Critical Issue: Implementing Curriculum, Instruction, and Assessment Standards in Mathematics

ISSUE: 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 Standards.

OVERVIEW: 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.

Susan 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 practice?

GOALS:

• 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 content.
• 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.

ACTION OPTIONS:

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 topics.
• 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.

Susan 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 is available.

• 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 mathematics.
• Continue to assess their own mathematics learning and progress.

Charlotte 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 (NCREL, 1991).

• Video
• Video with captions (audio description available)
• Text transcript (fastest download)

In order to facilitate students becoming mathematically literate, teachers must:

• 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.

Stephanie 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).

• Video
• Video with captions (audio description available)
• Text transcript (fastest download)
• 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.

Susan 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, 1991).

• Video
• Video with captions (audio description available)
• Text transcript (fastest download)
• 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 teaching procedures.

Susan 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 instructional approaches.

To support all students becoming mathematically literate, administrators must:

• 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 curricular goals.
• 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.

IMPLEMENTATION PITFALLS:

• 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.

DIFFERENT 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 is unnecessary.

ILLUSTRATIVE CASES:

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

Resources:

Math by All Means

Middle Grades Mathematics Project

The NCTM Addenda Series

The Quantitative Literacy Series

Used Numbers: Real Data in the Classroom

CONTACTS

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
E-mail: mailto:info@ncrel.org
WWW: http://www.ncrel.org/msc/msc.htm

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
E-mail: info@ENC.org

The EQUALS Project
Lawrence Hall of Science
University of California
Berkeley, CA 94720
510-652-1823; fax 510-643-5757
e-mail: equals@maillink.berkeley.edu
WWW: http://equals./lhs.berkeley.edu

Mathematical Sciences Education Board, National Academy of Science, and National Research Council
2101 Consitution Avenue, N.W.,
Washington, DC 20418
202-334-2000

National Council of Teachers of Mathematics
1906 Association Drive
Reston, VA 22091-1593
703-620-9840
Fax: 703-476-2970
e-mail: InfoCentral@nctm.org
WWW: http://www.nctm.org

References

Date posted: 1995