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Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
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OCR for page 1
A Rationale for Change
The basic premise of this
report is that the United
States must restructure the
mathematics curriculum-
both what is taught and the
way it is taught- if our chil-
dren are to develop the
mathematical knowledge
(and the confidence to use
that knowledge) that they
will need to be personally
and professionally compel
tent in the twenty-first centu-
ry. This restructuring involves
more than producing new
texts or retraining teachers.
Replacing parts is not suffi-
cient. What is required is a
complete redesign of the
content of school mathe-
matics and the way it is
taught.
Changing Conditions
We begin our analysis by reflecting on some of the major
changes affecting the context of mathematics education:
· Changes in the need for mathematics,
As the economy adapts to information-age needs,
workers in every sector from hotel clerks to secretaries,
from automobile mechanics to travel agents must learn
to interpret intelligently computer-controlled processes,
Most jobs now require analytical rather than merely
mechanical skills, so most students need more mathemat
OCR for page 2
2
Reshaping School Mathematics
ical power in school as preparation for routine jobs. Simi-
larly, the extensive use of graphical, financial, and statisti-
cal data in daily newspapers and in public policy discuss
signs compels a higher standard of quantitative literacy
for effective participation in a democratic society.
· Changes in mathematics and how it is used.
In the past quarter of a century, significant changes
have Qccurred in the nature of mathematics and the
way it is usecl. Not only has much new mathematics been
discovered, but the types and variety of problems to
which mathematics is applied have grown at an
unprececlented rate. Most visible, of course, has been
the development of computers and the explosive growth
of computer applications. Most of these applications of
computers have required the cleve~opment of new math-
ematics in areas where applications of mathematics
were infeasible before the acivent of computers (Howson
and K~ahane, 15861. Less visible, but equally important,
has been the enormous wealth of ideas generated in
several main branches of mathematics linked by unifying
concepts of wiclespre:ocl applicability (e.g., Boarcl on
Mathematical Sciences (BMS3, 1986~. Students must study
the mathematics usecl in such applications in order to
grasp the power of mathematics to solve real problems.
· Changes in the role of technology.
Computers ancl calculators have changed profoundly
the world of mathematics. They have affected not only
what mathematics is important, but also how mathemat-
ics is done (Rheinbolcit, 1985~. It is now possible to exe-
cute almost ail of the mathematical techniques taught
from kindergarten through the first two years of college
on hand-held calculators. This fact alone --I the fulfillment
in our age of the dream of Pascal-- must have significant
effects on the mathematics curriculum (Pea, 1987a).
Although most developments at the forefront of a disci-
pline cannot generally be expected to have a major
effect on the early years of eclucation, the changes in
mathematics brought about by computers and calcula-
tors are so profound as to require readjustment in the bal-
ance and approach to virtually every topic in school
mathematics.
· Changes in American society.
As mathematics has changecl, so has American soci-
ely. The changing demographics of the country and the
changing demancis of the workplace exert extraordinary
burdens on mathematics eclucation, burdens that we
OCR for page 3
A Philosophy and Framework
have not yet successfully borne (National Research
Council (NRCl, 19891. In the early years of the next centu-
ry, when toclay's school children will enter the work force,
most jobs will require greater mathematical skills (John-
ston ancl Packer, 1987~. At the same time, white
males-the fraclitional base of mathematically trained
workers in the United States-will represent a significantly
smaller fraction of new workers (Oaxaca and Reynolds,
1988~. Society's need for an approach to mathematics
education that ensures achievement across the clemo-
graphic spectrum is both compelling and urgent (Office
of Technology Assessment, 1988~.
· Changes in understanding of how stuclents ~earn.
Learning is not a process of passively absorbing infor-
mation ancl storing it in easily retrievable fragments as a
result of repeated practice and reinforcement. Insteacl,
students approach each new task with some prior knowl-
ecige, assimilate new information, ancl construct their
own meanings (Resnick, 1987~. Furthermore, ideas are not
isolated in memory but are organized ancl associated
with the natural language that one uses ancl the situa-
tions one has encountered in the past. This constructive,
active view of learning must be reflected in the way
mathematics is taught.
· Changes in international competitiveness.
Just as a global economy is emerging as a dominant
force in American society, many recent reports have
shown that U.S. students do not measure up in their math-
ematical accomplishments to students in other countries
(e.g., Stevenson et al., 1986; McKnight et al., 1987; Stigler
ancl Perry, 1988; Lapointe et al., 19891. Because of wicJeiy
different social contexts in which education takes place,
comparing the educational systems of different countries
is fraught with clangor. Nevertheless, the data are so
compelling that they cannot be ignorecl. In particular,
most other industrial countries have considerably different
expectations about topics taught and level of perfor-
mance than is common in American schools.
One implication of these reflections is the need for a new
practical philosophy of mathematics~education as a basis for
curricular reform. Each new generation neecis to step away
from current schooling practices, reflect on the mathematical
expectations for students ancl society, and restate the
assumptions upon which the system for teaching and learning
is basecl. Such reflection is especially important in a time of
rapid change.
OCR for page 4
4
Reshaping School Mathematics
Outdated Assumptions
The mathematical content of today's school curriculum is
about 500 years old. The core of this curriculum arithmetic,
geometry, and elementary algebra differs in only superficial
ways from the curriculum followed by tutors during the Renais-
sance. Advanced topics such as quadratic equations, loga-
rithms, and probability are of more recent vintage, but even
calculus as taught in today's schools and colleges is three
centuries old
The tremendous stability of today's curriculum depencis on
a guidance system controlled by two unwavering and outbat-
ed public assumptions:
· Mathematics is a fixed and unchanging body of facts
and procedures; and
· To do mathematics is to calculate answers to set prob-
lems using a specific catalogue of rehearsed techniques.
These principles are the gyroscopes of today's school mathe~
matics. Despite turbulence in schools and revolution in the
workplace, mathematics education maintains its course, fold
lowing a path little changed through the centuries,
To the Romans a curriculum was a rutted course that guicl-
ed the path of two-wheeled chariots. Toclay's mathematics
curriculum - a course of study-follows a deeply rutted path
directed more by events of the
past than by the changing neecis
of the present, Vast numbers of
specific learning objectives, each
with associated pedagogical
strategies, serve as mileposts
along the trail mapped by texts
from kindergarten until twelfth
grade. Problems are solved not
by observing and responding to
the natural landscape through
which the mathematics curricula
lum passes, but by mastering
time-testecl routines conveniently
placed along the path near
every anticipated problem. Stu-
dents who progress through this
curriculum develop a kind of
mathematical myopia in which
the goal is to solve artificial word
problems rather than realistic
world problems.
OCR for page 5
5
A Philosophy and Framework
Few have the stamina to survive the curriculum of mathe-
matics-at least not the way it is now delivered. Of 4 million
who begin, only 500,000 are still studying mathematics 12
years later. Most students receive little of lasting value from the
final mathematics course they study-typically high school
geometry or algebra 11. Many of those who cirop out harbor
life-long feelings of guilt or distaste for school mathematics.
Some of those who become disenchanted with mathematics
become teachers; others help decide educational and
research policy for the nation. Very few adults in the United
States have had the benefit of successful completion of a
mathematics curriculum,
Transitions to the Future
The price of stability is anachronism, Evidence is mounting
from many sources that our present curriculum must change
course if it is to serve society well in the twenty-first century,
Forces for change, which are growing increasingly powerful,
are beginning to redirect the mathematics curriculum in sever-
al important ways:
· The focus of school mathematics is shifting from a clualis-
tic mission minimal mathematics for the majority,
advanced mathematics for a few to a singular focus on
a significant common core of mathematics for all stu-
dents.
· The teaching of mathematics is shifting from an authori-
tarian model based on "transmission of knowledge" to a
student-centered practice featuring "stimulation of learn-
ing."
· Public attitudes about mathematics are shifting from inclif-
ference and hostility to recognition of the important role
that mathematics plays in today's society.
· The teaching of mathematics is shifting from preoccupa-
tion with inculcating routine skills to developing broad-
based mathematical power.
· The teaching of mathematics is shifting from emphasis on
tools for future courses to greater emphasis on topics that
are relevant to stuclents' present and future needs.
· The teaching of mathematics is shifting from primary
emphasis on paper-and-pencil calculations to full use of
calculators and computers,
These transitions, elaborated in Everybody Counts (NRC,
1989), are bringing about a substantial change in the way
OCR for page 6
6
Reshaping School Mathematics
mathematics is taught and learnecl. New strategies that
promise significant change are emerging in many districts and
states (e.g., Denham and O'Mally, 1985; Chambers, 1986; Alli-
goocl, 1989~. The ruts of the old curriculum are being eroded
by the waves of change sweeping across the landscape of
mathematics eclucation.
The following chaff from World Almanac gives speeds of animals in km/hr:
98 antelope
48 bear
48 cat
1 12 cheetah
14 chicken
69 coyote
48 deer
40 elephant
72 elk
67 fox
56 jacka/
5 1 giraffe
62 greyhound
77 horse
45 human
80 lion
18 pig
56 rabbit
24 turkey
48 warthog
64 zebra
· The human speed is listed as 45 km/hr. What do you think this means?
· Find the speed of the winner of the most recent Olympic 1500 meter run. How
does this compare with the speed listed for the human in the chart? What
explanation can you give for this?
· What is the typical speed for animals? How did you find this speed?
· How do the speeds differ? Are there any animals whose speeds are similar? Do
these animals have anything in common? Are there any animals whose speeds
are much different than the rest?
· Separate the animals into groups according to the kind of food they eat and
compare the speeds for each group. What conclusions can you make?
· Write a paragraph describing the results of your analysis.
Data
Anatysis
Representative terms from entire chapter:
mathematics curriculum
