Winter, 2000

Preliminary draft of points that will be made by the group of NYU and other math faculty participating in the Math Forum sponsored by Community School Board 2 in Manhattan.

Why we question the NCTM-approved programs being promoted by District 2

We believe that the math programs currently being put in place by District 2 suffer from severe deficiencies, and in the long run will work to the detriment of students and families in the district.  All of these programs—TERC, CMP, ARISE, and others—are based on a single radical educational philosophy and all share the following deficiencies.

1. They are Unbalanced.

Any reasonable program of math instructions should achieve a balance between

·        mastery of basic skills and extensive practice in problem solving techniques,

·        acquiring a firm grasp of mathematical concepts,

·        direct instruction and guidance by teachers knowledgeable in their subject, on the one hand and

·        individual and group activities involving discovery-based learning, on the other.

The programs being implemented in District 2 (TERC and CMP) in their pure forms focus almost entirely on having children learn math by discovery in group activities. Practice on basic problem solving skills is actively discouraged, and little meaningful homework is assigned that might strengthen those skills. The use of supplementary materials covering these topics is strongly discouraged, though a few principals allow supplementation, apparently in defiance of District orders. Teachers are strongly discouraged, and in some schools forbidden, to “instruct” or actively guide the discussions resulting from group activities.  Students are left with poor basic skills, and without a coherent understanding of what they have done, or the larger picture into which their activities fit.

All this is in accord with standards for new K-12 math curricula issued in 1989 by the National Council of Teachers of Mathematics (NCTM). Those standards firmly embraced the philosophy that the role of the teacher is not to teach, but to act as a mere “facilitator” for group efforts in which students are supposed to “discover math for themselves”. They strongly downplayed work on basic math skills and paid little attention to development of algebraic concepts. The District programs, based on the 1989 Standards, reflect those attitudes.

The 1989 NCTM Standards were substantially revised in year 2000, in response to rising criticism of the 1989 Standards by professional mathematicians, college teachers, and education authorities in states such as California. The year 2000 NCTM Standards placed much more emphasis on mastery of skills, but {\it none of the programs now in place reflect those changes.} Spokespersons for NCTM attempt to justify the existing programs by insisting that the original standards were “misinterpreted”, and that the revised standards merely “clarify” what was intended. A comparative reading of both Standards suggests otherwise.  The 1989 Standards were very clear in their de-emphasis of basic math and algebra skills, and in their zeal to replace traditional programs with new ones focused entirely on “learning by discovery”.

The programs TERC, CMP, ARISE being introduced in District 2 were created in the spirit of the 1989 Standards, and are inconsistent with the more moderate and realistic revised year 2000 Standards. Our District is being saddled with a set of programs that are already outmoded because they embody the serious faults of the 1989 Standards.

 2. They are Unworkable. The educational philosophy on which the District 2 programs are based appears to be fundamentally flawed.  These programs are unable to meet their own objectives, let alone the larger objective of providing sound training in math.

A certain level of discovery based learning in group activities is desirable, and can help students understand math concepts and make them feel more comfortable about math. However, this mode of learning is painfully slow. After all, it took the best thinkers of their times centuries to get algebra straight; do we want to have students spend 100% of their time re-inventing the wheel?

Typical District 2 programs consist of modules, between 9 and 11 units

to be covered in a year. Because of the time required to “rediscover

mathematics”, students do not in fact get to complete their yearly

programs. Based on our own interviews with in-service teachers in the

District it appears that

No teacher in any school has ever managed to cover more than 60\% of the units in the 1-year packages they have been given. The TERC and CMP programs cannot meet their own professed goals, because the pace of these activities is so very slow.

That means: each year, students fall about 40\% behind the materials they are supposed to cover. Moreover, what does get covered varies from school to school. Thus when students move from elementary to middle school, or to high school, there is no assurance they arrive with knowledge of any particular set of math concepts. The result can only be chaos.

All this should not surprise anyone who has been involved in such activities.  The remarkable thing is that proponents of the District 2 programs—which in their pure (unsupplemented) forms are 100\% group activity—have never acknowledged this gap between their professed goals and reality.  We doubt that these programs can ever be made to work without radical restructuring and a move away from instruction based on a single educational ideology.

3. They are Inconsistent with NY State Standards.

A grade-by-grade comparison of the New York State math standards, on which the new Regents A and B  exams are based, shows that TERC and CMP give short shrift to more than 30\% of the topics specified in those Standards—especially those related to basic math concepts and skills, and competence in algebraic reasoning. That would be bad enough, if teachers using the TERC and CMP programs could in fact cover the materials in each grade level program. They have not been able to this much in real life.

Failure to cover the mandated TERC and CMP materials is hardly the teachers’ fault, as we have noted in 2.Developers of the NCTM-approved programs have never acknowledged fundamental problems inherent in curricula that focus entirely on “discovery” projects.

It is impossible to master mathematics without serious attention to content and practice of skills.The deficiencies of the “constructivist” programs in these areas will have a serious

impact on families who hope to see their children advance through educational opportunities. We can expect children subjected to these programs to be at a disadvantage in: (i) statewide tests (grade 4) which are used for admission to desirable middle schools, (ii) citywide placement tests (grade 7) for admission to desirable high schools, (iii) the citywide specialized science high school entrance exam, (iv) achieving well in the NY State Regents tests, (v) succeeding in high school AP courses, (vi) performing well in college-entry SAT tests.

4. They Promote Inequities. About 75% of all high school students go on to college of some sort, even if not to a four-year program, and math is—after literacy—the most troublesome entrance hurdle.  In college they will face quantitative math and science requirements in a vast number of programs. District 2 curricula downplay basic problem solving skills and mastery of math concepts needed for success. Because group work proceeds so slowly, and because so much time is devoted to repetitive “math game” projects, these programs cover fewer math concepts than earlier programs did, and their coverage of these concepts is often superficial. As a result, many parents in the district have been resorting to extensive tutoring at their own expense. The District 2 programs will have a devastating effect on students from low-income families who cannot afford these extras.

Parents in a public school system deserve a level playing field, regardless of their economic circumstances.No parent should have to go to great expense to compensate for the built-in deficiencies of the math programs being promoted in District 2.

The inadequacies of the District 2 curricula are widening the gap between haves and have-nots.


5. They Promote a Failed Ideology. In the early 1990s TERC, CMP, and several other NCTM-approved “constructivist” math programs were implemented on a large scale in the state of California. There they proved such failures that in 1997 all NCTM-approved programs were decertified and new statewide math standards were formulated, this time with input from concerned mathematics professionals as well as members of the education establishment. Yet in District 2, and throughout New York City, school authorities seem determined to implement the same failed programs as if nothing ever happened.

In California the failure of these programs was evidenced by a steady decline in statewide math test scores, and by a dramatic increase in the need for math remediation among students entering the state college system. Furthermore, well documented nationwide studies extending over several years have demonstrated that direct instruction—allowing teachers to teach instead of relegating them to the role of “facilitators” of group investigations—is by far the most effective means of improving math skills, especially among low-income and minority K-12 students.[1]

6. They Ignore Proven Alternatives to NCTM-Based Curricula.The recent international TIMSS study of math instruction, involving over 500,000 students worldwide, demonstrated that by the time they reached 8th grade students from Singapore and Japan rate highest in math ability, while the U.S. students ranked 28th among 41 countries. By 12th grade, U.S. students ranked near the bottom, 19th out of 21 nations surveyed at that level, with performance comparable to those found in underdeveloped countries.  Proponents of the NCTM-approved programs claim that their curricula are closely modeled on the programs used in Singapore and Japan.  They are not. In fact, the NCTM-approved programs have distorted key tenets of the Asian programs beyond recognition, by focusing exclusively on just one aspect of those programs—discovery based learning by students working in groups.


The Singapore, Japanese, and European K-12 math curricula recognize the primary importance of a skilled teacher in math instruction. Their course materials strike an excellent balance between work on problem solving skills, direct instruction to convey math concepts, as well as group investigations designed to illuminate those concepts. As a result the Singapore materials, at every grade level, are far superior to those in the programs now being promoted in District 2, and student performance is far better. How could it be otherwise, when the District programs so thoroughly denigrate the role of the teacher?

The Singapore curriculum is particularly interesting. Singapore has a system of universal education, in English. Rich and poor alike were included in the TIMSS survey that placed Singapore first in the world. This curriculum consists of a series of English language texts and workbooks (including group projects), one for each grade level, that have been refined through more than a decade of use.  These materials are issued in inexpensive paperback editions, and are commercially available.[2] Why have these programs been so ignored by American educators?


1.Project Follow Through was conducted from 1967-1995, initially as an adjunct to the Head Start program. These studies were large scale and statistically sound, involving 700,000 students nationwide, and they compared direct instruction with various programs based on the same

“discovery” philosophy employed by TERC and CMP. Direct instruction methods proved clearly superior to all “constructivist” modes of instruction examined in this study. These studies have been largely ignored by the education establishment which showed, and continues to show, little interest in results counter to their “constructivist” prejudices

2.Texts for a single grade level cost about $20. Further information about the texts can be found on the internet: 


We have carefully examined videotapes of eight grade math classes in Japan and Germany, created as part of the TIMSS study. Proponents of NCTM-approved curricula often point to these tapes, claiming that they demonstrate the similarity between those successful programs and the “constructivist” programs being promoted in the United States. However, close examination of those tapes reveals startling discrepancies between the Asian and European programs, and those being implemented in the U.S.

The tapes show very skilled teachers at work; all have a firm grasp of the mathematical concepts they are teaching, and their classroom presentations are superb. On the tapes one sees those teachers spending more than 50% of their time in direct instruction,. Although classes involve a certain amount of group and individual effort on the day’s project, classes always begin with the teacher reviewing basic skills and concepts needed to solve the problems of the day; teachers actively intervene in guiding group discussions; finally, they spend considerable time at the end providing an overview of what has been accomplished, and reviewing the math concepts illustrated by the day’s project.

Teachers in the successful programs are hardly passive “guides on the side”—the role to which teachers have been confined in virtually all NCTM-approved programs.


The cornerstone of the constructivist philosophy of education, upon which the District 2 programs are based, is that more meaningful learning is supposed to take place when students teach each other in small peer-led groups and thereby “construct” their own knowledge. It is absurd to expect students to invent all of mathematics on their own, unaided, through the exclusive use of time consuming and wasteful group projects. The NCTM-approved programs in District 2 are a recipe for disaster, although this may not become clear until a generation of students has failed.  That was the pattern in California, and will be the pattern here unless something is done to modify these programs and acknowledge their flaws.

District 2 could begin to address these issues by recognizing that there may be serious problems with the “constructivist” curricula they have so ardently been promoting. What is needed is a meaningful dialog with teachers and concerned citizens about modifications and alternatives to these flawed curricula.

Many things could be done. As a first step, the best of the projects in the present curricula—those with substantial math content—could be kept, while the rest are discarded in favor of supplementation that restores some balance among the goals listed at the beginning of these remarks.  California has already endured the effects of the unadulterated constructivist programs (including TERC and CMP), and state authorities have now developed approved lists of more balanced instructional materials. We could take advantage of their experience by examining the texts on the California approved list. Portions of the English-language Singapore Curriculum might also be a useful resource. The Singapore texts are currently being used by a few schools in New Jersey and Maryland, and it would be interesting to learn more about their experience with these materials. (The entire Singapore text series is inexpensive and commercially available, and a new edition is being prepared that conforms to U.S. grade levels.) There is no lack of resources for supplementing the present programs, if the District has the will to acknowledge the deficiencies of those programs.

Finally, the most important step would be cease confining teachers to the role of passive “facilitators” of unworkable programs.

Let knowledgeable math teachers exercise their initiative in getting math concepts across, and shift the District’s focus toward getting more such teachers into our schools.