasis on these styles is pervasive in reform documents. Even reformers bemoan the fact their followers often carry out reform by adding a few new gimmicks to their bag of classroom tricks. The heavy emphasis on style quite naturally takes attention away from mathematical content. As teachers attend to implementing these processes, their evaluations of students become biased toward process and away from content. Mathematical learning will often take a back seat to artistic ability, cooperation, or even political correctness again blurring the distinctions between success and failure when it comes to learning mathematics.

With the demise of our ability to differentiate success from failure, the reform movement will claim broad successes. School systems in America have the uncanny ability to claim improvements and reforms year after year while the content is gradually leeched out of the system. Meanwhile, fewer students will suffer wounds to their self-esteem because their failures will go undetected. Such a system will identify fewer failures among poor and minority group students, so reformers will claim a victory for equity.

Unfortunately, success in this approach will have lost its value. The claims of success operate like social promotion as applied to education bureaucrats. We may gain some “equity” at the cost of achievement, but the more advantaged parents will continue to find ways to make sure that their children learn in spite the best efforts of the reform-minded. Meanwhile, the net effect of the reform will be further deterioration in the mathematical abilities of America’s youth. The majority of these students will not find alternative forms of education to make up this deficit. It is from this majority that we will draw our next generation of teachers.

 

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Note 1: Although not a communist conspiracy, there is some justification for the belief that some sort of conspiracy is at work. The reform designs are heavily promoted by the National Council of Teachers of Mathematics (NCTM). In turn, the educational branch of the National Science Foundation (NSF) then funds the development of curriculum materials that align with the NCTM dictates. The products of these efforts are then advertised by the U. S. Department of Education, while the NSF pushes for their adoption by states and districts.

Note 2: The interested reader should see Chapter 6, “Test Evasion,” in The Schools We Need and Why We Don’t Have Them by E. D. Hirsch, Jr., Doubleday, New York, 1996.

TUESDAY, MAY 23, 2000

How a new math program rose to the top

Critics say the process of giving ‘Core-Plus’ a top rating lacked rigor and evidence of long-term positive impact

Mark Clayton (claytonm@csps.com)
Staff writer of The Christian Science Monitor

BLOOMFIELD HILLS, MICH.

A plaque in Andover High School’s main office announces that this Bloomfield Hills, Mich., school is no ordinary place – it is ranked one of America’s “100 best” high schools.

Mathematics is a serious matter here. Andover students are drawn from a community of auto-industry engineers and business elites who expect their children to use high-level math skills in a variety of high-tech careers. More than 95 percent of students go on to college.

Andover’s math test scores soar above those of most other schools in the state. Despite that, the prestigious school stopped offering its traditional math curriculum to new students in 1994 and began an experimental program known as Core-Plus Mathematics, based on National Council of Teachers of Mathematics 1989 (NCTM) standards.

Several such programs have come online at all grade levels during the past decade. All have had an important goal: to boost US students’ performance and interest in math. But their method, which some say favors a broad conceptual approach that doesn’t adequately teach skills, has come under attack from mathematicians, scientists, and parents.

The story of Core-Plus is a tale of how such programs got high scores from the US Department of Education that encouraged their adoption in schools – a process that critics say is deeply flawed.

Core-Plus materials were not festooned with arcane math notation. Its texts focused on architectural design, manufacturing, and air-pollution problems. Instead of teaching algebra one year, geometry the next, then advanced algebra, trigonometry, and precalculus, Core-Plus wove together strands of each.

Guided by teachers, Andover students began working in small groups using powerful calculators, writing paragraphs to justify the mathematical rationale for their answers. It sounded promising.

 

High rating

 

Developing Core-Plus was a team effort, but it was Christopher Hirsch’s baby. So it was a happy moment for the professor of mathematics and math education at Western Michigan University at Kalamazoo, when he got the good news last fall: Education Secretary Richard Riley had named Core-Plus an “exemplary” program. Riley’s 16-member expert panel had sifted through a crop of 61 new programs. Core-Plus had popped to the top with nine others.

“We wanted to teach math as a whole – not this layer-cake approach,” says Professor Hirsch, a leading name in math education who helped write the 1989 NCTM standards the program followed. “Students advance along these strands so they develop a sense that mathematics is very connected.”

Andover became a Core-Plus pilot school in 1993, and the next year it became one of 36 “field test” high schools. The Education Department “exemplary” rating was vindication for that choice.

“Of course we were pleased,” says John Toma, Andover’s principal. “We really believed in what we were doing and had a strong belief in making the study of math applicable to daily life.”

 

A student’s view

 

But Melissa Lynn felt differently. A freshman at Andover in 1993, she became worried her Core-Plus class wasn’t very hard. She asked to switch into a traditional class. But she would have had to travel by bus to another school.

So she threw herself into the Core-Plus program and received straight A’s. She graduated in 1997 at the top of her class, with a 3.97 grade-point average.

Then she took the math placement test at the University of Michigan at Ann Arbor and bombed it. She found herself in “remedial math.” She’s still upset.

“I wasted precious time and money,” she wrote in a 1998 letter to a university professor. “I did receive an ‘A’ in my [university] precalculus class, but I did so in spite of Core-Plus.”

“Core-Plus has strong points,” she acknowledges now. She liked the real-life problem solving. Yet the emphasis on expanding math into other aspects of life was done “at the cost of teaching the basic algebraic manipulations,” she says.

Bloomfield Hills officials point out that Ms. Lynn did well in precalculus and later college math – showing that Core-Plus worked. She disagrees. “My eighth-grade math helped me out more in college than Core-Plus did,” she says.

 

A panel’s mandate

 

In summer 1996, a mathematician at University of Texas at San Antonio, Manuel Berriozabal, got a surprise phone call.

A Department of Education official wanted him to join a panel of experts whose congressional mandate was to identify top math programs.

“I was quite flattered,” he recalls. “I thought this was a first step that would help straighten out the math and science education system at the precollege, elementary and middle-school levels.”

Today, he is disappointed.

“The panel was a good idea,” Dr. Berriozabal says, “but we made some bad judgments. From the best I could tell, none of the programs we selected as ‘promising’ or ‘exemplary’ had any kind of long-term track record of achievement.”

After Berriozabal arrived in Washington, the panel began debating the criteria to determine a successful program. Berriozabal thought that long-term proof of achievement should top the list.

Most others on the panel wanted to require programs to conform to NCTM standards – then gauge achievement.

“These programs were just too new to require long-term impact studies,” says Steven Leinwand, co-chair of the expert panel on math. “To require that would have postponed any designation for years…. If we had built that criterion in, it would have been an uneven playing field, since many programs just haven’t been around long enough to have that kind of impact data.”

In 1997, after nearly a year of debate, the Education Department’s expert panel had decided on eight criteria – one of which required evidence of “a measurable difference in student learning” in order for a program to be “exemplary” or “promising.” But long-term evidence was not a factor.

Berriozabal abstained or voted against all 10 programs designated “exemplary” or “promising.”

 

On a mission for better math

 

Efforts to develop better ways to teach math emerged from concern about American students’ math performance. Especially after the results in 1997 of the Third International Math and Science Study, which showed US seniors lagging well behind their international counterparts, fears grew that America could one day lose its technological and economic lead. To help reverse the nation’s dwindling ranks of technical majors, the White House and Congress ramped up spending on K-12 math education in the early 1990s. Even the National Security Agency today sends speakers to schools to talk about the importance of math achievement, the basis for code breaking.

But one of the most influential powers in math education today is the National Science Foundation. And even before the report of the Education Department’s panel, the NSF had decided Core-Plus and others were winners.

Between 1990 and 1997, the Education and Human Resources (EHR) Division of the NSF put out calls for research proposals to explore new ways of teaching math. The division spent about $86 million in the past decade to fund 13 multiple-grade level math-curriculum projects and build four “implementation centers,” says John Bradley, EHR’s mathematics program officer.

At the elementary school level, approximately 2.5 million students are using NSF-funded math programs today, Bradley says. Another 5,000 middle schools use NSF-funded math programs. No numbers were available for the number of high schools involved in NSF math programs. But at least 500 high schools use Core-Plus, for example.

Still, the process troubled critics: Where was the independent evidence that they worked? For Connected Math, a middle school program, the NSF “outside evaluation” was done by a team that included Mark Hoover, now at the University of Michigan. For Core-Plus it was a team led by Harold Schoen, a University of Iowa professor.

 

Opening the floodgates of criticism

 

Those were questions on Norman Lynn’s mind. In fall 1997, four years after Core-Plus started at Andover, Dr. Lynn told his daughter’s story at a school board meeting, calling her and other Andover students “academic guinea pigs.” Many parents were shocked.

One of those was Gregory Bachelis, a mathematician at Wayne State University in Detroit. He decided to find out if Melissa’s math experience was unique. So he and a colleague put together a questionnaire to survey Melissa Lynn’s graduating class. Nearly half responded. Dr. Bachelis also surveyed graduates of nearby Lahser High School, the Andover rival that stuck with traditional math.

The results: 96 percent of Andover graduates who took Core-Plus and responded to the survey had taken remedial math in college, they said. Among Lahser High graduates who responded, 62 percent took remedial math.

What surprised Bachelis most, he says, were bitter comments from dozens of Andover graduates, including: “I have very few math skills, and none of them helped me with [math] in college.” Also, “I am … not the least bit confident with my math ability. I am upset that I was ever placed in a Core class.”

Not everyone was up in arms, though. Lisa Robinson, a freshmen at the University of Michigan, liked the new math program at Andover.

“I liked Core-Plus,” she says. “The math I’m doing here at U of M is the same kind of program. There’s a lot of calculator stuff. It’s the same thing.”

Several mathematicians reviewed Bechelis’s work and found it solid. Still, Bachelis and his survey were castigated by Bloomfield Hills and Core-Plus officials. A lawsuit was threatened.

“The Bachelis study is flawed,” wrote Bloomfield Hills superintendent Gary Doyle in a letter earlier this year to the American School Board Journal. “He repeatedly contacted students … especially if he believed that they were negative about the program.” Bachelis agrees he persisted to get a complete sample, but denies selecting negative views.

A study that rebutted Bachelis was soon unveiled. It said Andover graduates’ grades at the University of Michigan, Ann Arbor were “stronger” than years before the program was adopted, it said. But that study, too, is debated.

“What’s passing for educational studies these days is really embarrassing,” says David Symington, principal at Lahser High. “We tried Core-Plus. And I’ve been watching it for four years. I would not go to it and my math department wouldn’t either.”

Of the rebuttal study, he says, “They didn’t even have the correct year. They didn’t account for 100 students from Andover taking regular math classes here at Lahser, that made Core-Plus look better.”

Core-Plus director Hirsch dismisses the Bachelis survey as a manifestation of national “math wars.” Some of our critics will go to almost any length to marginalize the good that’s coming from these projects,” he says.

Yet from 1994 to 1998, the years when Core-Plus was Andover students’ only choice, math ACT scores at the school remained flat. Meanwhile, math ACT scores of rival Lahser High, and those of schools across Michigan and the US, rose about 6 percent, according to a study by R. James Milgram, professor of mathematics at Stanford University and a critic of some new NCTM programs.

Exactly what held Andover students back is not known. But the ACT scores, which Andover and Lahser provided to Milgram, are not in dispute, he says.

Colleen Zematis, mother of a student at Andover who went through three years of Core-Plus, decided action was needed. She and others rallied and circulated petitions until, last fall, Andover returned to offering a traditional math option.

Andover Principal Toma says the school likes Core-Plus and was never dissatisfied with it. “Parents’ wishes must be respected,” he says. Half of students now take traditional math, he says.

 

Opposition to Top 10 math programs

 

Meanwhile, pressure has been growing elsewhere. Last fall, the Education Department released its Top 10 list of math programs. Reaction was swift.

Within weeks, a full-page open letter to Secretary Riley protesting the department’s choices appeared in The Washington Post, signed by more than 200 mathematicians, physicists, and four Nobel laureates. Few math researchers were involved in the federal review, and there were many mistakes in the new textbooks, they charged.

Some also wondered whether the Education Department criteria were unduly biased toward NCTM standards. Others, whether the panel had relied on unbiased studies of student achievement.

There were other concerns as well. In congressional testimony last month, David Klein, a mathematician at California State University at Northridge, said conflicting interests on the expert panel were a key problem.

“This [expert panel’s] list includes some of the worst math programs you can find anywhere,” said Klein, who signed the open letter to Riley. “The minutes of the [Education Department’s] expert panel show that the panel was aware of the problem of conflicts of interest,” he continued. “They raised the issue – and then dismissed it.”

Education officials and panel members say conflict-of-interest guidelines were followed scrupulously. But they concede the appearance of vested interests.

Luther Williams, for instance, was appointed to the panel in 1996. By most accounts, Dr. Williams, director of the Education and Human Resources division of the NSF that funded math-curriculum development, played a minor role on the panel. He did not attend meetings and left the panel entirely in 1998 before it voted.

But some panel members were mystified and wondered whether having NSF officials on the expert panel opened the door to charges of vested interests.

“Not enough thought had gone into the makeup of the panel,” says James Rutherford, an adviser to the American Association for the Advancement of Science, who was on the panel, too. “I really wondered if Luther should have been there at all. After all, at the NSF he was directly involved in funding the very programs we were evaluating.”

Even after Williams left the panel, there was another NSF official on board. In the end, six of the 10 programs selected by the panel were NSF funded – a striking success rate since only 13 were NSF funded in the 1990s.

“We were just trying to get people with experience,” an Education Department official says. “The NSF had experience.” Both that official and Janice Earle, a program director in the EHR division of NSF, also on the expert panel, deny NSF programs were favored. “They weren’t my programs,” Ms. Earle says.

But critics say other issues raise questions about whether the process was as objective and thorough as it should have been:

 

• Programs did not have to show long-term evidence of achievement. But Leinwand says the programs are “too new” for that – and congressional pressure was building for action.
• Studies showing evidence of higher achievement did not have to be independently reviewed by being published in peer-reviewed journals before being submitted. Rutherford says the panel did a good job but relied on “very soft evidence.”

 

The dearth of solid research tended to show up most with programs tagged as “promising.” Take, for instance, Middle-school Mathematics through Applications. One of two impact reviews says: “Because the outside evaluation was not complete … the program did not submit sufficient data to substantiate its effect on student achievement…”

A second reviewer rated it “marginal for promising.” The panel rated it “promising.”

Another reviewer wrote of Everyday Math, a K-6 curriculum: “In reviewing all the evidence provided … it does not provide meaningful evidence for program effects….”

 

• Only three of the 96 reviewers had published a mathematical paper, says Richard Askey, a math professor at the University of Wisconsin at Madison and a reviewer. Most reviewers were not professional mathematicians, though many were math educators, he concludes. 

   

• Several panel members were affiliated with programs being judged. Most notable was the co-chairman of the panel, Mr. Leinwand, who also sat on advisory boards of three programs being judged, two of which were later selected as “exemplary” programs – Connected Mathematics Project and Interactive Mathematics Program. Mr. Leinwand and others say that he reported his affiliations and left the room for all discussions and voting on them. 

   

• Critics say the panel favored NCTM standards, which the panel mandated as “a filter,” according to meeting minutes. All 10 programs were based on the standards. But Leinwand denies his position on the NCTM board influenced that. Too, “those were the only national standards out there,” says panel member Jack Price, a math professor from California State Polytechnic University at Pomona, and former NCTM president. Forty-three states had NCTM-based standards.

 

Linda Rosen, adviser to Secretary Riley on math, says the recommendations are “just one tool.” “What [school districts] do with this tool … is up to them.”

But that’s small consolation for Robert Daitch, now a junior at the University of Michigan. He took four years of Core-Plus before graduating from Andover in 1998.

“Since I was a kid, I loved auto engineering,” he says. “So I attended a session for those who wanted go into engineering. When they asked how many of us took calculus, all the others raised their hands. I began to realize my dream was not going to happen.” So Mr. Daitch took remedial math at the university – and became a communications major.