From: Stanley Ocken, Professor of Mathematics, CCNY
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
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 investigation of the CSD 2
TERC Investigations K-5 curriculum, and careful study of
the mathematics education literature that has supported
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 career's 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 City College,a majority of students
in the calculus track have exhibited profound weaknesses
in their algebra preparation.
In recent years, my students' skills have shown even
further deterioration and, as a result, it has become
extremely difficult to teach and to test calculus students
in a meaningful way without failing a majority 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 they are based on the 1989 NCTM
Standards document, these curricula focus principally on
enhancing students' quantitative literacy skills rather
than on facilitating developmentof 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 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 students9pre-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 witha calculator. This ignorant
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 New York City and New York State, where
standardized tests in recent years have shown an alarming
tendency to de-emphasize or to omit entirely questions
that appropriately evaluate students' pre-algebra and
algebra skills.
The NCTM Standards-based curricula consistently
claim to enhance students9 conceptual understanding,
a supposed advance over traditional adherence to blind
rote manipulation. This is nonsense. When NCTM curricula
use the term understanding, they 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 curricula such as TERC 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. This reaction,
perhaps best characterized as symbol shock, is, in my
experience, a primary cause of students9 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 intense practice. Most importantly, students must be
trained to recognize legitimate patterns and processes of
symbol manipulation and to know why specious manipulations,
such as canceling the x's in the fraction (x + 5) / (x+1),
make no sense algebraically and must be avoided.
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, only to be 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 faddish
and demonstrably ineffective perspectives in mathematics
education. 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 Upper East Side. However, the proposed
high school, and existing CSD 2 High Schools as well,
will fail in their obligation to provide students with a
springboard to careers of choice if mathematics curricula
such as TERC, CMP, and ARISE are the only options offered
to CSD 2 students. In my view, such curricula provide
inadequate preparation not only for AP and college courses
in calculus, statistics, and physics, but also for some
of the SAT II Subject Tests in these subjects that are
frequently required of high school students who intend to
major in the sciences in college.
It may be the case that some students should be guided
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 any school
district that does not enthusiastically support rigorous
curricula offering all students the opportunity to pursue
mathematics-based careers.
Stanley Ocken
Professor of Mathematics
The City College of the City Universityof New York