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Man of Science Has a Problem With Real Math

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?

R James Milgram

1999 Conference on Standards-Based K-12 Education

California State University Northridge

Transcript of R. James Milgram
(edited by the speaker)
Biography

Return to conference page

Return to transcript of Norman Herr

Mr. Milgram: I would like to start by again thanking David Klein and Cal State Northridge for arranging and organizing this wonderful opportunity to get together and compare ideas on the incredibly challenging times ahead of us. Professor Wu brought up a number of critical points in his discussion and one of them that he mentioned — that this is a long term challenge — is particularly important.

I’d like to fill in somewhat what the problem is here. First of all, “long term” has generally been understood to be in the order of perhaps three years, and there seem to be real expectations of being able to meet the standards in that time frame. But this is very unrealistic!

A realistic long term is maybe 15 years. If we are lucky, in 15 years the average student may get near the standards if everything goes just right. In a shorter time than that, it is almost inconceivable to believe that this will happen. California today ranks just about at the bottom in the United States, in terms of the level of mathematical achievements of students in K-12. The United States ranks near the bottom among all the developed countries in the world in terms of math achievements of students. We have an incredibly long way to go because you have to remember that the new California Mathematics Standards were written to match the levels of the standards of the top achieving countries in the world. Meeting these standards is a daunting challenge and we had better take it seriously.

We now look at the reasons we clearly needed new standards in mathematics. They can be subsumed in three main areas.

REASONS FOR NEW
STANDARDS

The increasing failure of the present system to produce enough technically skilled graduates to meet national needs
Curricular problems which leave more and more students without the prerequisites needed for their majors, particularly in technical areas
Lack of a clear understanding – on the part of teachers and math educators – of the major goals of the mathematics component of K-12 education

The next three slides explain a little bit about how we see some of this so we cannot escape from these issues. The facts quoted in these slides come from recent newspaper articles for the most part.

INDICATIONS OF FAILURES

From 1990 to 1996 there has been a 5% decline in high-tech degrees — engineering, math, physics, computer science — in this country and the trend is continuing.
Of the decreasing number of high-tech degrees awarded a significant and growing proportion go to foreign nationals.
At the doctorate level 45% of high-tech degrees were granted to non-U.S. Citizens

From 1990-96, there’s been a 5% decline in high-tech degrees overall in this country. And the trend is continuing — in fact, the trend is accelerating. Even though the number of high-tech degrees is decreasing, it is vital to note that an ever increasing portion go to foreign nationals. At the doctorate level, for example, 45% of high-tech degrees are granted to non-U.S. Citizens and at Stanford, in the mathematics department, two thirds of our graduate students are foreign-born. Even 10 years ago, less than half were.

As a result of this situation it has been impossible to fill all our technical jobs with United States citizens. This is particularly true in Silicon Valley. To find qualified people to fill these positions Congress was intensely lobbied by Silicon Valley, and Congress was forced, much against their will, to provide 142,500 more visas for foreign nationals to fill jobs in Silicon Valley.

Currently it is estimated that the number of foreign-born residents of Silicon Valley is about 25% of the population.

Among all the states as I said in the beginning, California colleges showed the greatest decline in high tech degrees.

INDICATIONS OF FAILURES – II

Last year Congress was forced to provide 142,500 more visas for foreign nationals with high-tech skills
Currently it is estimated that the number of foreign born residents of Silicon Valley is about 25% of the population
Among all states, California’s colleges showed the greatest decline in high-tech degrees awarded.

So the first point is that the system today is simply failing to produce enough technically qualified graduates to meet national needs. The foremost problems and most dramatic declines are here in California.

Curricular problems are overwhelming here and leave more and more students without prerequisites needed for developing and learning technical skills in college. When they come to us, even at Stanford, more and more of them are just not able to become engineers and scientists, even though this is their original intent. They just don’t have the background any more. It is a dramatic change.

Finally, and sadly, because I have the utmost respect, and I think we all do, for the practicing teachers, the level of understanding on the part of teachers and above all of math educators — that is members of the educational schools throughout the country — that is required for teaching mathematics in K-12 is just not there any more.

Look at the effect of this lack of understanding on our students.

INDICATIONS OF FAILURES – III

The percentage of entering students in the California State University System who are place into remedial mathematics courses after taking the ELM placement exam is about 88%
Overall, well over 50% of entering students are placed into remedial mathematics courses.
The average level of the questions on the recent version of the ELM is about grade level 6.9 according to the new California Standards.

This 88% is a statistic that astounded me. And it is correct, differing from the failure rates commonly reported (which are bad enough). The percentage of entering students in the California State system who are placed into remedial mathematics courses after taking the ELM placement exam is 88%. Let me emphasize this: 88% of those students taking the exam fail it. Some of you may know a statistic of about 55% for the failure rate. Unfortunately, this is calculated by counting the 40% of the entering students who are not required to take the exam as having passed it.

These 40% are counted as passing it probably so the statistic will look reasonable. I reiterate that the actual statistic is 88% taking the ELM fail it, and it is not that hard an exam overall. In any case, well over 50% of entering students in the California State University system are placed into remedial math courses.

Those are some of the reasons for our current problems. They stare at us. We can’t avoid or deny them.

Now, I would like to give you an idea of the real complexity of the problem and the consequent difficulty with trying to fix it.

On our first slide the second problem with mathematics that I indicated is the lack of understanding of curricular development on the part of math educators.

Curricular development is a very complicated issue. As an illustration, I’m going to look at one topic, long division, now. Long division is something that a lot of professional math educators want to take out of curriculum. So let’s just look at why it is in the curriculum.

CURRICULAR PROBLEMS

The recent fashion of not teaching material like long division and factoring polynomials is based on claims that such skills are no longer useful.
This reflects a deep lack of understanding of the role of mathematics in fields like science, engineering and economics.
In mathematics many skills must be developed for many years before they can be used effectively or before applications become available.

First of all, I claim that taking — even asking to take it out of the curriculum — shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they’ve been introduced. Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples.

SOME SKILLS DIRECTLY
ASSOCIATED WITH LONG
DIVISION

Students cannot understand why rational numbers are either terminating or (ultimately) repeating decimals without understanding long division.
Long division is essential in learning to manipulate and factor polynomials.
Polynomial manipulation and factoring are skills critical in calculus and linear algebra: partial fractions and canonical forms.

So just to start, understanding that decimals represent rational numbers if and only if they are terminating or ultimately repeating — a skill that was requested be put into the standards by math educators — cannot be understood without long division. It is only in understanding of the process of taking the remainder in long division that you see the periodicity or termination happen.

I regard the repeating decimal standard as relatively minor, but some people seem to think it is important. The next topic is critical and almost everyone thinks it’s minor (Laughter). Long division is essential to learning to manipulate polynomials. Without it, you simply cannot factor polynomials.

So what, you ask? Again, this is a question that doesn’t come up until the third year in college. At this point the skills that have come from long division through handling polynomials become essential to things like partial fraction decomposition which is important in calculus but finds its main applications in the study of systems of linear differential equations, particularly in using Laplace transforms, which is the critical construction in control theory. It is also essential in linear algebra for understanding eigenvalues, eigenvectors, and ultimately, all of canonical form theory — the chief underpinning of optimization and design in engineering, economics, and other areas.

The previous slide indicated what I call the static applications of long division. The next slide illustrates some of the “dynamic” applications.

DYNAMIC SKILLS ASSOCIATED
TO LONG DIVISION

The process of long division is one of successive approximation, with the accuracy of the answer increasing by an order of magnitude at each step.
The skills associated with this process become more and more fundamental as students advance.
- They include all infinite convergence processes, hence all of calculus, as well as much of statistics and probability, to say nothing of differential equations.
Long division is the main application of the previously learned skills of approximation.

Long division is the only process in the K – 12 mathematics curriculum in which approximation is really essential. The process of long division is a process of repeatedly approximating and improving your estimates by an order of magnitude at each step. There is no other point in K – 12 mathematics where estimation comes in as clearly and precisely as this. But notice that long division is also a continuous process of approximation, the answer keeps getting more and more accurate and when the students learn how to do long division with decimals they learn to carry the process to many decimal places. This leads naturally — in a well conceived curriculum — to students understanding continuous processes, and ultimately even continuous functions and power series. The development of these skills are all contingent on a reasonable development of long division. I don’t know of any other or any better preparation for them.

What happens when you take long division out of the curriculum? Unfortunately, from personal and recent experience at Stanford, I can tell you exactly what happens. What I’m referring to here is the experience of my students in a differential equations class in the fall of 1998. The students in that course were the last students at Stanford taught using the Harvard calculus. And I had a very difficult time teaching them the usual content of the differential equations course because they could not handle basic polynomial manipulations. Consequently, it was impossible for us to get to the depth needed in both the subjects of Laplace transforms and eigenvalue methods required and expected by the engineering school.

But what made things worse was that the students knew full well what had happened to them and why, and in a sense they were desperate. They were off schedule in 4th and 3rd years, taking differential equations because they were having severe difficulties in their engineering courses. It was a disaster. Moreover, it was very difficult for them to fill in the gaps in their knowledge. It seems to take a considerable amount of time for the requisite skills to develop.

APPLICATION OF THE SKILLS
ASSOCIATED TO LONG DIVISION

The combination of these skills is used critically in economics, engineering and the basic sciences via Laplace transforms and Fourier Series.
Without a thorough grounding in these topics it is impossible to do more than routine work in most areas of engineering, the most active current areas of economics and generally, any area involving optimization.

So you see the problem. The problem is that the scope of things in mathematics is so long that an ordinary second, third, fourth grade teacher is not equipped to make a judgment about whether a subject is needed or not needed.

SCOPE IN THE MATHEMATICS

CURRICULUM

The long division story illustrates one of the chief problems with curricular development in mathematics. The period needed before a learned skill can be fully utilized can be as long as eight to ten years.
It takes real knowledge of mathematics as well as how it is applied to make judgements regarding curricular content.

I think the long division problem illustrates the problem described on the slide above very well. And I put that dragon up there advisedly.

EDUCATORS TELL US OF THE
NEED FOR CONCEPTUAL
UNDERSTANDING AND MATH
REASONING SKILLS IN OUR
STUDENTS

These skills ARE critical in todays technological society.
What many math educators tell us represent examples and exercises for developing these skills are NOT relevant and/or NOT correct.

The first slide mentioned a third aspect of the problem, which was the lack of knowledge of the subject on the part of math educators. To make it clear, I’m talking about math educators and not teachers. Teachers learn what they are told in the education schools and just hope that this background prepares them sufficiently. They do the best they can and have the most demanding job that I know of. As a group I believe they are the most dedicated people I know of. But if you do not provide teachers with the proper tools, they can’t do a proper job.

MATH EDUCATORS OFTEN
HAVE LIMITED KNOWLEDGE
OF MATHEMATICS

For example, three of the 14 problems originally proposed by the presidential commission on the eighth grade national mathematics text and/or the “solutions” they gave were INCORRECT. This commission included many of the best known math education experts in the country.
The next slides discuss one of these problems.

I just want to spend a few minutes, now, looking at some of the problems that we have seen in the last few years when we — as professional mathematicians — have looked at some of the things that math educators are trying to tell the world is mathematics. I will concentrate on problems that these people suggest for testing mathematical knowledge.

A PROBLEM FROM THE
NATIONAL EIGHTH GRADE
EXAM

We are given the following pattern of dots:

At each step more dots are added than were added at the last step.

How many dots are there at the twentieth step?

This is a problem from the original proposed 8th grade national exam, produced by a presidential commission including most of the best known math educators in the country. The problem appears to be simple and every person I’ve asked, who I haven’t warned to think hard and carefully about it, has answered immediately, “Oh, it’s of the form n times n plus 1, so you are looking at the 20th stage, therefore the answer is 20 times 21.”

But that’s not right. The words need to be read carefully.

The point is, the words tell you the only thing you are actually given — namely, that there are more dots added at each stage than the previous stage. That’s all you are given, and the picture is just a picture.

ANALYSIS OF THE PROBLEM

The answer given by the Presidential Commission on the National Eighth Grade Exam was

20 X 21 = 420

This is incorrect! The correct answer is that any number of dots is possible as long as there are at least 267.
As was pointed out, the Presidential Commission that proposed this problem included many of the best known math educators in the country.

ANALYSIS OF THE PROBLEM – II

This can be seen by considering that you must add at least seven dots to get to the fourth stage, eight to get to the fifth, nine to get to the sixth, and so on, but, of course, you can always add more.
So the formula for the number of dots at the n^th stage with n>2 becomes:
- any number at least as big as
  6+(6+7+8+…+(n+3)) which equals
- any number at least as big as
  (n+3)(n+4)/2 – 9 = 267

Hmm? Actually that problem was about as complicated as any problem I’ve seen at this level, and it was proposed for the 8th grade national exam! When you read it carefully, it is a problem a 12th grade senior would have trouble solving.

So what is the moral here?

If you want to learn mathematics, you must learn it precisely. Mathematics is precision and one of the first objectives in teaching K – 12 mathematics is for students to learn precise habits of thought.

The next slide presents a problem that Wu is very fond of (Laughter). It can be found in many sources, but in particular it was included as part of the original Mathematics Standards Commission’s proposed California Mathematics Standards.

A PROBLEM FROM THE
ORIGINAL STANDARDS COMMISSION
STANDARDS

You have a friend in another third grade class and want to determine which of your classrooms is bigger. How do you do it?

This problem is often proposed as an example which shows that “there is no single correct answer” since you could use perimeter or volume or area to measure size.

Of course, this is incorrect!

The trouble is that bigger is not precisely defined. And if every term is not precisely defined, your problem is not well posed. So technically this is not a well-posed problem. Of course, we realize that is a little technical. We have an idea that bigger has certain connotations — but unfortunately, a lot of them: perimeter, area, volume, and maybe even combinations of the three such as 3A + 2.4P + 7V.

ANALYSIS OF THE
COMMISSION PROBLEM

The difficulty here is that bigger is not precisely defined, and to do mathematics you generally have to know exactly what each term means.

However, mathematics does provide for the situation where terms can have different meanings. There is still a single “correct” answer. It consists of the set of all answers.

But since bigger can mean anything, the set of answers is uncountably infinite, and this problem is totally inappropriate for any but the most advanced high school students.

You see, when you put in a linear combination of the three, you get an uncountable number of possible definitions of bigger. That’s all right. Mathematics allows for this, as long as you can make some sense of the problem. Mathematics says the correct answer to the problem is all possible answers to the problem (Laughter). If you are going to take that problem at face value, you have to give me an uncountable number of answers.

NYC Honest Open Logical Debate (NYC HOLD)On Math Reform

NYC MATH WARS
NYC Honest Open Logical Debate (NYC HOLD)On Math Reform
Elizabeth Carson

November 27, 2000

New York, NY. NYU mathematicians speak out against controversial new math programs being taught in NYC’s premiere School District 2 in Manhattan. Community School Board 2 will hear the professors’ preliminary report at the calendar meeting on Tuesday, November 28, 6:30 pm at 333 7^{th Ave, Seventh Floor.}

^{A groundswell of parent concern over how math is being taught in District 2 schools led to a front page New York Times article last spring, “The New, Flexible Math Meets Parent Rebellion.” CBS Weekend News brought national attention to the local struggle last May in the segment, “New, New Math = Controversy.”}

^{District 2 parents have now gained the support of mathematicians at NYU and the University of Rochester in grieving their concerns to District officials. The professors are scheduled to take part in a Math Forum sponsored by Community School Board 2 scheduled for March 1, 2001.}

^{The controversy in District 2 has gained the attention of Schools Chancellor Harold O. Levy and members of his recently established Commission on Mathematics Education charged with investigating how math is being taught in schools across the city.}

^{District 2 pilots the latest wave of experimental math programs, called the “new, new math,” echoing an earlier failed reform in the 60’s called “new math.” Memorization of math facts is no longer emphasized. Children are encouraged to use language to describe solutions and the way they feel about math.}

^{Community response in NYC and across the country has erupted in what have become known as the “math wars.” Critical parents, joined by mathematicians and scientists advocate clarity and balance in math reform: urging the inclusion of grade by grade goals, explicit teaching of standard procedures, basic skill building and rigor along with the inclusion of some of the creative exercises in the new programs. The pendulum has swung too far and must be corrected.}

^{One parent, Mark Schwartz, in testimony last year before the House Education and Workforce Committee stated: “If medical doctors experimented with our kids in the same fashion school districts do they would be in jail.” The hearings were held to review the US Department of Education’s endorsement of 10 of the experimental programs.}

^{Over 200 of the nation’s top mathematicians, including seven Nobel Laureates,}winners of the Fields Medal ^{the department heads at more than a dozen universities including Caltech, Stanford and Yale, responded to the federal endorsements with an open letter of protest to Secretary of Education Richard Riley, published in the Washington Post in November, 1999.}

^{“These programs are among the worst in existence,” said David Klein, a Cal State Northridge professor who was one of the letter’s authors. “To recommend these books as exemplary and promising would be joke if it weren’t so damaging.” Several of the cited programs, Interactive Mathematics Project (IMP),Connected Mathematics Project (CMP) and Everyday Mathematics are now being used in NYC schools.}

^{The new wave of math reform is based on a “constructivist” teaching philosophy; emphasizing creative exercises, hands-on projects and group work, with far less attention given to basic skills. Students are asked to ‘construct’ their own solutions. The use of calculators is encouraged. Teachers are instructed to serve as “facilitators” and are discouraged from explaining to students the standard solutions of basic arithmetic. Practice and drill have been eliminated. In higher grades algebra is de-emphasized. Many of the programs have no textbooks.}

^{Members of Chancellor Levy’s Math Commission will consider the new programs, which are being used in over 60% of NYC schools, including roughly 50 of the city’s weakest, which comprise the Chancellor’s District . Plans are set to expand implementation into more schools. One of the most controversial programs, the Interactive Mathematics Project (IMP), will be mandated in Bronx High Schools beginning next year.}

^{Professor Richard Askey, who holds an endowed chair in math at the University of Wisconsin, explained his motives in co-authoring the protest letter to Secretary Riley, “I’m hoping to provide ammunition for teachers who are under pressure to adopt some of these programs.”}

^{High math scores remained stable in some of the privileged District 2 schools, last year; though some schools’ scores dropped significantly. Across the rest of the city, 75% of eighth grade students failed the state math test ; 66% failed the city math tests. Answers are critical at a time when graduation requires students pass a new math Regents exam.}

^{It was the introduction of Connected Mathematics Project (CMP) and Investigations in Number Data and Space (TERC), two of the District 2 programs, that sparked the initial parent revolt that led to the California Math Wars.}Six parents in Plano,Texas have filed suit in federal court against their local school district after parent requests for an alternative to CMP were denied.^{A nuclear physicist in Okemos Michigan led the local campaign against CMP. The use of TERC in one school system in Massachusetts prompted members of the Harvard Mathematics Department to issue a public protest. Parents in Reading Massachusetts fought the adoption of Everyday Math.}

^{The “new, new math” programs are based on the 1989 National Council of Teachers of Mathematics (NCTM) Standards. Frank Allen, past president of the NCTM and Emeritus professor of Mathematics at Elmhurst College comments: “NCTM leaders must admit that they have urged the application, on a national scale of highly controversial methods of teaching before they have been adequately debated or understood and before researchers have verified them by well-controlled and replicated studies.”}

^{=============}

November 27, 2000

To: Harold O. Levy, Chancellor

Dr Judith Rizzo, Deputy Chancellor for Instruction

Burton Sacks, Chief Executive for Community School District Affairs

Dr Irving Haimer, Member, Board of Education, Manhattan Representative

Trudy Irwin, Director of Education, Office of the Manhattan Borough President

From: NYC HOLD, Steering Committee

Re: Community School Board 2 Calendar Meeting November 28, 2000

We respectfully invite you to attend the calendar meeting of Community School Board 2 in Manhattan scheduled for Tuesday November 28, beginning at 6:30 pm. During the public session the Board will hear comment on District 2’s K-12 math programs. Distinguished faculty of the Courant Institute of Mathematical Sciences at New York University and concerned parents plan to speak.

The District’s mathematics reform is one component of the systemic instructional reform initiated by Anthony Alvarado in the previous decade. District 2 prides itself in being a leader in education reform in New York City. Our District piloted the New Standards Performance Standards in Mathematics which were adopted city-wide last year.

The implications of the relative success of our District’s math reform for the entire NYC school system should be evident.

In our District, parent concern with their children’s progress in mathematics is escalating; worries are being strongly voiced about aspects of the new math programs just as full implementation takes hold. School year 1999-2000 marked the first year the new K- 8 programs, Investigations in Number Data and Space (TERC) and Connected Mathematics Project (CMP) were mandated.

Parents are tutoring in record numbers. For many of their children, the new constructivist programs fail to provide a sound foundation in basic arithmetic. Parents recognize the value of the new programs’ creative, hands-on classroom and homework activities. However, many question the extensive time devoted to such activities, ostensibly at the expense of explicit teaching of the standard procedures and practice of skills. The absence of textbooks has exacerbated the situation.

Concerned parents have reached out to the NYU mathematics community in District 2 for analysis and opinion of our programs in hopes of finding ways to amend or extend the implemented programs to reach a better balance in the math instruction.

There are clearly implications in the course the District 2 community takes for the work of the newly appointed Commission on Mathematics Education to review math programs city-wide. We hope our experiences in District 2 can contribute to the assessment of the Committee.

Parents, teachers and administrators in District 2 share a well earned pride in our exemplary schools and strong educational community. Parents see on a daily basis, through their children’s work and enthusiasm for learning, the results of the skill and commitment of the teaching staff and administrators in our schools. Parents appreciate the formidable challenges district administrators face in guiding systemic reform in a large and diverse community school district.

We look forward to an ongoing and meaningful partnership in District 2.

We sincerely hope you will take an interest in our concerns and efforts to advance the course of math education for our children.

Sincerely,

Steering Committee
NYC HOLD

Elizabeth Carson
Christine Larson
Garry Dobbins
Margaret Hunnewell
Mary Somoza, Member, CSB #2
Granville Leo Stevens, Esq
Maureen McAndrew, DDS
Michael Weinberg

MATH LESSONS: BEYOND RHETORIC, STUDIES IN HIGH ACHIEVEMENT

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

Stand and Deliver Revisited July 2002

Stand and Deliver Revisited July 2002

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

By Jerry Jesness

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

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

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

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

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

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

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

Movie Magic

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

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

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

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

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

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

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

The Secrets to His Success

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Death of a Dynasty

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

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

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

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

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

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

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

Scattered Legacy

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

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

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

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

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

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

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

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

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

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

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

Barn Building

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

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

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

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

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

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

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

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

Music-Math Analogy

By Nakonia (Niki) Hayes
Columnist EdNews.org

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

He ignored them.

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

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

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

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

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

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

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

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

Published July 19, 2007

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

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

By David Klein

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

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

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

Common problems

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

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

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

The standard algorithms

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

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

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

572 = 5 (10²) + 7(10) + 2

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

5x² + 7x + 2.

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

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

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

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

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

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

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

Calculators

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

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

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

Specific shortcomings

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

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

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

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

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

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

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

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

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

How could they be so wrong?

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

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

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

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

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

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

Marks of a good mathematics program

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

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

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

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

For more information

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

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

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

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

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

Why Math Always Counts By Arthur Michelson

Why Math Always Counts

Respect Responsibility Readiness

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

Why Math Always Counts

By Arthur Michelson

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

It can open our minds to logic and beauty

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

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

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

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

by Douglas Carnine
04/01/2000

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

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

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

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

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

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

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

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

Introduction

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

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

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

Embracing Teaching Methods that Don’t Work

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A Large-scale Education Experiment

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Direct Instruction after Project Follow Through

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

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

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

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

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

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

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

Learning from Other Professions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Making Education a Mature Profession

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

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

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

Notes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

vAddress the relationship between young children and mathematics.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

APPENDIX

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

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

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

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

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

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

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

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

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

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

Mathwise

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Common problems

The standard algorithms

572 = 5 (10²) + 7(10) + 2

5x² + 7x + 2.

Calculators

Specific shortcomings

How could they be so wrong?

Marks of a good mathematics program

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Common problems

The standard algorithms

572 = 5 (102) + 7(10) + 2

5x2 + 7x + 2.

Calculators

Specific shortcomings

How could they be so wrong?

Marks of a good mathematics program

For more information

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

572 = 5 (10²) + 7(10) + 2

5x² + 7x + 2.