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7 Conclusion

The eight documents summarized in this report have been categorized as informing content, learning, teaching, assessment, and program changes. The Committee felt that organizing the recommendations in the same way would enable readers to understand the connections among the documents and illustrate common themes. Tables 7–1–5 highlight those recommendations that seem to cut across the eight publications. While there are diverse opinions in the mathematical community on how to improve mathematics education, the reports contain noticeable overlap in both the spirit and the specifics of the recommendations.

The tables do not contain all of the recommendations summarized in Improving Mathematics Education, let alone the entire set of recommendations across the eight documents. Empty cells in the table should not be construed as lack of support; the publication may actually address the particular issue but not make a direct recommendation. And, although some topics were common to several documents, this does not necessarily imply the recommendations were the same. Note that despite the increased interest in assessment and accountability throughout education, little attention seems to be paid to the topic across these publications.

The Committee identified several cross-cutting themes that emerged from the documents by considering the recommendations in each section and categorizing them across topics that emerged from the sets of recommendations. These common themes can provide a background to help frame the decisions we must make to improve school mathematics in the United States.

Change. The world is changing, society's expectations are changing, and our students are changing. It follows that the mathematics needed, its importance, and methods for teaching it are changing as well. As Before It's Too Late states, “It is abundantly clear from the evidence clearly at hand that we are not doing the job we should.” Five of the documents (see Table 7–5) reflect the need to change what we do, how we do it, and how for all students. For example, Adding It Up makes direct statements about



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Page 40 7 Conclusion The eight documents summarized in this report have been categorized as informing content, learning, teaching, assessment, and program changes. The Committee felt that organizing the recommendations in the same way would enable readers to understand the connections among the documents and illustrate common themes. Tables 7–1–5 highlight those recommendations that seem to cut across the eight publications. While there are diverse opinions in the mathematical community on how to improve mathematics education, the reports contain noticeable overlap in both the spirit and the specifics of the recommendations. The tables do not contain all of the recommendations summarized in Improving Mathematics Education, let alone the entire set of recommendations across the eight documents. Empty cells in the table should not be construed as lack of support; the publication may actually address the particular issue but not make a direct recommendation. And, although some topics were common to several documents, this does not necessarily imply the recommendations were the same. Note that despite the increased interest in assessment and accountability throughout education, little attention seems to be paid to the topic across these publications. The Committee identified several cross-cutting themes that emerged from the documents by considering the recommendations in each section and categorizing them across topics that emerged from the sets of recommendations. These common themes can provide a background to help frame the decisions we must make to improve school mathematics in the United States. Change. The world is changing, society's expectations are changing, and our students are changing. It follows that the mathematics needed, its importance, and methods for teaching it are changing as well. As Before It's Too Late states, “It is abundantly clear from the evidence clearly at hand that we are not doing the job we should.” Five of the documents (see Table 7–5) reflect the need to change what we do, how we do it, and how for all students. For example, Adding It Up makes direct statements about

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Page 41 Table 7–1 What Should We Teach? Recommendations in key categories Principles and Standards Adding It Up How People Learn Math Education of Teachers Educating Teachers Before It's Too Late High Stakes Every Child The documents that: Emphasize problem solving X X X X Emphasize mathematical reasoning X X X X X Suggest the use of technology in the development of mathematical proficiency X X X X X Advocate challenging mathematics for all students X X Advocate algebra and geometry early in the curriculum X X X Emphasize mathematics as sense making X X X Table 7–2 How Should We Teach? Recommendations in key categories Principles and Standards Adding It Up How People Learn Math Education of Teachers Educating Teachers Before It's Too Late High Stakes Every Child The documents that: Advocate that teachers build on what students know and on student thinking X X X X Emphasize the importance of understanding how to teach X X X X X Emphasize teaching so students learn with understanding X X X Support instruction that integrates skills and conceptual knowledge X X Emphasize the importance of having knowledge taught in a variety of contexts X X

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Page 42 Table 7–3 What Preparation and Support Do Teachers Need? Recommendations in key categories Principles and Standards Adding It Up How People Learn Math Education of Teachers Educating Teachers Before It's Too Late High Stakes Every Child The documents that: Advocate ongoing professional development X X X X X X Support programs for novice teachers X X X Emphasize the need for strong content knowledge on the part of teachers X X X X X X Promote K–12 partnerships to improve teacher education X X X Suggest that postsecondary educators assume more responsibility for preparing and developing teachers X X Table 7–4 How Do We Know That What We Are Doing Is Working? Recommendations in key categories Principles and Standards Adding It Up How People Learn Math Education of Teachers Educating Teachers Before It's Too Late High Stakes Every Child The documents that: Advocate using multiple ways to assess student understanding X X Emphasize the use of assessment to support the learning of mathematics X X X Caution about the implications of high-stakes assessments for instruction X

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Page 43 Table 7–5 What Must Change? Recommendations in key categories Principles and Standards Adding It Up How People Learn Math Education of Teachers Educating Teachers Before It's Too Late High Stakes Every Child The documents that: Emphasize the need to align curriculum, instruction, and assessment X X X X X Advocate expanding and using research to improve mathematics education X X X X X Support the use of math specialists or teacher leaders X X X X Emphasize high expectations for all students X X X X X Advocate engaging parents in improving mathematics education X X X Call for system changes X X X X X Stress coherent, important mathematics that is well articulated throughout the grades X X X we assess the impact of the changes to provide better instructional programs problems in the current system: “Our experiences, discussions, and review of the literature have convinced us the school mathematics demands substantial change” (p. 407). It also suggests coordination of curriculum, instructional materials, assessment, and professional development; school organization should drive the changes. Every Child Mathematically Proficient offers an action plan for changing mathematics education. The Mathematical Education of Teachers calls for “changing expectations for mathematical knowledge.” Equity. According to the documents, all students can and should be mathematically proficient. The reports consistently deliver the message that school mathematics can and must empower all students (see Table 7–5). For example, five of the documents emphasize high expectations for all students. Every Child Mathematically Proficient calls for ending “dead-end” tracks, and High Stakes calls for the end of tracking that discriminates

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Page 44against certain students. The content areas emphasized in the documents (see Table 7–1) underscore that “mathematics for all” does not mean a watered-down mathematics but an authentic, coherent, and functional mathematics that expects more of every student than in the past. For example, Every Child Mathematically Proficient recommends algebra and geometry for all students, and a majority of the documents emphasize mathematical reasoning. Content. Mathematical content is the centerpiece for teaching, instructional materials, assessments, teacher preparation, and professional development. The documents make strong statements about the importance of reasoning, problem solving, algebra, and geometry ( Table 7–1). For example, reasoning and proof is a content strand in Principles and Standards for School Mathematics, and Adding It Up includes the notion of adaptive reasoning as essential for mathematical proficiency. Six of the documents call for strong content knowledge on the part of teachers ( Table 7–3) and emphasize the need to align curriculum instruction and assessment ( Table 7–5). In particular, Before It's Too Late states “high-quality teaching requires that teachers have a deep knowledge of subject matter” (p. 22), while The Mathematics Education of Teachers suggests the specific number of content hours that should be part of undergraduate teacher preparation programs. If teachers do not fully understand fundamental mathematics, it is unlikely they will impart that knowledge to their students. Understanding. While research findings in education are always open to further interpretation, it is increasingly clear that learning is a complex cognitive process that builds on prior knowledge and requires active engagement with new situations. As described in How People Learn, a growing body of knowledge suggests that learning is best accomplished when accompanied by conceptual knowledge and making sense of new concepts, with an emphasis on understanding. Adding It Up makes this explicit in terms of learning mathematics, emphasizing the need for both conceptual understanding and procedural fluency. Four of the documents ( Table 7–2) support the use of student thinking to build understanding. The notion of learning as a sense-making process also has import for teaching and teacher preparation with recommendations that teachers be prepared to teach mathematics in ways that enable students to learn. For example, Educating Teachers of Science, Mathematics, and Technology suggests that teachers “regularly update the content knowledge and pedagogical tools needed to teach in ways that enhance student learning and achievement in these subjects” (p. 109). Research. The use of research to support mathematics education is a key element in majority of the documents ( Table 7–5). This ranges from How People Learn, which calls on existing research to substantiate claims made in the document, to High Stakes, which advocates advanced research on

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Page 45topics for which there are more questions than answers. Educating Teachers of Science, Mathematics, and Technology calls for enhanced links between research and practice, and Every Child Mathematically Proficient suggests such links should be translated into the development of curricular materials and professional development programs for teachers. The prominence of research in these documents suggests a shift in the way changes are made in mathematics education and has implications for how program changes are instituted. Collaboration. Making change in our complex education system is difficult, and improvement requires collaboration among diverse stakeholders who represent different parts of the system. Systemic change requires new forms of partnerships to make the system more productive and to provide solutions that cut across the system components. Educating Teachers of Science, Mathematics, and Technology and The Mathematics Education of Teachers make strong statements about forming K–12 and university partnerships, involving mathematicians and scientists to improve teacher preparation and development ( Table 7–3). Several of the documents suggest that parents as well as teachers must be brought into the process of change in school mathematics, including discussions of curricular goals and how teaching and assessments have changed in mathematics classrooms ( Table 7–5). Professional Development. Every report that addresses changing conditions and expectations eventually arrives at the need for more effective forms of professional development. Six of the reports ( Table 7–3) support the need for highly qualified teachers who take part in continuous learning that is part of the system for teachers: “High-quality teaching must be continually reshaped” (Before It's Too Late, p. 22). Given enhanced opportunities for professional growth and development, teachers will be able to implement the vision of mathematics instruction called for in these reports. Moreover, the professional development must be more than a “patchwork of courses” (Educating Teachers, p. 31) and must be related to the mathematics teachers are expected to teach and to how to helping students understand this mathematics. Technology. The documents recognize technology as a major factor driving the need for change, creating a different world with different expectations about the mathematics students will need ( Table 7–1). The reports, however, are not unanimous in their technology recommendations. Principles and Standards states that technology is essential to the teaching and learning of mathematics. Every Child Mathematically Proficient cautions that further study is needed to explore how technology should be used to enhance student learning in mathematics, and Adding It Up is also cautious, but suggests that technology offers much promise in the search for ways to improve what happens in classrooms.

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Page 46 Mathematics Specialists. As teachers learn new content, new methods of instruction, and new methods of assessment, they need resources and support. There is also immense pressure on elementary teachers, who have multiple subjects to teach. While math specialists are called for in several of the documents ( Table 7–5), the nature of such specialists and the recommendations vary. The Mathematics Education of Teachers and Every Child Mathematically Proficient call for math specialists to teach mathematics content by grade 5. Adding It Up recommends that mathematics specialists provide support for K–8 teachers, and Principles and Standards calls for teacher-leaders throughout the system. Closing Comments Considering the overall mathematical landscape and potential directions necessary for improvement, the Mathematical Sciences Education Board recommends that educators use reports such as the ones described in this document to help them make informed decisions. The committee also recommends that the decision making process include participation and input from across the community, with diverse contributions from a wide spectrum of individuals—in much the same spirit as these documents were developed. The committee recognizes that all the information necessary to make the many decisions facing educators is not available. The research-based findings and the recommendations in these documents, however, can be a rich resource for educators engaged in actions and initiatives intended to make real and positive change in the lives of students and teachers in mathematics classrooms across the nation. These documents can help educators as they struggle to make wise and defensible decisions and strive to meet the challenges of implementing truly excellent programs of mathematics instruction.