The Math and Science Wars
A Northeastern Faculty Member's Salvo from the Front
By Alan Cromer
In December 1997, the California State Board of Education published
a 100-page draft mathematics standard for kindergarten through grade twelve.
This document contained such statements as, "By the end of fifth grade,
students . . . [should be] proficient at multi-digit multiplication and
division of decimals and negative numbers, including long division with
multi-digit divisors." This rigorous-sounding standard drew unprecedented
condemnation from Luther Williams, assistant director of the National Science
Foundation (NSF) for education and human resources, who wrote to the president
of the California Board of Education that "the Board action is, charitably,
shortsighted and detrimental to the long-term mathematical literacy of
children in California."
He went on to say, "The wistful or nostalgic 'back-to-basics' approach
that characterizes the Board standards overlook [sic] the fact that the
approach has chronically and dismally failed . . . the Foundation cannot
support individual school systems that embark on a course that substitutes
computational proficiencies for commitment to deep, balanced, mathematical
learning. We view the Board actions in California with grave disappointment
. . . "
Among combatants in the mathematics and science wars, ongoing in Massachusetts
and many other states besides California, the Williams letter is viewed
as a nuclear attack. Why? What is the real issue? Williams feels that the
California standard will exclude "youngsters from engaging in genuine
mathematical thinking." But this is hardly
credible, since the standard was written by mathematics professors at
Stanford University.
The standard calls for integrated mathematics to end at seventh grade
and traditional subject matter mathematics-algebra, geometry, trigonometry,
and so on-to start in eighth grade. It defines these courses in terms of
linear and quadratic equations, geometric theorems, and trigonometric identities.
It is real mathematics as mathematicians and scientists understand it.
So what is the beef? Why is the head of education for the NSF agin it?
Is the standard too high? Williams doesn't accuse it of that. He says only
that it is "shortsighted and detrimental." This is meaningless
to a noncombatant. The only clue in the text to the real issue is in his
phrase "deep, balanced, mathematical learning." The word "balanced"
alludes to the mathematics standards published in 1992 by the National
Council of Teachers of Mathematics (NCTM).
The NCTM standard, which is vigorously endorsed by the NSF, calls for
a so-called balanced approach to mathematics: "Students should have
numerous and varied experiences related to the cultural, historical, and
scientific evolution of mathematics so that they can appreciate the role
of mathematics in the development of our contemporary society." Yet
mathematics itself is devalued: although "quantitative techniques
have permeated almost all intellectual disciplines . . . the fundamental
mathematical ideas needed in these areas are not necessarily those studied
in the traditional algebra-geometry-precalculus-calculus sequence."
Here, in a nutshell, is what the mathematics war is about. One group,
endorsed by the NSF, wants to define all the math out of mathematics. The
other group struggles to maintain mathematics as math. The California standard,
which was a victory for "the traditional algebra-geometry-precalculus-calculus
sequence," brought down the wrath of the NSF.
I believe that we have got to this sad state because of the convergence
of two erroneous ideologies. One, which has its origins in the writing
of Jean-Jacques Rousseau, maintains the romantic notion that everything
natural is better than anything man-made. Rousseau extended this idea to
education, which was not to be forced or structured, but was to emerge
from the natural inclinations of the child. This was taken up by nineteenth-century
German educators, most notably Friedrich Froebel, whose Kindergarten was
originally a full program of schooling along naturalistic lines. But the
methodology couldn't accommodate complex subjects such as science and mathematics,
and European educators resorted to more structured educational methods
by the end of the nineteenth century.
Americans, always the romantics, picked up the child-centered ideology
that the Europeans had found wanting and have made it the new orthodoxy.
As E. D. Hirsch Jr. explains in his 1996 book, The Schools We Need and
Why We Don't Have Them, the current reform movement in the United States
is merely a continuation of failed practices that have been taught in all
teachers' colleges since the 1920s.
The other erroneous ideology is the egalitarian belief that all children
can learn at the highest level. This bit of Lake Wobegon nonsense is actually
in the Massachusetts Common Core, much to the displeasure of John Silber,
chairman of the Massachusetts Board of Education. Now, you mix the idea
that students don't need knowledgeable teachers, but only facilitators
to help them construct their own knowledge, with the idea that all these
untaught children will miraculously achieve at the highest level, and you
get NCTM standards, the National Science Education standards, the Massachusetts
Curriculum Framework in Science & Technology, and all the other state
and national standards that purport to cure our educational problems without
anyone, especially the students, really trying.
Opposing these idealists are realists like Hirsch and me, who believe
that most of what a child learns in school from first grade on is nonintuitive.
Most children can't learn to read, write, multiply, and divide on their
own. They need extensive instruction in these subjects, and a lot of practice.
Contrary to the idealists, who believe that children should learn general
content-free problem-solving skills, realists believe that expert knowledge
is built on a large base of factual knowledge and specific problem-solving
procedures. For example, I never learned to play chess well, because I
approached it as a pure reasoning game. But it is far too complex for that.
Expert players study thousands of board configurations and specific moves
for specific situations. There are strategic principles, to be sure, but
those don't replace the need to know a vast number of moves. A good chess
player, like a good piano player, and a good student, gets that way through
countless hours of practice.
Practice is built into the traditional teaching of mathematics. It also
is central to the teaching of physics-my own field-which is built around
students solving many specific problems. The struggle, always, is to get
students to do enough problems to reach a critical level of proficiency.
There is, as Euclid told Ptolemy I twenty-three centuries ago, no royal
road to geometry. Everyone must travel the same road of continual practice.
The reform movement in education, which is dominated by idealists, never
mentions practice. Its royal road is "inquiry." The National
Research Council's National Science Education Standards says, "Inquiry
into authentic questions generated from student experiences is the central
strategy for teaching science. Teachers focus inquiry predominantly on
real phenomena . . . " The Massachusetts Curriculum Framework in Science
& Technology explains that "before there was science or technology,
there was inquiry." To idealistic educators, "inquiry" is
a diffuse and accommodating term that can, and does, include nonscientific
methodologies.
This can be seen in a unit titled "How Do Objects Fly?" given
as an example of inquiry in the Massachusetts Framework. The example contains
no scientific or technological inquiry at all. Students build paper airplanes,
but learn nothing about air flow, pressure differences, or the Bernoulli
principle. Instead, they "inquire" about the impact of air traffic
on people and organisms in communities near an airport. The unit diverges
away from its central question in an inherently unscientific fashion.
Actual scientific experiments on simple phenomena provide much deeper
insights into scientific principles. Last year students in a seventh-grade
science class stumped me with an authentic question generated from their
experiences with an activity I had developed for them. The activity, part
of a study of pressure, involved filling a container with water, putting
a sheet of paper over the top, and turning the container upside down. The
water stays in the bottle. Or not. The students' question was why the experiment
never worked with a particular plastic bottle used for a particular brand
of spring water. I repeated the experiment several times in front of the
students and satisfied myself that they were right. The water wouldn't
stay in that bottle.
Authentic questions of this sort frequently come up when investigating
real phenomena. I have published several papers explaining some puzzling
results from student experi
ments. But what is a publication for me is a bane to a science teacher,
who can't be expected to do original research on everything that occurs
in his or her classroom. I still don't have a good explanation for the
water-bottle phenomenon; I suspect it has something to do with the surface
properties of the plastic of which the bottle is made, but I have yet to
investigate it further. The bottle sits on my desk. Perhaps there is a
paper in it.
As interesting as the investigation of authentic questions can be, however,
they can't be the focus of a school science program. Only educators with
no idea of what a valid scientific investigation involves would suggest
such a thing. A school program must be built on the fundamental concepts
and principles of science. As far as possible, the study of these principles
should involve the study of real phenomena. For example, the study of balance
forces in seventh grade should involve appropriate experiments with forces
in equilibrium. Words alone can't convey the principles involved. Students
need direct experience with objects acted upon by opposing forces to connect
the words with observed events.
The science war is stalemated at present, because the idealists' belief
in the centrality of inquiry conflicts with the unreasonably large number
of topics that are listed in various state and national standards. The
Massachusetts Framework list fifty-eight separate areas of study that students
are to be tested on by eighth grade. This absurdity is the result of a
pattern of unholy compromises among competing forces. And in spite of millions
of taxpayer dollars spent by the NSF in the development of inquiry-based
middle-school science curricula, none has been judged satisfactory by either
idealists or realists.
For the past three years, I have been developing what I consider to
be a realistic science program for grades six, seven, and eight in Fall
River, Massachusetts. The work is being done under the auspices of Northeastern's
Professional Development Program. This is a unique situation; I have no
committee to report to, no government agency to pander to, and no ideology
to adhere to. My only constraint is that the curricular materials I write
be teachable by the teachers and learnable by the students. Topics are
chosen primarily for their sequential coherence, with an eye on their conforming
with the Massachusetts Curriculum Framework. The latter is no problem,
since the framework contains far more topics than can possibly be covered
by eighth grade.
Amazing as it sounds, state and national standards, for all their pretensions
of "reforming education," leave the critical job of sequencing
topics entirely up to teachers and school districts. In practice, this
involves the selection of new textbooks every five years. Fall River decided
to break this unsatisfactory pattern by developing a custom publishing
model in which lesson units, written for the system, are printed in workbooks
that are given to all students in grades six through eight. Each workbook
contains four to six units, and each unit contains about ten activities
that engage students with real phenomena selected to illustrate the unit's
central principles. Concepts, and the words used to describe them, are
used repeatedly in varying contexts to give students practice with them.
Schools can order workbooks with just the units they want. Each student
gets his or her own workbook, which is used to record the student's work.
Data, graphs, observations, and answers to questions all get written into
the workbook. Personal ownership of the schoolbook, which is common in
most European countries, is a major feature of this program, since it enhances
a student's connection to the subject.
Seventh graders in Fall River have been enthusiastic about their workbooks,
to judge from typical comments: "I really like our workbooks a lot
. . . My workbook explained things better than our regular book would.
I also like the workbook because of all the fun activities." "I
think these new workbooks are a really great idea . . . I find it easier
to understand information if we do hands-on experiments or if we make models
rather than just reading the information from the book." Many students
commented on the lightness of the workbooks compared with the textbook,
not an insignificant matter when you are twelve years old and carrying
several books around all day.
On a district-wide examination that included questions from an international
science test, the average Fall River
seventh grader who had used the workbook scored significantly higher
than the average Fall River eighth grader who had not used the workbook.
In fact, the Fall River seventh-grade average score was higher than the
international eighth-grade average score on questions given to students
in forty-one countries, whereas the Fall River eighth-grade average score
was below the international average. Since these questions did not bear
directly on the subjects covered by the workbooks, these performance differences
indicate that well-structured curricular materials can measurably increase
the general ability of students to engage productively in challenging intellectual
tasks.
The mathematics and science wars are fundamentally about the priorities
that educational programs should give to inquiry versus instruction. Most
of the recent mathematics and science programs developed with NSF funding
are seen by critics as spending too much time on student-
centered inquiry and too little time on focused instruction. Most students
are unable, from activities alone, to acquire core knowledge and skills.
Yet all the science and mathe-
matics standards to date have been written with the assumption that
by engaging in vague open-ended inquiry, students will learn college-level
subject matter by eighth grade.
It is time to reverse the whole standard-setting process. Before writing
standards, we need to develop focused curricular materials that are successful
in teaching some topics to a wide variety of students. Standards then can
be written to specify the exact knowledge and skills that students are
expected to acquire from such curricula. This is beginning to happen, as
standards committees look to working programs, like the one in Fall River,
for guidance in the writing of more realistic standards.
Alan Cromer is a professor of physics. A review of his latest book,
Connected Knowledge, appeared in the March issue of Northeastern University
Magazine. Editor's note: the workbooks used in Fall River schools can be
ordered from Ronjon Publishing at <www.ronjonpublishing.com>.
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