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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards Process for Designing a Curriculum Program The process of designing a curriculum program that includes components that meet the criteria described in the preceding section requires considerable time and commitment. Fortunately, the process does not have to be considered completely implemented for improvements in mathematics and science teaching and learning to be realized. Each stage of the process makes a contribution to these goals. This report assumes that a curriculum program design committee, with representatives of various stakeholders in the school system or district, will be responsible for the design process. This process — described in this section — will be a major professional development experience for the committee members. The process described in this section and illustrated by Figure 6 is not intended to be prescriptive but, rather, to suggest how to design a curriculum program. The boxes in Figure 6 represent key steps in the process. Factors that influence the process are represented by ovals. Early in the process of developing or revising a curriculum program, the committee should study the mathematics and science education context of the Figure 6. Process for Designing a Curriculum Program
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards local community, community priorities, state mandates and assessments, local and state educational system structures, and local history of educational practices and programs. It is important for the committee to become familiar with this context, particularly with local, state, and national standards. Other policy documents, such as goals, mission statements, course requirements, and curriculum guides, should be considered carefully in the initial part of the design process. In addition, the committee should not only study current practices, customs, and beliefs about education in the local schools but should compare these to the educational research literature on best practices in teaching, learning, and curriculum design. ESTABLISHING GOALS AND STANDARDS As the starting point in the development of an improved curriculum program, a district needs goals and a set of standards to guide the work of the curriculum program design committee, particularly in the important areas of creating a framework and selecting the core instructional materials. The previous section of the report, "Components of Coherent Mathematics and Science Education Curriculum Programs," lists criteria for goals and standards and indicates how national standards provide guidance for districts that are writing their own. In recent years, most states have adopted mathematics and science goals or standards (CCSSO, 1997). It is important for the design committee to base its work on state policy since that policy determines the extent to which state goals and standards must be used locally. Some states require local districts to follow the state standards, while others expect the standards to be used as guidelines only. In some cases, state content standards guide a state's assessment program. In these cases, districts — and their curriculum program design committees — will likely choose to focus on those standards so that their students will perform well on the state assessments. Where local or state-level standards do not exist or where state standards are optional or do not meet the criteria for high-quality standards given in the previous section of this report, design committees may want to use national standards. Many districts and states have used the following national standards as the basis for their own standards: The Curriculum and Evaluation Standards for Mathematics (NCTM, 1989); The Professional Standards for Teaching Mathematics (NCTM, 1991c);
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards The Assessment Standards for School Mathematics (NCTM, 1995); The National Science Education Standards (NRC, 1996b); and The Benchmarks for Science Literacy (AAAS, 1993). BUILDING A COMMON VISION Even with the availability of goals and a comprehensive set of standards, the curriculum program design committee needs to agree upon and articulate a common vision for the district in its own language. Teachers, administrators, and others on the committee should translate what is called for in national, state, and local standards into administrative and classroom policy and practice for their district. The committee will want to consult research literature and other sources on best practices in teaching and learning science and mathematics. Creating a common vision of what and how students will learn mathematics and science is an important component of the development of the curriculum program, regardless of whether most of the program's components are adopted or adapted from other programs or developed independently. A common vision helps focus all stakeholders on what the school district believes is important. The vision is critical for good communication, as it will help the committee describe what the practices and behaviors of students, teachers, administrators, and parents should be when the curriculum program is in place. In building a common vision, the design committee should describe what would be observable when the curriculum program is fully developed and implemented in terms of what students are learning and how they are learning it; what teachers are doing to support, encourage, and expect learning; the evidence to be used during assessment of student performance; and activities parents, administrators, businesses, and colleges and universities are engaged in to support and encourage high levels of student performance. Many approaches to this part of the design committee's work are possible as long as members engage in intellectual and focused discussion regarding issues of teaching and learning. One such discussion might include tracing the development of a particular concept or strand across several grade levels, and correlating this development with national and state standards documents. Examining sample instructional materials or student work also could help the committee clarify the nature of student
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards performances called for in the standards. Classroom examples paint pictures that communicate to teachers, administrators, supervisors, and members of the public how students engage in meaningful learning of mathematics or science. The classroom examples in the NSES have helped educators get a concrete sense of the national recommendations in practice. Other useful sources of examples include the NCTM's addenda to its Standards (NCTM, 1991a) and similar addenda in science, to be published in 1999 by the National Research Council for the Center for Science, Mathematics, and Engineering Education. In addition, committee members may want to participate in actual classroom lessons, either as teachers or as students. It is important at this stage in the development of a curriculum program for the committee members to have an opportunity to share and discuss their many, often diverse, views about the discipline itself (mathematics or science) and about how students learn that discipline. For example, teachers may not agree about the contributions and limitations of lecture versus inquiry or of small group versus large group instruction. These issues are best addressed explicitly in a positive, professional environment where relevant educational research literature can be accessed and reviewed. Ideally, when the vision statement is finalized, it will be a concrete statement that communicates the district's vision of what is important in mathematics and science education. It should illustrate how standards will be used and describe what a classroom should look like or what kind(s) of thinking a student should be able to do. It also should address the broader learning context, such as how teachers will teach, how students will be assessed, and how the district will support and be accountable for its students' learning. DRAFTING A K-12 CURRICULUM FRAMEWORK Once the goals and standards that guide the curriculum program have been identified and translated into a more specific vision of learning, teaching, assessing, and support, work on the curriculum framework can begin. This framework is meant to organize and sequence the content to create coherence in the curriculum program across all grades in all courses. It does this by assigning concepts to grade levels based on the growth and development of ideas and skills from grade to grade and by grouping the concepts to form units or courses. Whether the framework is developed locally or taken from an outside source, it should be reviewed carefully to ensure its coherence.
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards Although instructional materials are not part of the curriculum framework,10 the two elements must be aligned and interconnected. Therefore, drafting a framework is the beginning of an iterative process. First, a draft of the framework indicates at which grade level concepts and skills (these may be expressed as standards or benchmarks, depending on the document being used) might be taught. Then, as instructional materials are reviewed, this early framework will be refined until a final product emerges. It may be tempting to begin assigning concepts to grade levels and/or courses immediately, but it is wise to refrain from doing so too early in the process. If districts assign concepts to grade levels or courses before looking at instructional materials, they are likely to create a need for unique units or courses. This then places them in the position of having to write materials — a difficult and expensive process that most districts do not have the resources to accomplish. The design committee should have the following questions in mind while drafting the framework: Are some units so ingrained in the local curriculum that they must be retained? (If this is the case, the appropriate standards must be referenced in the framework.) Who are the intended audiences of the framework? How will these audiences use the framework? Will they understand it in its current form? What instructional resources are available locally in the community that should be built into the framework? (For example, an outdoor education laboratory school, a district greenhouse or planetarium, a museum of science and industry, or connections with a local industry may all complement a high-quality curriculum program.) IDENTIFYING CORE INSTRUCTIONAL MATERIALS Deciding What to Review. The goal of this step is to identify instructional materials that best support the standards as they are organized in the draft curriculum framework. As men 10 Because of the widespread influence of national standards in mathematics and science, increasing numbers of instructional materials and, in some cases, multi-year programs are available commercially that provide for instruction and the learning of content called for in the national standards. Both commercial publishers and curriculum development groups funded by the National Science Foundation have developed these materials. Some of the programs may address a number of the criteria for curriculum programs outlined above; therefore, local design process committees may use some aspects of these programs — as well as adjustments to their own local frameworks — to help them construct coherence within and across grades.
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards tioned earlier, the processes of identifying instructional materials and designing the curriculum framework go hand in hand. The draft framework indicates a possible assignment of standards to grade levels. Examining the instructional materials informs the committee as to what is available at each grade level, how the standards are presented in the materials, and the prior knowledge students need before beginning each unit. After the design committee reviews information about the materials, it may find a need to revise some portions of the framework. Committee members should be selective about their review of material because they will not have time to review everything. The committee may want to become acquainted with tools developed or under development by several national groups for the identification of ''exemplary" materials. For example, the National Science Foundation has made recommendations for middle-level science (NSF, 1997). The National Science Resources Center has done a similar study of materials for elementary- and middle-level science (NSRC, 1996 and 1998b). In addition, the NRC's Center for Science, Mathematics, and Engineering Education, AAAS's Project 2061, and the U.S. Department of Education all are working to analyze and identify or to help local decision makers analyze and identify exemplary mathematics or science materials (NRC, 1999c; AAAS, 1997; and DoEd, 1997). The design committee may want to consider reviewing the materials currently in use. Even though some members will be familiar with these materials, all materials that have the potential for use in the curriculum program should be reviewed using the same set of criteria. Then, if there is a decision to drop a widely used textbook, evidence will be available. Conversely, if a district chooses to continue using a program that is unpopular, how and why this decision was made may be more easily explained. Review Instruments and Procedures. With the introduction of content standards, most districts will need to revise their process for reviewing instructional materials, employing criteria similar to those outlined in the second section of this report. The review process should be clearly defined and involve the use of multiple techniques that give teachers and others an opportunity to analyze the materials under consideration. Supplemental materials may play a role in this process and will be discussed later in this section (beginning on pg. 37). A considerable amount of time is needed for careful review of materials. Setting up a two-stage process has some
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards advantages. The first stage can be used to eliminate the weakest instructional materials based on the criteria provided in this report and the design committee's vision of teaching and learning. First-stage "filtering" could involve analyzing the materials' instructional approach and quickly retaining those materials that support the skills and concepts needed for problem solving, but eliminating materials that, for example, emphasize confirmatory, pencil-and-paper activities or that would provide students with little or no experience in problem solving, data collection, and analysis. In this first stage, the committee also could exclude materials that are collections of unrelated activities or short units on single topics not connected to others. This would allow members to focus in the second stage on materials that have the most potential to support the development of a coherent curriculum program. In the second stage, reviewers need to begin by choosing a review instrument and then practicing with it. In the practice session, all reviewers should use the instrument to evaluate the same set of instructional materials. The reviewers should discuss and agree as a group on the interpretation of each criterion to increase the reliability of the resulting reviews (i.e., independent reviewers would arrive at similar judgments). Working through this process also will give reviewers further insight into the standards that they will be using to judge the materials. The results of the first and second stages of review should reveal strengths and weaknesses of each considered set of materials. The next step is to use this knowledge to refine the framework. REFINING THE CURRICULUM FRAMEWORK Finalizing Placement of Core and Supplemental Instructional Materials. The design committee, having worked with the policy documents (goals and standards), developed and described its vision of teaching and learning, analyzed the current program, and reviewed and selected instructional materials, is ready to begin refining the curriculum framework. Refining the framework involves 1) clarifying the growth of student understanding within and across years by assigning concepts to grade levels, and 2) identifying and addressing transitions and gaps in the framework, as follows: 1. Clarifying the growth of student understanding within and across years by assigning concepts to grade levels. Once instructional materials have been selected, concepts and skills can be assigned to grade levels.
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards When the design committee is focusing on the expected level of student understanding, they should describe performance expectations, not just specify topics. For example, a topic such as "introduction to photosynthesis" may be assigned to sixth grade. As written, this phrase can be interpreted in many ways. Some teachers may have their sixth-grade students memorize definitions; other teachers may go so far as to have students design experiments to collect evidence that plants need light; and yet others will ask students to memorize the equations that summarize the chemical reaction. In mathematics, a similar situation arises when a topic such as "length" is assigned to the first grade. Some teachers may feel that they should help students understand the concept of measuring length by using non-standard units; others may use standard units, such as centimeters or inches but not tools such as rulers and meter sticks; and yet others may go directly to measuring with rulers and meter sticks. As discussed in the previous section on curriculum program components, the most significant difference among frameworks is their level of coherence. To build coherence into a framework requires clear descriptions of the content. Necessary prior knowledge is identified, ideas sequenced, and connections among ideas developed. This process is facilitated by using "Growth of Understanding" tables similar to Figures 4 and 5 on pgs. 26 and 27 to identify the order of introduction of concepts, both within a given year and across years. To recap, Figures 4 and 5 illustrate how each year's instruction builds on that from prior years, with the concepts becoming progressively more complex. Such a progression helps insure that students will be able to develop a solid understanding of the "big ideas" of the discipline. Although this progression has to be considered for each major concept, skill, and ability, every concept should not be addressed every year. This would result in too many concepts being addressed each year without adequate time for students to develop an understanding of anything, exacerbating the problem described in TIMSS reports (Schmidt et al., 1998). Typical mathematics and science curricula attempt to address far too many topics per grade level. Unnecessary repetition can quickly stifle both student enthusiasm and understanding, leading to low expectations. One of the most important lessons from international studies and analyses cited in the introduction to this report is the value of tightening the focus of our curriculum to address fewer topics each year, allowing for much greater depth of learning. Ideas also must build within a year. Not only should there be a lesson-to
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards lesson progression, but units should also build from one to the next. Attention must be paid to the development of themes, concepts, and skills from the beginning to the end of a school year. The framework should be explicit about how skills and concepts interweave within individual units, across a year, and over a span of years. Without this purposeful interconnection, the curriculum program cannot be coherent. To illustrate, in science students too often read about a rigid "scientific method" at the beginning of the school year, then they read science content for the rest of the year without ever having the opportunity to experience how science concepts are derived from investigations. In mathematics, a parallel experience often occurs, when algorithmic procedures are taught as though they were important in and of themselves, rather than as tools for solving real-world problems. In such cases, the understanding that is essential to the eventual application of the algorithms is often omitted. Equally efficient procedures that are better understood — or that have been created by students — may be equally as acceptable and never considered. Even when efforts are made to address standards, concepts and skills often are separated. Students may study independent units on graphing, metric measurement, and microscope use or on plant classification, plant structure, and plant distribution, without ever seeing the connections among the concepts and skills in those units. The following questions can guide the development of a framework that supports a logical and sequential building of student understanding: What prior knowledge is needed for each concept? Is it addressed in an earlier unit? Can it be presented in the current unit? Is a logical or developmental sequence of concepts presented within each grade level? How will we assist students who have missed important prior knowledge and experiences? How does the framework convey the importance of interconnected skills and concepts? How will the connections and prior knowledge and skills called for in the framework be conveyed to teachers and others who are using the curriculum program? 2. Identifying and addressing transitions and "gaps" in the framework. The instructional materials selected to support the K-12 framework may come from several sources. Ideally, one curriculum series or program would suffice for a complete span of grades, such as K-5 or K-8. The challenge is to
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards figure out how to create links between different programs. For example, with series adopted for grades K-6, 7-8, and 9-12, the connections between grades 6 and 7 and between grades 8 and 9 must be examined closely. Does the middle-grades program pick up where the elementary grades program ends and adequately prepare students for the high-school learning goals? Linking different sources of materials makes the process of putting together a curriculum program complicated. Even when all the materials selected are of high quality, they usually differ in philosophy, program design, expectations of students, and attention to particular concepts. Some curriculum materials are written and marketed as a set of standalone units that schools or districts can mix and match to meet their needs. This situation is more typical of science than mathematics, for which complete programs are available that span several grade levels. Although the mix-and-match approach gives districts more flexibility, if districts choose units from a variety of sources, care should be taken to ensure that the units achieve the desired coherence in the curriculum program. After instructional materials have been selected, there may be places where the coherence is weak. The gaps in the curriculum program need to be identified, as well as inconsistencies in expectations of students when there are transitions from one set of materials to another. First, the big ideas (standards or benchmarks) in each program and flow between the units should be identified. If the understanding of the big ideas do not build progressively, then the remaining gaps must be filled and redundancies eliminated. It is equally important to track the skills that are expected of students, such as graphing, measurement, analysis of data, and use of tools. A design committee refining its curriculum framework could address gaps and transition problems by recommending that instructional materials be modified to create transition activities; new units be written where major gaps exist; supplemental units from other sources be identified; or professional development be provided for teachers that addresses the issues associated with the gaps. As mentioned earlier, modifying existing units or writing new units is an expensive and time-consuming solution to creating better transitions or covering gaps between the instructional materials from different sources. Finding units from other sources is an alternative way
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards to fill gaps. However, this approach may result in a "patchwork" program that creates additional inconsistencies. Professional development offers a flexible solution, in which teachers identify transition problems and consider ways to address them. They may supplement the content by using units from other programs or fill gaps or address transition problems with their own strategies. The process described here of refining the curriculum framework may seem overwhelming. The long-term goal — to design a high-quality curriculum program for mathematics and science — is well worth the additional commitment called for. In addition, keep in mind that this report does not mention timelines because the process never ends; there will always be room for improvement. The most difficult and time-consuming part of the process is the beginning, when the design committee and the district are learning to use standards to create coherence. Once this process is understood and a few examples developed across several grade levels, continuous improvement of the total curriculum program will become much easier. Pilot Testing the Proposed Curriculum Program. Although individual instructional materials adopted for the curriculum program may have been pilot tested in other school districts, the particular collection of materials now gathered in the design committee's framework may not have been. It needs to be. The purpose of pilot testing material is to identify problems and to correct them before full-scale implementation. Ideally, the total program would be evaluated over several years as a cohort of students moved through the program starting in kindergarten or first grade. Obviously, full-scale implementation cannot wait for this. Rather, the district should strive to answer questions such as those listed below by piloting the program over a short period of time in several schools that together cover the total K-12 grade span. How well do diverse students achieve the intended results (standards) using the new instructional materials? Teacher reports and some objective test data will be useful in comparing results across schools and against desired results. Do students have the expected prior knowledge and skills at the beginning of each unit/course so that the coherence designed into the curriculum program can be maintained? When students do not have the prior knowledge and skills, how do teachers
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards make the needed adjustments for these students? Are any problems severe enough to justify the revision of the framework? What professional development is needed so that teachers can maintain the coherence of the curriculum program without denying access to students who have not attained the prior knowledge and skills called for? What professional development is needed to support teachers and principals in the use of the instructional materials in the curriculum program? What resources (time, materials, instructional technology, facilities, and community resources) do teachers need to use the materials? What resources are needed to maintain durability and usefulness of the materials between adoption cycles? What resources are needed to keep parents informed? EVALUATING THE CURRICULUM PROGRAM The process of developing a coherent curriculum program is not complete until the design committee has formulated a method and schedule for the periodic evaluation and improvement of the curriculum program. Like other important programmatic changes in a district, designing a curriculum program should be considered to be an ongoing process, not a one-time event. In today's rapidly changing environment, mathematics and science programs can become outdated quickly, even if they had represented state-of-the-art thinking at the time they were designed. In making critical decisions about the nature of a new mathematics and/or science program, district policy makers and the committee will want to consider how drastic a change from their current program is desirable. As they compare their existing curriculum with national standards and their local vision, the benefits versus costs and risks will need to be weighed. An increasing number of districts also will need to take into account statewide standards of learning that are reinforced by mandatory statewide assessments. In the end, some communities may decide that they want to upgrade their mathematics and science programs dramatically to reflect the most innovative direction possible. Other communities, especially where the school district is large or where there are significant philosophical differences of opinion about what should be taught and how students should learn, may decide to proceed with more moderate change. Dramatic change and slower, more incremental change both have their
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards advantages and disadvantages. One advantage of dramatic change is that it requires teachers to learn new skills and ways of teaching. However, such change may be so radical that school district staff and the community reject it before there has been adequate opportunity to see results (Fullan, 1991). On the whole, a slower, more incremental approach may be more acceptable to staff and the community, but because teachers have the time and opportunity to modify new approaches, the innovation may be unacceptably compromised. Regardless of how drastic the change decided upon it should be noted that improvement in teaching and the subsequent improvement in student achievement will take time. The implementation of an innovative program requires that teachers learn new and different teaching strategies. Often, this takes as many as three years (Fullan, 1991). Indeed, the implementation of an innovative program may result in an achievement dip during the first year or two of implementation. Furthermore, because the program advocated in this report is connected across several years, significant improvement of students' achievement likely will result from their being in the program for more than one year. The district's evaluation plan should take into account the need for early data — as well as long-term data — on program effectiveness. The data gathered early will be helpful in deciding whether to move from pilot testing to full implementation. The data gathered over time will be critical to long-term evaluation and improvement decisions. One solution to this dual need for evaluation data is to collect enough to reach the consensus needed for formal approval of the program from the pilot sites that have been using the program for the greatest period of time and, simultaneously, to design a program evaluation process that gathers data over a more extended period of time from all schools. Such a process would provide, for example, a description of what constitutes a well-implemented curriculum program11; a means of identifying schools and 11 There are useful procedures for defining a well-implemented program. Three are mentioned here. First, the National Science Resources Center has developed a series of rubrics for this purpose (NSRC, 1997). Second, the Concerns-Based Adoption Model has developed a method of describing and mapping the progression of implementation of an innovative program (Hall & Loucks, 1978). Third, the National Science Foundation-funded Classroom Observation Protocols (Weiss et al., 1998; Horizon Research, 1999) provide criteria to determine effective classroom instruction and program implementation.
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards classrooms where curriculum programs have been well implemented; and a comparison of student achievement in classrooms and schools where curriculum programs have been well implemented for two or three years with student achievement in control classrooms in the same district or comparable districts. (The measures used should be aligned with the standards that were used to develop the program.) BUILDING CONSENSUS AMONG THE STAKEHOLDERS AND OBTAINING APPROVAL As indicated earlier, the committee responsible for designing the curriculum program will learn a great deal about curriculum, instructional materials, and other factors, such as pedagogy and assessment, that can affect the achievement of students. In the process, this committee will make a number of recommendations that will change the current program and that may impact teachers, administrators, and parents. The success of the new curriculum program ultimately will depend on the stakeholders outside the committee reaching a level of understanding and support for it comparable to that of the committee members. This requires that sustained and systematic communication be planned and executed to keep these stakeholders informed, to solicit their input, and to develop consensus and support for the committee's work. Specific strategies for the committee could include developing a two-way communication link with all stakeholders through status reports, newsletters, World Wide Web sites, newspaper articles, and presentations to school staffs and parent-teacher organizations; seeking input and response on a periodic basis through questionnaires, focus groups, use of external reviewers, and presentations to faculty and community groups; informing stakeholders of pilot testing results, how the feedback has been used, and the specific impact both are having on the committee's work; judging the degree to which the committee's work has been accepted by examining the various forms of feedback received (a committee that has communicated its work well will have a good sense of the degree to which its work will be supported); and convening meetings for the purpose of obtaining formal expressions of consensus. The next step is formal approval of the curriculum program by appropriate
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Designing Mathematics or Science Curriculum Programs: A Guide for Using Mathematics and Science Education Standards district decision makers. When the committee has achieved consensus with other stakeholders, it is ready to request this approval. Ideally the committee would simultaneously present its implementation plan12 and budget. 12 In order to translate the curriculum program into classroom practice, a number of implementation strategies, activities, and mechanisms must be in place. These include professional development for both teachers and administrators and development of a number of support mechanisms. See Appendix A for an overview of this topic.
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Representative terms from entire chapter: