NYC HOLD Honest Open Logical Debate on math reform
Are our school’s math programs adequate?
Experimental mathematics programs and their consequences
Professor of Mathematics
Courant Institute of Mathematical Sciences
I would just like to introduce myself, so members of the audience will know the issues I have been concerned with when we come to the Q/A period. During the past year I have spent quite a bit of time
Examining curricular materials for various NCTM-based middle and
high school programs -- CMP, ARISE, and to some extent the IMP
which is slated to go into effect throughout the
in Fall 2000.
Conducting numerous discussions and interviews with teachers
in District 2 and elsewhere in NYC
Participated in many discussions with colleagues at the Courant Institute
who have become concerned about the NCTM based math programs being
introduced in NYC schools.
Tonight I would like to comment on some of the flaws I see in these constructivist programs.
1. I've had a lot of experience with curriculum development, and that
includes serious efforts on programs for entering freshmen who are
not necessarily going to be math majors. Whether we're talking about
core programs in math and science literacy for liberal arts majors, or
regular math courses for science majors, business majors, and premed students, I have found that the single greatest obstacle to success for entering college students -- even in courses for non-majors -- is lack of proficiency in algebra. That means: being able to DO it, not just talk about it.
Most NCTM based programs I reviewed strongly downplay symbolic manipulation skills (which lie at the heart of real mathematics) in favor of ad hoc ``visualization'' techniques, and lengthy unguided projects in which students are supposed to ``discover math principles for themselves''. Now there is something to be said for including in a math curriculum some projects in which students are encouraged to ``learn by discovery'' -- I have often done this the courses I have developed. The problem is that most NCTM based programs being suggested for use in NYC are quite extreme in their emphasis on the process of “discovery'', at the expense of mastery of basic content and proficiency in basic skills. The NCTM based programs are quite unbalanced in their emphasis, and as a result are totally inadequate as preparation for eventual college level courses.
In a recently completed review of math programs being implemented in NYC, the Levy Commission conceded that the high school programs, such as those being implemented throughout the Bronx in Fall 2000, were not an adequate preparation for college level work, and went on to suggest that “choice'' be allowed beginning in grade 9, so college bound students might take courses with stronger content.
THAT IS TOO LATE! The underpinnings of proficiency in algebra are
laid in middle school, and even at the elementary level. For example,
learning to work with fractions, as fractions, is the precursor to algebra
It is not enough to deal with them as numbers punched up on a calculator, or
by comparing lengths of paper strips
Doors will be closed to students who aspire to college level work unless students are allowed to elect courses with stronger content, beginning in middle school at the latest, but preferably throughout the early grades.
Next I would like to comment on the view from the trenches: what do teachers think? In my interviews I found many math teachers willing to speak, as long as their anonymity was assured. I spoke to teachers at the elementary, middle and high school levels. Many complained that the NCTM based courses tend to be quite “dumbed down''. Here are some quotes:
``I've been teaching math for a long time, and am struck by how much less
math actually gets covered under the new programs, compared to what got
accomplished just a few years ago''
``Weaker students may benefit from these programs, but the effect on the
stronger students is going to be disastrous.”
ones who will really benefit from these programs are the
Kaplan tutorial centers -- for them it will be a godsend!''
One teacher described an hour-long training session in which kids colored an array of numbers. She asked the trainer, what was the point of spending so much time on this? What math concepts did it lead to? The reply:
`”Concepts don't matter. What counts is how the kids feel about it.''
I can't think of a better illustration of what the phrase ``dumbed down'' might mean.
Skilled teachers' hands are being tied by overzealous administrators and NSF-funded ``trainers'', who know about pedagogy but have very little knowledge of math CONTENT. Indeed some of the more zealous proponents of these constructivist programs have claimed that one doesn't really need to be proficient in content in the early grades, one simply has to know how to teach. Experienced math teachers are chastised -- even threatened with reprisals -- if they deviate from the mandated constructivist scripts. As one teacher put it:
``There is no longer any classroom autonomy. Teachers are being treated
as if expertise in one's subject, and personal teaching skills, are
irrelevant. Everyone is being forced to work from the same fairly
This atmosphere of micromanagement and intimidation is alienating many of the experienced teachers. Older, experienced teachers are contemplating
early retirement; younger ones look to other locations where ability to
teach math content is appreciated, and probably better paid.
The constructivist philosophy is flawed at its core. Educationists posit that really meaningful learning takes place when students teach each other in small peer-led group discussions, with teachers confined to roles as mere “facilitators'' of this process. It is absurd to expect students to invent major portions of math on their own, through extensive time consuming group projects. Discovery based learning has a valid place in the classroom, but the constructivist programs are extremely biased in this direction, and progress is very slow. In fact, many teachers report that:
``In our school, using program xxx, no teacher has ever managed to cover
more than 60\% of the year's material.''
The District replies ``We never expected to cover all of it''. Yet the District has repeatedly failed to offer guidance on which 60% is to be covered. Furthermore, if these curricula are to provide an adequate preparation for the State Regents exams, one would have to cover all of the course materials, and that is impossible.
It is time to acknowledge the serious flaws in the NCTM based programs
being promoted so zealously in District 2, so we can get on with the task of creating programs of math preparation adequate for today's world. What we really need are programs with some sense of balance, created with the involvement of mathematicians as well as educators.