A
Vision for School Mathematics
Imagine
a classroom, a school, or a school district where all students have
access to
high-quality, engaging mathematics instruction. There are ambitious
expectations for all, with accommodation for those who need it.
Knowledgeable
teachers have adequate resources to support their work and are
continually
growing as professionals. The curriculum is mathematically rich,
offering
students opportunities to learn important mathematical concepts and
procedures
with understanding. Technology is an essential component of the
environment.
Students confidently engage in complex mathematical tasks chosen
carefully by
teachers. They draw on knowledge from a wide variety of mathematical
topics,
sometimes approaching the same problem from different mathematical
perspectives
or representing the mathematics in different ways until they find
methods that
enable them to make progress. Teachers help students make, refine, and
explore
conjectures on the basis of evidence and use a variety of reasoning and
proof
techniques to confirm or disprove those conjectures. Students are
flexible and
resourceful problem solvers. Alone or in groups and with access to
technology,
they work productively and reflectively, with the skilled guidance of
their teachers.
Orally and in writing, students communicate their ideas and results
effectively. They value mathematics and engage actively in
learning it.
p.
3
The
vision for mathematics education described in Principles and
Standards for
School Mathematics is highly
ambitious. Achieving it requires solid mathematics
curricula, competent and knowledgeable teachers who can integrate
instruction
with assessment, education policies that enhance and support learning,
classrooms with ready access to technology, and a commitment to both
equity and
excellence. The challenge is enormous and È meeting
it is essential. Our students deserve
and need the best mathematics education possible, one that enables them
to
fulfill personal ambitions and career goals in an ever-changing world.
Since
the release in 1989 of the Curriculum and Evaluation Standards for
School
Mathematics followed
in 1991 by the Professional Teaching Standards for
School Mathematics and in
1995 by the Assessment Standards for School Mathematics the
National Council
of Teachers of Mathematics (NCTM) has remained committed to the view
that
standards can play a leading role in guiding the improvement of
mathematics
education. As an organization representing teachers of mathematics,
NCTM shares
with students, school leaders, and parents and other caregivers the
responsibility to ensure that all students receive a high-quality
mathematics
education. All interested parties must work together to create
mathematics
classrooms where students of varied backgrounds and abilities work with
expert
teachers, learning important mathematical ideas with understanding, in
environments that are equitable, challenging, supportive, and
technologically
equipped for the twenty-first century.
The
Need for Mathematics in a Changing World
We
live in a time of extraordinary and accelerating change. New knowledge,
tools,
and ways of doing and communicating mathematics continue to emerge and
evolve.
Calculators, too expensive for common use in the early eighties, now
are not
only commonplace and inexpensive but vastly more powerful. Quantitative
information available to limited numbers of people a few years ago is
now
widely disseminated through popular media outlets.
The
need to understand and be able to use mathematics in everyday life and
in the
workplace has never been greater and will continue to increase. For
example:
p.
4
* Mathematics
for life. Knowing
mathematics can be personally satisfying and empowering.
The underpinnings of everyday
life are increasingly mathematical and
technological. For instance, making purchasing decisions, choosing
insurance or
health plans, and voting
knowledgeably all call for quantitative
sophistication.
* Mathematics as
a part of cultural heritage. Mathematics
is one of the greatest cultural and
intellectual achievements of
human-kind, and citizens should develop an
appreciation and understanding of that achievement, including its
aesthetic
and
even recreational aspects.
* Mathematics
for the workplace. Just as
the level of mathematics needed for intelligent
citizenship has increased
dramatically, so too has the
level of mathematical
thinking and problem solving needed in the workplace, in professional
areas
ranging from health care to graphic design.
* Mathematics
for the scientific and technical community. Although
all careers require a
foundation of mathematical
knowledge, some are mathematics
intensive. More
students must pursue an educational path that will prepare them for
lifelong
work as mathematicians, statisticians, engineers, and scientists.
In
this changing world, those who understand and can do mathematics will
have
significantly enhanced opportunities and options for shaping their
futures.
Mathematical competence opens doors to productive futures. A lack of
mathematical competence keeps those doors closed. NCTM challenges the
assumption that mathematics is only for the select few. On the
contrary,
everyone needs to understand mathematics. All students should have the
opportunity and the support necessary to learn significant mathematics
with
depth and understanding. There is no conflict between equity and
excellence.
Principles
and Standards calls for
a common foundation of mathematics to be learned by all
students. This approach, however, does not imply that all students are
alike.
Students exhibit different talents, abilities, achievements, needs, and
interests in mathematics. Nevertheless, all students must have access
to the
highest-quality mathematics instructional programs. Students with a
deep
interest in pursuing mathematical and scientific careers must have
their
talents and interests engaged. Likewise, students with special
educational
needs must have the opportunities and support they require to attain a
substantial understanding of important mathematics. A society in which
only a
few have the mathematical knowledge needed to fill crucial economic,
political,
and scientific roles is not consistent with the values of a just
democratic
system or its economic needs.
The
Need for Continued Improvement of Mathematics Education
The
vision described at the beginning of this chapter is idealized. Despite
the
concerted efforts of many classroom teachers, administrators,
teacher-leaders,
curriculum developers, teacher educators, mathematicians, and
policymakers, the
portrayal of mathematics teaching and learning in Principles and
Standards is not the
reality in
the vast majority of classrooms, schools, and districts. Evidence from
a
variety of sources makes it clear that many students are not learning
the
mathematics they need or are expected to learn (Kenney and Silver 1997;
Mullis
et al. 1997, 1998; Beaton et al. 1996). The reasons for this deficiency
are
many: In some instances, students have not had the opportunity to learn
important
mathematics. In other instances, the curriculum offered to students
does not
engage them. Sometimes students lack a commitment to learning. The
quality of
mathematics teaching is highly variable. There is no question that the
effectiveness of mathematics education in the United States and Canada
can be
improved substantially.
p.
5
Standards
can play a central role in the process of improvement. The previously
released
NCTM Standards (NCTM
1989, 1991, 1995) have influenced state standards and
curriculum frameworks (Council of Chief State School Officers 1995;
Raimi and
Braden 1998), instructional materials (U.S. Department of Education
1999),
teacher education (Mathematical Association of America 1991), and
classroom
practice (Ferrini-Mundy and Schram 1997). As with any educational
innovation,
however, the ideas of the Standards have been
interpreted in many different ways
and have been implemented with varying degrees of fidelity. Sometimes
the
changes made in the name of standards have been superficial or
incomplete. For
example, some of the pedagogical ideas from the NCTM Standards such as
the emphases
on discourse, worthwhile mathematical tasks, or
learning through problem solving have
been enacted without sufficient attention to students' understanding of
mathematics content. Efforts to move in the directions of the original
NCTM Standards
are by
no means fully developed or firmly in place.
The
Role and Purpose of Standards
The
introduction to the 1989 Curriculum and Evaluation Standards noted
three reasons for
a professional organization to formally adopt standards: to ensure
quality, to
indicate goals, and to promote change. One way in which standards
documents
help meet these goals is by shaping conversations about mathematics
education.
As with the previous NCTM Standards, Principles and Standards offers
common language,
examples, and recommendations to engage many groups of people in
productive
dialogue. Although there will never be complete consensus within the
mathematics education profession or among the general public about the
ideas
advanced in any standards document, the Standards provide a guide for
focused,
sustained efforts to improve students' school mathematics education. Principles
and Standards supplies
guidance and vision while leaving specific curriculum
decisions to the local level. This document is intended to:
* set forth a
comprehensive and coherent set of goals for mathematics for all
students from
prekindergarten through
grade 12 that will orient
curricular, teaching, and
assessment efforts during the next decades;
* serve as a
resource for teachers, education leaders, and policymakers to use in
examining
and improving the quality of
mathematics instructional
programs;
* guide the
development of curriculum frameworks, assessments, and instructional
materials;
* stimulate ideas
and ongoing conversations at the national, provincial or state, and
local
levels about how best to help
students gain a deep
understanding of important
mathematics.
An
Overview of Principles and Standards
Principles
and Standards for School Mathematics builds on
and consolidates messages from the
previous Standards documents.
The document is organized into four main parts:
* Principles for
school mathematics (chapter 2)
* An overview of
the Standards for mathematics education in prekindergarten through
grade 12
(chapter 3)
* Standards for
four separate grade bands: prekindergarten through grade 2 (chapter 4),
grades
3-5 (chapter 5), grades 6-8
(chapter 6), and grades 9-12 (chapter 7)
* A discussion of
the steps needed to move toward the vision embodied in Principles
and
Standards (chapter
8)
p.
6
The
principles are statements reflecting basic precepts that are
fundamental to a
high-quality mathematics education. The discussions in chapter 2
elaborate on
the underlying assumptions, values, and evidence È on which
these Principles are founded. The
Principles should be useful as perspectives on which educators can base
decisions
that affect school mathematics. NCTM's commitment to mathematics for
all is
reaffirmed in the Equity Principle. In the Curriculum Principle, a
focused
curriculum is shown to be an important aspect of what is needed to
improve
school mathematics. The Teaching Principle makes the case that students
must
have opportunities to learn important mathematics under the guidance of
competent and committed teachers. The view of learning that is the
basis for
the document is taken up in the Learning Principle. The important roles
of
assessment and technology in school mathematics programs are discussed
in the
Assessment and Technology Principles.
Chapters
3-7 outline an ambitious and comprehensive set of curriculum standards
for all
students. Standards are descriptions of what mathematics instruction
should
enable students to know and do, statements of what is valued for school
mathematics education. Each of the ten curriculum standards proposed in
this
document spans the entire range from prekindergarten through grade 12.
Chapter
3 discusses each Standard in turn to convey its main ideas. In
addition, these
discussions give a sense of how the ideas encompassed in a Standard
develop
over all four grade bands, highlighting points at which certain levels
of
mastery or closure are appropriate. Chapters 4-7 present the Standards
in
detail for each grade band.
The
first five Standards describe mathematical content goals in the areas
of number
and operations, algebra, geometry, measurement, and data analysis and
probability. The next five Standards address the processes of problem
solving,
reasoning and proof, connections, communication, and representation. In
each
grade-band chapter, a set of "expectations" is identified and
discussed for each Content Standard. The appendix displays the Content
Standards and expectations in a chart that highlights the increasing
sophistication of ideas across the grades. Each grade-band chapter
discusses
what each Process Standard should "look like" in that grade band and
what the teacher's role is in supporting the development of that
process.
The
mathematical Content and Process Standards discussed in chapters 3-7
are inextricably
linked. One cannot solve problems without understanding and using
mathematical
content. Establishing geometric knowledge calls for reasoning. The
concepts of
algebra can be examined and communicated through representations.
One
purpose of this document is to offer teachers, curriculum developers,
and those
responsible for establishing curriculum frameworks a way to focus
curricula.
Focus is promoted through attention to the idea of "moving on."
School mathematics programs should not address every topic every year.
Instead,
students will reach certain levels of conceptual understanding and
procedural
fluency by certain points in the curriculum. Teachers should be able to
assume
that students possess these understandings and levels of fluency when
they plan
their mathematics instruction. Teachers and policymakers can then
fashion
instructional programs and curricular frameworks that develop
progressively
over the grades and that focus on important mathematical areas.
p.
7
Chapter
8 discusses what it will take to move toward the vision described in
the
previous chapters. In particular, it discusses critical issues related
to putting the
Principles into action and outlines the key roles played by various
groups and
communities in realizing the vision of the Principles and Standards.
As
We Move Forward
Attaining
the vision described at the beginning of this chapter will require the
talents,
energy, and attention of many individuals, including students,
teachers, school
administrators, teacher-leaders, policymakers, parents and other
caregivers,
mathematicians, mathematics educators, and the local community. It will
require
that the vision of this document be shared and understood and that all
concerned be committed to improving the futures of our children. The
task is
enormous and essential. All students need an education in mathematics
that will
prepare them for a future of great and continual change.