Debate Appendix:

Twelve Documents in the Math Wars, 1989-2006


In Chronological Order:


1.  The NCTM 1989 Curriculum and Evaluation Standards, generally known as the “NCTM Standards”, see for access to the full text, but for members of NCTM only.  Out of print but available in university libraries.


2.  The NCTM 2000 Principles and Standards for School Mathematics, generally known as “PSSM”, for access to the full text, but for members of NCTM only.  For sale by NCTM, and available in university libraries.


3.  Ten Myths About Math Education And Why You Shouldn’t Believe Them, generally referred to as “The Ten Myths”, a statement found at,  made public on May 4, 2005, by Karen Budd, Elizabeth Carson, Barry Garelick, David Klein, R. James Milgram, Ralph A. Raimi, Martha Schwartz, Sandra Stotsky, Vern Williams, and W. Stephen Wilson, in association with the advocacy groups NYCHOLD and Mathematically Correct.  Each “Myth” is followed by a statement called “Reality”, explaining how or why it is false, i.e., a myth, and then by some research references in support of that statement.  Many of the Myths are intended to be recognizable as doctrines of the National Council of Teachers of Mathematics, or closely allied with or support for their curricular and teaching recommendations.  For example, Myth #1 is “Only what students discover for themselves is truly learned.”


4.  Ten Myths (Maybe) About Learning Math, by Jay Mathews, a column printed in The Washington Post, May 31, 2005, in which he reprints the Ten Myths themselves and a brief reply (unsigned) to each from a representative from NCTM, as he had asked it to provide for publication in his column.  (Full text at


5.  Reaching for Common Ground in K-12 Mathematics Education, generally known as “Common Ground”, a paper publicized in June, 2005 as a working document in a project still underway; find it and sequels at, a web site maintained by the Mathematical Association of America.  Common Ground was written by Deborah Ball, Jeremy Kilpatrick, Joan Ferrini-Mundy, R. James Milgram and Wilfried Schmid, assisted in their deliberations by Richard Schaar as moderator.  (Ferrini-Mundy as chief editor of NCTM’s PSSM and Milgram as a leader in the writing of the 1998 California Mathematics Standards thus may be seen as representing the two poles of the recent controversies.)


6.  The NCTM Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, generally known at the “Focal Points”, is an official document of NCTM, released on September 12, 2006 and available at for free download.  For each of the ten grades mentioned in the title the paper lists three major topics for emphasis in mathematics, with some elaboration and some reference to earlier NCTM Standards publications.  There has been some dispute over whether the wording of the Focal Points represents a departure from those earlier NCTM recommendations, some mathematicians and journalists interpreting them as a reversal (see Documents 7, 8, and 10 below) and President Francis Fennell of NCTM disagreeing (Documents 9, 11, and 12).


7.   New Report Urges Return to Basics in Teaching Math:  Critics of ‘Fuzzy’ Methods Cheer Educators’ Findings; Drills Without Calculators; Taking Cues From Singapore, by John Hechinger in the Wall Street Journal, September 12, 2006. Page A1.  (Full text at


From the text:


The nation's math teachers, on the front lines of a 17-year curriculum war, are getting some new marching orders: Make sure students learn the basics.

In a report to be released today, the National Council of Teachers of Mathematics, which represents 100,000 educators from prekindergarten through college, will give ammunition to traditionalists who believe schools should focus heavily and early on teaching such fundamentals as multiplication tables and long division.

The council's advice is striking because in 1989 it touched off the so-called math wars by promoting open-ended problem solving over drilling. Back then, it recommended that students as young as those in kindergarten use calculators in class.

Those recommendations horrified many educators, especially college math professors alarmed by a rising tide of freshmen needing remediation. The council's 1989 report influenced textbooks and led to what are commonly called "reform math" programs, which are used in school systems across the country…

          Infuriated parents dubbed it "fuzzy math" and launched a countermovement. The council says its earlier views had been widely misunderstood and were never intended to excuse students from learning multiplication tables and other fundamentals.

Nevertheless, the council's new guidelines constitute "a remarkable reversal, and it's about time," says Ralph Raimi, a University of Rochester math professor…

Nearly 80 teachers and other experts spent 18 months writing and reviewing grade-by-grade guidelines, which cover preschool through eighth grade. The panel aims to give a roadmap to instructors, schools systems and states about exactly what children should be learning -- and to start a debate that could put the math wars to rest…

Supporters of the council's previous views worry that the new report may lead to a return to the kind of rote learning they say left many children without any understanding of concepts. ..

"The risk is that we end up with students who have no idea what math is all about or how to use it," says Joseph Rosenstein, a math professor at Rutgers University in New Jersey …




8.  Report Urges Changes in the Teaching of Math in the U.S. Schools, by Tamar Lewin in The New York Times, September 13, 2006.  (Full text at 


From the text:


“In a major shift from its influential recommendations 17 years ago, the National Council of Teachers of Mathematics yesterday issued a report urging that math teaching in kindergarten through eighth grade focus on a few basic skills.  If the report, “Curriculum Focal Points,” has anywhere near the impact of the council’s 1989 report, it could signal a profound change in the teaching of math in American schools… [The 1989 Standards] was incredibly influential,” said Chester E. Finn Jr., a Department of Education official in the Reagan administration… “This report is a major turnaround.”  Dr. Finn added, “This is definitely a back-to-basics victory…” 

In a way, the new report stands as a plea for consensus … And consensus may be at hand.  Some of the same math professors who  last year released a chart – aimed directly at the National council of Teachers of Mathematics – detailing the “10 myths” of “N.C.T.M. (Fuzzy)” math now find themselves generally in line with the new report.  “It represents an enormous evolution from the 1989 standards, from the perspectives and attitudes that were present in both camps then,” said R. James Milgram of Stanford, one of the “10 Myths” signers.


9.  An email letter to the entire membership of NCTM, from the President, September 15, 2006


[Reprinted here in full except, in the last paragraph, a list of links to related documents on the NCTM web site,]


From: Francis (Skip) Fennell (

Subject: Curriculum Focal Points Released

Date: Fri, 15 Sep 2006 22:19:47 -0400


Dear  NCTM Members:


I am pleased to announce that Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence as released on September 12.  The Curriculum Focal Points are the next step in the implementation of the Standards.  The focal points fully support the Council's Principles and Standards for School Mathematics.  The appendix in Curriculum Focal Points directly links the focal points to virtually all the expectations in Principles and Standards.


Curriculum Focal Points presents the most important mathematical topics for each grade level.  A focal point specifies the mathematical content that a student needs to understand thoroughly for future mathematics learning.  The focal points are compatible with the original Standards and represent the next step in realizing the vision set forth in Principles and Standards for School Mathematics in 2000.  The focal points are intended for use by mathematics leaders as they examine their state and local mathematics expectations and seriously consider what is important at each grade level.  This discussion, dialogue, or perhaps debate is designed to influence the next generation of curriculum frameworks, textbooks, and assessments.


Unfortunately, some of the media coverage has raised questions and caused concern among our members.  Despite several conversations with a reporter from the Wall Street Journal explaining what the Curriculum Focal Points are and are not, a September 12 Wall Street Journal article did not represent the substance or intent of the focal points.  The focal points are not about the basics; they are about important foundational topics.  The Council has always supported learning the basics.  Students should learn and be able to recall basic facts and become computationally fluent, but such knowledge and skills should be acquired with understanding. Unfortunately, some of the other news media have followed the Wall Street Journal's lead and have similarly misrepresented the Curriculum Focal Points.


The Council's goal is to support teachers in guiding students to learn mathematics with understanding.  Organizing a curriculum around a set of focal points can provide students with a connected, coherent, ever expanding body of mathematical knowledge.  The focal points describe what should be the focus of what students should know and understand thoroughly.


I encourage you to explore the complete …Curriculum Focal Points section of the NCTM Web site.




Francis (Skip) Fennell




10.  Teaching Math, Singapore Style, Editorial in The New York Times, September 18, 2006. 


Excerpt:   “Many people trace this unfortunate development to a 1989 report by an influential group, the National Council of Teachers of Mathematics. School districts read its recommendations as a call to reject rote learning. Last week the council reversed itself, laying out new recommendations that will focus on a few basic skills at each grade level.”


11.  A letter to the New York Times, from Francis Fennell  [President of NCTM], dated September 18, 2006.  (Full text at NCTM web site,




To the Editor:

Forty-nine of our 50 states have developed state curricular frameworks. Most of these have been influenced by the National Council of Teachers of

Mathematics' "Curriculum and Evaluation Standards" (1989), or the more

recent "Principles and Standards for School Mathematics" (2000). Close to 30 of the states have revised their curricular frameworks since 2003.


What some refer to as basic skills (for example, multiplication facts, and

fluency with the addition, subtraction, multiplication and division of whole

numbers) have always been a fundamental core of elementary school

mathematics. Always.


But we want more. We want children to understand the mathematics they are

learning and we want them to be able to solve problems, which is, in the

long run, why we do mathematics.


Our recently released "Curriculum Focal Points" identifies important

mathematical topics in each grade, from prekindergarten through eighth

grade. It identifies the mathematical content students need to understand

deeply and thoroughly for future mathematics learning.


It offers a framework to guide states and school districts as they design

and organize revisions of their expectations, standards, curriculums and

assessment programs.


This is not a change, but reflects what has been the council's commitment to

teaching and learning for more than 80 years.


Francis Fennell

President, National Council of Teachers of Mathematics

Reston, Va., Sept. 18, 2006


12.  Francis Fennell’s  Letter to the Wall Street Journal (Abridged; full text at NCTM web site

Reading, Writing and Troubled Arithmetic

September 27, 2006; Page A19


The Curriculum Focal Points released by the National Council of Teachers

of Mathematics are the next step in implementing the group's "Principles

and Standards for School Mathematics," published in 2000 ("Arithmetic

Problem: New Report Urges Return to Basics in Teaching Math," page one,

Sept. 12). Based on the NCTM standards, the Curriculum Focal Points

identify the most important mathematical topics from pre-kindergarten …


Contrary to the impression left in your article, learning the basics is

certainly not "new marching orders" from the NCTM, which has always

considered the basic computation facts and related work with operations

to be important. Nor is the new focal-points approach to curriculum

development a "remarkable reversal" for NCTM. As stated in NCTM's 1989

and 2000 standards, conceptual understanding and problem solving are

absolutely fundamental to learning mathematics. The council has never

promoted estimation "rather than precise answers." Estimation is a

critical component to the overall understanding and use of numbers.


Organizing curriculum around a set of focal points can provide students

with a connected, coherent, ever-expanding body of mathematical

knowledge. This isn't a change, but reflects what has been the council's

commitment to the teaching and learning of mathematics for more than 80



Francis (Skip) Fennell


National Council of Teachers of Mathematics

Reston, Va.



Footnote to the Twelve Documents:


To anyone reading this it should be well known that Documents 1 and 2 are the backbone of the current debate, for they are the stated policy of The National Council of Teachers of Mathematics (NCTM) on the subject of school mathematics curriculum, and #1, the 1989 “Standards”, is the genuinely seminal document.  The influence of the Standards was manifested over the following five to ten years by the textbooks and “programs” for elementary, middle and high schools written in their wake.  Elementary school mathematics doesn’t usually feature textbooks as such, but the published program material is given to the teacher, who has a book with exercises and other materials that can be reproduced and distributed to the students to take home or use in class; hence the word “program” rather than “textbook” in such cases, though “program” can apply even in cases where textbooks used by students exist, and at higher grade levels.  However presented, the math programs written to accord with the 1989 Standards all intend to inculcate a certain manner of teaching and learning (often called “constructivist”) in addition to the mathematical information  (“content”) to be learned.


In 2006 some of the more widely used reform programs, all written under NSF grants though published and sold commercially, books and materials continually being revised under subsequent NSF grants, are:


For primary grades,  Everyday Math and the TERC Investigations (sometimes merely called “TERC”);

For the middle schools, CMP (the Connected Math Program; and

For high school,  Core-plus and IMP (the Interactive Math Program. 


There may be a few reform programs that outsell these – I don’t really know the sales figures – but these are known to me as having generated the major outbursts of parents’ (and mathematicians’) indignation over the period since the 1989 Standards were published.  Yet the parents who have carried their objections to their school board meetings and their local newspapers have, except rarely, never heard of, let alone read, NCTM’s Standards and PSSM, the founding documents lying behind those programs, and only when they really began to engage their local school systems in debate over changing the curriculum have they become aware of the linked interests of NCTM, the publishing  industry, and, ultimately, national professional educational bureaucracy, aided by the National Science Foundation, that has been denying to their children what they expect of school math instruction.


          Two such parents’ advocacy groups have become much more than that since their founding, in that they operate web sites crowded with manifestos, statistics, opinions, reprinted newspaper and magazine articles and so on in support of their cause.  They are “Mathematically Correct” in California, with web site and NYCHOLD in New York City, with web site  There is much overlap in the things posted on these two, and cross-references, and they both espouse the same anti-“reform” point of view, but neither is superfluous and both serve as sources of material for smaller, local groups, mainly formed since about 2000, when disaffected parents in such places as Penfield, NY and Bellevue, WA have, in the course of their local fights with their local school districts, discovered they had company in other parts of the country. 


Most if not all of the more local uprisings take place in affluent suburbs of cities where the parents are by no means ignorant of mathematics and its uses.  Penfield is a suburb of Rochester, home of Kodak and Xerox; Bellevue, next to Seattle, has Boeing; Palo Alto is the intellectual center of “Silicon Valley”; and Montgomery County, Maryland is across the river from D.C.  These towns now have web sites of their own (Palo Alto’s was in fact the pioneer), detailing local problems and local education news but with internet links to relevant materials found on the two major sites, links that the local teachers and parents can use to educate themselves concerning the wider battle in whatever degree they have time for.  The material contained on the two major sites, from book reviews of reform math programs to advice on the political strategies of local education, is often, indeed usually, written by mathematicians and teachers, from knowledge and experience beyond what a local parents’ group could bring to its campaign unassisted.


Only in California, where HOLD (for “honest, open, logical debate”), the pioneer group in Palo Alto, soon followed by Mathematically Correct in San Diego, managed to bring the full weight of California’s education law, via extraordinary authority it grants to the California Board of Education and its subsidiary committees, onto their side – the anti-NCTM side – of the “math wars”, have local parents’ groups had any substantial success in changing the direction of math education in their districts.  Not even in New York City’s famed District 2, where with the aid of NYCHOLD they have gathered a most impressive team of mathematicians from CUNY and NYU to argue their case, first to the District authorities and then to the councils of the city itself, all without effect.


          A major obstacle to all such efforts has been the continuing financing, by the National Science Foundation’s division of Education and Human Resources (NSF-EHR), of the further spread of the reform curricula, far beyond what the publishers could have achieved by mere advertising, and certainly not the intrinsic values of their books and materials.  The NSF participation in the commercial success of the “reform” programs has been via the major schools of education, whose professors have applied for and received multi-million dollar grants for programs called “systemic initiatives” in earlier years and “math-science partnerships” (“MSTs”) more recently. 


A typical MST, entitled Deepening Everyone’s Mathematics Content Knowledge:  Mathematicians, Teachers, Parents, Students, & Community, is described in its abstract at the NSF site

Under the terms of this grant, a small team of professors in the Graduate School of Education of the University of Rochester cooperates with several small school districts in the Rochester, NY area “to develop effective ways to foster the mathematical content knowledge necessary for a successful implementation of reform mathematics curricula.  The three suburban Rochester districts are in various stages of the adoption and implementation of curricula while the rural districts are in the early stages of a curricular reform process.” 


The grant money is used to pay teachers’ coaches, to buy curricular materials, to reimburse professors and teachers for the expense of travel to conferences, to invite visiting lecturers and to help defray the University’s costs in having a school of education that can do such things.  It – and far from incidentally – makes it attractive to the school districts themselves, financially attractive, to adopt the programs being urged by the education professors, for they consequently obtain at no cost to the school district the services of these education professors in educating or re-educating the participating teachers (“professional development”, this is called) in both mathematics and pedagogy, all as a byproduct of showing them how to use these particular programs, which they are all assured, by both grant-holders and publishers, are the very finest of research-proved, “Standards-based”, modern math curricula for their children. 


The publishers’ salesmen sometimes like to call them “NSF-approved”, or “NCTM-approved”.   Certainly the acronyms “NSF” and “NCTM” appear largely in the publishers’ printed leaflets, flyers, and web-page descriptions, some of which are used by school districts to distribute to parents of children who will be affected by a new adoption of one of these  programs.  The CMP pamphlets, for example, offer easy-to-read graphs showing the improvement in math skills certain other districts have enjoyed, which have adopted CMP in earlier years. In strict truth, however, NSF does not formally approve the programs written with its financial aid, and NCTM says it never approves or disapproves any programs whatever.  A matter of principle.  However, see for a brief account of  an occasion, during a public dispute seven years ago between NCTM and its present adversaries, when NCTM found it convenient to violate this principle.


          Also in strict truth, there is no body of valid research literature showing that any of these programs is an improvement on any competing, “traditional”, program. (See, and pages 2 and 3 that follow, which form the Executive Summary of the Mathematical Sciences Education Board’s study, On Evaluating Curricular Effectiveness:  Judging the Quality of K-12 Mathematics Evaluations, published by the National Academies’ Press in 2004.  In the full report is given detailed evidence on the absence of such research literature, despite the blizzard of publishers’ propaganda that would have you believe these programs have a valid claim to superior performance.  In the time of these programs numerous mathematicians have pointed out the visible absence of good mathematics contained in them, but their advocates have cited studies showing that despite the absence of content (something merely denied) the programs produce good results.  It is now known that these studies, which have been so persuasive to those already persuaded, and so unpersuasive to the parents of children undergoing the reform programs, should in all honesty not be used against the latter.


Ralph A. Raimi

11 December 2006