New
Battles in the Math Wars

Wilfried Schmid

Harvard Crimson, May 4, 2000

What is 256 times 98? Can you do the multiplication
without using a calculator? Two thirds of Massachusetts’s 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! 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 computations.

Mathematicians are not convinced. By all means liven
up the textbooks, make the subject engaging, 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 them!

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
of the skeptics 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: the New York Times, in a front-page article on April 27, declares:
"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 to 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 basics".

The current fight echoes an earlier argument, over
the "New Math" of the Sixties and Seventies. 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
the Singapore approach were adopted in my daughter's school!

The curriculum, of course, is not the only reason
for Singapore's success, nor 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: if you
know how to teach, you can teach anything! It will take a heroic effort - by
math educators and mathematicians - to change the entrenched culture of teacher
training.

Wilfried Schmid is Dwight Parker Robinson Professor of Mathematics. Earlier this year, he served as mathematics advisor to the Massachusetts Department of Education.