__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__

[2] This is Document 3 of the Appendix.

[3] Document 1 of the Appendix

[4] Document 2 of the Appendix

[5] Document 6 of the Appendix

[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 http://www.math.rochester.edu/people/faculty/rarm/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.