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These standards are one
facet of the mathematics education community's response to the call
for reform in the teaching and learning of mathematics.(1)
They reflect, and are an extension of, the community's responses
to those demands for change.(2)
Inherent in this document is a consensus that all students need
to learn more, and often different, mathematics and that instruction
in mathematics must be significantly revised.
As a function of NCTM's
leadership in current efforts to reform school mathematics, the
Commission on Standards for School Mathematics was established by
the Board of Directors and charged with two tasks:
- Create a coherent vision
of what it means to be mathematically literate both in a world
that relies on calculators and computers to carry out mathematical
procedures and in a world where mathematics is rapidly growing
and is extensively being applied in diverse fields.
- Create a set of standards
to guide the revision of the school mathematics curriculum and
its associated evaluation toward this vision.
The Working Groups of the
commission prepared the Standards in response to this charge.
This report is organized
into six sections. This Introduction describes the need for standards,
discusses the need for new goals, and presents an overview of the
standards. The body of the report presents the standards themselves,
organized into four distinct sections: K-4, 5-8, 9-12, and Evaluation.
The concluding section outlines the steps necessary to accomplish
the needed reform of school mathematics.
Key terms used in the development
of this document include these three:
Curriculum.
A curriculum is an operational plan for instruction that details
what mathematics students need to know, how students are to achieve
the identified curricular goals, what teachers are to do to help
students develop their mathematical knowledge, and the context in
which learning and teaching occur. In this document, the term describes
what many would label as the "intended curriculum" or
the "plan for a curriculum."
Evaluation.
Standards have been articulated for evaluating both student performance
and curricular programs, with an emphasis on the role of evaluative
measures in gathering information on which teachers can base subsequent
instruction. The standards also acknowledge the value of gathering
information about student growth and achievement for research and
administrative purposes.
Standard.
A standard is a statement that can be used to judge the quality
of a mathematics curriculum or methods of evaluation. Thus, standards
are statements about what is valued.
Historically
there have been three reasons for groups to formally adopt a set
of standards: (1) to ensure quality, (2) to indicate goals, and
(3) to promote change. For NCTM, all three reasons are of equal
importance.
First, standards often
are used to ensure that the public is protected from shoddy products.
For example, a druggist is not allowed to sell a drug unless it
meets certain very rigid standards that include both the control
of how it was produced and evidence of its effectiveness. Standards
in this sense are minimal criteria for quality. They set necessary,
but not sufficient, conditions for producing desired results. There
is no guarantee that a drug will not be misused or will produce
expected results.
Second, standards often
are used as a means of expressing expectations about goals. Goals
are broad statements of social intent. For example, we can agree
that two goals for all tests are that they should be both valid
and reliable. The standards for tests developed by the American
Psychological Association in 1974 describe the kind of documentation
that publishers should provide about the reliability and validity
of each test.
Third, standards often
are set to lead a group toward some new desired goals. For example,
the medical profession has adopted and periodically updates standards
for the licensing of specialists based on changes in technology,
research, and so on. The intent is to improve or update practices
when necessary. In this sense, standards should be seen as "criteria
for excellence." They are based on an informed vision of what
should be done given current knowledge and experience.
Standards are needed for
school mathematics for all three purposes. Schools, teachers, students,
and the public at large currently enjoy no protection from shoddy
products. It seems reasonable that anyone developing products for
use in mathematics classrooms should document how the materials
are related to current conceptions of what content is important
to teach and should present evidence about their effectiveness.
For NCTM the development of standards as statements of criteria
for excellence in order to produce change was the focus. Schools,
and in particular school mathematics, must reflect the important
consequences of the current reform movement if our students are
to be adequately prepared to live in the twenty-first century. The
standards should be viewed as facilitators of reform.
Our vision of mathematical
literacy is based on a reexamination of educational goals. Historically,
societies have established schools to--
- transmit aspects of the
culture to the young;
- direct students toward,
and provide them with, an opportunity for self-fulfillment.
Thus, the goals all schools
try to achieve are both a reflection of the needs of society and
the needs of students.
Calls for reform in school
mathematics suggest that new goals are needed. All industrialized
countries have experienced a shift from an industrial to an information
society, a shift that has transformed both the aspects of mathematics
that need to be transmitted to students and the concepts and procedures
they must master if they are to be self-fulfilled, productive citizens
in the next century.
The Information Society.
This social and economic shift can be attributed, at least in part,
to the availability of low-cost calculators, computers, and other
technology. The use of this technology has dramatically changed
the nature of the physical, life, and social sciences; business;
industry; and government. The relatively slow mechanical means of
communication--the voice and the printed page--have been supplemented
by electronic communication, enabling information to be shared almost
instantly with persons--or machines--anywhere. Information is the
new capital and the new material, and communication is the new means
of production. The impact of this technological shift is no longer
an intellectual abstraction. It has become an economic reality.
Today, the pace of economic change is being accelerated by continued
innovation in communications and computer technology.
New Societal Goals.
Schools, as now organized, are a product of the industrial age.
In most democratic countries, common schools were created to provide
most youth the training needed to become workers in fields, factories,
and shops. As a result of such schooling, students also were expected
to become literate enough to be informed voters. Thus, minimum competencies
in reading, writing, and arithmetic were expected of all students,
and more advanced academic training was reserved for the select
few. These more advantaged students attended the schools that were
expected to educate the future cultural, academic, business, and
government leaders.
The educational system
of the industrial age does not meet the economic needs of today.
New social goals for education include (1) mathematically literate
workers, (2) lifelong learning, (3) opportunity for all, and (4)
an informed electorate. Implicit in these goals is a school system
organized to serve as an important resource for all citizens throughout
their lives.
1. Mathematically literate
workers. The economic status quo in which factory employees
work the same jobs to produce the same goods in the same manner
for decades is a throwback to our industrial-age past. Today, economic
survival and growth are dependent on new factories established to
produce complex products and services with very short market cycles.
It is a literal reality that before the first products are sold,
new replacements are being designed for an ever-changing market.
Concurrently, the research division is at work developing new ideas
to feed to the design groups to meet the continuous clamor for new
products that are, in turn, channeled into the production arena.
Traditional notions of basic mathematical competence have been outstripped
by ever-higher expectations of the skills and knowledge of workers;
new methods of production demand a technologically competent work
force. The U.S. Congressional
Office of Technology Assessment (1988) claims that employees
must be prepared to understand the complexities and technologies
of communication, to ask questions, to assimilate unfamiliar information,
and to work cooperatively in teams. Businesses no longer seek workers
with strong backs, clever hands, and "shopkeeper" arithmetic
skills. In fact, it is claimed that the "most significant growth
in new jobs between now and the year 2000 will be in fields requiring
the most education" (Lewis
1988, p. 468). Henry Pollak
(1987), a noted industrial mathematician,
recently summarized the mathematical expectations for new employees
in industry:
- The ability to set up
problems with the appropriate operations
- Knowledge of a variety
of techniques to approach and work on problems
- Understanding of the
underlying mathematical features of a problem
- The ability to work with
others on problems
- The ability to see the
applicability of mathematical ideas to common and complex problems
- Preparation for open
problem situations, since most real problems are not well formulated
- Belief in the utility
and value of mathematics
Notice the difference between
the skills and training inherent in these expectations and those
acquired by students working independently to solve explicit sets
of drill and practice exercises. Although mathematics is not taught
in schools solely so students can get jobs, we are convinced that
in-school experiences reflect to some extent those of today's workplace.
This is especially true given that the availability of such broadly
educated workers will be a major factor in determining how businesses
respond to today's changing economic conditions.
2. Lifelong learning.
Employment counselors, cognizant of the rapid changes in technology
and employment patterns, are claiming that, on average, workers
will change jobs at least four to five times during the next twenty-five
years and that each job will require retraining in communication
skills. Thus, a flexible workforce capable of lifelong learning
is required; this implies that school mathematics must emphasize
a dynamic form of literacy. Problem solving--which includes the
ways in which problems are represented, the meanings of the language
of mathematics, and the ways in which one conjectures and reasons--must
be central to schooling so that students can explore, create, accommodate
to changed conditions, and actively create new knowledge over the
course of their lives.
3. Opportunity
for all. The social injustices of past schooling practices can
no longer be tolerated. Current statistics indicate that those who
study advanced mathematics are most often white males. Women and
most minorities study less mathematics and are seriously underrepresented
in careers using science and technology. Creating a just society
in which women and various ethnic groups enjoy equal opportunities
and equitable treatment is no longer an issue. Mathematics has become
a critical filter for employment and full participation in our society.
We cannot afford to have the majority of our population mathematically
illiterate: Equity has become an economic necessity.
4. Informed electorate.
In a democratic country in which political and social decisions
involve increasingly complex technical issues, an educated, informed
electorate is critical. Current issues--such as environmental protection,
nuclear energy, defense spending, space exploration, and taxation--involve
many interrelated questions. Their thoughtful resolution requires
technological knowledge and understanding. In particular, citizens
must be able to read and interpret complex, and sometimes conflicting,
information.
In summary,
today's society expects schools to insure that all students have
an opportunity to become mathematically literate, are capable of
extending their learning, have an equal opportunity to learn, and
become informed citizens capable of understanding issues in a technological
society. As society changes, so must its schools.
New
Goals for Students. Educational goals for students must
reflect the importance of mathematical literacy. Toward this end,
the K-12 standards articulate five general goals for all students:
(1) that they learn to value mathematics, (2) that they become confident
in their ability to do mathematics, (3) that they become mathematical
problem solvers, (4) that they learn to communicate mathematically,
and (5) that they learn to reason mathematically. These goals imply
that students should be exposed to numerous and varied interrelated
experiences that encourage them to value the mathematical enterprise,
to develop mathematical habits of mind, and to understand and appreciate
the role of mathematics in human affairs; that they should be encouraged
to explore, to guess, and even to make and correct errors so that
they gain confidence in their ability to solve complex problems;
that they should read, write, and discuss mathematics; and that
they should conjecture, test, and build arguments about a conjecture's
validity.
The opportunity
for all students to experience these components of mathematical
training is at the heart of our vision of a quality mathematics
program. The curriculum should be permeated with these goals and
experiences so that they become commonplace in the lives of students.
We are convinced that if students are exposed to the kinds of experiences
outlined in the Standards, they will gain mathematical
power. This term denotes an individual's abilities to explore,
conjecture, and reason logically, as well as the ability to use
a variety of mathematical methods effectively to solve nonroutine
problems. This notion is based on the recognition of mathematics
as more than a collection of concepts and skills to be mastered;
it includes methods of investigating and reasoning, means of communication,
and notions of context. In addition. for each individual, mathematical
power involves the development of personal self-confidence.
Toward this end, we see
classrooms as places where interesting problems are regularly explored
using important mathematical ideas. Our premise is that what
a student learns depends to a great degree on how he or she
has learned it. For example, one could expect to see students recording
measurements of real objects, collecting information and describing
their properties using statistics, and exploring the properties
of a function by examining its graph. This vision sees students
studying much of the same mathematics currently taught but with
quite a different emphasis; it also sees some mathematics being
taught that in the past has received little emphasis in schools.
1. Learning to value
mathematics. Students should have numerous and varied experiences
related to the cultural, historical, and scientific evolution of
mathematics so that they can appreciate the role of mathematics
in the development of our contemporary society and explore relationships
among mathematics and the disciplines it serves: the physical and
life sciences, the social sciences, and the humanities.
Throughout the history
of mathematics, practical problems and theoretical pursuits have
stimulated one another to such an extent that it is impossible to
disentangle them. Even today, as theoretical mathematics has burgeoned
in its diversity and deepened in its complexity and abstraction,
it has become more concrete and vital to our technologically oriented
society. It is the intent of this goal--learning to value mathematics--to
focus attention on the need for student awareness of the interaction
between mathematics and the historical situations from which it
has developed and the impact that interaction has on our culture
and our lives.
2. Becoming confident
in one's own ability. As a result of studying mathematics, students
need to view themselves as capable of using their growing mathematical
power to make sense of new problem situations in the world around
them. To some extent, everybody is a mathematician and does mathematics
consciously. To buy at the market, to measure a strip of wallpaper,
or to decorate a ceramic pot with a regular pattern is doing mathematics.
School mathematics must endow all students with a realization that
doing mathematics is a common human activity. Having numerous and
varied experiences allows students to trust their own mathematical
thinking.
3. Becoming a mathematical
problem solver. The development of each student's ability to
solve problems is essential if he or she is to be a productive citizen.
We strongly endorse the first recommendation of An Agenda for
Action (National Council
of Teachers of Mathematics 1980): "Problem solving must
be the focus of school mathematics" (p. 2). To develop such
abilities, students need to work on problems that may take hours,
days, and even weeks to solve. Although some may be relatively simple
exercises to be accomplished independently, others should involve
small groups or an entire class working cooperatively. Some problems
also should be open-ended with no right answer, and others need
to be formulated.
4. Learning to communicate
mathematically. The development of a student's power to use
mathematics involves learning the signs, symbols, and terms of mathematics.
This is best accomplished in problem situations in which students
have an opportunity to read, write, and discuss ideas in which the
use of the language of mathematics becomes natural. As students
communicate their ideas, they learn to clarify, refine, and consolidate
their thinking.
5. Learning to reason
mathematically. Making conjectures, gathering evidence, and
building an argument to support such notions are fundamental to
doing mathematics. In fact, a demonstration of good reasoning should
be rewarded even more than students' ability to find correct answers.
In summary, the intent
of these goals is that students will become mathematically literate.
This term denotes an individual's ability to explore, to conjecture,
and to reason logically, as well as to use a variety of mathematical
methods effectively to solve problems. By becoming literate, their
mathematical power should develop.
This document presents
fifty-four standards divided among four categories: grades K-4,
5-8, 9-12, and evaluation. The four categories are arbitrary in
that they are not intended to reflect school structure; in fact,
we encourage readers to consider these as K-12 standards. In addition,
we believe that similar standards need to be developed for both
preschool programs and those beyond high school.
It was our task to prepare
the curriculum and evaluation standards that reflect our vision
of how the societal and student goals already articulated here could
be met. These standards should be seen as an initial step in the
lengthy process of bringing about reform in school mathematics.
Curriculum Standards.
When a set of curricular standards is specified for school mathematics,
it should be understood that the standards are value judgments based
on a broad, coherent vision of schooling derived from several factors:
societal goals, student goals, research on teaching and learning,
and professional experience. Each standard starts with a statement
of what mathematics the curriculum should include. This is followed
by a description of the student activities associated with that
mathematics and a discussion that includes instructional examples.
Mathematics.
The first consideration in preparing each standard was its mathematical
content. To decide on what is fundamental in so vast and dynamic
a discipline as mathematics is no easy task. John
Dewey's (1916) distinction between "knowledge" and
the "record of knowledge" may clarify this point. For
many, "to know" means to identify the basic concepts and
procedures of the discipline. For many nonmathematicians, arithmetic
operations, algebraic manipulations, and geometric terms and theorems
constitute the elements of the discipline to be taught in grades
K-12. This may reflect the mathematics they studied in school or
college rather than a clear insight into the discipline itself.
Three features of mathematics
are embedded in the Standards. First, "knowing" mathematics
is "doing" mathematics. A person gathers, discovers, or
creates knowledge in the course of some activity having a purpose.
This active process is different from mastering concepts and procedures.
We do not assert that informational knowledge has no value, only
that its value lies in the extent to which it is useful in the course
of some purposeful activity. It is clear that the fundamental concepts
and procedures from some branches of mathematics should be known
by all students; established concepts and procedures can be relied
on as fixed variables in a setting in which other variables may
be unknown. But instruction should persistently emphasize "doing"
rather than "knowing that."
Second, some aspects of
doing mathematics have changed in the last decade. The computer's
ability to process large sets of information has made quantification
and the logical analysis of information possible in such areas as
business, economics, linguistics, biology, medicine, and sociology.
Change has been particularly great in the social and life sciences.
In fact, quantitative techniques have permeated almost all intellectual
disciplines. However, the fundamental mathematical ideas needed
in these areas are not necessarily those studied in the traditional
algebra-geometry-precalculus-calculus sequence, a sequence designed
with engineering and physical science applications in mind. Because
mathematics is a foundation discipline for other disciplines and
grows in direct proportion to its utility, we believe that the curriculum
for all students must provide opportunities to develop an understanding
of mathematical models, structures, and simulations applicable to
many disciplines.
Third,
changes in technology and the broadening of the areas in which mathematics
is applied have resulted in growth and changes in the discipline
of mathematics itself. Davis and
Hersh (1981) claim that we are now in a golden age of mathematical
production, with more than half of all mathematics having been invented
since World War II. In fact, they argue that "there are two
inexhaustible sources of new mathematical questions. One source
is the development of science and technology, which make ever new
demands on mathematics for assistance. The other source is mathematics
itself ... each new, completed result becomes the potential starting
point for several new investigations" (p. 25). The new technology
not only has made calculations and graphing easier, it has changed
the very nature of the problems important to mathematics and the
methods mathematicians use to investigate them. Because technology
is changing mathematics and its uses, we believe that--
- appropriate calculators
should be available to all students at all times;
- a computer should be
available in every classroom for demonstration purposes;
- every student should
have access to a computer for individual and group work;
- students should learn
to use the computer as a tool for processing information and performing
calculations to investigate and solve problems.
We recognize, however,
that access to this technology is no guarantee that any student
will become mathematically literate. Calculators and computers for
users of mathematics, like word processors for writers, are tools
that simplify, but do not accomplish, the work at hand. Thus, our
vision of school mathematics is based on the fundamental mathematics
students will need, not just on the technological training that
will facilitate the use of that mathematics.
Similarly, the availability
of calculators does not eliminate the need for students to learn
algorithms. Some proficiency with paper-and-pencil computational
algorithms is important, but such knowledge should grow out of the
problem situations that have given rise to the need for such algorithms.
Furthermore, when one needs to calculate to find an answer to a
problem, one should be aware of the choices of methods (see
fig. 1). When an approximate answer is adequate, one should
estimate. If a precise answer is needed, an appropriate procedure
must be chosen. Many problems should be solved by mental calculation
(multiplying by 10, taking half). Some calculations, if not too
complex, should be solved by following standard paper-and-pencil
algorithms. For more complex calculations, the calculator should
be used (column addition, long division). And finally, if many iterative
calculations are required, a computer program should be written
or used to find answers (finding a sum of squares). Note in figure
1 that estimation can, and should, be used in conjunction with
procedures yielding exact answers to foreshadow any calculation
and to judge the reasonableness of results.
Fig. 1. Decisions
about calculation procedures in numerical problems
Contrary to the fears of
many, the availability of calculators and computers has expanded
students' capability of performing calculations. There is no evidence
to suggest that the availability of calculators makes students dependent
on them for simple calculations. Students should be able to decide
when they need to calculate and whether they require an exact or
approximate answer. They should be able to select and use the most
appropriate tool. Students should have a balanced approach to calculation,
be able to choose appropriate procedures, find answers, and judge
the validity of those answers.
Finally, in developing
the standards, we considered the content appropriate for all
students. This, however, does not suggest that we believe all students
are alike. We recognize that students exhibit different talents,
abilities, achievements, needs, and interests in relationship to
mathematics. The mathematical content outlined in the Standards
is what we believe all students will need if they are to be productive
citizens in the twenty-first century. If all students do not have
the opportunity to learn this mathematics, we face the danger of
creating an intellectual elite and a polarized society. The image
of a society in which a few have the mathematical knowledge needed
for the control of economic and scientific development is not consistent
either with the values of a just democratic system or with its economic
needs.
We believe that all students
should have an opportunity to learn the important ideas of mathematics
expressed in these standards. On the one hand, prior to grade 9,
we have refrained from specifying alternative instructional patterns
that would be consistent with our vision. On the other hand, for
grades 9-12, the standards have been prepared in light of a core
program for all students, with explicit differentiation in terms
of depth and breadth of treatment and the nature of applications
for college-bound students. At the same time, the mathematics of
the core program is sufficiently broad and deep so that students'
options for further study would not be limited. Our expectation
is that all students must have an opportunity to encounter typical
problem situations related to important mathematical topics. However,
their experiences may differ in the vocabulary or notations used,
the complexity of arguments, and so forth.
Student Activities.
The second aspect of each standard specifies the expected student
activities associated with doing mathematics. Two general principles
have guided our descriptions: First, activities should grow out
of problem situations; and second, learning occurs through active
as well as passive involvement with mathematics.
Traditional teaching emphases
on practice in manipulating expressions and practicing algorithms
as a precursor to solving problems ignore the fact that knowledge
often emerges from the problems. This suggests that instead of the
expectation that skill in computation should precede word problems,
experience with problems helps develop the ability to compute. Thus,
present strategies for teaching may need to be reversed; knowledge
often should emerge from experience with problems. In this way,
students may recognize the need to apply a particular concept or
procedure and have a strong conceptual basis for reconstructing
their knowledge at a later time.
Furthermore, students need
to experience genuine problems regularly. A genuine problem is a
situation in which, for the individual or group concerned, one or
more appropriate solutions have yet to be developed. The situation
should be complex enough to offer challenge but not so complex as
to be insoluble. In sum, we believe that learning should be guided
by the search to answer questions--first at an intuitive, empirical
level; then by generalizing; and finally by justifying (proving).
In many classrooms, learning
is conceived of as a process in which students passively absorb
information, storing it in easily retrievable fragments as a result
of repeated practice and reinforcement. Research findings from psychology
indicate that learning does not occur by passive absorption alone
(Resnick 1987). Instead,
in many situations individuals approach a new task with prior knowledge,
assimilate new information, and construct their own meanings. For
example, before young children are taught addition and subtraction,
they can already solve most addition and subtraction problems using
such routines as "counting on" and "counting back"
(Romberg and Carpenter 1986).
As instruction proceeds, children often continue to use these routines
in spite of being taught more formal problem-solving procedures.
They will accept new ideas only when their old ideas do not work
or are inefficient. Furthermore, ideas are not isolated in memory
but are organized and associated with the natural language that
one uses and the situations one has encountered in the past. This
constructive, active view of the learning process must be reflected
in the way much of mathematics is taught. Thus, instruction should
vary and include opportunities for--
- appropriate project work;
- group and individual
assignments;
- discussion between teacher
and students and among students;
- practice on mathematical
methods;
- exposition by the teacher.
Our ideas about problem
situations and learning are reflected in the verbs we use to describe
student actions (e.g., to investigate, to formulate, to find, to
verify) throughout the Standards.
Focus and Discussion.
Finally, our vision sees teachers encouraging students, probing
for ideas, and carefully judging the maturity of a student's thoughts
and expressions. Hence, each standard is elaborated on in a Focus
section followed by a discussion with examples, which is meant to
convey the spirit of this vision about both mathematical content
and instruction.
Another premise of the
standards is that problem situations must keep pace with the maturity--both
mathematical and cultural--and experience of the students. For example,
the primary grades should emphasize the empirical language of the
mathematics of whole numbers, common fractions, and descriptive
geometry. In the middle grades, empirical mathematics should be
extended to other numbers, and the emphasis should shift to building
the abstract language of mathematics needed for algebra and other
aspects of mathematics. High school mathematics should emphasize
functions, their representations and uses, modeling, and deductive
proofs.
The standards specify that
instruction should be developed from problem situations. As long
as the situations are familiar, conceptions are created from objects,
events, and relationships in which operations and strategies are
well understood. In this way, students develop a framework of support
that can be drawn upon in the future, when rules may well have been
forgotten but the structure of the situation remains embedded in
the memory as a foundation for reconstruction. Situations should
be sufficiently simple to be manageable but sufficiently complex
to provide for diversity in approach. They should be amenable to
individual, small-group, or large-group instruction, involve a variety
of mathematical domains, and be open and flexible as to the methods
to be used.
The first
three standards in each section are labeled Problem Solving, Communication,
and Reasoning, although details vary between levels with respect
to what is expected both of students and of instruction. This variation
reflects the developmental level of the students, their mathematical
background, and the specific mathematical content.
The fourth curriculum standard
at each level is titled Mathematical Connections. This label emphasizes
our belief that although it is often necessary to teach specific
concepts and procedures, mathematics must be approached as a whole.
Concepts, procedures, and intellectual processes are interrelated.
In a significant sense, "the whole is greater than the sum
of its parts." Thus, the curriculum should include deliberate
attempts, through specific instructional activities, to connect
ideas and procedures both among different mathematical topics and
with other content areas. Following the Connections standard, nine
or ten specific content standards are stated and discussed. Some
have similar titles, which reflects that a content area needs emphasis
across the curriculum; however, once again the concepts and processes
emphasized vary by level. Others emphasize specific content that
needs to be developed at that level.
The Evaluation Standards.
The evaluation standards are presented separately, not because evaluation
should be separated from the curriculum, but because planning for
the gathering of evidence about student and program outcomes is
different. The difference is most clearly illustrated in comparing
the curriculum standards titled Connections and the evaluation standards
titled Mathematical Power. Both deal with connections among concepts,
procedures, and intellectual methods, but the curriculum standards
are related to the instructional plan whereas the evaluation standards
address the ways in which students integrate these connections intellectually
so that they develop mathematical power.
We present fourteen evaluation
standards that can be viewed in three categories. The first set
of three evaluation standards discusses general assessment strategies
related to the curriculum standards. The second seven focus on providing
information to teachers for instructional purposes. They closely
parallel the curriculum standards--problem solving, communication,
reasoning, mathematical concepts, and mathematical procedures, in
addition to a separate standard on "mathematical disposition."
These seven standards are to be used by teachers to make judgments
about students and their mathematical progress. The final set of
four standards addresses the gathering of evidence with respect
to the quality of the mathematics program. These standards are to
be used by teachers, administrators, and policy makers to make judgments
about the quality of the mathematics program and the effectiveness
of instruction.
Such are the background,
the general focus, and the intent of our efforts. It is now left
to each of you concerned with the teaching and learning of mathematics
to read the standards, to share them with colleagues, and to reflect
on their vision. Consider what needs to be done and what you can
do, and collaborate with others to implement the standards for the
benefit of our students, as well as for our social and economic
future.
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