Imagine a classroom in which the Principles and Standards described
in this volume have come to life. Students of varied backgrounds and
abilities are working with their teachers to learn important mathematical
ideas. Expectations are high for all students, including those who need
extra support to learn mathematics well. Students are engaged by the
mathematics they are learning, study it every year they are in school,
and accept responsibility for their own mathematics learning. The classroom
environment is equitable, challenging, and technologically equipped
for the twenty-first century.
This vision of mathematics educationintroduced in chapter 1, given
focus in the Principles described in chapter 2, and more fully elaborated
in the discussions of the Standards in chapters 37is enticing.
But what would it take to realize this vision? Let us look beyond the
classroom to a broader context:
all mathematics teachers continue to learn new mathematics content and
keep current on education research. They collaborate on problems of
mathematics teaching and regularly visit one another's classrooms to
learn from, and critique, colleagues' teaching. In every school and
district, mathematics teacher-leaders are available, serving as expert
mentors to their colleagues, recommending resources, orchestrating interaction
among teachers, and advising administrators. Education administrators
and policymakers at all levels understand the nature of mathematical
thinking and learning, help create professional and instructional climates
that support students' and teachers' growth, understand the importance
of mathematics learning, and provide the time and resources for teachers
to teach and students to learn mathematics well. Institutions of higher
learning collaborate with schools to study mathematics
» education and to improve teacher preparation and professional
development. Professional mathematicians take an interest in, and contribute
constructively to, setting the content goals for mathematics in grades
K12 and for developing teachers' mathematical knowledge. Professional
organizations, such as the National Council of Teachers of Mathematics,
provide leadership, resources, and professional development opportunities
to improve mathematics education. And families, politicians, business
and community leaders, and other stakeholders in the system are informed
about education issues and serve as valuable resources for schools and
Making the vision
of mathematics teaching and learning a reality requires a strong system
of support at both
the local and the national levels. The National Council of Teachers of
Mathematics proposes that the Principles and Standardsgrounded in
decades of research and practice and refined in an extensive, collaborative
process of review and revisionserve as a basis for realizing the
Imagine a classroom in which the Principles and Standards described in this volume have come to life. Students of varied backgrounds and abilities are working with their teachers to learn important mathematical ideas. Expectations are high for all students, including those who need extra support to learn mathematics well. Students are engaged by the mathematics they are learning, study it every year they are in school, and accept responsibility for their own mathematics learning. The classroom environment is equitable, challenging, and technologically equipped for the twenty-first century.
This vision of mathematics educationintroduced in chapter 1, given focus in the Principles described in chapter 2, and more fully elaborated in the discussions of the Standards in chapters 37is enticing. But what would it take to realize this vision? Let us look beyond the classroom to a broader context:
Imagine that all mathematics teachers continue to learn new mathematics content and keep current on education research. They collaborate on problems of mathematics teaching and regularly visit one another's classrooms to learn from, and critique, colleagues' teaching. In every school and district, mathematics teacher-leaders are available, serving as expert mentors to their colleagues, recommending resources, orchestrating interaction among teachers, and advising administrators. Education administrators and policymakers at all levels understand the nature of mathematical thinking and learning, help create professional and instructional climates that support students' and teachers' growth, understand the importance of mathematics learning, and provide the time and resources for teachers to teach and students to learn mathematics well. Institutions of higher learning collaborate with schools to study mathematics » education and to improve teacher preparation and professional development. Professional mathematicians take an interest in, and contribute constructively to, setting the content goals for mathematics in grades K12 and for developing teachers' mathematical knowledge. Professional organizations, such as the National Council of Teachers of Mathematics, provide leadership, resources, and professional development opportunities to improve mathematics education. And families, politicians, business and community leaders, and other stakeholders in the system are informed about education issues and serve as valuable resources for schools and children.
Making the vision
of mathematics teaching and learning a reality requires a strong system
of support at both
the local and the national levels. The National Council of Teachers of
Mathematics proposes that the Principles and Standardsgrounded in
decades of research and practice and refined in an extensive, collaborative
process of review and revisionserve as a basis for realizing the
Putting the Principles into Action
The Principles in chapter 2 offer perspectives that can guide decision
makers in mathematics education. If essential supports of good mathematics
classrooms are missing, not all students can learn the mathematics they
need. Teachers need to work in environments where they can act, and continue
to develop, as professionals. Mathematics teaching and learning should
take place in a broader context that embraces and supports high-quality
mathematics instruction. The following sections highlight each of the
Principles in turn, showing how they can shape answers to important questions
in mathematics education.
How can all students
have access to high-quality mathematics education?
A major area of policy that affects students' access to mathematics education is "tracking," which is the long-term, often permanent, placement of students in classes, courses, or groups that offer different curricula according to the students' perceived mathematical abilities. Historically, tracking has consistently resulted in a select group of students being enrolled in mathematics courses that challenge and enrich them while othersoften poor or minority studentsare placed in mathematics classes that concentrate on remediation or do not offer significant mathematical substance (Wheelock 1992). For example, many middle-grades and high school students have been excluded from experiences in which they could learn significant amounts of algebra, instead spending much of their time reviewing the mathematics content studied in the elementary grades. As a result, these students are unable to experience a full program of high school mathematics across the range of content areas. Principles and Standards takes a strong stance: All students should have a common foundation of challenging mathematics, whether those students will enter the workplace after high school or pursue further study in mathematics and science. »
Taking this stance necessitates addressing the unique mathematical needs of all students. Students with exceptional promise in mathematics and deep interest in advanced mathematical study need appropriate opportunities to pursue their interests. Students with special learning needs in mathematics must be supported both by their classroom teachers and by special education staff. Special-needs educators responsible for mathematics instruction should participate in mathematics professional development, which will allow them to collaborate with classroom teachers in assessing and analyzing students' work in order to plan instruction.
Teachers and school and district leaders face complex decisions about how best to structure different curricular options. One traditional way for students to learn additional mathematics in which they have a particular interest is differential pacingallowing some students to move rapidly through the mathematical content expected of all so that they can go on to additional areas. However, some alternatives to differential pacing may prove advantageous. For example, curricula can be offered in which students can explore mathematics more deeply rather than more rapidly. This model allows them to develop deep insights into important concepts that prepare them well for later experiences instead of experiencing a more cursory treatment of a broader range of topics. Or schools can offer supplementary mathematics opportunities in areas not studied by all students or in extracurricular activities such as mathematics clubs or competitions.
Schools face difficult decisions about groupingwhether students should
be offered mathematics instruction in homogeneous or heterogeneous groups.
Students can effectively learn mathematics in heterogeneous groups if
structures are developed to provide appropriate, differentiated support
for a range of students. Structures that exclude certain groups of students
from a challenging, comprehensive mathematics program should be dismantled.
All such efforts should be monitored and evaluated to ensure that students
are well served.
Choices of mathematics
instructional materials can be controversial. Teachers should be prepared
to work with new curricular materials, and they need considerable time
to "live with" curricula in order to discover their strengths and weaknesses.
Only then can they develop the kinds of knowledge necessary to make materials
work well in particular contexts. The selection of curriculum and materials,
therefore, needs to be a long-term collaborative process involving teachers,
teacher-leaders, and administrators. Extensive field-testing should be
conducted, with information and interactions at the district level so
that choices are made wisely and support structures are put in place.
If instructional materials are not consistent with the expectations of families and community members or do not seem reasonable to them, serious difficulties can arise. For that reason, teachers and administrators should help families understand the goals and content of curricular materials, and community members should be consulted and informed about decisions regarding curricula and materials. Choices of instructional materials should be based on a community's agreed-on goals for » mathematics education. Principles and Standards, together with province, state, and local frameworks, offers proposals for such goals.
Developers of instructional materials and frameworks should draw on research
in their efforts to implement the ideas of standards. We urge them to
use Principles and Standards as a guide when making the many
decisions involved in creating curricula. Similarly, through the evaluation
and study of curricular efforts and through discussions of the ideas in
this document, the mathematics education community can continue to develop
a base of knowledge to guide the direction of mathematics education in
prekindergarten through grade 12.
Teachers need to know and use "mathematics for teaching" that combines mathematical knowledge and pedagogical knowledge. They must be information providers, planners, consultants, and explorers of uncharted mathematical territory. They must adjust their practices and extend their knowledge to reflect changing curricula and technologies and to incorporate new knowledge about how students learn mathematics. They also must be able to describe and explain why they are aiming for particular goals.
Preservice preparation is the foundation for mathematics teaching, but it gives teachers only a small part of what they will need to know and understand throughout their careers. No matter how well prepared teachers are when they enter the profession, they need sustained, ongoing professional development in order to offer students a high-quality mathematics education. They must continue to learn new or additional mathematics content, study how students learn mathematics, analyze issues in teaching mathematics, and use new materials and technology. Teachers must develop their own professional knowledge using research, the knowledge base of the profession, and their own experiences as resources. Preservice education, therefore, needs to prepare teachers to learn from their own teaching, from their students, from curriculum materials, from colleagues, and from other experts.
Unfortunately, the preparation today's teachers have received is in many instances
inadequate for the needs of tomorrow. The reality is simple: unless teachers
are able to take part in ongoing, sustained professional development,
they will be handicapped in providing high-quality mathematics education.
The current practice of offering occasional workshops and in-service days
does not and will not suffice.
Most mathematics teachers work in relative isolation, with little support for innovation and few incentives to improve their practice. Yet much of teachers' best learning occurs when they examine their teaching practices with colleagues. Research indicates that teachers are better able to help their students learn mathematics when they have opportunities to work together to improve their practice, time for personal reflection, and strong support from colleagues and other qualified professionals (see, e.g., Brown and Smith ; Putnam and Borko ; Margaret Smith [forthcoming]). The educational environment must be characterized by trust and respect for teachers and by patience as they work to develop, analyze, and refine their practice. Too often we place the responsibility for change solely on the shoulders of teachers and then blame them when things do not work as expected. We need instead to » address issues in a systemic way, providing teachers with the resources they need for professional growth.
Such shifts in the system are feasible. The typical structures of teachers' workdays
often inhibit community building, but structures can be changed. In some
cultures, shared discussions of students and teaching are the norm. In
Japan and China, the workdays of teachers include time for meeting together
to analyze recent lessons and to plan for upcoming lessons (Ma 1999; Stigler
and Hiebert 1999). During this "lesson study," teachers plan the lesson,
teach the lesson with colleagues watching, revise the lesson collaboratively,
teach the revised lesson, evaluate and reflect again, and share the results
in written form. Part of the planning includes predicting what groups
of students will do when presented with particular problems and tasks.
The ongoing analysis of practice is thus built into the fabric of teaching,
not treated as an added task that teachers must organize themselves. Although
this level of professional collaboration may be hard for U.S. and Canadian
teachers to imagine within the constraints of the prevailing professional
culture and system, it illustrates the potential power of learning communities
to improve mathematics teaching and learning. Finding ways to establish
such communities should be a primary goal for schools and districts that
are serious about improving mathematics education.
Learning mathematics with understanding requires consistent access to high-quality mathematics instruction. In the elementary grades, students should study mathematics for at least an hour a day under the guidance of teachers who enjoy mathematics and are prepared to teach it well. Achieving this objective takes thoughtful administrative arrangements, such as orchestrating shared teaching responsibilities or using mathematics specialists.
Every middle-grades and high school student should be required to study the equivalent of a full year of mathematics in each grade. Ways of organizing programs will vary according to local goals and situations. Some schools are using such organizational alternatives as block scheduling. The impact of such alternatives on students' learning needs further study. It is not clear, for example, whether long intervals between periods of intensive study benefit or detract from students' learning or whether mathematics-intensive workplace activity can support students' mathematical engagement and growth. All middle-grades and high school students should be expected to spend a substantial amount of time every day working on mathematics outside of class, in activities ranging from typical homework assignments and projects to problem solving in the workplace.
A significant challenge to realizing the vision portrayed in Principles
and Standards is disengagement. Too many students disengage from
school mathematics, which creates a serious problem not only for their
teachers but also for a society that increasingly depends on a quantitatively
literate citizenry. Students may become uninvolved for various reasons.
Many, for example, find it difficult to sustain the motivation and effort
required to learn what can be a challenging school subject. They may find
the subject as taught to be uninteresting and irrelevant.
Disengagement is too often reinforced in both overt and subtle ways by the attitudes and actions of adults who have influence with students. » Some parents and other authority figures, as well as societal influences like the media, convey the message that not everyone is expected to be successful in mathematics and thus that disengagement from school mathematics is acceptable. Such societal tolerance makes it less likely that all students will be motivated to sustain the effort needed to learn mathematics, which in turn makes the job of their teachers even more challenging. Some teachers also believe that many students cannot learn mathematics, which supports those students in their beliefs that they cannot learn mathematics, which then leads to further disengagement. Thus, a vicious cycle takes hold. It affects school mathematics in profound ways and is especially prevalent in the middle grades and high school.
Although the challenge presented by disengagement is formidable, it is not insurmountable.
Teachers need to uphold high expectations that all children should learn
with understanding, including children of minorities or from poor communities.
Many teachers have found that if they teach mathematics in ways similar
to those advanced in Principles and Standardsfor example,
by approaching traditional topics in ways that emphasize conceptual understanding
and problem solvingmany apparently uninterested students can become
High-stakes assessmentsfrom nationally normed achievement tests to state, province, or locally developed measures of students' performanceare a particular concern for educators. If they are not aligned with school and community goals for mathematics education and with the curriculum, teachers and students are left in a precarious position. If teachers are committed to pursuing goals and practices consistent with those in Principles and Standards, satisfying the sometimes contradictory requirements of the local, state, or province assessment system is challenging. It is not realistic for teachers simply to ignore the pressure of these tests. Students may be penalized if they do not perform well, staff or school evaluations may depend on demonstrating progress, and decisions about resource allocation and salaries may be tied to test scores. Yet "teaching to the test"a political reality when the consequences of test scores are significantcan undermine the integrity of instruction. To put teachers in the position of deciding between what they believe best enhances their students' learning and what is required to survive in the educational system puts them in an untenable position. High-stakes assessments must be closely linked to the goals teachers are being asked to achieve; where they are not, teachers must be supported in the decisions they make.
The assessment of students' understanding can be enhanced by the use of multiple
forms of assessment, such as portfolios, group projects, and writing questions.
However, students and parents alike may find these forms unfamiliar in
the mathematics classroom. Teachers need support from administrators in
helping students and parents understand the utility and purpose of such
approaches in improving mathematics instruction.
The Technology Principle
To make technology an essential part of classrooms, the technological tools must be selected and used in ways that are compatible with » the instructional goals. When technological tools are considered essential instructional materials for all students, then decisions about resources must reflect this view, despite the costs of purchases and upgrades. Schools, districts, or provinces that integrate technology in mathematics teaching and learning face challenging issues of equity. The need for high-quality technology is as great in urban and rural settings as in suburban schoolsperhaps greater.
Decisions to incorporate new technology also require that teachers be prepared
and supported in using it to serve instructional goals. Teachers must
themselves experience how technology can enhance the learning of significant
mathematics and explore models for incorporating it in their classroom
practice. Moreover, technology must be embedded in the mathematics program
rather than be treated as just another flashy add-on. Without coherent,
comprehensive implementation plans, the incorporation of new technology
is likely to fall short in improving mathematics teaching and learning.
The sections that follow discuss the kinds of commitments and actions
that various communities must make to realize the vision of Principles
and Standards. The role of teachers, of course, is central. The choices
that mathematics teachers make every day determine the quality and effectiveness
of their students' mathematics education. But teachers alone do not make
all the decisionsthey are part of a complex instructional system.
Othersstudents themselves; mathematics teacher-leaders; school,
district, and state or province administrators; higher-education faculty;
families, other caregivers, and community members; and professional organizations
and policymakershave resources, influence, and responsibilities
that can enable teachers and their students to be successful.
Mathematics teachers must develop and maintain the mathematical and pedagogical
knowledge they need to teach their students well. One way to do this is
to collaborate with their colleagues and to create their own learning
opportunities where none exist. They should also seek out high-quality
professional development opportunities that fit their learning needs.
By pursuing sources of information, building communities of colleagues,
and participating in professional development, teachers can continue to
grow as professionals.
Mathematics teachers generally are responsible for what happens in their own classrooms and can try to ensure that their classrooms support learning by all students. For example, whether or not their school has implemented tracking, teachers must challenge and hold high expectations for all their students, not just those they believe are "gifted." Elementary school students need at least an hour of mathematics instruction each day. The decisions teachers make in the classroom about how to offer all students experiences with important mathematics and how to accommodate the wide-ranging interests, talents, and experiences of » students are essential to giving all students access to mathematics. Although many matters bearing on their classrooms are beyond teachers' sole control, they need to take the initiative in discussing trends and opportunities in mathematics education with administrators.
Teachers must help students be confident, engaged mathematics learners. In the elementary grades, convincing students that they can do mathematics and helping them enjoy it are important goals. Typically, student disengagement has been a serious problem in the middle grades and high school. Teachers at those levels should work to keep students involved in relevant classroom activities, assign projects that make connections between mathematics and students' daily lives, and allow students multiple avenues to display what they have learned. Learning experiences based in the workplace have also proved effective in motivating students who are at risk of becoming disengaged from school.
Mathematics teachers can foster reinforcement of their efforts by families and other community members by maintaining dialogue aimed at the improvement of mathematics education. Communicating about mathematics goals, students' learning, teaching, and programs helps families and other caregivers understand the kind of mathematics learning in which children are engaged. Giving them opportunities to ask questions, express concerns, and experience classroom activities can be very useful in shaping improvements. Many groups of teachers organize "Math Nights" at least once a year. At such events, usually held during the evening for the convenience of parents, students and their parents work together on engaging mathematics activities. Newsletters, homework assignments that involve family collaboration, and other means can help maintain communication between home and school. To do all of this well, teachers need to understand their mathematical goals and their perspectives on mathematics education and be able to articulate them in compelling ways.
Mathematics teachers ultimately control the range of mathematical ideas made
available to their students. They have the responsibility to ensure that
a full range of mathematical content and processes, such as those described
in this document, are taught and that mathematical emphases fit together
into a coherent whole. They can do so by using the available textbooks,
support materials, technology, and other instructional resources effectively
and tailoring these resources to their particular situations so that their
goals for mathematics instruction and their students' needs are met. Teachers
need to seek out support and professional development as they implement
current or new curricula. They should constantly evaluate curricular materials
and offer suggestions to teacher-leaders and administrators, and they
should find ways to be involved in choosing the instructional materials
for their school or district.
Integrating assessment into instruction and using a variety of sources of evidence
to evaluate the learning of each student is challenging. It may be especially
difficult in the face of mandated high-stakes assessments. Although teachers
may at times feel trapped between their goals for their students' mathematics
learning and those of high-stakes tests, they are not powerless. If teachers
recognize that required tests are not aligned with meaningful instructional
goals, they should voice their concerns to their teacher-leaders and administrators
and seek ways to participate in decisions about testing.
Learning mathematics is stimulating, rewarding, and at times difficult. Mathematics students, particularly in the middle grades and high school, can do their part by engaging seriously with the material and striving to make mathematical connections that will support their learning. If students are committed to communicating their understandings clearly to their teachers, then teachers are better able to plan instruction and respond to students' difficulties. Productive communication requires that students record and revise their thinking and learn to ask good questions as part of learning mathematics.
Beyond the classroom, students need to build time into their days to work on
mathematics. They need to learn how to use resources such as the Internet
to pursue their mathematical questions and interests. As students begin
to identify potential careers, they can take the initiative in researching
the mathematics requirements for those careers and investigate whether
their school programs offer the necessary preparation.
There is an urgent and growing need for mathematics teacher-leadersspecialists
positioned between classroom teachers and administrators who can assist
with the improvement of mathematics education. The kinds of roles and
influences that such leaders can have, as well as the nature of their
position and responsibility, will vary widely. In recent years, schools
have increasingly turned to teachers in the system as potential leaders.
Sometimes leaders are teachers on special assignment, released from the
classroom for an extended period to work in one or several school buildings.
In other situations, leaders are released from a portion of their classroom
teaching so that they can work directly with other teachers. No matter
what the particular arrangement may be, mathematics teacher-leaders should
take responsibility for focusing on mathematics through their work with
teachers, administrators, and families and other community members.
Teacher-leaders can have a significant influence by assisting teachers in building their mathematical and pedagogical knowledge. Leaders face the challenge of changing the emphasis of the conversation among teachers from "activities that work" to the analysis of practice. Teacher-leaders in some settings work with their colleagues to design professional development plans for individual teachers, for a school, or for a larger system. They can arrange collaborative investigations or discussion groups with teachers at a school site, encourage participation in workshops at the school or district level, promote attendance at professional conferences, organize the study of professional resources such as Principles and Standards and articles in professional journals, recommend Internet sites that discuss mathematics teaching, and provide information about in-service programs or graduate courses. Teachers can benefit greatly from the knowledge and support of peers and mentors as they move in the directions recommended in Principles and Standards. Teacher-leaders' support on a day-to-day basisranging from conversations in the hall to in-classroom coaching to regular grade-level and departmental seminars focused on how students learn mathematicscan be crucial to a teacher's work life. »
Teacher-leaders should have the knowledge and expertise to play a role in the design of curriculum frameworks and the selection of instructional materials, and they should ensure that teachers are involved in these processes, too. Schools and districts can also rely on mathematics teacher-leaders to organize and lead the piloting and implementation of new instructional materials. They might help teachers in a school begin working through the materials themselves, reading and discussing any commentary about student learning found in the materials, analyzing students' work, and designing and critiquing lessons. In this way, the materials can serve as a means for teachers to learn.
Teacher-leaders also have a role in working with administrators and policymakers
to help guide their decisions about the improvement of mathematics education.
They can help ensure that teachers and administrators develop and share
a common perspective about goals for mathematics teaching and learning.
Doing so involves keeping everyone informed of new directions and emphases
in mathematics education and facilitating substantive discussions and
planning sessions aimed at reaching common goals. Finally, like all teachers,
mathematics teacher-leaders must themselves engage in ongoing learning
and professional development.
Administrators at every level have responsibilities for shaping the instructional
mission in their jurisdictions, providing for the professional development
of teachers, designing and implementing policies, and allocating resources.
To deploy their influence well on behalf of mathematics education, administrators
who understand the goals of mathematics instructionincluding those
described in Principles and Standardscan work to create
institutions in which teachers have access to human and material resources
that will help them attain those goals. In particular, administrators
can identify individuals or teams of teachers as leaders in establishing
mathematics communities, and they can ensure that such learning communities
develop, flourish, and grow beyond a few teachers. When administrators
themselves become part of the mathematics learning community, they develop
deeper understandings of the goals of mathematics instruction. They can
understand better what they are seeing in mathematics classrooms (Nelson
1999) and can make more-informed judgments about the curricular, technological,
and pedagogical resources teachers need.
If they are truly committed to the improvement of mathematics education, administrators at all levels should ensure that mathematics expertise and leadership are developed in their schools or systems. Administrators can influence the quality of mathematics education by supporting the professional growth of mathematics teachers. They can arrange for meaningful professional development workshops, provide libraries and Web access to instructional and other materials, and foster cross-school conversations about goals and instructional practices. Administrators can help arrange teachers' work schedules so that meaningful collaboration with colleagues is part of the school day. They can shape work environments so that they are conducive to productive professional interactions, they can include such interactions as part of teachers' work, and they can establish a program of mathematics teacher-leaders within the school or system. »
Perhaps the most important sphere of influence for administrators is the area of structures and policies. Through their decisions about hiring, teaching assignments, evaluation, and mentoring for new teachers, administrators have powerful opportunities to strengthen the focus on mathematics. They also have a role in guaranteeing equitable access for students and in arranging time and space for effective mathematics instruction. They can work to align curricular materials, technology, and assessments at all levels with agreed-on goals for mathematics education, such as those represented in Principles and Standards.
Administrators can support the improvement of mathematics education by establishing effective processes for the analysis and selection of instructional materials. These processes, whether at the classroom, school, or system level, should involve wide consultation with teachers and teacher-leaders and a deep and careful analysis of the materials. Many districts pilot-test one or more programs before making a final decision. No matter how decisions about the selection of instructional materials are made, they should always be guided by school, state or province, and national goals for mathematics instruction. Furthermore, the adoption of new instructional materials is only a beginning. No matter how well curricular materials may be designed, they are unlikely to lead to the continuing improvement of instruction unless plans for implementation and professional development are formulated along with the plan for adoption.
Finally, to make long-term progress in improving students' learning, administrators
and policymakers must carefully consider the impact of high-stakes assessments
on the instructional climate in schools, and they must understand what
can be learned from assessments and what cannot. If a test focuses primarily
on the acquisition of superficial skills rather than on the deep mathematical
understandings described in this document, its use in making decisions
that promote constructive change will be limited and perhaps even counterproductive.
Decisions about placing students in different instructional situations
and evaluations of teachers' effectiveness should never be based on a
single test score, especially when that test has been designed to measure
how well students carry out routine procedures. Implementing and sustaining
assessment policies and practices that support high-quality mathematics
education is a difficult but essential part of administrators' responsibilities.
Faculty in two-year and four-year colleges and universities have a significant
impact on school mathematics, primarily through their work with students
who will become teachers. They have considerable influence on whether
teachers enter the profession with the strong knowledge of the mathematics
needed to teach pre-K12 mathematics, of student learning, and of
mathematics teaching. They can also model the effective practices they
believe teachers should employ.
The first few years in teachers' careers are critical to their persistence in mathematics teaching and to their dispositions toward continued professional growth and learning. The in-service education and professional development of teachers, especially in content knowledge, are not the exclusive mission of any single type of institution, so a significant leadership role is available for higher education. Teachers need » in-service and graduate education that help them grow mathematically and as practitioners.
Faculty members in institutions of higher education should be partners in the development of school-based mathematics communities. Teacher educators, mathematicians, and practicing teachers working together can create a rich intellectual environment that will promote veteran teachers' growth and demonstrate to new teachers the value of learning communities. Mathematics education researchers and teacher educators can collaborate with classroom teachers to investigate research questions based in classroom practice or to look at mathematics as it occurs in classrooms. In such contexts, higher-education faculty can serve as resources to mathematics teachers at all levels and also learn from them.
In recent decades, research in mathematics education has coalesced as a powerful
field of intellectual study. The efforts of education and mathematics
faculty can result in increased knowledge and the improved preparation
of teachers, teacher-leaders, administrators, and researchers. Such ongoing
efforts, in collaboration with school personnel, are a major aspect of
improving mathematics teaching and learning.
Teachers and administrators should invite families, other caregivers, and community members to participate in examining and improving mathematics education. All partners in this enterprise need to understand the changing goals and priorities of school mathematics, as expressed in the Principles in chapter 2. Families need to know what options are available for their children and why an extensive and rigorous mathematics education is important. When parents understand and support the schools' mathematics program, they can be invaluable in convincing their daughters and sons of the need to learn mathematics and to take schooling seriously. Families become advocates for education standards when they understand the importance of a high-quality mathematics education for their children.
Families can establish learning environments at home that enhance the work initiated at school. Respect shown to and for teachers is often carried over from parent to child. By providing a quiet place for a child to read and attend to homework and by monitoring students' work, families can signal that they believe mathematics is important. Such attention and appreciation of mathematics is not lost on students.
If families and other members of the public do not understand the intent of,
and rationale for, improvements in mathematics education, they can halt
even the most carefully planned initiatives. Principles and Standards
was written with the hope that the conversations it engenders will
ultimately generate a widespread commitment to improving mathematics education.
As part of this effort, it is the responsibility of the education community
to inform the general public and its elected representatives about the
goals and priorities in mathematics education, thereby empowering them
to participate knowledgeably in its improvement.
Professional organizations can provide national and regional leadership and expertise in support of the continuing improvement of mathematics » education. The National Council of Teachers of Mathematics (NCTM), using Principles and Standards as a focus, will offer its members many means of professional development, including conferences, classroom resources, research publications, and Web-based materials. NCTM is also engaged in a variety of efforts to educate the public and to support parents and other caregivers as they encourage their children in mathematics. (See the NCTM Web site at www.nctm.org for details.) Professional organizations like the NCTM are positioned to promote policies that support high-quality mathematics education. For instance, they can work to establish certification and accreditation requirements that include high expectations for teachers' knowledge of content, teaching, and students. Through their members, publications, and meetings, professional organizations can focus attention on mathematics education issues.
High-quality mathematics education is vital to the health of mathematics as a discipline, and mathematical professional societies have a clear stake in this enterprise. Mathematics education serves as the pipeline for future mathematicians, statisticians, and mathematics teachers as well as scientists, engineers, and all professionals who use mathematics. The public needs to be well informed about, not intimidated by, mathematics if it is to support ongoing research, development, and funding in mathematics and related fields. The nature of undergraduate mathematics programs is closely intertwined with K12 mathematics education, and efforts to improve mathematics teaching and learning from kindergarten through graduate school can be coordinated across professional organizations. Issues about the mathematical preparation of teachers are currently being addressed by the Conference Board of the Mathematical Sciences (CBMS) in its "Mathematical Education of Teachers" project (CBMS, 2000). All professional organizations concerned with the mathematical sciences can help improve mathematics education through coordinated, collaborative efforts.
Policymakers at national and local levels are in a unique position to view the
broad range of influences on mathematics education and to make decisions
that promote improvements in the field. They can allocate the funding
and resources needed to continue the study and implementation of improvements.
They can also examine teacher-certification standards and accreditation
requirements to ensure that teachers have the strong and deep content
knowledge needed today. In the same way that standards need ongoing examination
and revision, so do local, state, and provincial curricular frameworks
and standards. With Principles and Standards available as a
resource to identify key issues in contemporary mathematics education,
policymakers can ensure that that process occurs and can promote programmatic
activity that is designed to further address those issues.
Principles and Standards for School Mathematics has both an immediate and a long-term role in realizing the vision of improved mathematics » education. First, it sets out a carefully developed and ambitious but attainable set of expectations for school mathematics. Educators, families, policymakers, and others can use the ideas contained in this volume to guide the decisions they make about mathematics education, from classroom practice to establishing local and state education standards and frameworks. The interest generated by NCTM's original Curriculum and Evaluation Standards for School Mathematics (1989) demonstrates the extent to which various groups with a stake in mathematics education are committed to its improvement. Principles and Standards is intended to build on and extend that commitment. It represents a collective judgment, based on research, practice, and an extended consultative process, about what students need to learn in order to be prepared for the future.
Principles and Standards is also a tool for better understanding the
issues and challenges involved in improving mathematics education. It
offers information and ideas that those with responsibility for mathematics
educationwhether at the local, state or provincial, or national
levelneed in order to engage in constructive dialogues about mathematics
teaching, curricula, and assessment. Any vision of school mathematics
teaching and learning needs ongoing examination; it needs to be refined
continually in light of the greater understanding achieved through practice,
research, and evidence-based critiques. The process that NCTM put in place
for developing Principles and Standards reflects a commitment
to ongoing discussion and reflection. This document, therefore, should
be seen as part of a work in progress that can help guide decision makers
in developing excellent mathematics programs, not as a prescription to
be rigidly imposed on others. (See Kilpatrick and Silver  and Ferrini-Mundy
 for additional discussion.)
Achieving high standards in mathematics education calls for clear goals. It calls for the active participation of teachers, administrators, policymakers, higher-education faculty, curriculum developers, researchers, families, students, and community members.
Principles and Standards is provided as a catalyst for the continued improvement of mathematics education. It represents our best current understanding of mathematics teaching and learning and the contextual factors that shape them. It was created with the input and collaboration of members of all the communities mentioned above. It articulates high but attainable standards.
Realizing the vision of mathematics education described in this document requires the continued creation of high-quality instructional materials and technology. It requires enhanced preparation for teachers and increased opportunities for professional growth. It requires the creation of assessments aligned with curricular goals. Realizing the vision depends on the participation of all the constituencies mentioned above in reflecting on, supporting, and improving educational practice. We should not underestimate the difficulty of the task, but it can be done. Now is the time to undertake the collaborative efforts that will make the vision come alive. We owe our children nothing less.
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Copyright © 2000 by the National Council of Teachers of Mathematics.