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Deciding on the content
of school mathematics is the initial step in the necessary change
process. So that the next steps proceed in harmony with the Standards,
both the nature of the changes needed and the strategy implied for
change should be understood. Given the overwhelmingly positive response
to the Working Draft of the Standards, the commission members
and writers are convinced that hundreds of teachers and other mathematics
educators are eager to change school mathematics. In fact, we are
optimistic that such changes can and will be accomplished.
The Nature of Change.
In any field, when systems are not working, those involved in them
must decide whether the problem is the result of a lapse in quality
control, a design flaw, or a combination of the two. Quality-control
solutions improve the efficiency and effectiveness of what is being
done without disturbing its basic features. Design solutions fundamentally
alter the organization of the systems themselves. Both of these
solutions are applicable to the subject of school reform. Quality-control
changes in education, for example, have included "recruiting
better teachers and administrators, raising salaries, allocating
resources more equitably, selecting better textbooks, adding (or
deleting) content or course work, scheduling people and activities
more efficiently, and introducing new versions of evaluation and
training" (Cuban 1988, p.
342). However, design changes in education "introduce new
goals, structures, and roles that transform familiar ways of doing
things into new ways of solving persistent problems" (Cuban
1988, p. 342).
Given this distinction
about strategies for change, it should be obvious that we see the
Standards as an initial step in a design-change process.
The changes advocated in the Standards should lead to a fundamental
restructuring of the mathematics curriculum and instruction. However,
the redesign we envision cannot be done in a mechanical fashion.
Instead, teachers and other educators must come to a consensus and
work collaboratively to bring about the changes needed. The remaining
steps, and there are many, in the redesign of school mathematics
are based on the following strategy for change.
The Professional
Development Change Strategy. In the past most educational
changes have been approached through a "top down" managerial
strategy borrowed from industry. Managers and experts design new
parts or procedures and then train workers to use them. The change
strategy being advocated here, however, is based on professional
development rather than administrative directives.
Professional development
implies the direct involvement of professional organizations and
their members. The vision of the mathematics curriculum
and the goals in the Standards are offered by NCTM without
a prescription for achieving them. The approach taken is one that
will empower teachers and other educators, through their professional
organizations, to make the changes. Thus, it is the responsibility
of all those involved and interested in school mathematics to implement
reform.
The next steps toward change
should not be considered as linear or exhaustive but rather as steps
along many paths headed in the same direction. Professionals in
different areas will follow different paths to redesign components
for a new system of school mathematics. Some of those paths follow:
Curriculum Development.
The Standards is a framework for curriculum development.
However, it contains neither a scope-and-sequence chart nor a listing
of topics by specific grade level. This is deliberate; a coherent
network of relationships exists among the identified topics, and
multiple paths are available throughout this network. What we have
done is to identify the primary elements, or nodes, of the network
to be included in a quality mathematics curriculum. One possible
next step is for teachers and mathematics educators to develop curricula
based on the standards.
For example, the secondary
school mathematics curriculum has typically been separated into
courses with a specific subject orientation (e.g., algebra, geometry,
statistics). This sequence provides teachers and students with a
single-course focus. We now challenge educators to integrate mathematics
topics across courses so that students can view major mathematical
ideas from more than one perspective and bring interrelated ideas
to bear on new topics or problems. Similarly, texts in grades K-8
often include a variety of mathematical topics but primarily stress
arithmetic. We favor instead a truly integrated curricular organization
in all grades to permit students to develop mathematical power more
readily and to allow the necessary flexibility over time to incorporate
the content of these standards. This integration is intended both
among mathematical topics and with their use in other subjects as
expressed in the "Connections" standards at each level.
Textbooks and Other
Materials. We are aware that the curriculum in many schools
is geared to textbooks, and we expect the standards to be used as
criteria for measuring text content. However, we do not believe
the standards can be met simply by altering current texts (e.g.,
appending a genuine problem at the end of each chapter or inserting
a chapter on statistics). Those are quality-control solutions to
a problem that demands a design solution. Nor do we believe that
textbooks should drive instruction. Rather, other materials that
support the standards, such as manipulatives and courseware, must
be developed, in addition to new textbooks. As an initial step on
this path, NCTM is developing addenda to the standards that will
contain many exemplary instructional activities.
Tests. Tests
have an influence on what actually gets taught in a classroom, especially
in urban areas where teachers know that the test results will be
used, rightly or wrongly, as an evaluation of them. New tests
must be developed to assess problem solving, reasoning, and so on,
in a valid way to ensure that the mathematics intended in the Standards
is taught in all classrooms. Such new testing strategies
must be applied not only to standardized tests but to state testing
programs as well. Finally, until tests provide for the appropriate
use of calculators, many teachers will continue to prohibit their
use in the classroom. Without changes in how mathematics is assessed,
the vision of the mathematics curriculum described in the standards
will not be implemented in classrooms, regardless of how texts or
local curricula change.
Instruction.
The spirit and vision of the Standards cannot be achieved
if instruction is inconsistent with its underlying philosophy. Specifying
the content for a quality mathematics program is impossible without
addressing the accompanying instructional conditions. Thus, the
elaboration of each standard deliberately contains implications
for instruction and includes expectations about teachers' actions,
such as the use of a variety of sequences, grouping procedures,
instructional strategies, and techniques for evaluation. However.
the Standards was not developed as a set of instructional
standards. NCTM has formed several task forces to develop materials
to assist with the delivery of instruction.
Teacher In-Service
Programs. Although we are confident that many teachers are
now ready to teach the kind of mathematics program outlined in the
Standards, many others will need and demand additional training
or refresher courses. These programs must be developed in collaboration
with the teachers. They must include mechanisms for sustained collegial
interaction, links between staff development and classroom practice,
and the participation of administrators to ensure support for the
proposed changes. We challenge states, provinces, and school districts
to use their in-service resources to help teachers in this manner.
Again, NCTM has created an implementation committee to coordinate
and orchestrate the development of in-service materials and programs.
Teacher Education.
Prospective teachers must be taught in a manner similar to how they
are to teach--by exploring, conjecturing, communicating, reasoning,
and so forth. Thus, colleges of education and mathematical sciences
departments should reconsider their teacher preparation programs
in light of these curriculum and evaluation criteria. Teacher education
programs now being redesigned to follow the recommendations of such
groups as the Holmes Group and the Carnegie Commission must be compatible
with the Standards. To implement these standards, all teachers
need an understanding of both the historical development and current
applications of mathematics. Furthermore, they should be familiar
with the power of technology. The NCTM is developing professional
standards for teaching mathematics.
Technology.
Throughout each standard, we have assumed that appropriate technology
is available for classroom instruction. Calculators, computers,
courseware, and manipulative materials are necessary for good mathematics
instruction; the teacher can no longer rely solely on a chalk-board,
chalk, paper, pencils, and a text. Criteria for measuring the adequacy
of materials and equipment must be developed. Note, however, that
simply providing teachers with these materials will not produce
a new program; teachers must also know how to integrate this technology
into a quality mathematics program.
Students with Different
Needs and Interests. The consequences of dealing with students
with different talents, achievements, and interests have led to
such practices as grouping and tracking and to special programs
for gifted or handicapped students who need and deserve special
attention. However, we believe that all students can benefit
from an opportunity to study the core curriculum specified in the
Standards. This can be accomplished by expanding and enriching
the curriculum to meet the needs of each individual student, including
the gifted and those of lesser capabilities and interests. We challenge
teachers and other educators to develop and experiment with course
outlines and grouping patterns to present the mathematics in the
Standards in a meaningful, productive way.
Equity. As
a pluralistic, democratic society, we cannot continue to discourage
women and minority students from the study of mathematics. We believe
that current tracking procedures often are inequitable, and we challenge
all to develop instructional activities and programs to address
this issue directly. One reviewer of the Working Draft of the Standards
suggested the establishment of some pilot school mathematics programs
based on these Standards to demonstrate that all students--including
women and underserved minorities--can reach a satisfactory level
of mathematical achievement and urged that the success of these
students be widely publicized.
Working Conditions.
In too many schools, teachers will find it difficult to teach the
mathematical topics or create the instructional environments envisioned
in these standards because of local constraints, such as directives
about which chapters or pages to cover, inadequate time for instruction,
and the administration of tests. In many grades too little time
is spent on mathematics instruction. Teachers and students should
spend an hour a day on mathematics at all grades and take advantage
of the many opportunities to connect mathematics to other school
subjects.
Teachers also lack the
necessary resources, the time to reflect, and the opportunities
to share ideas with other teachers. Under such conditions, it is
difficult to create a sense of exploration, curiosity, or excitement
in the classroom. Although new standards alone cannot alter these
conditions, they implicitly argue for everyone to make the work
environment for teachers support professional activities.
Research.
The Standards is based on a set of values, or philosophical
positions, about mathematics for students and the way instruction
should proceed. These values both are consistent with current research
findings and establish a new research agenda. In the redesign of
school mathematics, much careful research is needed. Instead of
dealing solely with the study of what is happening in the
teaching and assessment of mathematics instruction, research should
deal more with what ought to be. For example, the Standards
offers curricular and pedagogical support for students as they engage
in mathematical thinking and problem solving. Although considerable
research has dealt with mathematical problem solving, very little
of it has examined some of the main components of problem solving,
such as conjecturing and problem formulation, described here. Therefore,
an examination of these more generative aspects of mathematical
thinking is needed.
In summary, our system
of schooling needs to be redesigned. We challenge all readers to
act, examine these and other constraints of the schooling system,
and work toward aligning them with the Standards.
Today, what happens in
America's classrooms is being given lots of attention and scrutiny.
The reactions and responses to the recent reports on education offer
the mathematics education community a rare opportunity to shape
school mathematics during the next decade. Public interest and concern,
when combined with changing technology and a growing body of research-based
knowledge, are the ingredients necessary for real reform. The NCTM
Standards is a vehicle that can serve as a basis for improving
the teaching and learning of mathematics in America's schools.
The reactions to the Working
Draft of the Standards have convinced us that many educators
are eager to reform school mathematics. Through their professional
organization, NCTM, which best reflects their interests and the
mathematical learning of their students, knowledgeable teachers
and other mathematics educators should assume responsibility for
leading the reform effort.
There are, of course, barriers
to the implementation of these standards, the most important being
the strongly held beliefs, expectations, and attitudes of all people
in education about specific aspects of the reform. A teacher who
believes that speed in paper-and-pencil calculation is most important
will be reluctant to let children use calculators. The administrator
who has charted group scores on a standardized test for years will
be reluctant to replace it. Parents who expect students to do mathematics
homework on paper at a desk rather than by gathering real data to
solve a problem will be surprised. The best way to bring about reform
is to challenge directly the perceptions held by many about the
content of mathematics, what is important for students to learn,
the job of teaching, what constitutes the work of students, and
the professional roles and responsibilities of teachers and administrators.
It is all too easy to agree with the rhetoric of reform but still
maintain long-held beliefs or practices inconsistent with intended
reform practices. Likewise, it is easy to agree but at the same
time claim that it "won't work here." We challenge readers
to recognize their beliefs and practices and test them against the
standards we have proposed.
Another barrier relates
to the political framework within which schools operate. Policy
decisions about schooling are made in the context of pressure, consensus,
conflict, and compromise by well-meaning elected representatives
at the federal, state, provincial, and local levels. These decisions
are then made operational by administrative directives. Many of
these standards can be fully implemented only by changing directives
about the selection of texts, mandated testing, and so forth, in
consultation with professionals in mathematics and mathematics education.
We challenge policy makers to change the rules.
Still another barrier to
reform is cost. Excellence costs money. Most schools, like the communities
they serve, are surviving but not thriving. To be successful, any
reform requires a considerable commitment of time and resources,
and our proposals for school mathematics are no exception. Many
resources are scarce, yet they must be found and used judiciously.
These and other barriers
to change can be viewed as insurmountable or as challenges to be
met and overcome. The Standards was produced by working groups
confident that if the recommendations are followed, a new school
mathematics program can be developed and implemented. The content
that should be included in a school mathematics program has been
specified. Such materials as texts, courseware, and tests can be
produced so that constructive learning will take place in classrooms.
However, let it be understood that we hold no illusions of immediate
reform. We believe a new program can be developed, but it will be
accomplished only by hard work. Our hope and expectations are that
a sufficient number of persons are willing to work to accomplish
the reform.
Now that you have read
this document and deliberated on its vision and recommendations,
we reemphasize the following points:
The National Council of
Teachers of Mathematics has created a vision of--
- mathematical power for
all in a technological society;
- mathematics as something
one does--solve problems, communicate, reason;
- a curriculum for all
that includes a broad range of content, a variety of contexts,
and deliberate connections;
- the learning of mathematics
as an active, constructive process;
- instruction based on
real problems;
- evaluation as a means
of improving instruction, learning, and programs.
If we keep these points
in mind, collectively we have a rare opportunity to provide the
kind of leadership that will make real, substantive changes in school
mathematics. These changes will ensure that all students possess
both a suitable and a sufficient mathematical background to be productive
citizens in the next century.
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