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others both orally
and in writing the
results of mathe-
matical investiga-
tions and problem solving. The mathematics curriculum,
therefore, must provide appropriate contexts in which
students can learn to read, write, and speak about math-
ematics.
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Principle 2: Calculators and computers should be used
throughout the mathematics curriculum.
Students will achieve mathematical power only if they
see mathematics as a modern, relevant subject. New
curricular materials must be designed in the expectation
of continuous change resulting from further scientific And
technological clevelopments. In mathematics, under-
stancling cannot generally be achieved without active
participation in the actual process of mathematics-in
conjecture and argument, in exploration and reasoning,
in formulating and solving, in calculation and verification,
Calculators function like "fast pencils," so the mathemati-
cal process can be made more useful and efficient than
with paper and pencil. Computers, similarly, enable stu-
clents to quickly calculate, graph, or simulate processes
that are simply impossible to carry out by any other
means. Instruction based on calculators and computers
has, therefore, the potential to lead to more understand-
ing than does traditional instruction,
Calculators and computers also appeal to teachers
because they introduce excitement and inventiveness to
otherwise routine courses. Of course, technology should
not be used just because it is appealing. But it must be
used when it can enhance the teaching and learning of
mathematics. There are very few portions of the curricu-
lum where such improvement is not possible.
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Reshaping School Mathematics
Principle 3: Relevant applications should be an integral part
of the curriculum.
Students need to experience mathematical ideas in
the context in which they naturally arise-from simple
counting and measurement to applications in business
and science. Calculators and computers make it possible
now to introduce realistic applications throughout the
curriculum.
The significant criterion for the suitability of an applica-
tion is whether it has the potential to engage students'
interests and stimulate their mathematical thinking.
Appealing applications should be cirown from the world
in which the child lives, from community events, or from
other parts of the curriculum-and not just from science,
but also from business, geography, art, and other sub-
jects.
The primary goal of instruction should be for students to
learn to use mathematical tools in contexts that mirror
their use in actual situations, Mathematical ideas should
always be presented and developed in the context of
meaningful mathematical activities.
Principle 4: Each part of the curriculum should be justified on
its own merits.
Mothematics offers such a rich array of interesting and
useful ideas that choices are necessarily difficult, Howev-
er, no concept or skill should remain in the curriculum just
because it is there now. Although there is much that is
timeless in the present curriculum, we can no longer
afford as the chief justification for a topic that it is in the
curriculum already. We need, instead, a "zero-based" cur-
riculum process in which no idea is immune from careful
scrutiny,
Revision of curriculum should not be just an exercise in
adding more topics. It should be, rather, a discipline of
establishing priorities. Some emphases should be
dropped, others added, and some retainecl. Even for
important priorities that do remain, modern applications
or technology may suggest quite different approaches.
Often a fresh approach can avoid the rigidity of thought
that inhibits desirable change,
In many clistricts, the secondary mathematics curricu-
lum especially is already full of topics, many treated too
quickly, Nevertheless, there is much mathematics that
could be made accessible and interesting to students,
Unless traditional topics contribute directly to curricular
goals, they clutter the curriculum, Neither the number of
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39
A Philosophy and Framework
topics nor the nature of the topics is as important as a
curriculum that instills in students firm command of math-
ematical thinking.
Principle 5: Curricular choices should be consistent with con-
temporary standards for schoo/ mathematics.
The new Curriculum and Evaluation Standards for
School Mathematics (National Council of Teachers of
Mathematics (NCTM), 1989) exemplify the kind of broach
curricular standards that should be used as criteria for
assessing the merit of topics in school mathematics, Simi-
lar stanclarcis are expressed in many recent state docu-
ments, for example, California (Denham and O'Malley,
1985) and Wisconsin (Chambers, 1986), Although differing
in many cletails, these various clocuments establish signifi-
cant new goals for effective school mathematics, Local
choices should be made in a manner consistent with
these curricular stancdarcis, The pace of change is so
great that even current curriculum guides are likely to be
inadequate for tomorrow's needs, Curricular change
requires sustained effort by people of vision, rooted in the
reality of schools, yet with objectives firmly set on the
future.
Principle 6: Mathematics instruction at all levels should foster
active student involvement.
The proper use of technology requires new approach-
es to teaching mathematics in which students will be
much more active learners. Quite asicle from technology,
research on how students learn suggests more effective
ways to teach mathematics. Mathematics teaching must
adapt to both of these clevelopments. It will no longer be
appropriate for most mathematics instruction to be in the
traditional mode where teachers present material to a
class of passive students.
No single teaching method nor any single kind of learn-
ing experience can develop the varied mathematical
abilities implied under the definition of mathematical
power (Fey, 1979; Mathematical Sciences Education
Board (MSEB], 1987~. What is needed is a variety of activi-
ties, including discussion among pupils' practical work,
practice of important techniques, problem solving, appli-
cation of everyday situations, investigational work, and
exposition by the teacher.
Teachers should be catalysts who help students learn
to think for themselves. They should not act solely as train-
ers whose role is to show the "right way" to solve problems.
In aciclition, classroom activities should provide ample
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Reshaping School Mathematics
opportunity for students to communicate with each other
using the language of mathematics in both written and
oral form,
A useful metaphor is that of the teacher as an intellec-
tual coach. At various times, this will require that the
teacher be:
· A role model who demonstrates not just the right
way, but also the false starts and higher-orcler think-
ing skills that lead to the solution of problems;
· A consultant who helps inclividuals, small groups, or
the whole class to decide if their work is keeping to
the subject and making reasonable progress;
· A moderator who poses questions to consider, but
leaves much of. the decision making to the class;
· An interlocutor who supports students during class
presentations, encouraging them to reflect on their
activities and to explore mathematics on their own;
· A questioner who challenges students to make sure
that what they are doing is reasonable and purpose-
ful, and ensures that students can defend their con-
clusions.
All these roles serve well the most important aim of
education, namely, to wean students from their teachers.
Mathematics education must aim to make students self-
sufficient, so that they can use mathematics effectively
without a teacher present, as they must once they leave
the school environment. To do this, schools must foster suf-
ficient independence in students that they can function
mathematically on their own while still in school, or there
will be no possibility of their doing so after leaving school.
Goals
By themselves, general principles provide insufficient direc-
tion to help focus curriculum development. More specific
goals, related to the new Curriculum and Evaluation Stan-
clarcis for School Mathematics (NCTM, 1989), must be built on
the foundation provided by our expression of a practical phi-
losophy of mathematics as a language and science of pat-
terns, and on the related technological and research per-
spectives. These goals offer a more constructive framework for
the process of curricular change.
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A Philosophy and Framework
Mathematics education needs to be viewed as an integrat-
ed whole, progressing continually from primary school through
graduate school. Students learn in different ways and at differ-
ent rates, in different directions and at different depths. Such
differences cut across grade levels and school-level bound-
aries. Many areas of mathematics, not just arithmetic, algebra,
and geometry, should be seen as lengthy strands to be woven
throughout all of school mathematics.
Notwithstanding the continuity of mathematics, goals for
different levels of school must reflect different stages in chil-
dren's development of mathematical power. As new curricula
are developed to meet the challenges of a changing society,
they must strive to achieve certain broad goals that form an
effective framework for school mathematics:
A primary goal of elementary school mathematics is to clevel-
op number sense.
Student abilities to reason effectively with numerical
information requires experience with:
· Representation-the ability to use numbers to
express quantitative data and relations,
· Operations mastery of single-digit arithmetic;
ability to determine appropriate arithmetic proce-
dures; facility in estimation; experience in select
ing appropriate means to carry out complex cal-
culations.
Interpretatio~the ability to draw inferences from
data and check both the data and the infer-
ences for accuracy and reasonableness.
Elementary school mathematics should use con-
crete materials, computer software, and calculators. It
should emphasize mental arithmetic, particularly for
estimating the results of multidigit computations. At the
same time there should be a sharp reduction in time
devoted to teaching the traditional written methods of
calculation for multidigit numbers, fractions, and deci-
mals.
An elementary school curriculum that approaches
arithmetic from this perspective will be strikingly differ-
ent from the arithmetic commonly taught today. The
central mathematical task of today's elementary
school is to develop manual skill in a wide variety of
operations on whole numbers, rational fractions, and
decimals. Reducing emphasis on these topics while
increasing opportunities for reasoning, for discovering
patterns, for identifying correct procedures, and for
drawing inferences will require a fundamental change
in the conditions of teaching. A school mathematics
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Reshaping School Mathematics
program with this kind of emphasis offers the promise of
impressive progress in the level of quantitative reason-
ing.
Elementary school mathematics should provide an effective
foundation for all aspects of mathematics.
If students are to be better prepared mathematical-
ly for vocations as well as for everyday life, the elemen-
tary school mathematics must include substantial sub-
ject matter other than arithmetic
· Geometry, including properties of two- and three-
dimensional objects, symmetry and congruence,
constructions of geometric figures, and transfor-
mations of geometric figures;
· Measurement, including units of measure, telling
time, reading temperatures, and computing with
money;
· Data analysis, including collection, organization,
representation, and interpretation of data; con-
struction of statistical tables and diagrams; and
the use of data for analytic and predictive pur-
poses;
Probability, introduced with simple experiments
and data-gathering;
· Discrete mathematics, including basic combina-
torial thinking and the use of graphs to model
problems.
Each of these topics can play a distinctive role in
making the elementary school mathematics curriculum
more interesting and relevant to stuclents. Geometry
provides an obvious window on the physical world,
now enhanced through computer graphics, In mathe-
matics as in life, a picture is worth a thousand words,
Measurement provides meaningful applications even
to very young children, as well as reinforcement of
number concepts. Data analysis provides a source of
interesting and relevant problems, as does probability,
which can also be related to familiar games, Concepts
from algebra can introduce students to simple aspects
of abstraction, while discrete mathematics provides
topics to relate mathematics to many areas, particular-
ly computers.
Moreover, instruction should be integrated so that
relations among different areas will be perceived and
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A Philosophy and Framework
reinforcecl. For example, teachers should stress the use
of arithmetic in geometry and probability, and the use
of geometric concepts in the representation of data.
Calculators should be available in all instructional and assess-
ment situations.
Calculators should be used in school mathematics
from kindergarten on as crevices that children use to
develop and discover number relationships and to
solve problems. The replacement of most paper-and-
pencil cirills with calculator-based instruction will not
itself be a panacea. Although it is just as possible to
assign minciless clri~ls with calculators as with paper and
pencil, young children can instead be given activities
with calculators that emphasize discovery and explo-
ration in ways not possible or practical with paper and
pencil.
It is as important as ever for children to learn when
and how to use addition, subtraction, multiplication,
and division. But there is no evidence that drill-and-
practice on standard algorithms leads to understand-
ing. Mathematics educators must take advantage of
ca~culator-based instruction as a tool to help students
to achieve this unclerstancling.
Students /earning mathematics should use real objects and
real data.
Observation is as fundamental to mathematics as to
science. Young chilciren need to manipulate real
objects as they learn to count and to explore arith-
metic, To develop sound intuition for length, area, vol-
ume, and shape, chilciren studying mathematics must
draw, cut, fold, construct, pour, and measure.
Children of ail ages must constantly explore the rela-
tion between the relatively pristine patterns studied in
school mathematics and the messier reoiity of worldly
clata. Real data are more convincing than contrived
data. The act of gathering data-whether by mea-
surement, counting, polls, experiments, or computer
simulation-enriches the chilcl's engagement in learn-
ing. Moreover, the inevitable dialogue that emerges
between the reality of measurement and the reality of
calculation-between the experimental and the theo
retical-captures the whole science of mathematics.
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Reshaping School Mathematics
Middle school mathematics should emphasize the practical
power of mathematics.
If instruction is to give students mathematical power,
then problem solving needs to be emphasized
throughout all grades. Students need to perceive
mathematics as more than the subject matter
itself-as, in fact, a discipline of reasoning that enables
them to attack and solve problems of increasing diffi-
culty and complexity, A focus on problems rather than
just on exercises is important throughout the curriculum.
Broadening the elementary school curriculum has
important implications about entry into secondary
school mathematics. The middle school grades should
not be viewed as a time for consolidation or as a
pause for rest, but as an essential part of a child's
mathematical development. Its focus should be on
mathematics for everyday life, a theme rich in motiva-
tion that leads naturally to many important mathemati-
cal topics (e.g., data analysis, geometric measure-
ment, interest rates, and spreadsheet analysis).
Understanding the concepts of elementary school
mathematics is essential for the study of secondary
school mathematics; however, proficiency in the pro-
cedures of hand arithmetic computation should no
longer be the critical factor in judging student readi-
ness for advanced study.
Mathematics in school should reinforce other school subjects,
and vice versa.
Much of the motivation for the development of
mathematics-both historical and personal- is related
to science, yet in school there are precious few honest
links between mathematics and any of its applications.
The applications of mathematics extend far beyond
the natural sciences-to business, social science,
geography, and various vocational and commercial
areas. Young children can learn much mathematics in
the context of explorations: experience with data,
practice with arithmetic, and exposure to shapes and
change. High school students need to experience
applications in their mathematics classes as well as to
use mathematics extensively in other classes.
Since mathematics is both the language of science
and a science of patterns, the special links between
mathematics and science are far more than just those
between theory and applications. The methodology of
mathematical inquiry shares with the scientific method
a focus on exploration, investigation, conjecture, evi
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A Philosophy and Framework
dence, and reasoning. Firmer school ties between sci-
ence and mathematics should especially help
strengthen students' grasp of both fields.
A major goal of the secondary mathematics curriculum
should be to develop symbol sense.
The transition from elementary to secondary mathe-
matics is characterized by a shift from concrete
objects to abstract symbols. Developing fluency with
symbols and other abstract entities which can be
geometric, algebraic, or algorithmic-must be a cen-
tral aim of secondary school mathematics. Student
ability to reason effectively with symbols requires expe-
rience with:
· Representation-the ability to represent mathe-
matical problems in symbolic form and to use
these symbolic representations in relations, expres-
sions, and equations;
· Operations--the ability to determine appropriate
symbolic procedures and to select appropriate
means to solve problems expressed in symbolic
form;
· Interpretatio~the ability to ciraw inferences by
reasoning with symbolic systems to check these
results for accuracy and reasonableness.
Computers and calcula-
tors have, of course, an
important role to play in
the development of sym-
bol sense. Since powerful
calculators will have just as
profound an effect on how
symbolic manipulations
are done as they have
had on how arithmetic is
done, the current empha-
sis in secondary school on
manipulative skills will need
to be replaced by a larger
emphasis on unclerstand-
ing and problem solving. A valuable impact of tech-
nology on the secondary curriculum will surely be the
development of sophisticated software that will enable
students to discover patterns rather than just to manip-
ulate symbols,
What type of common household phenomenon is
represented by the following graph:
E
a,
4 -
3500 Be/
/
time
Mystery
Graph
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Reshaping School Mathematics
Secondary school mathematics should introduce the entire
spectrum of mathematical sciences.
Secondary school mathematics must prepare stu-
dents for the workplace, for college, and for citizen-
ship, To meet these objectives, the curriculum must
include a broad range of topics reflecting the full
power of the mathematical sciences:
· Algebra, including general algorithms and families
of functions (polynomial, trigonometric, exponen
tial, logarithmic).
· Geometry, including transformational geometry,
vector geometry, solid geometry, and analytic
geometry.
· Data analysis, including measures of uncertainty,
probability and sampling distributions, and inferen-
tial reasoning.
Discrete mathematics, including combinatorics,
graph theory, recurrence relations, and recur-
sion-all emphasizing algorithmic thinking.
Optimization, including mathematical modelling,
"what if" analysis, systems thinking, and network
flows.
Stressing general algorithms in a computer context
will make algebra and trigonometry more interesting.
Despite its reputation as a subject that is boring and
irrelevant, geometry has always been a subject of
great potential interest because of its associations with
the physical world. Data analysis can easily be related
to interesting and significant applications, as can dis-
crete mathematics and optimization.
In teaching mathematics it is important to illustrate
the unity and integrity of the discipline. For example,
fractal geometry is quite accessible to high school stu-
dents and involves aspects of algebra, geometry, and
discrete mathematics, as well as providing fascinating
uses of computers, Data analysis Reacts directly to alge-
braic and geometric methods, while algebra and
geometry themselves are joined in analytic geometry.
The ties that bind topics to each other are often as
important as the topics themselves.
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. . .
A Philosophy and Framework
Students should apprehend that
in mathematics, reasoning is the
standard of truth.
Learning to under-
stancl and construct logi-
cal, coherent mathe-
matical arguments is a
major goal of school
mathematics. Euclidean
geometry, however, is
not the only vehicle for
teaching students about
reasoning. Both algebra
and discrete mathemat-
ics provide excellent
opportunities for argu-
ments expressed in oara-
graph form;
,
even
flowcharts and spread-
sheets can be used to
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OCR for page 48
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Reshaping School Mathematics
disciplines. Mathematics, like writing, is a subject that
should be regularly taught "across the curriculum."
Advocating that all students take essentially the
same core mathematics may appear unrealistic, given
the uneven preparation students bring to high school
mathematics. Indeed, the United States separates stu-
dents by ability earlier-and into more tracks-than
any other industrialized country. Many students, under
this system, are tracked out of real secondary school
mathematics,-being relegated to cleacl-end courses
like "general mathematics." Too little has been expect
ecl for too long from too many stuclents. Recognizing
this, many give up much too early on studying signifi-
cant amounts of mathematics in secondary school.
More must be expected of our students than is the
case now. Heightened expectation will surely lead to
better performance.
All high schools should offer four full years of mathe-
matics appropriate for all students. It is particularly
important for students to take mathematics throughout
their last year of high school. All too open, mathemat-
ics students who complete all of the mathematics
available to them in the eleventh grade find that a
year away from mathematics leaves them ill-prepared
for college mathematics or for job-related require-
ments.
Enabling Conditions
One cannot separate curriculum and instruction from the
brooder context of education, To improve mathematics edu-
cation, change must occur simultaneously in curriculum, in
teaching, in professional development, in textbooks, and in
assessment practices. Although our focus in this Framework is
principally on the content of the curriculum, there are impor-
tant implications of our recommendations for other parts of
the educational context.
Professional Development
No significant curriculum reform will be possible without an
effective program of professional development for mathemat-
ics teachers' As teachers implement important, timely, and
exciting changes, they will require continuing programs of pro.
fessional support. Such programs will require a commitment
from local school districts as well as leadership and funding at
the state and federal levels.
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A Philosophy and Framework
Although the evidence is more anecdotal than analytical, it
is likely that many elementary school teachers would welcome
an opportunity to increase their confidence about teaching
mathematics. Increased confidence would encourage alter-
native styles of teaching that diminish emphasis on formalism
and drill-and-practice, Any curriculum developed according
to this framework will be more challenging to teach than the
present curriculum. Therefore, particular attention must be
paid to in-service training of elementary school teachers, as
well as to the use of specialist mathematics teachers through-
out the elementary school Oracles.
Secondary and middle school teachers who are already
prepared to some extent with a specialization in mathematics
will need extensive continuing education both in new areas of
content that are not part of their present repertoire, and in
styles of teaching better suited to active student participation.
Programs that provide vertical integration of teaching, learn.
ing, and research experiences are well suited to this purpose
since they enable teachers to experience for themselves the
kind of open environment for learning that they should
attempt to create for their stuclents.
However, neither specialist teachers nor special programs to
introduce a new curriculum can by themselves ensure suc-
cess, Mathematics teachers, like other professionals, must
engage in life-long programs of professional development. As
professionals who must keep up with a rapidly changing and
technically complex field, mathematics teachers especially
need time and opportunity to read, to reflect, to plan, and to
exchange ideas with other mathematics teachers. Further-
more, for teachers to succeed with a technology-based cur-
riculum, they will need properly equipped classrooms and
appropriate rewards for the special effort and innovative
teaching that will be required. Teachers' working environments
must support teachers' professional lives: an improved profes-
sional climate for teachers is absolutely critical for improve
ment in mathematics education.
Instructional Materials
In the overwhelming majority of classrooms, the content of
the textbook determines what is taught and how it is taught,
Teachers may skip topics in textbooks, but they will seldom
give significant attention to topics not included in texts, Nei-
ther will most teachers approach a topic differently from the
way it is treated in the text Therefore, real curriculum change
is possible only if it is accompanied by new curricular materials,
New textbooks must be designed and written to reflect the
important principles of mathematics curricula: genuine prob-
lems; calculators and computers; relevant applications; read-
ing and writing about mathematics; and active strategies for
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Reshaping School Mathematics
learning. It is not sufficient for publishers to provide ancillary
software or supplementary materials to be used with a text-
book but that are not discussed in it. Neither will it be accept-
able to relegate new material to separate sections,
Publishers need to assess the entire structure and philosophy
of current textbooks in light of research findings on how chil-
ciren construct mathematical images and how they learn to
make sense of formal procedures. If textbooks began to inte-
arate relevant insights from cognitive research, they would
~ ~ _ ,
begin to exert a positive influence for change In school matn-
ematics, Publishers also need to recognize that mathematics
education will undergo significant and continuous change
well info the next century. As a consequence, textbooks will
become out-of-date more rapidly than in the past. This short-
ening of the useful life of a textbook will require considerable
adjustment in publishers' plans and in school districts' pur-
chase policies.
Textbooks must reflect fully a new conception of mathemat-
ics education, integrating into the main subject matter of the
text all important principles of mathematics curricula and edu-
cational research. Unless textbooks help teachers to use
actively calculators, computers, and genuine problems, new
emphases such as technology and applications will continue
to have insignificant impact on the curriculum, Unless text-
books contain engaging projects and group activities, few
teachers will have time to create them on their own, And
unless textbooks include suitable assignments to enhance stu-
dents' experience with reading and writing in the language of
mathematics, students will remain deficient in their ability to
communicate effectively.
. . . . .
Assessment
Textbooks circumscribe what topics may be taught, but
tests determine what topics will be taught. Too many of
today's standardized tests stray far from both the available
and the adopted curriculum; none even gets near the ideal
curriculum. Just as new text materials must be developed in
parallel with the new curriculum, so also should new strategies
of and standarcis for assessment be clevelopecl as the curricu-
lum is defined.
A curriculum fitting this framework will require methods of
assessment quite different from current ones. Guidelines for
effective assessment are discussed in the Curriculum and Eval-
uation Standarcis for School Mathematics (NCTM, 19891.
Assessment must shape and guide instruction and not remain
separate from it; it must determine not just what students do
not know, but what they do know and how they think. Diag-
nostic materials that probe student understanding can pro-
vicle a springboard for improved instruction. Assessment must
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51
A Philosophy and Framework
permit full use of calculators ancl, where appropriate, of com-
puters. Instruments can be developed to assess mathematical
power rather than merely mathematical skill. But just as careful
assessment of writing cannot be accomplished without having
students write real essays, neither can mathematical power be
assessed unless students have to solve nonroutine problems.
Mathematics for the Future
We are entering a decade in mathematics education of
transition from entrenched precomputer traditions to new
structures appropriate to the twenty-first century. These transi-
tions will inclucle:
Greater breacith of mathematical sciences.
· More students who take more mathematics.
· Increased use of technology.
· More active learning.
· Enhancecl professionalism for teachers.
· Increased need for higher-order thinking skills.
More sophisticated means of assessment.
Effective change requires significant movement in each
area, coordinated and sustained for an indefinite period.
Efforts to orchestrate this change have only just begun, but
must be continued.
The tapestry of mathematics in the twenty-first century will
be woven not just from the ancient threads of algebra and
geometry, but also from more contemporary themes such as
uncertainty, symmetry, data, algorithm, and computation. As
applications expand the variety of roles played by mothemat-
ics, and as computers reduce the role of routine calculations,
the balance and connections among different parts of math-
ematics will change significantly.
Important threads in the tapestry extend throughout the
entire range of the mathematics curriculum, providing rich
opportunity for sustained development of a chilcl's mathemat-
ical intuition and power. They lead to deep themes of contem-
porary mathematics; they point to ideas that explain and unify
the process of mathematical cliscovery; anGI they provide a
secure base for mathematics' many applications. The chal-
lenge for those who develop new mathematics curricula is to
emphasize themes that both advance the power of mathe-
matics and at the same time offer developmental opportunity
for children's mathematical education.
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Representative terms from entire chapter:
elementary school
