His role is to demonstrate, provide practice, and check on their progress. (1996). In the experimental projects described above, calculator use was accompanied by instruction on number combinations and traditional written algorithms and by an emphasis on mental calculations. Slavin, R.E. Cohen, D.K., & Ball, D.L. The pace of the lesson is carefully controlled to allow students time to think but with enough momentum to engage and maintain their interest. Ms. Kaye’s lesson is an example of how practice can be embedded in problem-solving activity. Although Mr. Angelo and Ms. Lawrence get some idea of how students are doing by circulating around the room, they use the questions they ask during class as their primary mode of assessment during the lesson. ED 373 969). It takes sensitive account of students’ current knowledge and ways of thinking as well as ways in which those develop. Students speak only in response to closed questions calling for a short answer, and students do not interact with one another. Mr. Angelo designs the work to progress from simple examples (multiplying by 10, 100, and 1,000), to more complex ones (multiplying by multiples of powers of 10), to multiplying by powers of 10 less than one.4 He stages the examples so that the procedure he is trying to teach covers more and more cases, thus leading to a more general rule usable for multiplication by any power of 10 other than 10°=1. Ability grouping and student achievement in elementary schools: A best-evidence synthesis. 545–615). These teachers are committed to their plans and have difficulty making midcourse adjustments. Mr. Angelo gives 40 practice problems, whereas Ms. Lawrence uses only four. Expectations and actions. Although existing research does not provide clear guidelines for teaching mathematics to children with severe learning difficulties, existing evidence and experience suggest that the same teaching and learning principles apply to all children, including special-needs children. For example, a teacher could not effectively define a prime number if her students did not already possess some understanding of factoring. Leder, G.C. Another approach distributes the practice over a longer period: On any one day, only a few of the exercises assigned might address the lesson topic, and the rest would address topics studied earlier in the year. (in press). Later learning builds on earlier successful accomplishment; new ideas are constructed using those already known. Describing these changes in terms of “adding zeros” or “moving the decimal point” stays at the surface level of changes in written symbols and does not go beneath to the numbers themselves and what it means to multiply them. ...or use these buttons to go back to the previous chapter or skip to the next one. Instead, when a number is multiplied by a power of 10 other than one, each digit can be thought of as shifting into a new decimal place. She wants to build on her students’ mathematical understanding. Finally, a core feature of these classrooms is the recognition that the authority for whether something is both correct and sensible lies in the logic and structure of the subject rather than the status of the teacher or the popularity of the person making the argument. Mainstreaming is an instructional tool that can be used wisely or not. Good, T.L., & Brophy, J.E. Mr. Angelo works on developing fluency with the procedures without a focus on their underlying meanings or justification. Journal for Research in Mathematics Education, 17, 83–99. Number without language? Cooperative grouping is one of many instructional practices that teachers may choose to use at times. 231–263). She seeks to make the procedure make sense by asking for and providing explanations. (4) Adaptive reasoning is the
Some of the students are asked to share their solutions with the rest of the class. A number of tools— including counters of various kinds, plastic coins, and base-10 blocks—are available in the corner of the room. Baroody, A.J. Knapp M.S., Shields, P.M., Turnbull, B.J. Rewards can also have undesirable side effects if they are tied to competitions that create winners and losers. While the children work, Ms. Kaye talks with individual children about their solutions. An interactive perspective on teaching and learning has been discussed by a number of people, including Piaget, Vygotsky, Bauersfeld, Steier, Voigt, Hawkins, Gravemeijer, Easley, Cobb, and von Glaserfeld. Although the focus of the lesson is not on strategic competence, when she asks students to estimate the sum of two fractions, she is helping them become sensitive to strategies they might use. Mosteller, F., Light, R.J., & Sachs, J.A. Fuson, K.C. (1996). He is careful to connect what are often two disjointed fragments: a rule for adding zeros when multiplying by powers of 10 greater than one and a different rule for moving the decimal point when multiplying by powers of 10 less than one. They need to investigate not just whether calculators or other resources are used, but how they are used.70 Research that looks across countries can provide a sharper picture of what matters in instruction aimed at developing proficiency. Otherwise, the practice can automatize incorrect procedures, which are then difficult to correct. Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. These are the powers of 10 that are greater than one, right?”, “You just add the same number of zeros to the end of the number as are in the number you are multiplying by.”. (1997). Linchevski, L., & Kutscher, F. (1998). Any gains that might accrue to the high achievers are more than offset by losses to the low achievers and by the resultant perpetuation of social class, racial, and ethnic inequities in schooling.39, This controversy highlights a second point about grouping: Many studies on grouping have been conducted over the years (including studies on grouping for mathematics instruction), but the results concerning effects on achievement have been both weak and mixed.40 The findings indicate that overall mathematical achievement is likely to be similar whether students are grouped homogeneously or heterogeneously, especially if the same curriculum is provided to all groups. We view the teaching and learning of mathematics as the product of interactions among the teacher, the students, and the mathematics. (1988). Sustained inquiry in education: Lessons from skill grouping and class size. Thompson, P.W., & Lambdin, D. (1994). School Psychology Quarterly, 6, 174–185. Moreover, instruction takes place in contexts. ), The development of mathematical skills (pp. The first is to design for success by assigning tasks on which students can succeed if they invest reasonable effort. Bandura, A. It is worth noting that Alfred Binet devised the IQ test and advocated segregated instruction for low-ability students for the most humane of reasons. You're looking at OpenBook, NAP.edu's online reading room since 1999. He lists specified portions of these areas—as a percentage of the total, a fraction of the total, a decimal fraction of the total, or a specific number of squares— and the students are to shade that portion. “Is that right?”, “Yes, you explained it well,” says Ms. Lawrence. When the students arrive at a numerical answer, he asks questions such as “Can you explain what that number refers to?”, To wrap things up for today, Mr. Hernandez summarizes the different strategies presented. Teacher student interaction: A case study. If students merely see different representations without explicit attention to their correspondences, the lesson he is teaching will not produce the learning that he is striving for. (1987). 31–40). See, for example, Campbell, 1996; Carpenter, Fennema, Peterson, Chiang, and Loef, 1989; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997; Knapp, 1995; Silver and Stein, 1996. Thornton, C.A., & Bley, N.S. Beyond competence: The significance of performance for conceptual development. Students interpret and respond differently to the same mathemati-, Box 9–1 The Instructional Triangle: instruction as the interaction Among Teachers, Students, and Mathematics, in Contexts. ), Research on teaching: Concepts, findings, and implications. Cobb, P., Boufi, A., McClain, K., & Whitenack, J. Argys, L.M., Rees, D.I., & Brewer, D.J. 2, pp. See, for example, the report by Fey, 1979, or the more recent TIMSS video study (Stigler and Hiebert, 1999). These projects also demonstrate how instructional emphasis in a calculator-inclusive environment can shift from computational procedures to problem solving and mental arithmetic. Common mistakes in their instruction include (a) not assessing, fostering, or building on their informal knowledge; (b) overly abstract instruction that proceeds too quickly; and (c) instruction that relies on memorizing mathematics by rote. Studies of how U.S. teachers plan show that they tend to focus on the activities in which students will be engaged and how those activities will be organized.16 Teachers’ plans seldom elaborate the content that the students are to learn through their engagement with the proposed activities.17 Other research suggests that teachers who make detailed plans can sometimes be relatively inflexible when students encounter difficulties or raise thoughtful questions. Ms. Lawrence wants the practice that she provides to require the students to think and not merely follow the algorithm blindly. Lisbon, Portugal: PME Program Committee. Reston, VA: National Council of Teachers of Mathematics. A third feature of classrooms that function as communities of learners is an appreciation of the value of mistakes as sites of learning for everyone. Chief among them is that the task is made routine in one of two ways: The students may start pressing the teacher to reduce the challenge by specifying explicit procedures or steps for them to perform, or the teacher may take over the demanding aspects of the task when the students encounter difficulty by either telling them or demonstrating what to do. (NRC, 2001,
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