A teacher must interpret students’ written work, analyze their reasoning, and respond to the different methods they might use in solving a problem. Strutchens, M.E., & Silver, E.A. (1996). To represent a problem accurately, students must first understand the situation, including its key features. Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently. We each get pieces of pizza. (1995). Ball, D.L. It is important for computational procedures to be efficient, to be used accurately, and to result in correct answers. Understanding teachers’ self-sustaining, generative change in the context of professional development. In D. A.Grouws (Ed. Conference Board of the Mathematical Sciences, 2000. At least some of the teachers continued the process of learning mathematics by examining the mathematical work of their own students in their own classrooms. Each is discussed in more detail below. Regardless of the domain of mathematics, conceptual understanding refers to an integrated and functional grasp of the mathematical ideas. The predictable failure of educational reform: Can we change course before it’s too late? These study groups focus on the development and refinement of one specific mathematical lesson, called a “research lesson.” Teachers work together to consider a specific difficulty entailed in teaching some important piece of mathematics. ), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. For example, the lesson on the division of fractions is part of a larger agenda that includes understanding division, its relationship to fractions and to multiplication, and the meaning and representation of operations. Teachers are expected to explain and justify their ideas and conclusions. The “‘Standards’-like” role of teachers’ mathematical knowledge in responding to unanticipated student observations. 33–59). Using the lesson as the unit of analysis and improvement, the teachers are encouraged to improve their knowledge of all aspects of teaching within the context of their own classrooms—knowledge of mathematics, of students’ thinking, of pedagogy, of curriculum, and of assessment. Education and learning to think. That proficiency should enable them to cope with the mathematical challenges of daily life and enable them to continue their study of mathematics in high school and beyond. Although within most countries, positive attitudes toward mathematics are associated with high achievement, eighth graders in some East Asian countries, whose average achievement in mathematics is among the highest in the world, have tended to have, on average, among the most negative attitudes toward mathematics. 45–72). Fuson, K.C., & Briars, D.J. In those programs, teachers engage in analyses in which they are asked to provide evidence to justify claims and assertions. For example, prospective elementary school teachers may take a mathematics course that focuses, in part, on rational numbers or proportionality rather than the usual college algebra or calculus. (1999). The instructor walks around, watching them work, and occasionally asking a question. The data do not indicate, however, whether the students thought they could make sense out of the mathematics themselves or depended on others for explanations. Franke, M.L., Carpenter, T.P., Fennema, E., Ansell, E., & Behrent, J. [July 10, 2001]. Used by permission of the authors. They are most proficient in aspects of procedural fluency and less proficient in conceptual understanding, strategic competence, adaptive reasoning, and productive disposition. Steinberg, R.M. At the workshop, the teachers share their findings with the other participants. The learning gap: Why our schools are failing and what we can learn from Japanese and Chinese education. National Research Council. But they also have to know how to use both kinds of knowledge effectively in the context of their work if they are to help their students develop mathematical proficiency. A linear function (y=1.30x+2.50) fits the three values, and one can use constant differences or a graph to find this function (although that is not necessary to answer the two questions). For work in psychology, see Baddeley, 1976; Bruner, 1960, pp. Connected with procedural fluency is knowledge of ways to estimate the result of a procedure. The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency. We describe what students are capable of, what the big obstacles are for them, and what knowledge and intuition they have that might be helpful in designing effective learning experiences. The teachers focus on children’s thinking about a critical mathematical idea. Results from the second mathematics assessment of the National Assessment of Educational Progress. Teaching and Teacher Education, 14(1), 81–93. Thousand Oaks, CA: Corwin Press. (1988). In 1980 the populations of Town A and Town B were 5,000 and 6,000, respectively. This type of knowledge is gained through experience in classrooms and through analyzing and reflecting on one’s own practice and that of others. In this report, we present a much broader view of elementary and middle school mathematics. One pair of students has a different problem: I have cups of sugar. Although teachers need a range of routines, teaching is very much a problem-solving activity.30 Like other professionals, teachers are constantly faced with decisions in planning instruction, implementing those plans, and interacting with students.31 Useful guidelines are seldom available for figuring out what to teach when, how to teach it, how to adapt material so that it is appropriate for a given group of students, or how much time to allow for an activity. But many tasks involving mathematics in everyday life require facility with algorithms for performing computations either mentally or in writing. Shannon, A. Without sufficient procedural fluency, students have trouble deepening their understanding of mathematical ideas or solving mathematics problems. To become proficient, they need to spend sustained periods of time doing mathematics—solving problems, reasoning, developing understanding, practicing skills—and building connections between their previous knowledge and new knowledge. For helping thousands of aspirants, our exam experts are researching for best book for economics exam preparation. Cobb, P., Yackel, E., & Wood, T. (1995). 106–116). Washington, DC: National Academy Press. 597–622). Hiebert, J., & Wearne, D. (1986). Ball, D.L., & Bass, H. (2000). So, stay tuned with Eduncle to get all the updates regarding IIT JAM Economics Books If you like the above content, stay connected with Eduncle for more quality info. A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers. Fuson, K.C., Carroll, W.M., & Landis, J. Franke, Carpenter, Levi, and Fennema, in press. The strong connection between economic advantage, school funding, and achievement in the United States has meant that groups of students whose mathematics achievement is low have tended to be disproportionately African American, Hispanic, Native American, students acquiring English, or students located in urban or rural school districts.73 In the NAEP assessments from 1990 to 1996, white students recorded increases in their average mathematics scores at all grades. MyNAP members SAVE 10% off online. In R.I.Charles & E.A.Silver (Eds. Novice problem solvers are inclined to notice similarities in surface features of problems, such as the characters or scenarios described in the problem. Mathematics that whets the appetite: Student-posed projects problems. Conventional wisdom holds that a teacher’s knowledge of mathematics is linked to how the teacher teaches. Learning to understand arithmetic. divorced from practice. Before the teachers studied the case and the accompanying materials, they solved the mathematical problem themselves. Leinhardt and Smith, 1985; Putnam, Heaton, Prawat, and Remillard, 1992. Reston, VA: National Council of Teachers of Mathematics. They design a lesson, and one member of the group teaches it while the others watch. Available: http://www.maa.org/cbms/metdraft/index.htm. Students need to see that procedures can be developed that will solve entire classes of problems, not just individual problems. If students are to understand the algorithm, they also need experience in explaining and justifying it themselves with many different problems. In both places the teachers engage in inquiry to gain a deeper understanding of mathematics, students’ thinking about that mathematics, and how to plan their instruction so as to foster the development of students’ mathematical thinking. But pitting skill against understanding creates a false dichotomy.12 As we noted earlier, the two are interwoven. Working directly on improving teaching is their means of becoming better teachers. Steele, 1997; and Steele and Aronson, 1995, show the effect of stereotype threat in regard to subsets of the GRE (Graduate Record Examination) verbal exam, and it seems this phenomenon may carry across disciplines. Barnett, C. (1998). We also raise the standard for success in learning mathematics and being able to use it. Lubinski, C.A., Otto, A.D., Rich, B.S., & Jaberg, P.A. For adults, division is an operation on numbers. New York: Macmillan. The prospective teachers stare at the board, trying to figure out what the instructor is asking them to do. Other programs use students’ mathematical thinking as a springboard to motivate teachers’ learning of mathematics. Help her figure out how much it would most likely cost to ship a 1-kg package and how much each additional kilogram would cost. (1998). Bloomington, IN: Agency for Instructional Television. For teachers who have already achieved some mathematical proficiency, separate courses or professional development programs that focus exclusively on mathematics, on the psychology of learning, or on methods of teaching provide limited opportunities to make these connections. Problem-solving transfer. workshop on children’s development of addition and subtraction concepts taught problem solving significantly more and number facts significantly less than did teachers who had instead taken two 2-hour workshops on nonroutine problem solving. In programs focusing on children’s mathematical thinking, teachers learn to recognize and appreciate the mathematical significance of children’s informal methods for solving problems, how these methods evolve into more abstract and more powerful methods, and how the informal methods could serve as a basis for students to learn formal concepts and procedures with understanding. Acquiring proficiency takes time in another sense. To understand the sense that children are making of arithmetic problems, teachers must understand the distinctions children are making among those problems and how the distinctions might be reflected in how the children think about the problems. ), The teaching of arithmetic (Tenth Yearbook of the National Council of Teachers of Mathematics, pp. Journal for Research in Mathematics Education, 25, 279–303. In building a problem model, students need to be alert to the quantities in the problem. (1992). Knowing mathematics for teaching also entails more than knowing mathematics for oneself. Begle, E.G. Davis, R.B., & Maher, C.A. The discussion generates insights about how children are thinking and what teachers can learn by listening to their students. [July 10, 2001]. In S.Berenson, K.Dawkins, M. Blanton, W.Coulombe, J.Kolb, K.Norwood, & L.Stiff (Eds. Although there is not a perfect fit between the strands of mathematical proficiency and the kinds of knowledge and processes identified by cognitive scientists, mathematics educators, and others investigating learning, we see the strands as reflecting a firm, sizable body of scholarly literature both in and outside mathematics education. She also asks them to try to connect what they have done in class today with the familiar algorithm of “invert and multiply.”. The research, however, does suggest that proposals to improve mathematics instruction by simply increasing the number of mathematics courses required of teachers are not likely to be successful. In J.Brophy (Ed. Understanding the mathematics of the domain being studied is a prerequisite to understanding children’s thinking in that domain. Tharp, R.G., & Gallimore, R. (1988). 19–38. 1–48). ), Proceedings of the Twentieth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (vol. Teachers analyze classes not to figure out how they can do what the teacher in the case example did; instead, the case discussions provide models for inquiry that teachers may apply to analyze their own students’ mathematical thinking and their own teaching practices. [July 10, 2001]. Philadelphia: University of Pennsylvania, Consortium for Policy Research in Education. For example, students who understand place value and other multidigit number concepts are more likely than students without such understanding to invent their own procedures for multicolumn addition and to adopt correct procedures for multicolumn subtraction that others have presented to them.9. For example, an initial procedure for 86–59 might be to use bundles of sticks (see Box 4–3). Begin with 8 bundles of 10 sticks along with 6 individual sticks. New York: Macmillan. Similarly, for each of the five toppings on the hamburger, there are two options: include the topping or exclude it. When sustained work is focused on mathematics, on students’ thinking about specific mathematical topics, or on the detailed work of designing and enacting instruction, the resources generated for teachers’ own practice are greater than when there is less concrete focus. Student achievement data were based on items developed for NAEP. (2000, April). [January 3, 2001]. The promise of educational psychology. Now we turn to specific issues that arise in the context of the components. (Eds.). Despite the finding that many students associate mathematics with memorization, students at all grade levels appear to view mathematics as useful. A dozen teachers are gathered around a table. In J.Boaler (Ed. In the course of their work as teachers, they must understand mathematics in ways that allow them to explain and unpack ideas in ways not needed in ordinary adult life. Thompson, A.G. (1992). Comparative Education Review, 40, 139–157. (1999). Baddeley, A.D. (1976). For this reason, summer leadership training programs have been used to develop mathematics specialists. She asks them to interpret what each student did and to compare the two solutions. However, such professional development requires the marshalling of substantial resources. Research findings on what children know about numbers by the time they arrive in pre-K and the implications for mathematics instruction. In D.Grouws (Ed. Consequently, they are likely to need experience and practice in problem formulating as well as in problem solving. A beginner who happens to forget the algorithm but who understands the role of the distributive law can reconstruct the process by writing 268×47=268×(40+7)=(268×40)+(268×7) and working from there. (Eds.). Many algorithms for computing 47×268 use one basic meaning of multiplication as 47 groups of 268, together with place-value knowledge of 47 as 40+7, to break the problem into two simpler ones: 40×268 and 7×268. Alexander, P.A., White, C.S., & Daugherty, M. (1997). This information, we believe, reveals how to improve current efforts to help students become mathematically proficient. In the first case the answer is the number of groups; in the second, it is the number in each group. Schoenfeld, A.H. (1992). Other models of mathematics specialists are used, particularly in elementary schools, which rarely are departmentalized. Programs that provide readymade, worked-out solutions to teaching problems should not expect that teachers will see themselves as in control of their own learning. Cited in Wearne and Kouba, 2000, p. 186. People sometimes assume that only the brightest students who are the most attuned to school can achieve mathematical proficiency. Carpenter, T.P., Fennema, E., Franke, M.L., Empson, S.B., & Levi, L.W. Research on whole number addition and subtraction. the spot, teachers need to find out what a student knows, choose how to respond to a student’s question or statement, and decide whether to follow a student’s idea. The assumption of constant differences is one suggested by the problem and common in situations like those involving shipping costs, but it is not necessarily always warranted. When a second student offers the sesame cracker problem, most nod again, not noticing the difference. Unfortunately, most university teacher preparation programs offer separate courses in mathematics, psychology, and methods of teaching that are taught in different departments. For discussion of learning in early childhood, see Bowman, Donovan, and Burns, 2001. For one-semester courses in Finite Mathematics. Washington, DC: National Academy Press. During the second phase, teachers design the targeted lessons (often just one, two, or perhaps three lessons). The final task for the group is to prepare a report of the year’s work, including a rationale for the approach used and a detailed plan of the lesson, complete with descriptions of the different solution methods students are likely to present and the ways in which these can be orchestrated into a constructive discussion. Washington, DC: Author. To search the entire text of this book, type in your search term here and press Enter. If teachers are going engage in inquiry, they need repeated opportunities to try out ideas and approaches with their students and continuing opportunities to discuss their experiences with specialists in mathematics, staff developers, and other teachers. 9). Elementary mathematics curricula as a tool for mathematics education reform: Challenges of implementation and implications for professional development. NAEP findings regarding race/ethnicity and gender: Affective issues, mathematics performance, and instructional context. Hatano, G. (1988, Fall). Shulman, L.S. The group briefly discusses some ways to vary the problem to make it either simpler or more complex. Hilgard, E.R. The relation between conceptual and procedural knowledge in learning mathematics: A review. NAEP 1996 mathematics report card for the nation and the states. Saxe, G. (1990). Cognitively Guided Instruction (CGI) is a professional development program for teachers that focuses on helping them construct explicit models of the development of children’s mathematical thinking in well-defined content domains. Teachers whose learning becomes generative see themselves as lifelong learners who can learn from studying curriculum materials35 and from analyzing their practice and their interactions with students. American Psychologist, 41, 1040–1048. The five strands are interwoven and interdependent in the development of proficiency in mathematics. And while carrying out a solution plan, learners use their strategic competence to monitor their progress toward a solution and to generate alternative plans if the current plan seems ineffective. Routine problems require reproductive thinking; the learner needs only to reproduce and apply a known solution procedure. Borko, H., Eisenhart, M., Brown, C.A., Underhill, R.G., Jones, D. & Agard, P.C. Journal for Research in Mathematics Education, 31, 524–540. (2000). Small groups of teachers form within the school around areas of common teaching interests or responsibilities (e.g., grade-level groups in mathematics or in science). Conceptual and procedural knowledge: The case of mathematics. They also need to know reasonably efficient and accurate ways to add, subtract, multiply, and divide multidigit numbers, both mentally and with pencil and paper. “Doesn’t dividing make numbers smaller?” she asks. This is a 168 hour lab course. The program, modeled after Cognitively Guided Instruction (CGI), which has proven to be a highly effective approach,41 assists teachers in understanding how to help their students reason about number operations and relations in ways that enhance the learning of arithmetic and promote a smoother transition from arithmetic to algebra.42 This particular workshop was directed at illuminating students’ misconceptions about equality and considering how those misconceptions might be addressed. Campbell, 1996; Carpenter, Fennema, Peterson, Chiang, and Loef, 1989; Cobb, Wood, Yackel, Nicholls, Wheatley, Tragatti, and Perlwitz, 1991; Fennema, Carpenter, Franke, Levi, Jacobs, and Empson, 1996; Silver and Stein, 1996; Villasenor and Kepner, 1993. They may attempt to explain the method to themselves and correct it if necessary. Part of developing strategic competence involves learning to replace by more concise and efficient procedures those cumbersome procedures that might at first have been helpful in understanding the operation. © 2021 National Academy of Sciences. Knowledge of students and how they learn mathematics includes general knowledge of how various mathematical ideas develop in children over time as well as specific knowledge of how to determine where in a developmental trajectory a child might be. In short, teachers need to muster and deploy a wide range of resources to support the acquisition of mathematical proficiency. Knapp, M.S., Shields, P.M., & Turnbull, B.J. The workshop described in Box 10–2 forms part of a professional development program designed to help teachers develop a deeper understanding of some critical mathematical ideas, including the equality sign. (Eds.). Carpenter, T.P., Fennema, E., Peterson, P.L., Chiang, C.P., & Loef, M. (1989). For example, as students build strategic competence in solving nonroutine problems, their attitudes and beliefs about themselves as mathematics learners become more positive. Box 4–3 Subtraction Using Sticks: Modeling 86–53=? Journal for Research in Mathematics Education, 23, 194–222. New York: Columbia University Press. By the same token, a certain level of skill is required to learn many mathematical concepts with understanding, and using procedures can help strengthen and develop that understanding. ), Handbook of research on mathematics teaching and learning (pp. It is the ability to apply knowledge to solve problems.82 For students to be able to compete in today’s and tomorrow’s economy, they need to be able to adapt the knowledge they are acquiring. Schon, D. (1987). In addition to providing tools for computing, some algorithms are important as concepts in their own right, which again illustrates the link between conceptual understanding and procedural fluency. Data from the NAEP student questionnaire show that many U.S. students develop a variety of counterproductive beliefs about mathematics and about themselves as learners of mathematics. Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Gregory, K.D., Garden, R.A., O’Connor, K. M., Chrostowski, S.J., & Smith, T.A. View our suggested citation for this chapter. Backer, A., & Akin, S. Although the items in the NAEP assessments were not constructed to measure directly the five strands of mathematical proficiency, they provide some useful information about these strands. Using that body of evidence, researchers have also. 334–370). A comprehensive guide for designing professional development programs can be found in Loucks-Horsley, Hewson, Love, Stiles, 1998. Similarly, many of the measures of student achievement used in research on teacher knowledge have been standardized tests that focus primarily on students’ procedural skills. Shifting the emphasis to learning with understanding, therefore, can in the long run lead to higher levels of skill than can be attained by practice alone. Hiebert, J. Many children subtract the smaller from the larger digit in each column to get 26 as the difference between 62 and 48 (see Box 4–2). (1995). Rational numbers. Erlwanger, S., & Berlanger, M. (1983). Results from the seventh mathematics assessment of the National Assessment of Educational Progress.
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