In Refutation of the NCTM comment on Myth #2

On A Debate Between NCTM and the Civilized World

In the Washington Post, Tuesday, May 31, 2005, appeared an article by Jay Mathews[1], entitled

10 Myths (Maybe) About Learning Math,

and beginning:

I love debates, as frequent readers of this column know. I learn the most when I am listening to two well-informed advocates of opposite positions going at each other.

The article continues with a quotation of the complete Ten Myths About Math Education and Why You Shouldn’t Believe Them, a manifesto[2] against certain propositions (the “myths”) that the authors considered representative of the educational doctrines of the “reform” curricular program of the National Council of Teachers of Mathematics (NCTM), as illustrated by their 1989 Standards[3] and its revision of 2000, called Principles and Standards for School Mathematics[4] (PSSM). Each Myth is followed by some evidence in support of the authors’ view that it is indeed a myth, i.e., false, though attractive enough to be a dangerous guide to action. The authors of this document are associated with an advocacy group called NYCHOLD, initially made up of mathematicians, teachers and parents in the New York City area but now increasingly national in scope.

Mathews had sent the “Ten Myths” manifesto to representatives of NCTM asking them for comment, and in the article of May 31, 2005 he printed all the Myths (including brief accompanying explanations), each followed by the NCTM response. He ended the column by inviting the public to send him their own opinions for use in future columns. All this was well over a year ago, and while the general debate, called “the math wars”, is still going on as 2006 draws to a close, the Myths themselves, as an inferential statement of NYCHOLD attitudes, have not had much currency recently. Other developments have taken the lead, in particular the NCTM publication of a document called “Focal Points”[5], formally presented as official NCTM doctrine.

These Focal Points were composed by a small team comprising both professors of mathematics and of mathematics education, and it selected certain topics for each elementary school grade as preeminent. The emphases of the Focal Points included special attention to arithmetic and a much diminished role for data analysis in the early grades, and generated favorable commentary from several observers who were known opponents of much that NCTM had been saying since 1980. The initial major newspaper accounts of their reception included such quotations as “a remarkable reversal” and “a major shift”.

However, NCTM soon replied, saying that the Focal Points are no reversal at all, but rather a clarification of priorities, and that the newspapers and commentators were quite wrong in depicting them as a revolution of some sort. NCTM stands by all its earlier policies, explained its President, Francis Fennell[6], including the 1989 Standards which have long been the centerpiece of sometimes acrimonious debate between NCTM and the parents and mathematicians who deplore them and the ill effect they have had on school math education.

Fennell was not necessarily saying this particular document, the Focal Points, was being misinterpreted; he was saying that the Focal Points did not represent a change in NCTM policy. NCTM was by no means confessing past error in saying what it did in the language of the Focal Points, even if there were those who thought so. The newspapers were deceived. NCTM had always advocated the primacy of the items of mathematics content brought to the fore in the Focal Points, and merely believed that these particular instructional ambitions for each of the grades (Pre-kindergarten through the 8^th) needed emphasizing, so that they would be given more time and attention than some other valuable topics mentioned in the Standards and PSSM. Recent experience had shown that schools were in some cases devoting unwarranted amounts of time to relatively minor topics, he explained, and the Focal Points were designed to restore a proper perspective.

Here then, I thought, was but the latest statement in a long series from NCTM, saying in effect that I, and all of us who had been struggling since 1989 to free our schools of what we considered a dangerous doctrine in mathematics education, had been wrong in our reading of what the NCTM had actually been saying. There was no “math war” except in our imaginations. Either we were reading the 1989 Standards wrong or we were reading the Focal Points wrong. Or there was some failure in our logic, to think the two documents were in conflict.

Earlier NCTM statements replying to attacks from critics who read its Standards as removing genuine content from school mathematics, in favor of “discovery” and irrelevant exercises, often were not quite to the point, as when one President of NCTM said, all around the nation, the “of course” NCTM advocated “learning the multiplication tables”, as if our criticisms went only that deep. With the 2005 publication of the Ten Myths by Mathews in the Washington Post, however, the NCTM rejoinders gave us a little more to think about, and something more substantial to respond to. The 2005 responses to the Myths could not be directly related to the Focal Points as such, for these had not been written yet, but their spirit was similar. It is therefore worth while to examine at least one of them, if only as preparation for a further examination, later on, of the NCTM defense of the Focal Points as if they were a consistent restatement of certain old NCTM policies.

In its response to Mr. Mathews’s 2005 query, then, NCTM had said that the so-called Myths were incorrect (inferential) portrayals of NCTM documents. NCTM went on to refute, point for point, the notion that the Myths as worded by us were anything like what NCTM was or had been saying in its 1989 Standards, its 2000 PSSM, or anywhere else[7].

Now, we can easily show that some of the math curricula and textbooks written with National Science Foundation assistance and disastrously being used by increasingly many schools, and which in their own advertising state they are being written in alignment with NCTM standards, do in fact lean on a philosophy of mathematics education that largely takes the Myths as truths. (We have also shown in many book reviews and articles[8] that these same programs are visibly in conflict with some of the Focal Points in that they do not merely fail to emphasize some of the curricular demands made explicit there, but they often omit them entirely, leaving the students ignorant of the content demanded by those “points” and ignorant of their ignorance. Their teachers likewise. We will return to the Focal Points later, however.)

Now the fact that a program, such as the TERC Investigations or the CMP middle school program, claims NCTM philosophy as a background doesn’t mean that it got that philosophy right, and that NCTM was at fault for the result. The NCTM answer to Myth # 4 pointed this out, and was thereby able to evade the intended questioning of the efficacy of those programs that clearly are based on NCTM dogmas. For example, if --- as was the case in 1999 at least --- CMP discouraged or omitted all the standard algorithms of arithmetic (failing entirely, for example, to consider the topic of division of fractions), and if it claimed NCTM Standards alignment, and if a sometime President of NCTM was one of its authors, can one really say it does not follow that CMP represented NCTM’s values, but that we only know that CMP said it did?

In strict logic, yes; but in this case there is another piece of evidence. In an open letter[9] of November 30, 1999 by John Thorpe, Executive Director of NCTM, to Richard Riley, Secretary of Education, in response to a petition protesting Mr. Riley’s endorsement of ten new math programs as “exemplary” or “promising”, Thorpe, speaking for NCTM, did give explicit NCTM endorsement of those programs, CMP among them, as having been particularly beneficial to thousands of American children. Mr. Thorpe further represented the signers of the petition as having been for unknown reasons “bothered” by the prospect of further such “positive effects” on the youth of the nation.

Well, there certainly is a clash of opinion here, John Thorpe for the NCTM on one side (along with the authors of, for example, the CMP middle school program), and on the other the writers of Ten Myths, who include or represent a good number of mathematicians. We were all referring to the same documents. We know CMP represented NCTM policy as to content and method, and that Mr. Thorpe found it was good. And we know some other certified examples that have found NCTM favor. It is therefore time to go to the Myths, and later to the Focal Points as well, and see what the actual words are, and whether the writers of the Myths document were mistaken in their implied characterization of NCTM policy, or mistaken in declaring some of it mistaken.

A comparison of one of the Myths, of what NCTM said to Mr. Mathews about its being a misstatement of NCTM’s position, with what NCTM has itself said in its writing of earlier years, and with what can be seen in practice via some of the programs written in pursuit of NCTM dicta, can serve as a particularly well focused example. (It should also, in passing, indicate why so many of us consider the wording of the new Focal Points a genuine departure in NCTM policy, even if that organization itself wishes to be seen as consistent over the years.)

I shall comment on NCTM's answer to Myth #2 and what the “Advocates” said about it, the Advocates being the authors of the Myths document and more generally the opponents of NCTM curriculum policy, myself among them. Here it is as printed in the Mathews column:

Myth #2 -- Children develop a deeper understanding of mathematics and a greater sense of ownership when they are expected to invent and use their own methods for performing the basic arithmetical operations, rather than being taught the standard arithmetic algorithms and their rationale, and given practice in using them.

Advocates: Children who do not master the standard algorithms begin to have problems as early as algebra I.

The snubbing or outright omission of the long division algorithm by NCTM-based curricula can be singularly responsible for the mathematical demise of its students. Long division is a pre-skill that all students must master to automaticity for algebra (polynomial long division), pre-calculus (finding roots and asymptotes), and calculus (e.g., integration of rational functions and Laplace transforms.) Its demand for estimation and computation skills during the procedure develops number sense and facility with the decimal system of notation as no other single arithmetic operation affords.

The NCTM response, as reported in the Mathews article, was:

NCTM has never advocated abandoning the use of standard algorithms. The notion that NCTM omits long division is nonsense. NCTM believes strongly that all students must become proficient with computation (adding, subtracting, multiplying, and dividing), using efficient and accurate methods.

Regardless of the particular method used, students must be able to explain their method, understand that other methods may exist, and see the usefulness of algorithms that are efficient and accurate. This is a foundational skill for algebra and higher math.

I selected this particular myth and response both because the matter of NCTM’s attitude towards the algorithms of arithmetic has been important in itself and because the NCTM response illustrates the way NCTM has been responding to other attacks of the kind seen in the Ten Myths.

First, the strong and flat-out denial: NCTM has never advocated abandoning the use of standard algorithms. We’ll see in a moment what NCTM has had to say in this connection in the past. However, even in this very paragraph, NCTM does not quite go on to say that it does advocate the use of the standard algorithms, or the their teaching. No, it is more ambitious than that. It says students should be proficient with computation (and even names the four arithmetic operations for us, in case we have forgotten), and insists on “efficient and accurate methods”. Why does it not mention “long division” by name here, for example? Why does it go on about “regardless of the method used”? What is this flurry of words in seeming explanation of the opening denial?

Well, in truth, never in either the 1989 Standards nor in the PSSM of 2000, its revision, did NCTM advocate learning what we all call "long division", or any other particular algorithm. Yes, of course they are right in saying "NCTM has never recommended the abandoning of standard algorithms." Very logical: They don't even mention them in the 1989 Standards; how could they have advocated their abandonment? What did they recommend? Here they can point to page 94 of the 1989 Standards:

In grades 5-8 the mathematics curriculum should develop the concepts underlying computation and estimation in various contexts so that students can compute with whole numbers, fractions, decimals, integers, and rational numbers... [and] select and use an appropriate method for computing from among mental arithmetic, paper-and-pencil, calculator, and computer methods."

There: They can choose between a calculator and a pencil. Nothing is forbidden. But nowhere in the 1989 Standards is it explained just what good that pencil could be. On the following page the Standards explains in more detail:

The greatest revisions to be made in the teaching of computation include the following

* Fostering a solid understanding of, and proficiency with, simple calculations

* Abandoning the teaching of tedious calculations using paper-and-pencil algorithms in favor of exploring more mathematics

* Fostering the use of a wide variety of computation and estimation techniques . . . ranging from quick mental calculation to those using computers . . . suited to different mathematical settings

* Developing the skills necessary to use appropriate technology and then translating computed results to the problem setting

* Providing students with ways to check the reasonableness of computations (number and algorithmic sense, estimation skills)

Thus the only thing NCTM has advocated “abandoning” is the teaching of “tedious calculations using paper-and-pencil algorithms”. If, in response to Myth # 2 they are correct in their looking backwards to see that they were in favor of teaching long division all this time we had imagined them to be opposed to it, then what this “revision” imagines is that long division, which the text never mentions by name, can and indeed should be taught without “tedious calculations”. Any teacher, whether or not he favors teaching long division, will tell you that this tour de force is impossible. Still, just how tedious must we be? How much should we make the children suffer? Surely we all agree that there is a limit to the tedium we should inflict on our young.

Perhaps we are misinterpreting the 1989 Standards when we imagine the above quoted text constitutes an outright ban on “long division”? Here now is the immediately following text, where examples are given to amplify the “revision” NCTM had in mind in 1989 as to the algorithms of arithmetic:

The ability to compute 0.17 X 45 correctly is not as interesting for its own sake as it is for estimating the number of times a certain result will occur on a spinner in a game or in determining a discount when buying a new tennis racquet. Students should know when it is appropriate to multiply 0.17 X 45 in problem-solving situations and how to multiply 0.17 X 45 with a calculator.

Let us emphasize what they have said here, by separating the two phrases that serving as objects of verb “know” in “Students should know…”:

(1) Students should know when it is appropriate to multiply 0.17 X 45 in problem-solving situations; and

(2) Students should know how to multiply 0.17 X 45 with a calculator.

As to (1), when to do it, few would disagree; but they must also know (2) how to do it. How? With a calculator. There is no statement here – or anywhere – that they should know how to do it by hand. How could NCTM have forgotten that method? Slipped their mind? One can understand their recommendations about spinners and the occasional need to multiply (e.g.) 0.17 by 45, but once they mention how this should be done, and list only the calculator explicitly as the method, with no suggestion that it is only an example of a method, one must consider this a discouragement of "long multiplication", even for two-digit numbers.

The Standards text goes on (still on p.95):

Despite these fundamental revisions, certain aspects of computation continue to be important. A knowledge of basic facts and procedures is critical in mental arithmetic and estimation. Knowing that 8X7 = 56 is a basis for finding 8X700 mentally, multiplying (+8)X(-7), estimating 824 x 689, and estimating 8.24 X 6.89.

Valuable class time should not be devoted to developing students' proficiency in calculating 824 X 6.89 with paper and pencil, since these exercises can be done more readily with a calculator.

There we have it: “Certain aspects” of computation continue (after the NCTM revisions) to be important, for example, the 9 by 9 one-digit tables as might be useful for mentally finding approximate results when more complicated numbers turn up. Certain aspects of computation, but not computation itself. The multiplication of three-digit numbers by hand is not forbidden here, but valuable class time should not be devoted to developing students' proficiency in it. Is there such a thing as learning to multiply two three-digit numbers without developing proficiency in it? Should the schools, every time valuable class time comes into play, hide the pencils? Does NCTM believes the algorithm can be taught and learned without tedium?

Thousands of years of evidence point the other way. Nobody learns to play a real sonata without first being able to play arpeggios, rapidly, accurately, no matter how much music theory he has studied. And as everyone knows, one is not able to play even the arpeggios on first acquaintance, even when they are conceptually understood. They must be practiced again and again, and then again, until they are right.

In effect, the 1989 NCTM is recommending music lessons without arpeggios. Too tedious. Arpeggios don’t occur in real music, anyhow. If the student needs a sonata let him use a record player.

It is true that most people who want to hear Beethoven do so by going to concerts or buying records. Similarly, most people needing the results of good mathematical skill hire engineers and scientists to do the relevant work, and do not themselves attain the skills the scientist must have. But is it not unconscionable that NCTM should outline a program for the middle schools that absolutely prevents everyone’s attaining the skill in arithmetic that some of them would ultimately need, in college or in later life? If not everyone can be an engineer, shall we therefore arrange our public educational system so that nobody can?

While NCTM in 1989 used guarded language, sufficiently guarded so that its President, seventeen years later, can deny it has said what it meant, the textbooks written with NSF financial support in the early 1990s did illustrate what it meant, for their authors truly understood the NCTM intent and followed it, as its Executive Secretary unguardedly stated in his 1999 letter. In one direction, NCTM can present a lawyer’s brief to the effect that it never advocated the abandoning of long division, but in another direction, by 1991 a team of math education specialists, including two sometime NCTM Presidents, was already composing (with financing from the NSF) a demonstration that long division was in fact abandoned, the middle school program called “Connected Math” (often referred to as CMP, for “Connected Math Project”, the group that produced it), a program that is still expanding its presence in Grades 6-8 across the country. CMP not only “claimed to be” consonant with NCTM standards, but, as one of several later called “exemplary” by the Secretary of Education, was explicitly defended as such by the Executive Secretary of NCTM[10].

Today, seven years later, parents without expertise in “mathematics education” are still discovering that their children aren’t being taught arithmetic, whether via CMP or its allied programs, programs still being pressed on children by educators even today financed by NSF to enter the schools and, under the title of “professional development”, to coach teachers in their use.

These educators knew and know full well that the children subject to their tutelage are not being taught what any sensible person or mathematician will call arithmetic. The teachers being coached in their use are being taught, furthermore, that this old-time arithmetic is not necessary to later study in mathematics or to daily life. On an explicit theoretical basis they are being taught to avoid the algorithms and “tedious” computations which can in no other way inculcate that facility with decimal and fraction notation and the mental structures needed for analogous algebraic operations – and much else. The authors of these “reform” textbooks or programs did their work according to a theory of education that had them persuaded that only what was valuable in mathematics for all students should be offered to any students, and that in arithmetic at least need only be as much as would turn up as a byproduct of the kind of “problem solving” they were placing at the center of their programs.

Well, the byproduct is not enough. Not only are the needs of future scientists left out of these programs, but much of what is needed by everyone else is also missing. Parents who have no particular desire to see their children become engineers or scientists, let alone mathematicians, are nonetheless forming advocacy groups in opposition to programs they see as mathematically inadequate. The swell of objection which now reaches across the country, and which has its voice in two well-known national web sites as well as in local ones, is gradually being heard by local school officials, who find themselves embarrassed to have to defend programs they had been assured by education professionals were research-based (which they were not) and efficacious. But the most embarrassed are the education professionals themselves, those who have to defend this spotted past in public. Among these is, naturally enough, the President of NCTM, who believes (though here I have to be guessing) that he is required by his office to defend all NCTM doctrines, past and present.

If he believes so, he is making a mistake. It is to its credit that NCTM in company with the authors of the recent “Focal Points” is making a “remarkable reversal”, but it would be more honest and it would advance that reversal by a number of years if NCTM would say so, or at least – which is closer to the whole truth – that the Focal Points are a step in that direction. Instead, when called on to counsel the nation’s teachers and textbook writers, NCTM instead takes refuge in legalisms: “The Focal Points are compatible with the original Standards”, and “The Council has always supported learning the basics.”

There are always, unfortunately, obstacles in the path of truth and justice, even for the well-intentioned. CMP and its allies in the “reform” textbook business are now commercial products, whose publishers are loath to announce they have been mis-educating a generation. The NSF financing of the writing of so many of these misguided programs has for many years now been a source of honor and airplane trips to faraway conferences; even NSF cannot be persuaded to stop the golden flow overnight. Supervisors of mathematics in local school districts have been making life hard for teachers who for these past fifteen years been teaching long division behind closed doors; how shall we tell them to take down that wall?

How shall we tell young teachers, newly minted in colleges of education, who for so long have been “misinterpreting” NCTM’s stance on the algorithms of arithmetic, that their elders were poor readers, and somehow had missed the point, and that NCTM really meant to say that “calculating 824 X 6.89 with paper and pencil” should be done by “long multiplication” when learning arithmetic in school, and by calculator when out in the world needing an answer for commercial purposes and not trying to improve their mathematics?

Today we learn from NCTM that “the ability to compute 0.17 X 45 correctly is not as interesting for its own sake as it is for estimating the number of times a certain result will occur on a spinner in a game or in determining a discount…” was a misprint in the 1989 Standards, which had meant to write “the ability to compute 0.17 X 45 correctly is not as interesting for its own sake as it is for its exhibiting the nature and structure of the decimal notation, of the number system itself, and of other algebraic systems – some of which some students will necessarily encounter in later life; and this is why every student must have repeated practice in making such calculations via the standard multiplication algorithms.”

One could go on, item by item, including the arithmetic of fractions, which has not yet been mentioned here. For even the revised PSSM of 2000, which was said to have corrected some of the items in the 1989 document that might have been “misinterpreted”, contains (page 219) an explicit warning against the "invert and multiply" algorithm for fractions, followed by a half-page exposition of how one can use a stone-age device to get around it, though in a marvelously simple case, viz.

The PSSM example divides 5 yards (of ribbon) by 3/4 yards (ribbon needed per bow), in order to find out how many bows the 5 yard ribbon contains. The solution is 5/(3/4) of course, which by “invert and multiply” is equal to 5 X 4/3, or 20/3, or 6 with 2/3 bows left over. PSSM cleverly avoids the dreaded algebra by pictorially laying off copies of segments of length 3/4 against a picture of a segment of length 5, and find there are 6 full 3/4 yard segments and 1/2 yard left over. See? The quotient is six and a bit more. (By some miracle they are able to see that the 1/2 yard remaining amounts to 2/3 of a bow, a feat requiring the division of 1/2 by 3/4, actually, though PSSM does it by eyeball.) A good-looking method, but one that leads to serious tedium if applied to, say, dividing 5280 feet per mile by 1/12 feet per inch, to find out how many inches there are in a mile. Even with no remainder to interpret.

The earlier, 1989, Standards handled fractions and decimals in more or less the same breath, without even considering a problem as difficult as 5/(3/4) except as something to be done by calculator. On page 96 it gives its definitive statement, which Mr. Fennell, en passant, thinks himself required to defend with all the rest:

The mastery of a small number of basic facts with common fractions (e.g., 1/4 + 1/4 = 1/2; 3/4 + 1/2 = 1-1/4, and 1/2 X 1/2 = 1/4) and with decimals (e.g., 0.1 + 0.1 = 0.2 and 0.1 X 0.1 = 0.01) contributes to students’ readiness to learn estimation and for concept development, and problem solving. This proficiency in the addition, subtraction, and multiplication of fractions and mixed numbers should be limited to those with simple denominators that can be visualized concretely or pictorially and are apt to occur in real-world settings; such computation promotes conceptual understanding of the operations. This is not to suggest, however, that valuable instruction time should be devoted to exercises like 11/24 + 5/18 or 5-3/4 X 4-1/2, which are much harder to visualize and unlikely to occur in real-life situations. Division of fractions should be approached conceptually. An understanding of what happens when one divides by a fractional number (less than or greater than one) is essential.

(Here I have used the notation “5-3/4” to represent the number usually pronounced “Five and three-fourths”, and so with other mixed numbers.)

The line about division of fractions being “approached conceptually” would be quite mysterious to most people, as it was to me until the publication of PSSM in 2000, when I read the “conceptual” account of the ribbon problem, and was able to understand what it meant: A/B is the integer you get when laying off lengths B against a picture of a line of length A, as many times as you can, and never mind the bit left over. As a concept, as a partial definition, this is a beginning, but as a method? Well, it was really not intended as a method by PSSM; the calculator is the method, and the picture the concept.

All the other recommendations concerning fractions also ignore the central role the manipulation of fractions plays in algebra, where the algorithms are necessary if a student (or scientist) is to make sense of the formulations that occur in mathematical sciences, including economics and engineering. The quoted NCTM recommendations were for Grades 6-8, and such “reform” programs as were consequently written for middle schools followed them sedulously. If now we ask what NCTM advocated for algebra at the high school level, where these manipulative skills are thought to be necessary, at least by scientists and financiers, we find on page 127, among the “Topics to Receive Decreased Attention”, under Algebra,

…*The simplification of radical expressions

*The use of factoring to solve equations and to simplify rational expressions

*Operations with rational expressions

*Paper-and-pencil graphing of equations by point plotting …

Well! With such inattention prescribed for the high school program, the omissions at the middle school level make sense.

In all its documents, NCTM has been careful to advocate very little

explicitly. NCTM does advocate "efficient and accurate methods" of computation, however, and so has made itself a bit of wiggle-room. With the Focal Points we see some of that wiggling. Here are some excerpts from the Focal Points as published September 12, 2006:

From Grade 4:

…[Students] develop fluency with efficient procedures, including the standard algorithm, for dividing whole numbers, understand why the procedures work (on the basis of place value and properties of operations) …

From Grade 5:

… Students apply their understandings of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators… They develop fluency with standard procedures for adding and subtracting fractions and decimals.

From Grade 6:

Students use the meanings of fractions, multiplication and division, and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions and explain why they work. … They multiply and divide fractions and decimals efficiently and accurately. They multiply and divide fractions and decimals to solve problems, …

Grade 7:

Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. … They use the arithmetic of rational numbers as they formulate and solve linear equations …

We should be grateful for that much, and if we get a little more I will try hard not to say they are lying about their past. But neither I nor anyone else who has used fractions in his work or conversation can believe that these “Focal Points” announcements are consistent with the 1989 advice to confine instruction in fractions to the simple fractions only, those that can be visualized pictorially, fractions like 1/2 and 3/4.

The Focal Points are a departure. They are not complete; they do not reverse enough of the past to be satisfactory as they stand. They employ in too many places the unpleasant jargon of the education world, jargon often used to conceal more than to explain. Some future users of the Focal Points are sure to have scope for sabotage of the good intentions of some of the writers, who have had to compromise to get the document written at all. But others will find in them the authoritative statements needed in their fight to establish adequate math programs in their localities.

Well, this is politics. As with the lands newly released from dictatorship and tyranny, we really need a reconciliation commission to end the cycle of guilt and retribution. But it is hard to do this with no confessions, and the Focal Points as defended by Mr. Fennell does not qualify, even though as a document it is a step in the right direction.

Meanwhile, the "Standards-based" math programs continue to avoid the teaching of the standard algorithms of arithmetic, among other difficult mathematical topics (among them the deductive structure of Euclidean geometry), because they have read the NCTM literature correctly, despite the new (partial) disclaimers. It is not possible to reconcile the Focal Points, or even NCTM’s denials of some of the more crucial of the “Ten Myths” theses, with TERC’s Investigations, or the programs certified by NCTM’s Executive Director in 1999, that were written in consequence of the vision of school mathematics put forward by NCTM in 1989.

The NSF-financed “reform” programs of the 1990s really were, as their advertising claimed, based on NCTM doctrines of the time. Those 1989 doctrines are today really changing, even if NCTM itself denies it. It probably has to deny it, but though I cannot say I feel sorry for its having so difficult a political problem to solve I can offer the usual solace for what it is worth: Time alone can solve many problems if you leave them alone. With time, as 1989 recedes into a more shadowy past, the more sensible attitude heralded by the Focal Points – though not yet fully laid out – can come to seem more natural and less of a novelty to quarrel about. Will the writers of the next generation of textbooks, and the teachers of the next generation of teachers, then provide our next generation of children with a better chance to succeed in mathematics than did those of the past? I believe this will require a deliberate downplaying of past doctrine, or at least a decent silence, rather than a continued effort, as NCTM is now making, to portray the new doctrine as a logical continuation or clarification of the old.

Ralph A. Raimi

5 December 2006

Corrected and slightly revised 21 December 2006

[1] This is Document 4 in the lengthy Appendix to this paper, which itself has the title Twelve Documents in the Math Wars, 1989-2006

[6]See http://www.nctm.org/focalpoints.aspx?d=286 for Fennell’s September, 2006 Letter to the membership of NCTM, almost all of it reproduced as Document 9 of the Appendix; see also Documents 11 and 12, quoting at length from his letters to the New York Times and the Wall Street Journal.

[7] With two or three exceptions. Myth # 7 found NCTM in full agreement that it was a myth, and Myth #8 similarly, though in this case NCTM’s agreement is not evidenced in the actual textbooks and programs prepared with NCTM guidance.

[8] see the NYCHOLD web page http://www.nychold.com

[9] See endorse.htm for an account of the affair. For other details see the references in Footnote 9.

[10] For the letter itself and its list of signatories, see http://www.mathematicallycorrect.com/riley.htm. For some mathematicians’ reviews of CMP, detailing its principled omission of mathematical information, including the most elementary arithmetic computations, see http://www.nychold.com/cmp.html.