How should we teach so students learn? What students learn is related to how they learn. What do we know about how students process ideas and how they put them together to make sense of the mathematics they are studying? How is this different from current practice? As educators decide what mathematics programs they want for their students, they should consider not only what content is important but also what research can tell us about how students learn and how this should inform the curriculum they put in place and the instructional processes used to deliver that curriculum. These questions should drive decision making:
How do students learn mathematics?
What are the implications of what we know about how students learn for curriculum and instruction?
What is the nature of teaching practice supported by research in cognitive science?
Adding It Up: Helping Children Learn Mathematics, developed by the National Research Council's Mathematics Learning Study Committee, 2001.
How People Learn: Brain, Mind, Experience, and School, developed by the National Research Council's Committee on Developments in the Science of Learning and the Committee on Learning Research and Educational Practice, 2000.
These two resources address issues of research related to student learning. Adding It Up sets forth what we know about children's learning of mathematics,
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Page 10 3 How Should We Teach? How should we teach so students learn? What students learn is related to how they learn. What do we know about how students process ideas and how they put them together to make sense of the mathematics they are studying? How is this different from current practice? As educators decide what mathematics programs they want for their students, they should consider not only what content is important but also what research can tell us about how students learn and how this should inform the curriculum they put in place and the instructional processes used to deliver that curriculum. These questions should drive decision making: How do students learn mathematics? What are the implications of what we know about how students learn for curriculum and instruction? What is the nature of teaching practice supported by research in cognitive science? RESOURCES AVAILABLE Adding It Up: Helping Children Learn Mathematics, developed by the National Research Council's Mathematics Learning Study Committee, 2001. How People Learn: Brain, Mind, Experience, and School, developed by the National Research Council's Committee on Developments in the Science of Learning and the Committee on Learning Research and Educational Practice, 2000. OVERVIEW OF THE RESOURCES These two resources address issues of research related to student learning. Adding It Up sets forth what we know about children's learning of mathematics,
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Page 11 particularly number, in grades pre-K–8, as well as the implications this knowledge has for teaching. How People Learn calls on recent findings in brain research to address issues of learning in general. In particular, it describes how skill and understanding in key subjects are acquired and discusses our growing knowledge about complex reasoning and problem solving. Adding It Up: Helping Children Learn Mathematics Adding It Up is a research-based examination of pre-K–8 mathematics that focuses on what the mathematical content is with which students must develop proficiency, how instruction can help students develop this proficiency, and, most significant, the research that undergirds why these positions are taken. The report was developed by the Mathematics Learning Study Committee of the National Research Council. It arose out of concerns on the part of the National Science Foundation and the U.S. Department of Education about the shortage of reliable information on the learning of mathematics that could guide best practice. The developers of Adding It Up based their findings and recommendations on research that is “relevant to important educational issues, sound in shedding light on the questions it sets out to answer, and generalizable in that it can be applied to circumstances beyond those of the study itself.” They also “looked for multiple lines of research that converge on a particular point” (p. 3). Using the analysis of this research, much of which is synthesized in the report, the committee states that “All young Americans must learn to think mathematically, and they must think mathematically to learn” (p. 16). “Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics” the report chooses the phrase mathematical proficiency to capture “what it means for anyone to learn mathematics successfully.” Mathematical proficiency is then defined as having five interwoven and interdependent strands (p. 5): conceptual understanding—comprehension of mathematical concepts, operations, and relations; procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately; strategic competence—ability to formulate, represent, and solve mathematical problems; adaptive reasoning—capacity for logical thought, reflection, explanation, and justification; and productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy.
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Page 12 While the report focuses on curriculum and learning, some of the discussion relates to instruction and to issues of professional development. A key message is that the report endorses no single instructional approach but contends that “instruction needs to configure the relations among teachers, students, and mathematics in ways that promote the development of mathematical proficiency. Under this view, significant instructional time is devoted to developing concepts and methods; carefully directed practice, with feedback, supports learning. Discussions build students' thinking, attend to relationships between problems and solutions and to the nature of justification and mathematical argument as the strands of proficiency grow in a coordinated, interactive fashion” (p. 11). Finally, explaining why so much of the report focuses on the domain of number, it notes that “most of the controversy over how and what mathematics should be taught in elementary and middle school revolves around number” (p. 20), including questions such as the following: Should children learn computational methods before they understand the concepts involved? Should they be introduced to standard algorithms for arithmetic computation, or should they be encouraged to develop their own algorithms first? How proficient do children need to be at paper-and-pencil arithmetic before they are taught algebra and geometry? Thoughtful discussion about these and similar controversial questions is provided in this report, which considers the mathematical knowledge children bring to school and how students develop proficiency with numbers and in other mathematical areas. The report also discusses teaching for mathematical proficiency, describes instruction as “interactions among teachers and students around content” (p. 313), and outlines what it takes to be proficient at mathematics teaching. Adding It Up emphasizes two points: “Our experiences, discussions, and review of the literature have convinced us that school mathematics demands substantial change” (p. 407); and “…[T]hroughout the grades from pre-K through 8 all students can and should be mathematically proficient” (p. 409). Adding It Up comes to the following conclusion: School mathematics in the United States does not now enable most students to develop the strands of mathematical proficiency in a sound fashion. Proficiency for all demands fundamental changes be made concurrently in curriculum,
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Page 13instructional materials, classroom practice, teacher preparation, and professional development. These changes will require continuing, coordinated action on the part of policymakers, teacher educators, teachers, and parents. Although some readers may feel that substantial advances are already being made in reforming mathematics teaching and learning, we find real progress toward mathematical proficiency to be woefully inadequate, (p. 409) These observations lead to five general recommendations and a series of specific recommendations that detail the policies and practices needed if all children are to become mathematically proficient. The five general recommendations are as follows (pp. 409–410): “The integrated and balanced development of all five strands of mathematical proficiency should guide the teaching and learning of school mathematics. Instruction should not be based on extreme positions that students learn, on the one hand, solely by internalizing what a teacher or book says or, on the other hand, solely by inventing mathematics on their own. Teachers' professional development should be high quality, sustained, and systematically designed and deployed to help all students develop mathematical proficiency. Schools should support, as a central part of teachers' work, engagement in sustained efforts to improve their mathematics instruction. This support requires the provision of time and resources. The coordination of curriculum, instructional materials, assessment, instruction, professional development, and school organization around the development of mathematical proficiency should drive school improvement efforts. Efforts to improve students' mathematics learning should be informed by scientific evidence, and their effectiveness should be evaluated systematically. Such efforts should be coordinated, continual, and cumulative. Additional research should be undertaken on the nature, development, and assessment of mathematical proficiency.” Among the 58 specific recommendations made to help move the nation toward the change needed in school mathematics, Adding It Up urges the following with respect to student learning and its relation to curriculum: “Mathematics programs in the early grades should make extensive use of appropriate objects, diagrams, and other aids to ensure that all children understand and are able to use number words and the base-10 properties of numerals, that all children can use the language of
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Page 14 quantity (hundreds, tens, and ones) in solving problems, and that all children can explain their reasoning in obtaining solutions.” (p. 412) “Instructional materials and classroom teaching should help students learn increasingly abbreviated procedures for producing number combinations rapidly and accurately without always having to refer to tables or other aids.” (p. 413) “For addition, subtraction, multiplication, and division, all students should understand and be able to carry out an algorithm that is general and reasonably efficient.” (p. 414) “The basic ideas of algebra as generalized arithmetic should be anticipated by activities in the early elementary grades and learned by the end of middle school.” (p. 419) “Problem solving should be the site in which all the strands of mathematics proficiency converge. It should provide opportunities for students to weave together the strands of proficiency and for teachers to assess students' performance on all of the strands.” (p. 421) With respect to assessment, Adding It Up recommends (pp. 423–424): Assessment, whether internal or external, should be focused on the development and achievement of mathematical proficiency. The results of each external assessment should be reported so as to provide feedback useful for teachers and learners rather than simply a set of rankings. With respect to instruction, Adding It Up recommends the following: “A significant amount of class time should be spent in developing mathematical ideas and methods rather than only practicing skills.” (p. 425) “Questioning and discussion should elicit students' thinking and solution strategies and should build on them, leading to greater clarity and precision.” (p. 426) “Discourse should not be confined to answers only but should include discussion of connections to other problems, alternative representations and solution methods, the nature of justification and argumentation, and the like.” (p. 426) “In all grades of elementary and middle school, any use of calculators and computers should be done in ways that help develop all strands of students' mathematical proficiency.” (p. 427) With respect to teacher preparation, the report makes the following recommendations (p. 429): To provide a basis for continued learning by teachers, their preparation to teach, their professional development activities, and the instructional
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Page 15 materials they use should engage them, individually and collectively, in developing a greater understanding of mathematics and of student thinking, and in finding ways to put that understanding into practice. All teachers, whether preservice or inservice, should engage in inquiry as part of their teaching practice. Teachers of grades pre-K to 8 should have a deep understanding of the mathematics of the school curriculum and the principles behind it. Mathematics specialists—teachers who have special training and interest in mathematics—should be available in every elementary school. How People Learn: Brain, Mind, Experience, and School How People Learn is an example-laden review of what is known about learning and its implications for teaching. In the medical profession, research-based practice and revisions to practice based on newer research are the norm. Education needs to develop a similar culture and similar expectations for using research to inform and direct practice. The development of a “science of learning” and the translation of this science into practice can be critical for enhancing educational productivity. As noted early in the report: “the new science of learning is beginning to provide knowledge to improve significantly people's abilities to become active learners who seek to understand complex subject matter and are better prepared to transfer what they have learned to new problems and settings…. The emerging science of learning underscores the importance of rethinking what is taught, how it is taught, and how learning is assessed” (p. 13). Learning research suggests a need to change current practice; “there are new ways to introduce students to traditional subjects…and that these new approaches make it possible for the majority of individuals to develop a deep understanding of important subject matter” (p. 6). The scientific achievements synthesized in the report “include a fuller understanding of (1) memory and the structure of knowledge, (2) problem solving and reasoning, (3) the early foundations of learning, (4) regulatory processes that govern learning, including metacognition, and (5) how symbolic thinking emerges from the culture and community of the learner” (p. 14). This overview of the research generates three key findings highlighted in the report: “Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp the new concepts and information that are taught, or they may learn them for purposes of a test but revert to their preconceptions outside the classroom.” (p. 14)
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Page 16 “To develop competence in an area of inquiry, students must (a) have a deep foundation of factual knowledge, (b) understand facts and ideas in the context of a conceptual framework, and (c) organize knowledge in ways that facilitate retrieval and application.” (p. 16) “A ‘metacognitive' approach to instruction can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them.” (p. 18) The implications of these findings for teaching are summarized similarly: “Teachers must draw out and work with the pre-existing understandings that their students bring with them.” (p. 19) “Teachers must teach some subject matter in depth, providing many examples in which the same concept is at work and providing a firm foundation of factual knowledge.” (p. 20) “The teaching of metacognitive skills should be integrated into the curriculum in a variety of subject areas.” (p. 21) These findings are then translated into four interrelated attributes of learning environments: “Schools and classrooms must be learner centered.” (p. 23) “To provide a knowledge-centered classroom environment, attention must be given to what is taught (information, subject matter), why it is taught (understanding), and what competence or mastery looks like.” (p. 24) “Formative assessments—ongoing assessments designed to make students' thinking visible to both teachers and students—are essential. They permit the teacher to grasp the students' preconceptions, understand where the students are in the ‘development corridor' from informal to formal thinking, and design instruction accordingly. In the assessment-centered classroom environment, formative assessments help both teacher and students monitor progress.” (p. 24) “Learning is influenced in fundamental ways by the context in which it takes place. A community-centered approach requires the development of norms for the classroom and school, as well as connections to the outside world, that support core learning values.” (p. 25) Each of the findings, implications and attributes listed above is supported and elaborated upon in great detail in the report. In lieu of specific recommendations, How People Learn concludes with a detailed set of summary findings and conclusions and 23 categories of recommended research and development for future research. Recommendations for practice include the following:
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Page 17 People's ability to transfer what they have learned depends upon a number of factors: — “People must achieve a threshold of initial learning that is sufficient to support transfer.” (p. 235) — “Spending a lot of time (“time on task”) in and of itself is not sufficient to ensure effective learning.” (p. 235) — “Learning with understanding is more likely to promote transfer than simply memorizing information from a text or a lecture.” (p. 236) — “Knowledge that is taught in a variety of contexts is more likely to support flexible transfer than knowledge that is taught in a single context.” (p. 236) — “Students develop flexible understanding of when, where, why, and how to use their knowledge to solve new problems if they learn how to extract underlying themes and principles from their learning exercises.” (p. 236) — “All learning involves transfer from previous experiences.” (p. 236) — “Sometimes the knowledge that people bring to a new situation impedes subsequent learning because it guides thinking in the wrong direction.” (p. 236) Teachers need expertise in both subject-matter content and in teaching; need to develop an understanding of the theories of knowledge that guide the subject-matter disciplines in which they work; need to develop an understanding of pedagogy as an intellectual discipline that reflects theories of learning, including knowledge of how cultural beliefs and the personal characteristics of learners influence learning; and need to develop models of their own professional development that are based on lifelong learning, rather than on an “updating” model of learning (p. 242). “Computer-based technologies hold great promise both for increasing access to knowledge and as a means of promoting learning” (p. 243). ACTIONS EDUCATORS MIGHT CONSIDER As comprehensive summaries of what we currently know about learning in general, about learning mathematics in particular, and about the implications of this knowledge for teaching and creating school environments conducive to learning, Adding It Up and How People Learn can serve as resources for professional discussions, for course and seminar content, and for guiding future research. What we know about how students learn suggests that educators, researchers, and policy makers can do the following: Draw from the extensive findings in these reports to educate parents and the community at large about, and build support for, changes in school mathematics programs.
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Page 18 Analyze current practice in light of research findings about learning and work to enable more research-based practices in schools and classrooms. Support additional research along the lines proposed in Chapter 11 of How People Learn, forming teams that combine the expertise of researchers and the wisdom of practitioners. This research should focus on curriculum materials, formative assessment, the use of technology, and on the alignment and effectiveness of professional development programs. Analyze, and adjust the curriculum, instructional practices, and assessments that are used, in light of the evidence presented throughout Adding It Up. Create study groups and other professional development opportunities that bring together teachers, administrators, and teacher educators; use selected chapters from How People Learn and Adding It Up as catalysts for discussion and for planning improvements.