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.
Furthermore, the TERC curriculum omits completely the standard algorithms
for multiplication and division, apparently heeding the frequent assertion
in the mathematics education literature that traditional algorithms are
obsolete because they produce answers more easily obtained with a calculator.
This uninformed and specious argument fails to recognize the standard
algorithms’ critical importance to students’ mathematical development, for these
algorithms encapsulate fundamental computational and algebraic ideas and
experiences necessary for children’s future success in algebra and calculus.
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 in
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 Investigations use the term
“understanding,” they often refer merely to the obvious and pedagogically
useful technique of furnishing concrete models for simple arithmetical
examples, e.g. by using fraction strips to picture fractions such as 3/4 and
2/3. Every competent parent or educator knows that this is a good way to start.
Unfortunately, a principal failing of Standards-based curricula is that
students never move beyond, and so are forced to rely on, simple models and
representations. As a result, when students confront purely symbolic
representations that are not attached to physical models, they simply freeze.
Their reaction, perhaps best characterized as “symbol shock,” is, in my
experience, a primary cause of students’ failure to succeed in college
mathematics.
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 x's in the fraction (x + 5) / (x +1), make
no sense algebraically and must be instinctively avoided. This and similar algebra
rules must be appreciated on their own terms, as referring to symbols rather
than to things. You can’t draw a picture of x!
As a professor of long standing in the CUNY system, I have never ceased to
wonder why so many students enter college seemingly endowed with the ability to
succeed at mathematics, but are subsequently prevented from doing so by their
lack of mastery of basic algebra skills. The mathematical training of our
city’s most precious resource, its children, has for too long been driven by
the faddish and demonstrably ineffective policies and perspectives of some
mathematics educators. I for one, together with many of my university
colleagues, am prepared to devote considerable effort to the true reform of
mathematics education in this city.
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, it is unconscionable for the Board of Education to consider
establishing a new high school in, or to continue promoting the mathematics
programs of, any school district that does not enthusiastically support
rigorous mathematics curricula offering all students the opportunity to pursue
mathematics-based careers.
Professor of Mathematics
The City College of the City University of New York
stevock@aol.com
212-650-5139