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OCR for page 7

A Practical Philosophy
Foundations for an improved
mathematics curriculum must
rest on analysis of the nature of
mathematics and the goals of
mathematics eclucation. Even
the most superficial observation
shows unequivocally that the
nature of mathematics is chang-
ing,-that the goals of mathemat-
ics education are expancling,
and that schools are in the micist
of major transitions.
Expanding Goals
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We teach mathematics to serve several very different goals
that reflect the diverse roles that mathematics plays in society:
· A Practical Goal: To help individuals solve problems of
everyday life.
· A Civic Goal: To enable citizens to participate intelligently
in civic affairs.
· A Professional Goal: To prepare stuclents for jobs, voca-
tions, or professions.
· A Cultural Goal: To impart a major element of human
culture.
The mathematical knowledge needed to achieve these gocis
has changed ciramaticoi~y in the twentieth century and is
changing more rapicily now than ever before.

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8
Reshaping School Mathematics
Perhaps most obvious is the change needed for everyday
life. Whereas daily activities once required a considerable
amount of paper-and-pencil calculation, virtually all routine
household arithmetic is now done either mentally or with an
inexpensive hancl-held calculator, It is not just that the kinds of
solutions we have at our disposal have changed, but so too
have the problems. Today to be mathematically literate one
must be able to interpret both quantitative and spatial infor-
mation in a variety of numerical, symbolic, and graphical con-
texts. These changes provide an unprecedented opportunity
to redirect much of current elementary school mathematics to
more fruitful and important areas especially to the new world
of sophisticated electronic computation, As calculators and
computers diminish the role of routine computation, school
mathematics can focus instead on the conceptual insights
and analytic skills that have always been at the heart of math-
ematics.
The changes in mathematics needed for intelligent citizen-
ship have been no less significant. Most obvious, perhaps, is
the need to understand data presented in a variety of differ-
ent formats: percentages, graphs, charts, tables, and statisti-
cal analyses are commonly used to influence societal deci-
sions. Largely because data are now so widely available, daily
newspapers employ a considerable variety of quantitative
images in ordinary reporting of news events. Citizens who cans
not properly interpret quantitative data are, in this day and
age, functionally illiterate.
It is, however, the professional and vocational needs for
mathematics that have changed most rapidly. Mathematics is
essential to more disciplines than ever before. The explosive
growth of technology in the twentieth century has amplified
the role of mathematics. By increasing the number and variety
of problems that can be solved, calculators and computers
have significantly increased the need for mathematical knowl-
edge and changed the kind of knowledge that is needed.
Computers have moved many vocations (e.g., farming) to
become more quantitative and thus more productive. The
result is that people in an expanding number of vocations and
professions need to know enough mathematics to be able to
recognize when mathematics may be helpful to them.
Because of society's preoccupation with the practical and
professional roles of mathematics, schools rarely emphasize
cultural or historical aspects of mathematics. Like all subjects,
mathematics is dehumanized when divorced from its cultural
contributions and its history. To the extent that these subjects
are discussed at all, students are likely to get the impression
that mathematics is static and old-fashioned. While it is com-
monplace for school children to become familiar with modern
concepts in the sciences such as DNA and atomic energy,

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9
A Philosophy and Framework
rarely are children introduced to any mathematics (such as
statistics or topology) discovered less than a century ago. Chil-
dren never learn that mathematics is a dynamic, growing dis-
cipline, and only rarely do they see the beauty and fascina-
tion of mathematics. The mathematics curriculum can no
longer ignore the twentieth century.
Fundamental Questions
To realize a new vision of school mathematics will require
public acceptance of a realistic philosophy of mathematics
that reflects both mathematical practice and pedagogical
experience. One cannot properly constitute a framework for a
mathematics curriculum unless one first adciresses two funda-
mental questions:
· What is mathematics?
· What does it mean to know mathematics?

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10
Reshaping School Mathematics
Although few mathematicians or teachers spend much
time thinking about these philosophical questions, the unstat-
ed answers that are embecided in public ancl professional
opinion are the invisible hancis that control mathematics edu-
cafion. No change in education can be effective until the hicl-
den influences of these creep issues are redirected toward
objectives more in tune with today's woricl.
Yet even as forces for change are providing new directions
for mathematics eclucation, the public instinct for restoring tra-
clitional stability remains strong. Unless the 3uiclance system for
mathematics education is permanently reset to new and
more appropriate gocis, it will surely steer the curriculum back
to its Al path once present pressure for change abates.
Answers to these fundamental questions would help clarify
for both eclucators and the pubic what mathematics is really
about-what it stuclies, how it operates, what it is good for
(Romberg, 1988~. Appropriate answers would provide a con-
vincing platform on which to erect a new mofhe-matics cur-
riculum of the twenty-first century in which children would be
introclu£ecJ not only to the traditional themes of number and
space, but also to many newer themes such as logic, chance,
computation, and statistics. From these answers would flow a
pragmatic philosophy of mathematics that could help explain
the creative tension that bincis the two funclamenfal poles of
mathemofical reality:
· Theory: That in mathematics, reasoning is the test of truth.
· Applications: That mathematical models are both apt
and useful.
One might think that the many definitions of mathematics
provided by scholars in centuries past would suffice for this
task. But in the past few years, as computers have begun to
unfold new potentials of mathematical systems, we have
been able to see mathematics in a significantly broadened
context. As the Apollo missions for the. first time enabled peo-
ple to see and describe the back side of the moon, so com-
puters have now enabled us to grasp a much richer land-
scape of the mathematical sciences. It is now time to reshape
mathematics education to reflect both the significant role of
computers in the practice of mathematics ant] the frans-
formed role played by mathematics in modern society.
Describing Mathematics
We begin with a simple approximation: mathematics is a
science. Observations, experiment, discovery, and conjecture

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11
A Philosophy and Framework
are as much part of the practice of mathematics as of any
natural science. Trial ancl error, hypothesis and investigation,
and measurement and classification are part of the mathe-
matician's craft and should be taught in school. Laboratory
work and fieldwork are not only appropriate but necessary to
a full understanding of what mathemofics is ancl how it is used.
Calculators ancl computers are necessary tools in this mathe-
matics lab, but so too are sources of real data (scientific
experiments, demographic clata, opinion polls), objects to
observe ancl measure (clice, blocks, balls), ancl tools for con-
struction (rulers, string, protractors, clay, graph paper).
As biology is a science of living organisms ancl physics is a
science of matter ancl energy, so mathematics is a science of
patterns. This description goes back at least to Descartes in a
slightly different form (he called mathematics the "science of
order"), and has been refined by physicist Steven Weinberg
who used it to explain the uncanny ability of mathematics to
anticipate nature (Steen, 1988~. A sirni~ar view of mathematics
as the science of "patterns and relationships" forms the basis
for the expression of mathematics in Science for All Americans
(American Association for the Acivancement of Science
(AAAS), 19891. By classifying, explaining, ancl describing pat-
terns in all their manifestations-number, data, shape, arrange-
ments, even patterns themselves mathematics ensures that
any pattern encountered by scientists will be explained some-
where as part of the practice of mathematics.
Patterns are evident in every aspect of mathematics. Young
children learn how arithmetic clepencis on the regularity of
numbers; they can see order in the multiplication table and
wonder about clisorder in the pattern of primes. The geometry
of polyhedra exhibits a regularity that recurs throughout nature
and in architecture. Even statistics, a subject which studies dis-
order, depencis on exhibited patterns as a yardstick for assess-
ing uncertainty.
As a science of patterns, mathematics is a mode of inquiry
that reveals fundamental truth about the order of our world.
But mathematics is also a form of communication that com-
plements natural language as a tool for describing the world
in which we live. So mathematics is not only a science, but
also mathematics is a language. It is, as science has revealed,
the language in which nature speaks. But it is also an apt lan-
guage for business ancl commerce.
From its beginnings in ancient cultures, the language of
mathematics has been widely used in commerce: measure-
ment ancl counting-geometry ancl arithmetic-enabled
trade and regularized financial transactions. In recent cen-
turies, mathematics (first calculus, then statistics) provided the
intellectual end inferential framework for the growth of sci-
ence. The mathematical sciences (inclucling statistics) are now
the founclation disciplines of natural, social, ancl behavioral
sciences, Moreover, with the support of computers ancl woricl

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12
Reshaping School Mathematics
wide digital communication, business and industry depend
increasingly not only on traditional but also on modern mathe-
matical methods of analysis.
Mathematics can serve as the language of business and
science precisely because mathematics is a language that
describes patterns. In its symbols and syntax, its vocabulary
and idioms, the language of mathematics is a universal means
of communication about relationships and patterns, It is a lan-
guage everybody must learn to use,
Knowing Mathematics
If mathematics is a science and language of patterns, then
to know mathematics is to investigate and express relation-
ships among patterns: to be able to discern patterns in com-
plex and obscure contexts; to understand and transform rela-
tions among patterns; to classify, encode, and describe
patterns; to read and write in the language of patterns; and to
employ knowledge of patterns for various practical purposes.
To grasp The diversity of patterns-indeed, to begin to see pat-
terns among patterns -it is necessary that the mathematics
curriculum introduce and develop mathematical patterns of
many different types. As the patterns studied by mathematics
are not limited to the rules of arithmetic, so the patterns stucl-
ied in school mathematics must break the bonds of this artifi-
cial constraint.
A person engaged in mathematics gathers, discovers, cre-
ates, or expresses facts and ideas about patterns. Mathemat-
ics is a creative, active process very different from passive
mastery of concepts and procedures. Facts, formulas, and
information have value only to the extent to which they sup-
port effective mathematical activity. Although some funda-
mental concepts and procedures must be known by all stu-
dents, instruction should persistently emphasize that to know
mathematics is to engage in a quest to understand and com-
municate, not merely to calculate, By unfolding the funda-
mental principles of pattern, mathematics makes the mind an
effective tool for dealing with the world. From these views can
flow an effective and dynamic school curriculum for the next
century.
Practical Effects
The practical test of a philosophy is the effect it should have
on practice-in this case, on the teaching of mathematics.

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A Philosophy and Framework
~ ABCD -EF`H=2
AIR ~ :( ~
ABED ' FHJK= ~ En.
ICKY
D Draw
Label
E Erase
M Measure ~ Repeat
S $~e change N New shape
Q Quit I;
~ Education Dsvelopm~l Cenlor. 1989
The Geometric Supposer is a set of software learning environments
deliberately designed to change school plane geometry from a
closely guided museum tour (where the guide points out certain arti-
facts to be "proven") to an active process of building and exploring
conjectures. For example, a student who constructs the three medi-
ans of a triangle and notices that they all intersect in a point might
wonder if this is a fluke, or whether it might hold for other triangles. By
using a repeat feature, the student can quickly execute the same
construction on a series of triangles, either generated at random by
the computer or produced by the student in a way designed to stress
the conjecture in some particular way (e.g., on a long, thin obtuse tri-
angle).
In the six figures above, a student uses the Supposer to generalize
a basic construction where corresponding points of adjacent sides of
a square are joined and the ratio of the area of the square and the
interior figure are calculated. From left to right, the construction and
calculation are repeated on different quadrilaterals. In the top row,
the sides of each figure are divided into two equal parts; in the bot-
tom row, the sides of each figure are divided into three equal parts.
What conjectures emerge? How can these conjectures be justi-
fied?
The
Geometric
Supposer

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14
Reshaping School Mathematics
Many important ideas follow from the view of mathematics as
a science and language of patterns.
· By expressing a broad view of the mathematical sci-
ences, this proposed philosophy encompasses all tracli-
tional topics covered by school mathematics. Arithmetic
ancl geometry, algebra and calculus are richly endowed
with patterns of number, shape, and measure-patterns
that will supporl much of the traclitiona~ curriculum.
· By suggesting that mathematics encompasses all kincis of
patterns wherever they arise, this perspective compels a
broader vision of school mathematics that inclucles, for
example, mathematical structures in probability ancl
statistics, in discrete mathematics ancl optimization.
· By stressing that mathematics is a science, this philosophy
supports a style of instruction that rewarcis exploration,
encourages experiments, ancl respects conjectural
approaches to solving problems.
· By recognizing that mathematics is an apt language of
business ancl science, this view underscores the universal
importance of mathematics as a subject that all students
must learn to use.
· By invoking the metaphor of science in which experiment
complements theory, the perspective of mathematics as
a science of patterns helps bridge the gap between
"pure" and "appliecl" mathematics. The patterns studied
by mathematicians are, for all practical purposes, as real
as the atomic particles studied by physicists.
By emphasizing that mathematics is a process rather
than a set of facts, this perspective makes clear that stu-
dents need to experience genuine problems-those
whose solutions have yet to be developed by the stu-
dents (or even perhaps by their teachers). Problem situa-
tions should be complex enough to offer challenge, but
not so complex as to be insoluble, Learning should be
guided by the search to answer questions-first at an intu-
itive, empirical level; then by generalizing; ancl later by
justifying (proving).
By making clear that mathematics is the study of patterns
rather than merely a craft for calculation (or an art with
no evident purpose), this pragmatic view highlights the
philosophical basis for using calculators in school
mathematics: as microscopes are to biology ancl tele-
scopes to astronomy, calculators ancl computers have
become essential tools for the study of patterns.
· By recognizing that practical knowledge emerges from
experience with problems, this view helps explain how

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15
A Philosophy and Framework
experience with problems can
help develop students' ability
to compute. This recognition
contrasts sharply with the pre-
vailing expectation in schools
that skill in computation should
precede encounter with word
problems. Present strategies
for teaching need to be
reversecl: students who recog-
nize the need to apply partic-
ular concepts have a stronger
conceptual basis for recon-
structing their knowledge at a
later time.
By stressing mathematics as a
language in which students express
ideas, we enable students to devel-
op a framework that can be cirawn
upon in the future, when rules may
have been forgotten but the struc-
ture of mathematical language
remains embecicled in memory as
a foundation for reconstruction.
Learning the language of mathe-
matics requires immersion in situa-
tions that are sufficiently simple to
be manageable, but sufficiently
complex to offer diversity: incliviclu-
a~, small-group, or large-group instruction; a variety of mathe-
matical domains; and open and flexible methods.
By affirming the importance of mathematics as a language
and science of patterns we reset the gyroscopes that guide
school mathematics. instead of being viewed as an
immutable collection of absolute truths, mathematics will be
seen as it is-as an evolving, pragmatic discipline that seeks to
understand the behavior of patterns in science, in society, and
in everyday life.
Philosophical Perspectives
Changing the public philosophy of school mathematics is
an essential step in effecting reform of mathematics educa-
tion. An effective practical philosophy of mathematics can be
based on two considerations:

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16
Reshaping School Mathematics
· That mathematics is a science and language of patterns;
· That to know mathematics is to investigate and express
relationships among patterns.
Nothing in this approach implies that these are unique or
necessary considerations. They are, however, sufficient to
meet certain important criteria that any effective philosophy
of mathematics education must satisfy:
· They encompass new as well as traditional topics;
· They provide a substantive rationale for using calculators
and computers in school mathematics;
· They encourage experience with genuine problems;
· They stimulate exploration, use of real data, and appren-
ticeship learning;
· They help bricige the gap between pure and applied
mathematics;
· They emphasize active modes of learning;
· They are understandable to a broad segment of the public.
The framework for mathematics education that follows from
this practical philosophy provides an environment to support
present efforts at curricular reform. Other philosophies can also
provide similar support, and surely many others will emerge in
the process of national curricular change. The counterpoint
between a philosophy and a framework of mathematics edu-
cation will continue as long as the process of change remains
vigorous.