From:
Stanley Ocken, Professor of Mathematics, The City College of CUNY

To: Members of the Board of Education and other Interested Parties

Subject: Implications of CSD 2 mathematics programs for the new

Upper
East Side High School

Date:
December 12, 2001

At its December 12th meeting, the Board will consider the Chancellor’s
Resolution to establish in Community School District 2 (CSD 2) a new high
school that is supposed to offer a “challenging college preparatory program,”
according to the description in Item 16. **I am writing to address the
question of whether mandated K-11 mathematics programs in CSD 2 offer
appropriate preparation for Advanced Placement and college courses in calculus,
statistics, and physics. In my view, the answer to this question is an
unequivocal NO. **In support of this conclusion I shall draw on my experience
as Professor of Mathematics at the City College of New York since 1971, close
examination of the CSD 2 TERC *Investigations* K-5 curriculum, and
careful study of the mathematics education literature that has supported
National Council of Teachers of Mathematics (NCTM) Standards-based curricular
reform in recent years.

My analysis is based on the role of pre-algebra and algebra in the mathematics
programs that are being implemented in CSD 2. Algebra, the manipulation and
solution of expressions and equations involving numbers, equations, and
variables, is the fundamental tool of calculus. In turn, calculus is a
prerequisite for careers in engineering, science, architecture, medicine, and
computer science. The cohort of students pursuing these careers is a
significant part of the college population.
Unfortunately, American high schools have a poor track record of
ensuring that their students graduate with algebra skills adequate for success
in college mathematics. Indeed, during the thirty years I have been
teaching at

In recent years, my students’ skills, as well as those of my colleagues at
other branches of CUNY, have shown even further deterioration. As a result, it
has become extremely difficult to teach and to test my calculus students in a
meaningful way without failing a large portion of the class**. The grim fact
is that students who enter college with inadequate basic algebra skills seldom
make up for the deficiencies imposed on them by their earlier mathematics
education.** **Their ladder of opportunity to pursue rewarding careers is
cut off at the base. Among the groups most severely impacted are the
children of non-English speaking immigrants, children who traditionally have
entered the mainstream of American society by pursuing careers that emphasize
mathematical, as opposed to linguistic, competency.**

New mathematics programs being tested in CSD 2 are of course not to blame for
current college students’ poor algebra skills. Unfortunately, indiscriminate
implementation of these curricula will significantly exacerbate students’
algebra deficiencies. Since these curricula are based on the 1989 NCTM
Standards document, they focus principally on enhancing students’ quantitative
literacy skills rather than on facilitating development of the formal
mathematical skills crucial to success in calculus. Although the anti-formal
bias of the 1989 NCTM standards was attenuated somewhat in their Year 2000
Revision, it is unclear to what extent, if at all, CSD 2 programs such as TERC,
CMP, and ARISE will be modified to incorporate a greater emphasis on formal
skills.

The three curricula just mentioned are among those alluded to as follows in the
draft document of the report issued recently by a commission of experts
convened by Schools Chancellor Harold Levy and chaired by CUNY Chancellor
Matthew Goldstein:

“Whenever an emphasis is placed on ensuring that applications are made to ‘real
world’ situations …less emphasis is placed on arithmetical or mathematical
ideas and the formal, abstract contextual settings sought particularly by
students who will go on to become scientists, engineers, mathematicians,
computer scientists, physicians, and educators of mathematics.”

“Despite their many strengths, *the NCTM standards do not contain the
rigor, algorithmic approach, formal methods, and logical reasoning which are
required of this small ** but
critically important portion of the population.* [Emphasis added]”

The Commission Report further asserts that quantitative literacy skills should
be developed as a supplement to, rather than as a substitute for, pre-calculus
skills. In my view, this warning will *ipso facto* be ignored by any
school district that forces students into programs based on the 1989
NCTM-Standards.

Among the new CSD 2 programs, the TERC K-5 curriculum exhibits the most
egregious deficiencies. Although this curriculum sometimes achieves its goals
of engaging students and providing interesting in-class activities, it fails
utterly to provide the firm bedrock of hands-on computation and symbol
manipulation that serve as a foundation for students’ pre-algebra and algebra
skills. One chilling fact:** fewer than twenty computations in the entire set
of TERC K-5 student materials require the use of the multiplication facts 6 x
6, 6 x 7, 6 x 8, 6 x 9, 7 x 7, 7 x 8, 7 x 9, 8 x 8, 8x 9, 9 x 9, either as
standalone problems or as part of multi-digit multiplication problems. The
bias toward easy rather than hard multiplication facts is pervasive and cannot
be due to chance.**

The TERC teacher manuals offer scenarios in which children develop their own strategies for multi-digit multiplication and division problems. Unfortunately, nearly all of the demonstration problems are carefully chosen to be amenable to trial-and error or other non-systematic solution methods. The showcased methods are in fact unsuitable for solving truly representative problems.

Having referenced one misguided argument in the mathematics education literature, I feel obligated to note others as well:

There is an expanding body of misleading data and obfuscation in much of the “research” that is marshaled to support NCTM Standards-based curriculum initiatives. Among these is the questionable relevance of scores on standardized testing instruments, including those used

NCTM Standards-based curricula consistently claim to enhance students’ conceptual understanding, a goal typically touted as a revolutionary advance over traditional adherence to “blind rote manipulation.” This is nonsense. When NCTM curricula such as TERC’s

It cannot be overemphasized that the essential methodology of college mathematics and science is to analyze a real-life problem, then to represent it by symbolic expressions, and finally to simplify and transform those expressions into a solution to the original problem. Successfully implementing this agenda requires a tremendous repertoire of purely technical skills that must become second nature and that can become so only with intensive practice. For example, students must be trained to recognize legitimate patterns and processes of symbol manipulation and to explain why specious manipulations, such as canceling the

I applaud the Board’s intent to establish a new high school on the

It may be desirable to guide some students with special needs toward a curriculum that de-emphasizes preparation for college-level mathematics, although, in my view, such tracking should be undertaken only as a last resort. However

Professor of Mathematics

The City College of the City University of New York

stevock@aol.com 212-650-5139