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

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

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


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

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

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

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

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


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


Cheating Our Children

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

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


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

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

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

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

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

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


Effect on Higher Education

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


The Lineage of “Reform”

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


The Significance of Structure

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

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


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

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

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

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

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


An Analysis of the Present Situation and How it Developed

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

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

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


Moving the Goal Posts

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

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




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



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




Indictment of the Theoreticians

If the past has nothing to say to the present then the present has nothing to say to the future.
Frank B. Allen 1909-200? AD


Pity the NCTM today
A worthy group that’s gone astray
A group completely under the sway
Of theoreticians, far away
From schoolroom events of everyday
Who conduct research in a curious way
It matters little what they say
This is the message their deeds convey:

Standardized tests are an awful bane,
They reveal little or negative gain,
And we regard them with disdain.
A little logic might cause some pain,
From proof that’s tough we will abstain.
We’ll appeal to the hand instead of the brain.
Subject teacher time to a terrible drain,
With an assessment system that’s hard to explain.
We’ll repeat sixth grade, like an old refrain,
Recycling the facts all over again.

If you disagree with us at all
You are a Neanderthal.”

If we can’t stop them then let us pray
For secondary math in the USA.