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CHANGE
...mobiliziJ'g for curricular reform
Since the publication in 1983 of A Nation at Risk, Amer
icans have known that fundamental changes must occur
throughout all parts of our educational system in order to:
· Raise performance levels significantly in our nation's
schools and colleges;
· Prepare young people for lifelong learning;
· Educate all students well, not only those identified as col
lege bound;
· Create learning environments better suited to the needs of
disadvantaged groups.
The future of our country depends strongly on our ability
to bring about these fundamental changes in mathematics
education.
C~ e ~  e e ·  ~ ~ ~ e ~ ~ ~ e ~ ~ e
ontinual change is a natural and essential
characteristic of mathematics education
Because mathematics is one of the pillars of education, re
form of education must include significant change in the way
mathematics is taught and learned. As mathematics and so
ciety change continuously, so must mathematics education.
Change is a natural state for education, not just a transition
between old and new To ensure continuous responsiveness
in the future, mathematics education must adopt structures
that will make change permanent; mathematics education
must always respond to changes in science, in society, and
in mathematics itself.
Challenges
Mathematics education in the United States is facing ma
jor challenges on nearly every front:
· Far too many students, disproportionately minority, leave
school without having acquired the mathematical power
necessary for productive lives.
"If an unfriendly foreign
power had attempted to
impose on America the
mediocre edFucatio1'al
performance that exists
today, we migh' well have
viewed it as an act of war.
As it stands, we have al
lowed this to happen to
ourselves."
 A Nation al Risk
73
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Change
"We have drifted into a
curriculum by default,
a curriculum of mini
mum expectations that
resists the changes needed
to keep pace with the d~e
ma1'ds of prepari1'gstu
de1tts for contemporary
life."
John A. Dossey
74
· The shortage of qualified mathematics teachers in the
United States is seriousmore serious than in any other
area of education and affects all levels from elementary
school to graduate school.
· At a time when the percentage of minority students is in
creasing, the shortage of new minority teachers of mathe
matics is particularly acute.
· On average, U.S. students do not master mathematical
fundamentals at a level sufficient to sustain our present
technologically based society.
· When compared with other nations, U.S. students lag far
behind in level of mathematical accomplishment; the re
sulting educational deficit reduces our ability to compete
.
.
.

in international arenas.
Public attitudes, which are reflected and magnified by
the entertainment industry, encourage low expectations in
mathematics. Only in mathematics is poor school perfor
mance socially acceptable.
Curricula and instruction in our schools and colleges are
years behind the times. They reflect neither the increased
demand for higherorder thinking skills, nor the greatly
expanded uses of the mathematical sciences, nor what we
know about the best ways for students to learn mathemat
iCS.
Calculators and computers have had virtually no impact
, . .  · . . .
on mathematics Instruction In spite ot their great poten
tial to enrich, enlighten, and expand students' learning of
mathematics.
· Common methods of evaluation especially standardized,
paperandpencil, multiplechoice tests of"basic skilIs"
are themselves obstacles to the teaching of higherorder
thinking skills as well as to the use of calculators and com
puters.
· Undergraduate mathematics is intellectually stagnant,
overgrown with stale courses that fail to stimulate the
mathematical interests of today's students.
The information ~ 1 ~  ~
morrow's scientist and engineer will need extensive math
ematics education, tomorrow's citizen will need a very
different type of mathematical education to deal with
age Is a mathematical age. Even as to
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...mobilizing for curricular reform
mathematicsbased tools, equipment, and techniques which
will permeate the workplace. Far more than most citizens
currently appreciate, mathematics education will play a sub
stantial role in determining which doors are open and which
are closed as students leave school and enter the world of
work.
Counterproductive Beliefs
It is mistakenly thought, even by otherwise wellinformed
adults, that the mathematics they learned in school is ade
quate for their children. Parental and legislative pressures
in the past few years, driven largely by frustration over de
clining test scores, have led to many rash actions:
· Increased numbers of required courses where there is no
agreement on what the added courses should contain or
where capable teachers are to be found to teach them;
· Increased use of standardized tests where there is very
little understanding of what the tests contain or what they
.
_ ~ O
are capable of testing;
Increased use of test scores, especially for teacher and
school accountability where there is little recognition
that the tests reflect only a small part of curricular ob
Octaves.
The nation is in the grip of a testing mystique that has led
to widespread misuse of standardized tests. Public pressures
for "backtobasics" stem from a very limited understanding
of the challenges we face. Carried to the extreme, these pres
sures will rob our children of the opportunity to learn what
they will need to know of mathematics in their adult lives.
Too often, what results from such actions are watered
down curricula, unreliable tests, and diminished morale.
The only elective way in which these relatively illinformed
policies can be combated is through a systematic effort to
develop in the public a deeper understanding of what works
and what does not.
It will not be easy to develop better understanding. Of
ten, public discussion about mathematics education masks
Myth: Increased requirements
yield better prepared students.
Reality: Motivation almost al
ways works better than require
ments. Often, increased require
ments have an eject quite the op
posite of what was intended. In
Wisconsin, for example, when
the university increased from
two to three years the number
of courses required for admission
and also increased the minimum
grade point requirement, in some
schools the number of students
who elected four years of high
school mathematics dropped.
Once the threeyear requirement
was met, students skipped se
nior mathematics to protect their
grade point averages. In Florida,
increased requirements for grad
uation from high school have
caused an increase in the num
ber who drop out.
75
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Change
Cultural Context
Competition and individualism
ingrained parts of traditional
American culture, are reflected
in typical mathematical courses
where students work alone to
solve set problems. Other cul
tures, including many which are
now a growing part of the Amer
~can scene, stress teamwork and
group problemsolving. To the
extent that mathematics instruc
tion in the United States contin
ues to stress individualism and
competition over cooperation and
teamwork, to that extent we con
tinue to introduce unnecessary
counterproductive practices for
many in our multicultural nation.
Adult Attitudes
Too many Americans seem to be
lieve that it does not really matter
whether or not one learns math
ematics. Only in America do
adults openly proclaim their ig
norance of mathematics ("I never
was very good at math") as if it
were some sort of merit badge.
Parents and students in other
countries know that mathemat
ics matters.
76
a hidden agenda of values that have traditionally been car
ried forward by the school study of mathematics. Since the
demise of Latin as a required school subject, it is to math
ematics that many look as a vehicle to teach such qualities
as precision, discipline, neatness, and accuracy. Mathemat
ical truth in popular caricature is certain, absolute, un
changing, eternal. Mathematics appears to many to be a safe
harbor of calm in a turbulent sea of social and educational
change.
Proposals to change mathematics education appear to
threaten timehonored values that are deeply embedded in
the public image of mathematics. The need for change in
mathematics education is too great to allow stereotypes of
mathematics to impede reform.
It is important that the public learn not only about the
need for change, but also about how the essential qualities
of mathematics are conveyed by contemporary as well as
traditional views of the field. As an active partner on the
rapidly advancing frontier of science, mathematics is con
stantly expanding and changing. Mathematics education, in
contrast, has been constrained by societal forces to such a
degree that it has hardly changed at all. This contrast in the
pace of change virtually ensures that mathematics education
is perpetually out of date.
N..........
alive nol~c~es
rooted in myth impede
reform of mathematics education.
As a subject with an extensive and substantial history,
mathematics more than any other science has been taught
as an ancient discipline. A nation that persists in this view
of mathematics is destined to fall behind scientifically and
economically. Parents who persist in this view deny their
children the opportunity to develop and prosper in the in
~ .
formation age.
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...mobilizing for curricular reform
The American Way
The development of more effective strategies for revital
izing mathematics education must be based in part on an
understanding of why it is so difficult in the United States
to bring about change in education. The truth we shrink
from confronting is that most previous reform efforts have
failed. A properly skeptical public will rightly ask why any
new effort is more likely to succeed. Part of the difficulty
we face in mathematics education is a natural reflection of
our constitutional dilemma: to reconcile local authority with
national need.
Most other countries have either national curricula or na
tionwide curricular guidelines. Curricular development is
typically a routine function of a ministry of education which
taps the best brainpower in the nation to develop complete
sets of texts and other resource materials for classroom use.
Specific daytoday syllabi and teacher guides are often pro
vided to schools; in some cases, these syllabi are actually
mandated by a ministry of education. In many countries,
all children in the same grade study essentially the same
material in almost the same way. Such practice, common
around the world, reveals a strong tradition of a "topdown"
approach in education.
in the United States, with our traditional and legal decen
tralization of education, we go about things very differently.
Every summer, thousands of teachers work in small teams
for periods ranging from one week to two months, charged
by their school districts to write new mathematics curric
ula. These teacher teams usually have little training in the
complicated process of curricular development, little or no
help in coping with changing needs, and little to fall back on
except existing textbooks, familiar programs, and tradition.
The consequence usually is the unquestioned acceptance of
what already exists as the main body of the new curricu
lum, together with a little tinkering around the edges. Many
school districts simply adopt series of textbooks as the cur
riculum, making no effort to engage the staff in rethinking
curricula; in those places, the status quo certainly reigns.
International Expectations
Average students in other coun
tries often learn as much mathe
matics as the best students learn
in the United States. Data from
the Second International Math
ematics Study (1982) show that
the performance of the top 5 per
cent of U.S. students is matched
by the top 50 percent of stu
dents in Japan. Our~very best
students the top 1 percent
scored lowest of the top 1 percent
in all participating countries. All
U.S. students whether below, at,
or above averagecan and must
learn more mathematics.
77
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Change
"Most students seem to
think that mathematics
courses are chiefly
designed to winnow out
the weak and grind down
the ungifted. We need a
change in attitude."
Edward E. David, Jr.
78
T· · · . . . . . · . . . . . . · . . . · . . . . . . . . . . . . .
raditional U.S. approaches to curricular
change make reform impossible.
The American process of curricular reform might be de
scribed as a weak form of a grassroots approach. The record
shows that this system does not work. It is not our teachers
who are at fault. In fact, teachers should play a dominant
role in curricular decisionmaking. But teachers who work
in summer curricular projects are being given an unrealis
tic task in an impossible time frame, with only the familiar
status quo to guide them.
In static times, in periods of unchanging demands, perhaps
our grassroots efforts would suffice to keep the curriculum
current. In today's climate, in which technology and research
are causing unprecedented change in the central methods and
applications of mathematics, present U.S. practice is totally
inadequate. International comparisons of student perfor
mance in mathematics for example, the Second Interna
tional Mathematics Study show that U.S. students lag far
behind their counterparts in other industrialized countries.
The topdown systems have beaten us hands down.
Modern Mathematics
Curricular reforms undertaken in the two decades from
1955 to 1975 under the slogans of "modern mathematics"
or "new math" left a mixed legacy to American mathemat
ics education. The movement sprang from many roots and
took on many different (and sometimes opposing) forms.
Implementation was quite uneven, as were results.
Looking back, one can identify several important areas of
success and failure:
· Certain important seeds sowed during this period (for ex
ample, renewed emphasis on geometry, probability, and
statistics) have taken root and are now on the verge of
blossoming.
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...mobilizing for cllrricuiar reform
· Too often, the proposed means to achieve deeper under
standing (for example, sets and commutative law) became
ends in themselves, thus opening mathematics education
to public ridicule.
· Innovative applications of mathematics to nontraditional
fields (for example, to biology and business) became ac
cepted as part of the content of school mathematics.
· By moving some parts of school curricula into unfamiliar
areas, mathematics educators lost the confidence of their
most important ally parents.
Both educators and parents can learn from the experiences
of the modern mathematics era, but the lessons are not so
simple as conventional wisdom often suggests.
Lessons from the Past
The history of the past twentyfive years of curricular re
form gives us only negative examples from which to learn.
Few traces remain of the expensive major curricular devel
opment projects so prominent in the 1960's and 1970's.
These projects tried to develop, on a national scale, com
plete curricula (including instructional materials) that could
be adopted by school districts. But the theorists and planners
who developed these curricula were naive about the process
of change; big curricular projects failed to take root in Amer
ican schools because they were transplanted fully grown into
an environment better suited to locally grown methods.
Where teachers were not directly a part of the develop
ment procedure, where their ownership of the product was
not ensured, where teachers considered district acceptance of
the curriculum as a topdown imposition, the revised pro
grams did not last. Where parents could not (or did not)
understand the need for change or the reasons new curric
ular emphases were chosen, resentment and anger resulted
and a solid conviction set in that if the "old math" was good
enough for parents, it was good enough for their children.
As the United States enters a new period of change in
mathematics education, we can benefit from several lessons
drawn from these previous attempts. First, freestanding,
fullservice curricular development projects adopted intact
79
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Change
80
by school districts do not work. Second, a superficial,
districtbydistrict approach to curricular overhaul is poten
tially disastrous, given the demanding nature of what educa
tors face.
U......
=ffective reform requires strong leadership
by teachers, parents, professionals, and
. · ~
po Cans.
Third, any successful effort to improve mathematics cur
ricula and instruction in the schools will require an extensive
public information campaign that reaches all the varied con
stituencies of mathematics education. These diverse publics
must be convinced in understandable language that a very
different mathematics education is both better and neces
sary for their children and for the country. Effective change
requires a great deal from the public:
Conviction of the need for change;
Consensus on highquality mathematics education for ev
eryone;
· Skepticism of "quick fixes" and simplistic solutions;
· Awareness of the general nature of needed changes;
· Support for investment of necessary resources;
· Recognition of the need for continuing leadership at the
national level.
General reaction to the many recent calls for school reform
has been uneven and fragmented. The pattern of unfocused
reaction shows that it is not enough just to get the public's at
tention. Public concern is a necessary, but by no means su~
cient, condition for meaningful educational change. Too of
ten, a partially informed public becomes a poorly informed
electorate. The time is ripe for a new approach to curricu
lar reform, one that establishes appropriate national expec
tations supported by broad public support among parents,
teachers, and taxpayers.
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...mobilizing for curricular reform
T· ~
ransoms
In order to meet the challenges of our time, mathematics
education is already beginning to negotiate several Biscuit
transitions which will dominate the process of change during
the remainder of this century. Only gradually, by extensive
experience, will teachers find the most effective point along
each transition path. Although no one can say in advance
where the best balance lies, it is quite clear that present prac
tice is at an ineffective extreme.
Transition I: The focus of school mathematics is shiftingirom
a dualistic mission minimal mathematics for the majority,
advancer! mathematics for a few to a singular focus on a
significant common core of mathematics for ad students.
The needs of industry for a quantitatively literate work
force compel schools to provide more mathematical educa
tion to more students than ever before. Accomplishing this
will pose significant challenges to:
· Develop a core of mathematics appropriate for all students
throughout each year of school;
· Educate well a significantly larger fraction of the popula
tion;
.
Stimulate able students with the excitement and challenge
of mathematics;
· Differentiate instruction by approach and speed, not by
curricular goals;
· Select topics and approaches of broad interest and effec
tiveness.
Transition 2: The teaching of mathematics is shifting from
an authoritarian mottle! based on "transmission of knowledge"
to a studentcentered practice featuring "stimulation of ~learn
ing'
In both schools and colleges, classrooms of passive stu
dents who are expected to sit and absorb rules which appear
as arbitrary dicta from on high are gradually giving way to
learning environments that:
· Encourage students to explore;
· Help students verbalize their mathematical ideas;
Voice of Experience
"Math Achievement through Prob
lem Solving is an activityoriented
process that uses small groups to
focus on nonroutine problems. It
was designed for students with bad
work habits who seem to get very
little out of traditional high school
algebra. We've founds out that a
lot of these students know much
more than we thought, and many
know less. We've been surprised
at the high level of thinking that
goes on in some of these students."
Jon Brace
81
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Charge
82
· Show students that many mathematical questions have
more than one right answer;
· Provide evidence that mathematics is alive and exciting;
· Teach students through experience the importance of care
ful reasoning and disciplined understanding;
· Build confidence in ad students that they can learn math
ematics.
Transition 3: Public attitudes about mathematics are shifting
from ir~di~erer~ce and hostility to recognition of the important
role that mathematics Allays in today's society.
Although the burden of unfavorable school experiences
continues to color public opinion about mathematics, con
temporary events are sending different messages which are
gradually being heard:
· In other nations where more is expected, more mathemat
ics is learned;
· As the role of science and technology expands, so does the
importance of mathematics;
· To function as an informed citizen, numerary is as impor
tant as literacy.
As attitudes about the importance of mathematics improve,
so will expectations for mathematics education.
Transition 4: The teaching of mathematics is shifting from
preoccupation with inculcating routine skills to developing
broadbased mathematical power.
Mathematical power requires that students be able to dis
cern relations, reason logically, and use a broad spectrum
of mathematical methods to solve a wide variety of non
routine problems. The repertoire of skills which now un
dergird mathematical power includes not only some tradi
tional paperandpencil skills, but also many broader and
more powerful capabilities. Today's students must be able
to:
· Perform mental calculations and estimates with profi
c~ency;
· Decide when an exact answer is needed and when an esti
mate is more appropriate;
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...mobilizing for curricular reform
· Know which mathematical operations are appropriate in
particular contexts;
· Use a calculator correctly, confidently, and appropriately;
· Estimate orders of magnitude to confirm mental or calcu
lator results;
· Use tables, graphs, spreadsheets, and statistical techniques
to organize, interpret, and present numerical information;
· Judge the validity of quantitative results presented by oth
ers;
· Use computer software for mathematical tasks;
· Formulate specific questions from vague problems;
· Select effective problemsolving strategies.
Transition 5: The teaching of mathematics is shifting from
emphasis on tools for future courses to greater emphasis on
topics that are relevant to students' present and future needs.
Most mathematics should be presented in the context of its
uses, with appreciation of mathematics as a deductive logical
system built up slowly through the rising levels of education.
Examples of areas deserving greater emphasis are:
Probability, which facilitates reasoning about uncertainty
and assessment of risk;
· Exploratory data analysis and statistics, which facilitate
reasoning about data;
Modelbuilding, which facilitates systematic, structured
understanding of complex situations;
· Operations research, which facilitates planning of complex
tasks and achieving performance objectives;
· Discrete mathematics, which facilitates understanding of
most applications of computers.
These new topics imply that observation and experimenta
tion will be important in future mathematics programs and
that school mathematics will draw closer to other school sub
jects, especially to science.
Transition 6: The teaching of mathematics is shifting from
primary emphasis on paperandpencil calculations to fuR use
of calculators and computers.
Mathematics teachers at all levelsfrom elementary
school to university are adapting their teaching methods
Voice of Experience
"I approach each problem as if I
didn't already know the conven
tional solution. The students are
much more involved and excited!.
They become creators. It's as de
scribedt by Felix Klein: The math
ematician himself does not work
in a rigorous, deductive manner,
but rather uses fantasy."
Kenneth Cummins
83
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Change
"Over the long term, ba
sic skills only give you the
right to compete against
the Third Worl~for
Third World wages."
Marc S. Tucker
84
to include both new approaches to instruction as well as
new subject matter appropriate to futureoriented curricula.
Calculators and computers make new modes of instruction
feasible at the same time that they inject into the learning en
vironment a special sense of wonder which goes with healthy
development of mathematical power.
Calculators and computers should be used in ways that
anticipate continuing rapid change due to technological de
velopments. Technology should be used not because it is
seductive, but because it can enhance mathematical learning
by extending each student's mathematical power. CaTcula
tors and computers are not substitutes for hard work or pre
cise thinking, but challenging tools to be used for productive
ends.
Transition 7: The public perception of mathematics is shifting
from that of a f xed body of arbitrary rules to a vigorous active
science of patterns.
Mathematics is a living subject which seeks to understand
patterns that permeate both the world around us and the
mind within us. Although the language of mathematics is
based on rules that must be learned, it is important for mo
tivation that students move beyond rules to be able to express
things in the language of mathematics. This transformation
suggests changes in both curricular content and instructional
style. It involves renewed effort to focus on:
· Seeking solutions, not just memorizing procedures;
· Exploring patterns, not just learning formulas;
· Formulating conjectures, not just doing exercises.
As teaching begins to reflect these emphases, students will
have opportunities to study mathematics as an exploratory,
dynamic, evolving discipline rather than as a rigid, absolute,
closed body of laws to be memorized. They will be encour
aged to see mathematics as a science, not as a canon, and to
recognize that mathematics is really about patterns and not
merely about numbers.
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