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OCR for page 43
CURRICULUM
Mathematics is one way we make sense of things. It en
ables us to perceive patterns, to comprehend data, and to
reason carefully. Truth and beauty, utility and application
frame the study of mathematics like the muses of Greek
theater. Together, they define mathematical power, the ob
jective of mathematics education.
The transformation of mathematics from a core of ab
stract studies to a powerful family of mathematical sciences
is reflected poorly, often not at all, by the traditional math
ematics curriculum. One can hardly blame students for not
becoming interested in mathematics if they rarely see evi
dence of its full power and richness.
M ...............................
athematics curricula at all levels must
introduce more of the breadth and power of the
mathematical sciences.
As mathematics is more than calculation, so education in
mathematics must be more than mastery of arithmetic. Ge
ometry, chance, and change are as important as numbers
in achieving mathematical power. Even more important is
a comprehensive flexible view that embodies the intrinsic
unity of mathematics: estimation supplements calculation;
heuristics aid algorithms; experience balances innovation.
To prepare students to use mathematics in the twentyfirst
century, today's curriculum must invoke the full spectrum
of the mathematical sciences.
Philosophy
Virtually all young children like mathematics. They do
mathematics naturally, discovering patterns and making con
jectures based on observation. Natural curiosity is a power
fu! teacher, especially for mathematics.
...developing mathematical power
43
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Curriculum
" 'To some extent, every
body is a mathematician'
. ..Schoo! mathematics
must endow all students
with a realization of this
fundamental fruth."
NCTM Standards
44
Unfortunately, as children become socialized by school
and society, they begin to view mathematics as a rigid sys
tem of externally dictated rules governed by standards of
accuracy, speed, and memory. Their view of mathematics
shifts gradually from enthusiasm to apprehension, from con
fi(lence to fear F.ventil~liv most ct,~Antc learn Phi_
~ ~^~ ~ ^~ v ~ ~ ~ ~ ~ 1~ ~ ~ 111CI L11~111~ L
. . 
arcs under duress, convinced that only geniuses can learn it.
Later, as parents, they pass this conviction on to their chil
dren. Some even become teachers and convey this attitude
to their students.
Doing mathematics is much like writing. In each, the fi
na~ product must express good ideas clearly and correctly,
but the ideas must be present before the expression can take
form. Good ideas poorly expressed can be revised to im
prove their form; empty ideas well expressed are not worth
revising. A mathematics curriculum that emphasizes compu
tation and rules is like a writing curriculum that emphasizes
grammar and spelling; both put the cart before the horse.
Excessive emphasis on mechanics of mathematics not only
inhibits learning, but also leads to widespread misconcep
tions among the public concerning strengths and limitations
of mathematical methods. Because early school experiences
suggest that all mathematics problems have a single correct
answer, the public to its great risk tends to believe un
critically any expert who employs mathematical arguments.
Such a misconception can lead to disastrous consequences
when naive mathematical modeling is used as the basis for
economic, military, or social planning. Too few of those
who use mathematical models are well enough prepared to
~7
_ me, ~ ~ . ~ ~ ~ . .
appreciate the limitations of the models they use.
Mathematics instruction must not reinforce the common
impression that the only problems amenable to mathemati
cal analysis are those that have unique correct answers. Even
more, it must not leave the impression that mathematical
ideas are the product of authority or wizardry. Mathematics
is a natural mode of human thought, better suited to certain
types of problems than to others, yet always subject to con
firmation and checking with other types of analyses. There
is no place in a proper curriculum for mindless mimicry
mathematics.
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...developing mathematical power
~ Selfconfidence built on success is the
most important objective of the mathematics
curriculum.
The ability of individuals to cope with mathematics wher
ever it arises in their later lives" whether as wageearners,
parents, or citizens depends on the attitudes toward math
ematics conveyed in school and college classes. Above all,
mathematics curricula must avoid leaving a legacy of mis
understanding, apprehension, and fear.
Standards
Independence is the hallmark of U.S. education. Educa
tional policy is set not by the U.S. Department of Education,
but by fifty states and thousands of independent school dis
tricts. Local control of education is embedded in the Amer
ican body politic as deeply as anything, a legacy of consti
tutional authority that reserves to the states all matters not
expressly granted to the federal government.
Yet this independence is largely a myth, especially for
mathematics education. Effective control comes not from
Washington, but from invisible local or state committees that
approve textbooks and from anonymous officials who select
standardized tests. Few facts stand undisputed in educa
tional research, but the dependence of teachers on textbooks
and of students on tests is as firm a finding as exists in this
amorphous discipline; especially in mathematics, teachers
teach only what is in the textbook and students learn only
what will be on the test.
In practice, although not in law, we have a national cur
riculum in mathematics education. It is an "underachieving"
curriculum that follows a spiral of almost constant radius,
reviewing each year so much of the past that little new learn
ing takes place. Some states (for example, California, Texas,
Wisconsin, and New York) have recently promulgated new
standards for mathematics education, often with surprising
Myth: What was good enough
for me is good enough for my
child.
Reality: Today's world is more
mathematical than yesterday's,
and tomorrow's world will be
more mathematical than today's.
As computers increase in power,
some parts of mathematics be
come less important while other
parts become more important.
While arithmetic proficiency may
have been "good enough" for
many in the middle of the cen
tury, anyone whose mathematical
skills are limited to computation
has little to offer today's society
that is not done better by an in
. ~ .
expensive machine.
45
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Curriculum
Myth: Arithmetic is the major
goal of elementary school mathe
matics.
Reality: Number sense builds on
arithmetic as words build on the
alphabet. Numbers arise in mea
surement, in chance, in data, and
in geometry, as well as in arith
metic. Mathematics in elemen
tary school should weave all these
threads together to create in chil
dren a robust sense of numbers.
46
consequences. In California, new standards led to initial
rejection of ad mathematics textbook series submitted for
authorized adoption. California wanted to stress as emphat
ically as possible that, in the future, mathematics textbooks
must be designed to develop student capabilities to address
and solve complex, subtle, and unpredictable problems.
A,,
erica needs to reach consensus on
national standards for school mathematics.
We must judge schools not by remembrances of things
past, but by necessary expectations for the future. Students
must learn not only arithmetic, but also estimation, measure
ment, geometry, optimization, statistics, and probability
all of the ways in which mathematics occurs in everyday life.
In the process, they must gain confidence in their ability to
communicate and reason about mathematics; they should
become mathematical problemsolvers.
Elementary Education
Elementary school is where children learn the mathemat
ical skills needed for daily life. Formerly, shopkeeper arith
metic was an adequate objective since, for most people,
mathematics in daily life required little more than arith
metic.
This is no longer true. Calculators now do most of
the arithmetic needed for
dominated society requires
of chance, of reasoning, of form, and of pattern. while the
goal of elementary education has not changed, the mathe
matical objectives appropriate to this goal are very different
now from what they were half a century ago.
The major objective of elementary school mathematics
should be to develop number sense. Like common sense,
daily life, while a technology
that everyone have a good grasp
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...dereloping mathematical power
number sense produces good and useful results with the least
amount of effort. It is not mindlessly mechanical, but flexible
and synthetic in attitude. It evolves from concrete experi
ence and takes shape in oral, written, and symbolic expres
sion. Links to geometry, to chance, and to calculation should
reinforce formal arithmetic experience to produce multiple
mental images of quantitative phenomena.
Developing number sense will move children (and teach
ers) beyond narrow concern for schoolcertified algorithms
for arithmetic. Even in the absence of calculators, neither
children nor adults make much use of the specific arithmetic
techniques taught in school. School children do, however,
need to learn how to use mathematics for common tasks
making change, measuring quantities (food, lumber, fabric),
planning schedules, estimating chances but the particular
means that they use must be appropriate to the task. They
need to learn not only how to estimate and calculate, but
also how to decide whether to estimate or calculate. Good
number sense includes common sense about how to find an
answer as well as a range of choices of methods.
Appropriate use of calculators enhances
children's understanding and mastery of
arithmetic.
using calculators intelligently is an integral part of num
ber sense. Children should use calculators throughout their
school work, just as adults use calculators throughout their
lives. More important, children must learn when to use them
and when not to do so. They must learn from experience
with calculators when to estimate and when to seek an exact
answer; how to estimate answers to verify the plausibility of
calculator results; and how to solve modest problems men
tally when neither pencil nor calculator is convenient.
Calculators create whole new opportunities for ordering
the curriculum and for integrating mathematics into science.
Back to School
"Target Addition" is a calcula
tor game that reinforces mental
arithmetic. Two children with a
calculator agree on a target num
her such as 21. They take turns
adding a number from 1 to 5 into
the memory. The goal is to make
the memory number match the
target number.
47
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.
Curriculum
Myth: Early use of calculators
will prevent children from learn
ing basic facts of arithmetic.
Reality: Children learn to solve
arithmetic problems by a variety
of means just as adults do in
ordinary activities. Sometimes,
one calculates mentally (figur
ing tips); other times, one esti
mates (buying groceries). Still
other times, one calculates with a
pencil (doing carpentry) or with
a calculator (preparing taxes).
Learning basic arithmetic re
quires that one learn all these
approachesnot only how to
do them, but, more importantly,
when to do them. Calculators are
an essential part of the repertoire
of arithmetic skills that any child
or adult needs to use.
48
No longer need teachers be constrained by the artificial re
striction to numbers that children know how to employ in the
paperandpencil algorithms of arithmetic. Decimals can be
introduced much earlier since they arise naturally on the cal
culator. Real measurements from science experiments can
be used in mathematics lessons because the calculator will
be able to add or multiply the data even if the children have
not yet learned how. They may learn first what addition and
multiplication mean and when to use them, and only later
how to perform these operations manually in all possible
cases.
Elementary school mathematics should reinforce a child's
natural curiosity about patterns. Children must be encour
aged to perceive mathematics in the world around them.
Shapes, numbers, chance the foundations of geometry,
arithmetic, probability will emerge from careful guided ob
servation. Science study will lead naturally to mathematics,
following the paradigm of data, deduction, and observation.
Many adults fear that early introduction of calculators will
prevent children from learning basic arithmetic "properly,"
as their parents learned it. The experiences of many schools
during the last fifteen years show that this fear is unfounded.
Students who use calculators learn traditional arithmetic as
well as those who do not use calculators and emerge from el
ementary school with better problemsolving skills and much
better attitudes about mathematics. Although mindless cal
culation can be as destructive as mindless arithmetic, proper
use of calculators can stimulate growth of a realistic and
productive number sense in each child.
Secondary Education
Secondary education is where students begin to learn the
mathematics they will need for careers as well as the mathe
matics required for elective citizenship. Whereas, tradition
ally, secondary school has been characterized by the intro
duction of algebra as an extension of arithmetic, contempo
rary society requires much greater breadth from secondary
school mathematics.
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...developing mathematical power
The focus of the secondary school curriculum remains
as it should on the transition from concrete to conceptual
mathematics. As students' understanding moves from num
bers to variables, from description to proof, from special
cases to general equations, they learn the power of mathe
matical symbols. In a very real sense, the major objective of
secondary school mathematics is to develop symbol sense.
All students need to leave secondary school well prepared
mathematically tor leading intelligent lives as productive
citizens since even many of those who go on to higher
education will take little or no further mathematics. High
school graduates need to know enough about chance to
understand health and environmental risks; enough about
change and variability to understand investments; enough
about data and experiments to understand the grounds for
scientific conclusions; enough about representation to inter
pret graphs; and enough about the nature of mathematics to
be supportive parents to their children who will learn aspects
of mathematics that their parents never studied.
Students who enter the work force directly from high
school will be expected to be able to read documents re
plete with technical languagecomputer guides, shop man
uals, financial reports. They need to be able to comprehend
threedimensional images (assembly diagrams) and logically
intricate instructions (tax code); they need to be able to read
symbolladen text (computer manuals) and to plan complex
systems (purchasing property). They need enough mathe
matics and enough confidence to be able to learn what they
need as they need it.
A
I} high school students should study a
common core of broadly useful mathematics.
_ · ,
Typically, secondary school mathematics curricula are
dominated by a philosophy of preparation for college calcu
lus. Few natural applications emerge from secondary school
Back to School
Investigate the relation between
automobile mileage and age by
gathering data from cars in the
school parking lot.
AIDS Tests
How is it possible for an AIDS
test that is 95 percent accurate to
be misleading 90 percent of the
time when used in mass testing?
Issues such as this are central to
public policy, yet require a basic
knowledge of probability to be
understood properly.
49
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Curriculum
Myth: The primary goal of
mathematics education is to
educate mathematicians.
Reality: School mathematics
is part of universal education.
Three of every four college ma
jors require students to study
collegelevel mathematics. To
cope with a technological age, all
students should study mathemat
ics every year they are in school.
50
mathematics since the traditional topics are merely tools for
the applications of calculus. Even worse, students rarely
learn mathematics appropriate to enlightened citizenship or
to the needs of the workplace.
These deficiencies are not inherent in secondary school
mathematics, but historical accidents arising from the teleo
logical influence of calculus. New mathematics with broader
applications can offer much greater appeal to all students. In
stead of tracking students by curricular objective (for exam
ple, commercial, general, or precollege), school mathematics
should provide for all students a core of mainstream math
ematics in which different student groups are distinguished
not by curricular goals, but only by speed, depth, and ap
proach.
For students planning to enter college, still more is needed:
thorough grounding in mathematical methods required for
calculus, statistics, and computer science. As these colle
giate subjects change in response to the impact of comput
ers, the prerequisites will shift away from formal algebra
to more integrative. Droblemsolvina approaches. Versatil
~ , ~ ~ ~ ,~ .
ity, confidence, experience, reasoning, and communication
about mathematics are skills that will be valued as prerequi
sites to college mathematics.
A ................................
11 students should study mathematics
every year they are in school.
. · ~
The gradual mathematization of society has increased con
tinuously the amount of mathematics that students must
learn. Universal elementary education has been replaced,
for all practical purposes, by universal secondary education.
As a consequence, students' study of mathematics must con
tinue throughout secondary school.
· ~.
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...d~eveloping mathematical power
Higher Education
Most students who enter postsecondary education must
study further mathematics, either as part of a general degree
requirement or as a prerequisite to their particular course of
study. in this regard, vocationaltechnical institutes, com
munity colleges, continuing education, and adult education
programs are no different from fouryear colleges and uni
versities. Most courses of study have some kind of mathe
matical prerequisite and most students find that they need
additional study of mathematics to meet these prerequisites.
Over three quarters of the degree programs at most uni
versities require courses in calculus, discrete mathematics,
statistics, or other comparable mathematics. These subjects
are required because they introduce students to functions,
to relations among variables. The language of change and
chance is conveyed by the symbolism of functions. College
students need this level of mathematical literacy in order
to understand with precision the mathematical ideas that
form the foundations for science, business, and engineering
courses.
Learning mathematics entails a gradual progression from
the concrete to the abstract, from the specific to the general.
Numbers lead to symbols, to names for variables; relations
among symbols lead to functions, to the links among objects
expressed in symbols. if it does nothing else, undergraduate
mathematics should help students develop function sense a
familiarity with expressing relations among variables.
Nine of every ten mathematics course enrollments in
higher education are in elementary calculus, in elementary
statistics, or in courses that are prerequisites to these sub
jects. Prerequisite courses are normal parts of the secondary
school curriculum and should, if at all possible, be studied
there rather than in college. Elementary statistics is not of
ten taught in secondary schools today, but should become
a regular part of secondary school mathematics. Calculus,
taught both in high school and in college, is the principal
gateway through which most students must pass if they are
to prepare for mathematicsbased careers.
Although discrete mathematics and statistics provide nec
Undergraduate Mathematics
THOUSANDS OF ENROLLMENTS
FALL SEMESTER
2,800
2,400
2,000
1,600
1 ,200
800
400
Advanced Course
Enrollments
~3,

1965 1970 1975 1980 1985
Each term, nearly 3 million students
enroll in postsecondary mathematics
courses. About 60 percent study elemen
tary mathematics and statistics below the
level of calculus, while 30 percent take
calculuslevel courses. The remaining
10 percent study higher (postcalculus)
mathematics.
51
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Curriculum
Symbolic Computer Systems
As calculators have surpassed hu
man capacity for arithmetic cal
culations, so now are symbolic
computer packages overtaking
human ability to carry out the
calculations of calculus. Until re
cently, computers could only op
erate numerically (with rounded
numbersJ and graphically (with
visual approximations). But now
they can operate symbolically just
as people do, solving equations in
terms of x and y just as we teach
students in school mathematics.
Symbolic computer systems com
pel fundamental rethinking of
what we teach and how we teach
it.
52
essary foundations for computer, engineering, and social sci
ences, calculus remains the archetype of higher mathematics.
It is a powerful and elegant example of the mathematical
method, leading both to major applications and to major
theories. The language of calculus has spread to all scientific
fields; the insight it conveys about the nature of change is
something that no educated person can afford to be without.
Unfortunately, calculus as presently taught has little in
common with the way calculus is used. Many students who
enroll never complete the course; many of those who do fin
ish learn little beyond a series of memorized techniques now
more commonly performed by computers. Because of the
central importance of calculus for all scientific study and
research, the National Science Foundation has launched an
. . . . .
nltlatlve to improve instruction in calculus, which is now
studied by nearly one million students every year.
~ Successful calculus is essential to healthy
mathematics and science.
The quality of calculus instruction is a barometer of math
ematics education. Since preparation for calculus has been
the organizing principle of high school mathematics, calcu
lus receives the inheritance of school practice. Changes in
calculus reverberate throughout secondary school curricula,
just as changes in school mathematics are magnified by the
challenge of calculus. Although other courses need improv
ing as much as calculus, and although many courses are as
important or as practical, the unique position of calculus as
the gateway from school to college mathematics imposes on
it a special burden to be attractive, compelling, and intellec
tually stimulating.
For the past two decades, enrollments in mathematics ser
vice courses in colleges and universities have been rising,
yet throughout most of that period the number of math
ematics majors has been declining. Part of this paradox is
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...d~eveloping mathematical power
due to career competition from attractive disciplines that are
new users of mathematics, notably computer science, eco
nomics, and now, increasingly, biology. But another part of
the problem lies with the mathematics major itself, which in
many institutions has continued a narrow focus on graduate
school preparation, ignoring the many students who might
(and should) major in mathematics as a versatile preparation
for careers in other disciplines.
Undergraduate mathematics provides a
powerful platform for careers in many fields.
Building and sustaining a mathematics major with broad
appeal require commitment to integrity in the undergraduate
program:
· Freshman and sophomore courses (especially calculus and
linear aIgebra) should be taught by the most able instruc
tors who can motivate students to study mathematics.
.
Introductory courses must be taught in a manner that re
flects the era in which we live, making full use of comput
ers as an integral too! for instruction and for mathematics.
· Upperdivision offerings should be designed to represent
broadly the several mathematical sciences, introducing
students in appropriate ways to applications, computing,
modeling, and modern topics.
.
Diverse opportunities outside formal classroom work
should be provided so that students can engage In math
ematics through projects, research, teaching, problem
solving, or independent study.
Broad undergraduate mathematics programs will attract
more students to extended study of mathematics, will offer
these students appealing opportunities to explore mathemat
ics applicable to many fields, and will engage good students
in exploring and learning mathematics on their own. Such
Continuing Education
Almost as many persons study
mathematics outside traditional
school structures as inside them.
Whereas mathematics curric
ula have been conceived in the
traditional structure of continu
ous gradelevel education from
childhood to early adulthood, to
day much mathematics is stud
ied by older adults. Some large
businesses actually operate mini
school districts just for the con
tinuing education of employees;
many universities and colleges
especially community colleges
attract large numbers of adults
both to regular degree programs
and to special short courses for
professional growth or cultural
enrichment. Every issue of pur
pose, quality, and electiveness
concerning traditional mathe
matics education also affects con
tinuing mathematics education.
The need for renewal in the non
traditional forums is just as ur
gent as it is within the walls of
traditional schools.
53
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Curriculum
54
an approach, while intended as a foundation for students in
many different majors, will inevitably attract good students
to careers in mathematics.
Demographic data and degree trends show that it is vi
tally important for undergraduate mathematics departments
to offer effective, broadbased curricula. The United States
needs more American graduate students in the mathematical
sciences; more mathematics graduates who can teach in sec
ondary schools; and more students with better preparation
in mathematics entering graduate programs in science and
engineering. Redressing the serious shortage of mathemat
ically educated college graduates is a significant and urgent
challenge to our mathematics faculties.
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