in The City: A View from the College Classroom
Stanley Ocken & Robert Feinerman
Brown’s recent article, “Math Adds up at CCNY Teacher Training
Program,” [Education Update, Nov. 2002] paints a warm picture
of Prof. C. T. Fosnot’s Mathematics in the City teacher
training program. We are writing as professors of mathematics,
engaged in both teaching and research at the City University of
New York. We are profoundly disturbed by the philosophy of mathematics
education proffered in that article, by the weak content of many
curricula that have been implemented on the basis of that philosophy,
and by the effect of both on students who will be entering college
mathematics classrooms in the next decade.
The curricula we refer to, developed during the last decade and
now being implemented nationally, are those based on the vision
of the 1989 NCTM Standards. That vision (later revised in the
Year 2000 update) called for a de-emphasis of the formal algorithmic
and algebraic component of the curriculum, a call that was heeded
in the most extreme way by curriculum developers and graduate
schools of education. Concerning such curricula, the draft report
of the Commission on Mathematics Education, convened by former
New York City Schools Chancellor Harold Levy, asserts: “despite
their many strengths, the NCTM standards do not contain the rigor,
algorithmic approach, formal methods, and logical reasoning which
are required” of students who will go on to become scientists,
engineers, mathematicians, computer scientists, physicians, and
educators of mathematics.”
We fully concur. Judging by their product, the developers of NCTM
standards-based curricula were motivated by sentiments expressed
in Ms. Brown’s article, each of which is italicized below and
followed by our reactions.
are submerged in a mathematics environment where math is not a
Mathematics used in college courses is formulated in a difficult
symbolic language. To succeed in those courses, students need
twelve years of carefully structured instruction in order to learn
the language fluently and to use it to solve hard problems. Those
who lack fluency will be shut out of careers listed above, with
the greatest negative consequences for children of immigrants,
a group whose entry into the mainstream of American society has
historically been facilitated by demonstration of mathematical
rather than linguistic competence.
Children need to understand the meaning behind the math.
In much of mathematics education literature, the “meaning” referred
to is provided by reference to a concrete or pictorial model.
That’s not enough. If the K-12 curriculum does not offer a coherent
path for moving from concrete/pictorial to symbols (and many NCTM-inspired
curricula do not) then students will be totally unprepared for
advanced mathematics courses. The world’s highest performing students
use the Singapore curriculum, whose driving principle is that
children must begin with “the concrete and pictorial stages, followed
by the abstract stage to enable them to learn mathematics meaningfully.”
Just understanding rules doesn’t enable you to do the math.
Understanding rules, knowing when and how to apply them, does
enable you to do the math. Students are flunking advanced algebra
and calculus courses not because they don’t understand the “meaning”
of mathematics, but rather because they are afraid and unable
to deal with symbolic expressions used to represent real-world
problems. Their failure rate will only increase if they are raised
on a K-12 diet that is deficient in algebra and mathematical formalism.
is about ongoing observation of the world around you.
Mathematics is about many things, including modeling the outside
world. Bridge design requires deep mathematics. One certainly
has to calculate, to a very high degree of precision, exactly
how long the bridge should be. Unfortunately, prominent mathematics
educators disparage the idea that math problems have “single answers.”
Bridge-builders, take note!
They also wanted to help them [the teachers] see themselves as
We are unfamiliar with the above usage of the term “mathematicians,”
which ordinarily refers to researchers at the frontiers of mathematics
knowledge who publish their work in refereed journals. The objection
here is not a quibble about redefining membership in our profession,
but rather is conditioned by the attempts of mathematics educators
to redefine the nature of mathematics itself. Fosnot’s unconditional
rejection of all of the views:
meaning can be passed on to learners via symbols; that whole concepts
can be broken into discrete subskills; that concepts can be taught
out of context;” while formulated as a vision of pedagogy, forces
a redefinition of mathematics content diametrically at odds with
the needs of college mathematics and science students.
Our nation’s K-12 classrooms need not mathematicians, but rather
mathematics teachers who are knowledgeable in content as well
as pedagogy. Revitalizing the teaching workforce can
be accomplished only when curriculum developers recognize their
responsibility to prepare students for the rigors of college mathematics,
and when university mathematics departments play a substantive
role in teacher training and professional development programs.#
Stanley Ocken is Professor of Mathematics at the City College
of the City University of New York, where he is working, together
with preservice teachers, to develop a text and software resources
for their introductory math content course.
Robert Feinerman is Professor of Mathematics and Computer Science
at Lehman College of CUNY, Department Chair, Chair of the CUNY
Math Chairs, was CUNY’s representative to Chancellor Levy’s Commission
on Mathematics Education and is a member of Community School Board
#10 in the Bronx.
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