Date: May 4, 2000 11:27 AM
Author: Anne Wheelock
Subject: Op-Ed by W.Schmid in Harvard Crimson
This op-ed may be of interest. It's posted at
Published on Thursday, May 04, 2000
New Battles in the Math
By WILFRIED SCHMID
What is 256 times 98? Can you do the multiplication
without using a calculator? Two thirds of Massachusetts
fourth-graders could not when they were asked this
question on the statewide MCAS assessment test last year.
Math education reformers have a prescription for raising
the mathematical knowledge of schoolchildren. Do not
teach the standard algorithms of arithmetic, such as long
addition and multiplication, they say; let the children find
their own methods for adding and multiplying two-digit
numbers, and for larger numbers, let them use calculators.
One determined reformer puts it decisively: "It's time to
acknowledge that continuing to teach these skills (i.e.,
pencil-and-paper computational algorithms) to our
students is not only unnecessary, but counterproductive
and downright dangerous."
Mathematicians are perplexed, and the proverbial man on
the street, when hearing the argument, appears to be
perplexed as well: improve mathematical literacy by
downgrading computational skills?
Yes, precisely, say the reformers. The old ways of teaching
mathematics have failed. Too many children are scared of
mathematics for life. Let's teach them mathematical
thinking, not routine skills. Understanding is the key, not
Mathematicians are not convinced. By all means, liven up
the textbooks, make the subject engaging and include
interesting problems. But don't give up on basic skills!
Conceptual understanding can and must coexist with
computational facility--we do not need to choose between
The disagreement extends over the entire mathematics
curriculum, kindergarten through high school. It runs right
through the National Council of Teachers of Mathematics
(NCTM), the professional organization of mathematics
teachers. The new NCTM curriculum guidelines,
presented with great fanfare on April 12, represent an
earnest effort at finding common ground, but barely
manage to paper-over the differences.
Among teachers and mathematics educators, the
avant-garde reformers are the most energetic, and their
voices drown out those skeptical of extreme reforms. On
the other side, among academic mathematicians and
scientists who have reflected on these questions, a clear
majority oppose the new trends in math education. The
academics, mostly unfamiliar with education issues, have
been reluctant to join the debate. But finally, some of them
are speaking up.
Parents, for the most part, have also been silent, trusting
the experts--the teachers' organizations and math
educators. Several reform curricula do not provide
textbooks in the usual sense, and this deprives parents of
one important source of information. Yet, also among
parents, attitudes may be changing. A recent front-page
headline in the New York Times declares that "The New,
Flexible Math Meets Parental Rebellion."
The stakes are high in this argument. State curriculum
frameworks need to be written, and these serve as basis for
assessment tests; some of the reformers receive substantial
educational research grants, consulting fees or textbook
royalties. For now, the reformers have lost the battle in
California. They are redoubling their efforts in
Massachusetts, where the curriculum framework is being
revised. The struggle is fierce, by academic standards.
Both sides cite statistical studies and anecdotal evidence
support their case. Unfortunately, statistical studies in
education are notoriously unreliable--blind studies, for
example, are difficult to construct. And for every
charismatic teacher who succeeds with a "progressive"
approach in the classroom, there are other teachers who
manage to raise test scores dramatically by "going back to
The current fight echoes an earlier argument, over the
"New Math" of the '60s and '70s. Then, as now, the old
ways were thought to have failed. A small band of
mathematicians proposed shifting the emphasis towards a
deeper understanding of mathematical concepts, though on
a much more abstract level than today's reformers. Math
educators took up the cause, but over time, most
mathematicians and parents became unhappy with the
results. What had gone wrong? Preoccupied with
"understanding," the "New Math" reformers had neglected
computational skills. Mathematical understanding, it
turned out, did not develop well without sufficient
computational practice. Understanding and skills grow
best in tandem, each supporting the other. In most areas of
human endeavor, mastery cannot be attained without
technique. Why should mathematics be different?
American schoolchildren rank near the bottom in
international comparisons of mathematical knowledge.
Our reformers see this as an argument for their ideas. But
look at Singapore, the undisputed leader in these
comparisons: their math textbooks try hard to engage the
students and to stimulate their interest. In early grades,
they present mathematical problems playfully, often in the
guise of puzzles. Yet the textbooks are coherent,
systematic, efficient, and cover all the basics--worlds
apart from the reform curricula in this country. How I
wish Singapore's approach were adopted in my daughter's
The curriculum, of course, is not the only reason for
Singapore's success, nor is it even the most important
reason. The teachers' grasp and feeling for mathematics:
that is the crucial issue, already for teachers in the early
grades. Here, it turns out, many of the reformers agree
with the critics. Teacher training in America has
traditionally and grossly stressed pedagogy over content.
The implicit message to the teachers is: If you know how
to teach, you can teach anything! It will take a heroic
effort--by mathematicians and math educators--to change
the entrenched culture of teacher training.
Mathematicians do not want to invade the educators' turf.
We are not qualified to do their work. Yet we are
qualified as critics of reforms in math education. We
should call attention to reforms we see as well meaning,
but hectic and harmful. Most music critics would not do
well as orchestra musicians. They do have acute hearing
for shrill sounds from the orchestra.
Wilfried Schmid is Dwight Parker Robinson Professor of
Mathematics. Earlier this year, he served as a
mathematics advisor to the Massachusetts Department of