Critical Issue: Implementing Curriculum, Instruction, and Assessment
Standards in Mathematics
National educational standards are being developed in nearly all subject
areas. The National Council of Teachers of Mathematics has recently developed
standards for mathematics. Their Curriculum and Evaluation Standards
for School Mathematics reflect a concern that students in the U.S.
are not mathematically literate and often fail to see the relationship
between mathematics learned in school and real-life situations.
The NCTM Standards are primarily expressed as goals
for students. It is up to schools and communities to determine how to make
sure all students reach these goals. Schools are faced with a huge challenge:
to offer curriculum content, instructional methods, and forms of assessment
that are aligned with the goals described in the Curriculum and Evaluation
Most U.S. students learn basic mathematical facts and formulas but many
of them are unable to use this knowledge to solve everyday problems! Results
of the National Assessment of Educational Progress (NAEP) help us understand
the extent to which students in our country can solve problems with their
mathematical knowledge. Employers
tell us about the skills they wish public school graduates would bring
to the workforce.
One way that the United States has addressed this situation in math
as well as other subject areas is to develop a nationwide system of standards
and assessments. The Goals
2000 legislation and the reauthorized Elementary
and Secondary Education Act, known as Improving America's Schools Act of
1994, have formalized this system. Both laws indicate that compliance
to these standards is voluntary, although states applying for Goals 2000
funds from the Department of Education must agree to develop and implement
higher standards. Furthermore, many states are developing curriculum frameworks
based on national standards.
Teachers participated in the creation and review of the standards for
math through their professional organization, NCTM. Many teachers and many
schools communities--both prior to and subsequent to the release of the
standards in 1989--have tried to provide the meaningful
learning experiences implicit in the standards. Other teachers and
school communities find it a challenge to design curriculum, instruction,
and assessment that reflect national standards as well as curriculum frameworks
developed by their state. Teachers, whether they specialize in teaching
mathematics or teach it along with other subjects, may lack confidence
in the progress of their own mathematical knowledge. The challenge to apply
the standards may be greatest for school personnel who see the standards
movement as just another wave of school reform such as the New Math movement
of the 1960s.
Although the Standards have been described as the largest mathematics
reform effort in the United States since 1900, many of us vividly remember
a reform effort that dates from the 1960s, popularly referred to as New
Math. The modern math or New Math movement was the outgrowth of the Cold
War and a pervasive perception that the United States was not training
enough mathematicians and scientists. There was significant pressure to
increase the capability of our institutions of higher education and to
redefine the secondary school curriculum as well as altering the way in
which mathematics was taught. New Math created temporary changes in the
secondary school curricula, but the process of teaching mathematics courses
has changed very little over the last few decades. The New Math movement
was not successful at bringing about overall improvement in mathematics
teaching and learning.
The New Math movement and today's challenge as articulated in the Standards,
although both focused on improving students' understanding of mathematics,
are significantly different. New Math was aimed at developing student understanding
through mathematical structure and a focus on abstractions, appealing to
students' intellect. Meaning was imposed out of the structure of mathematics.
The Standards recommend developing understanding by using realistic
contexts and applications as well as concrete pictorial models, appealing
more to students' intuitive sense. Therefore, meaning is constructed out
of prior knowledge and experiences.
Gehn, a first-grade teacher in Cottage Grove, Wisconsin, talks about why she
uses modeling and problem-solving activities in her mathematics curriculum rather
than the memorization of facts. [252k audio file] Excerpted from the video
series Schools That Work: The Research Advantage, videoconference #2,
Children as Problem Solvers (NCREL, 1991). A text
transcript is available.
Despite the difference, we can learn several lessons about implementation
of large-scale reform by revisiting New Math. Rather than blaming the mathematics
content of the New Math movement, many critics believe that the real problem
with its lack of success lay in the implementation process. As efforts
to produce large-scale curriculum dissemination began, instructional issues
were addressed tangentially, if at all. Only small-scale, short-term training
was made available to teachers, and no real implementation plans were designed.
For a reform effort to be successful, all classroom teachers affected by
it need to be involved in all implementation stages:
- Building awareness and a vision of the reform changes.
- Designing and developing curriculum.
- Implementing new curricula and instructional practice in the classroom.
- Reflecting upon practice.
- Refining the implementation process and continuing to learn and grow.
The key question to be addressed in this issue is: What actions must
be taken to integrate the spirit of the Standards into classroom
- All students are becoming mathematically literate. NCTM (1989) has carefully
articulated five general goals for all students
that help to define mathematical literacy.
- All students are developing a strong, working knowledge of mathematical
- All teachers are continuing to learn about mathematics content and
the teaching and learning processes that facilitate the implementation
of new curricula and instructional practices that foster mathematical literacy.
- All administrators are supporting and encouraging mathematics teachers
to work among themselves and with other teachers and community members
toward the goal of enabling all students to become mathematically literate.
- All parents and community members are meaningfully engaged in: learning
about the new curriculum, instruction, and assessment practices recommended
in the Standards documents; helping the school community rethink
the learning and teaching of mathematics; making decisions about instructional
approaches and the mathematics program; providing resources for supporting
this new view of learning and teaching; and modeling this new vision of
teaching and learning in their interactions with students.
- The school's curriculum reflects the relevancy of mathematics to other
curricular subjects and everyday life.
The goals above present a challenge to students, teachers, administrators,
parents, and community members to create and implement a comprehensive
and long-term plan for the improvement of mathematics curriculum, instruction
and assessment in the education system. The goals were developed to ensure
that all student become mathematically literate. The following action options
are divided into implications for each of the significant stakeholders
within the educational system as they move toward ensuring mathematical
literacy for all students.
In order to become mathematically literate, students must:
- Experience mathematics in the context of the guidelines outlined
in the NCTM K-12 Standards.
- Develop a conceptual understanding of the full range of mathematical content
- Model, ground, and enrich their understanding of concepts by linking them
to multiple forms of representation (e.g., diagrams, graphs, tables, simulations/computer
visualizations, manipulative and symbolic expressions).
- Participate actively in the creation of mathematical knowledge.
Gehn, a first-grade teacher in Cottage Grove, Wisconsin, discusses how students
should have an active role in shaping the learning that takes place in her
class. [180k audio file.] Excerpted from the video series Schools
That Work: The Research Advantage, videoconference #2, Children as
Problem Solvers (NCREL, 1991). A text transcript
- Engage in rich mathematical discourse.
- Develop skills and processes of problem solving,
reasoning, communicating and connecting concepts within and outside of mathematics.
- Make appropriate and ongoing use of technology,
including but not limited to calculators and computers, as tools for learning
- Continue to assess their own mathematics learning
Higuchi, third and fourth grade teacher at Farmdale Elementary School in East
Los Angeles, California, talks about involving her students in parent conferences.
[423k QuickTime slide show] Excepted from Schools That Work: The Research
Advantage, Program #4, Alternatives for Measuring Assessment
In order to facilitate students becoming mathematically literate, teachers
- Develop their own positive attitudes toward mathematics and continue to
build upon their own mathematics content knowledge.
- Develop an understanding of a comprehensive mathematics curriculum and consider
the range of concepts and processes that are outlined in the Mathematics
Standards section of McRel's Standards and Benchmarks database,
when planning for curriculum and instruction.
- Participate in selecting and/or adopting curricula, textbooks,
and instructional materials/programs, and in decisionmaking relative to assessment
issues (e.g., grading and reporting).
- Establish high standards for all students and be certain that every effort
is made to provide equitable learning opportunities for each of them.
Marshall, Executive Director of the Illinois Math and Science Academy, Aurora,
Illinois, discusses the importance of having high expectations for all students
in mathematics. [639k QuickTime slide show] Excerpted from the video series
Schools That Work: The Research Advantage, videoconference #2, Children
as Problem Solvers (NCREL, 1991).
- Create a climate that facilitates students communicating
with one another in the language of mathematics, carefully listening to
each others' ideas, taking effort to express clearly their own ideas, and
showing mutual respect for each other.
Gehn, a first-grade teacher in Cottage Grove, Wisconsin, describes how she
persuades students to communicate how they solved problems. Additionally,
she explains why it is important for her, as a teacher, to listen to the different
methods her students use to arrive at their answers. [630k QuickTime slide
show.] Excerpted from the video series Schools That Work: The Research
Advantage videoconference #2, Children as Problem Solvers (NCREL,
- Find innovative and creative ways to connect to the existing network of
ideas and experiences of students, thereby teaching
mathematics in the context of meaning and application rather than merely
Gehn, a first-grade teacher in Cottage Grove, Wisconsin, tells how using story
problems in her class was one instructional and curricular change that helped
to spark an interest in her students and herself in mathematics. [324k
audio file] Excerpted from the video series Schools That Work: The Research
Advantage, videoconference #2, Children as Problem Solvers (NCREL,
1991). A text transcript is available.
- Plan lessons that involve students in the crucial elements of mathematical
discovery and engage them in an active process of learning in which they create
and discover mathematical concepts through the
use of manipulative materials and models.
- Focus on planning for meaningful connections
between mathematical content and specific problem-solving processes and strategies.
- Seek ways to relate mathematics learning to
other disciplines to form mathematical connections and to structure teaching and learning through problem-centered methodologies.
- Utilize technology to enhance and extend classroom experiences.
- Use alternative forms of assessment, including
performance-based assessment tasks that are consistent
with the instructional goals and the conceptually oriented aims of the curriculum,
to gain information about what students know and are able to do.
- Use research-based assessment strategies for diagnosing
each student's strengths and weaknesses, and effective diagnosis results
to create remediation plans/activities.
- Examine their own current practice and take advantage of opportunities for
ongoing learning of new instructional strategies linked to effective practice,
such as participating in workshops, attending professional meetings, reading
professional journals, and conducting classroom research.
- Seek opportunities to work and plan with others in their learning communities
in the co-development of supplementary curriculum units and sharing
To support all students becoming mathematically literate, administrators
- Become informed about the changes necessary for reforming mathematics
curriculum, instruction, and assessment.
- Articulate a new vision of curriculum, instruction, and assessment
in mathematics consistent with a schoolwide view of engaged learning.
- Provide the leadership and resources to support teachers in curriculum
development and planning, collegial interaction, and ongoing, intensive
professional development, all necessary for bringing about changes in curriculum,
instruction, and assessment in mathematics.
- Work with and involve teachers centrally in designing
and evaluating programs for professional development.
- Create different working models and implement changes in current school
structures to support and significantly enhance students' mathematical
power and teachers' professional growth.
- Garner the support and engagement of parents and community members
in reform efforts in mathematics curriculum, instruction, and assessment.
To enhance the learning of students, parents and community leaders must:
- Participate in the process of articulating the new vision of engaged
learning in mathematics and help to work out its implications for changing
curriculum, instruction, and assessment in mathematics.
- Enhance mathematics teaching and learning by creating partnerships
between schools, community facilities, and businesses. Include activities
such as mentorships, experiential learning opportunities, and co-investigations
whereby students and community members solve community problems.
- Support teachers and schools that are seeking to align assessment with
- Support local efforts to secure funds for educational reform.
- Provide learning opportunities in the home, allowing students to participate
in activities that stimulate reasoning and problem solving, and share those
activities with other parents.
- The area where reformers are most likely to run aground in bringing
about change is the task of building consensus on the beliefs and values
that should guide the teaching and learning of mathematics. Educators need
to learn how to communicate their goals clearly to others, and should initially
include a broad-based group of individuals in the process, to attain a
critical mass of people committed to reform. These individuals can become
important spokespeople for the cause of reform.
- The other area where difficulty arises is in the implementation of
the plan. As educational change agents, most of us are all too familiar
with the plan that gathers dust. We often underestimate the complexity
of change required, the length of time it takes, the intensity and frequency
of professional development required, and the significant reforming of
the school culture. If we accept this reality, we can better muster the
patience, commitment, energy, and time it will require to bring about the
deep and genuine transformation of learning and teaching in mathematics.
It is necessary to create and to periodically revisit, evaluate, and revise
a detailed plan for implementation.
- Testing and assessment programs that
are not coordinated with the instructional aims and curricular goals
reflected in the NCTM Standards can also be significant obstacles
to implementing reform in curriculum, instruction, and assessment. Teachers
and school district administrators often feel significant pressure for
their students to perform well on these tests, yet the tests usually monitor
the achievement of computational skills rather than the reasoning and problem-solving
abilities of students. There are major efforts underway to change these
assessment practices, and teachers and administrators can lead in advocating
for such changes.
- An additional implementation pitfall is related to the pervasive practice
of teaching the procedures of mathematics detached from the meaning and
applications of these procedures. There are a host of reasons that support
the emphasis on teaching procedures. Schools and classrooms moving toward
more of an emphasis on teaching students both what to do and why will want
to explore the roadblocks to this practice. Burns (1992) discusses the
reasons for teaching procedures and the reasons
for teaching arithmetic in this context of meaning and application.
POINTS OF VIEW:
- Some educators dismiss the idea of standards and feel it runs counter
to the goal of enhanced professionalism, since it takes the decision-making
power over what is taught out of the hands of teachers and into the hands
of a national standards-setting body. Standards, at least when used in
a rigid and highly prescriptive manner, they feel, are anathema to professional
autonomy. Thus, the attempt to set forth standards in mathematics curriculum,
instruction, and assessment is misguided.
- Questions have also been raised about whether the NCTM Standards
are the right standards. Some dissenters claim that the Standards
have an inadequate research base, that fluency with algorithms and computational
procedures is not adequately emphasized, and that the expansion in content
The Illustrative cases listed as samples below have been described as
such by many teachers and curriculum developers because of their usefulness
as models and frameworks for developmentally sequenced curriculum units.
Problem Solving and Critical Thinking in Mathematics:
K-8 Professional Development and Interdisplinary Curriculum Development
The Algebra Project
Math by All Means
Middle Grades Mathematics Project
The NCTM Addenda Series
The Quantitative Literacy Series
Used Numbers: Real Data in the Classroom
Midwest Consortium for Mathematics and Science Education
North Central Regional Educational Laboratory
1120 Diehl Road, Suite 200
Naperville, IL 60563-1486
(630) 649-6500, fax (630) 649-7600
Eisenhower National Clearinghouse for Mathematics and Science Education
The Ohio State University
1929 Kenny Road
Columbus OH 43210-1079
614-292-7784, fax 614-292-2066
Lawrence Hall of Science
University of California
Berkeley, CA 94720
510-652-1823; fax 510-643-5757
Mathematical Sciences Education Board, National Academy of Science,
and National Research Council
2101 Consitution Avenue, N.W.,
Washington, DC 20418
Council of Teachers of Mathematics
1906 Association Drive
Reston, VA 22091-1593
This Critical Issue Summary was researched and written by Cathy J.
Cook, mathematics education and professional development specialist, Midwest
Consortium for Mathematics and Science Education, North Central Regional
Date posted: 1995
Copyright © North Central Regional Educational Laboratory. All rights reserved.
Disclaimer and copyright information.