9
TEACHING FOR MATHEMATICAL PROFICIENCY

Previous chapters have described mathematical proficiency as the integrated attainment of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Effective forms of instruction attend to all these strands of mathematical proficiency. In this chapter we turn from considering what there is to learn and what is known about learning to an examination of teaching that promotes learning over time so that it yields mathematical proficiency.

Instruction as Interaction

Our examination of teaching focuses not just on what teachers do but also on the interactions among teachers and students around content.1 Rather than considering only the teacher and what the teacher does as a source of teaching and learning, we view the teaching and learning of mathematics as the product of interactions among the teacher, the students, and the mathematics in an instructional triangle (see Box 9–1).

We view the teaching and learning of mathematics as the product of interactions among the teacher, the students, and the mathematics.

Certainly the knowledge, beliefs, decisions, and actions of teachers affect what is taught and ultimately learned. But students’ expectations, knowledge, interests, and responses also play a crucial role in shaping what is taught and learned. For instruction to be effective, students must have, perceive, and use their opportunities to learn. The particular mathematical content and its representation in instructional tasks and curriculum materials also matter for teachers’ and students’ work, but teachers and students vary in their interpretations and uses of the same content and of the same curricular resources. Students interpret and respond differently to the same mathemati-



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Adding + It Up: Helping Children Learn Mathematics 9 TEACHING FOR MATHEMATICAL PROFICIENCY Previous chapters have described mathematical proficiency as the integrated attainment of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Effective forms of instruction attend to all these strands of mathematical proficiency. In this chapter we turn from considering what there is to learn and what is known about learning to an examination of teaching that promotes learning over time so that it yields mathematical proficiency. Instruction as Interaction Our examination of teaching focuses not just on what teachers do but also on the interactions among teachers and students around content.1 Rather than considering only the teacher and what the teacher does as a source of teaching and learning, we view the teaching and learning of mathematics as the product of interactions among the teacher, the students, and the mathematics in an instructional triangle (see Box 9–1). We view the teaching and learning of mathematics as the product of interactions among the teacher, the students, and the mathematics. Certainly the knowledge, beliefs, decisions, and actions of teachers affect what is taught and ultimately learned. But students’ expectations, knowledge, interests, and responses also play a crucial role in shaping what is taught and learned. For instruction to be effective, students must have, perceive, and use their opportunities to learn. The particular mathematical content and its representation in instructional tasks and curriculum materials also matter for teachers’ and students’ work, but teachers and students vary in their interpretations and uses of the same content and of the same curricular resources. Students interpret and respond differently to the same mathemati-

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Adding + It Up: Helping Children Learn Mathematics Box 9–1 The Instructional Triangle: instruction as the interaction Among Teachers, Students, and Mathematics, in Contexts SOURCE: Adapted from Cohen and Ball, 1999, 2000, in press. cal task, ask different questions, and complete the work in different ways. Their interpretations and actions affect what becomes the enacted lesson. Teachers’ attention and responses to students further shape the course of instruction. Some teachers may not notice how students are interpreting the content, others may notice but not investigate further, and still others may notice and respond by reiterating their own interpretation. Moreover, instruction takes place in contexts. By contexts we mean the wide range of environmental and situational elements that bear on instruction—for instance, educational policies, assessments of students and teachers,

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Adding + It Up: Helping Children Learn Mathematics school organizational structures, school leadership characteristics, the nature and organization of teachers’ work, and the social matrix in which the school is embedded. These matter principally as they permeate instruction—that is, whether and how they enter into the interactions among teachers, students, and content.2 Hence, what goes on in classrooms to promote the development of mathematical proficiency is best understood through an examination of how these elements—teachers, students, content—interact in contexts to produce teaching and learning. Much debate centers on forms and approaches to teaching: “direct instruction” versus “inquiry,” “teacher centered” versus “student centered,” “traditional” versus “reform.” These labels make rhetorical distinctions that often miss the point regarding the quality of instruction. Our review of the research makes plain that the effectiveness of mathematics teaching and learning does not rest in simple labels. Rather, the quality of instruction is a function of teachers’ knowledge and use of mathematical content, teachers’ attention to and handling of students, and students’ engagement in and use of mathematical tasks. Moreover, effective teaching—teaching that fosters the development of mathematical proficiency over time—can take a variety of forms. To highlight this point, we use excerpts from four classroom lessons and analyze what we see going on in them in light of what we know from research on teaching. Four Classroom Vignettes The pedagogical challenge for teachers is to manage instruction in ways that help particular students develop mathematical proficiency. High-quality instruction, in whatever form it comes, focuses on important mathematical content, represented and developed with integrity. It takes sensitive account of students’ current knowledge and ways of thinking as well as ways in which those develop. Such instruction is effective with a range of students and over time develops the knowledge, skills, abilities, and inclinations that we term mathematical proficiency. The four classroom vignettes we present below offer four distinct images of what mathematics instruction can look like. Each vignette configures differently the mathematical content and the roles and work of teachers and students in contexts; hence, each produces different opportunities for mathematics teaching and learning. Two points are important to interpreting and using these vignettes. First, to provide a close view, each vignette zooms in on an individual lesson. Effective instruction, however, depends on the

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Adding + It Up: Helping Children Learn Mathematics coherent connection over time among lessons designed collectively to achieve important mathematical goals. For example, some of these teachers may be attempting to develop students’ productive disposition toward mathematics and as mathematics learners, but it is difficult to pinpoint isolated attempts in a single lesson since that development takes place gradually—over months rather than minutes. Second, rather than seeking to argue that one of these lessons is “right,” our analysis probes the possibilities and the risks each affords. The instructional challenge in any approach to teaching and learning is to capitalize on its opportunities and ward off its pitfalls. The first example (Box 9–2) is typical of much teaching that many American adults remember from their own experience in mathematics classes.3 Note how the teacher, Mr. Angelo, constructs the lesson in a way that structures the students’ path through the mathematics by tightly constraining both the content and his students’ encounters with it. The approach used by Mr. Angelo structures and focuses students’ attention on a specific aspect of the topic: multiplying by powers of 10. He has distilled the content into an integrated “rule” that his students can use for all instances of multiplication by powers of 10. Box 9-2 Mr. Angelo— Teaching Eighth Graders About Multiplying by Powers of 10 After a conducting a short warm-up activity and checking a homework assignment that focused on multiplying by 10, Mr. Angelo announces that the class is going to work on multiplying by powers of 10. He is concerned that students tend to perform poorly on this topic on the spring tests given by the school district, and he wants to make sure that his students know what to do. He reviews briefly the idea of powers of 10 by showing that 100 equals 102, 1000 equals 103, and soon. Going to the overhead projector, he writes the following: 4×10= 45×100= 450×100= “Who knows the first one?” Mr. Angelo asks. “Luis?” “Forty,” replies Luis. Nodding, Mr. Angelo points to the second, “And this one?” Sonja near the front offers, “Forty-five hundred.” “That’s right—forty-five hundred,” affirms Mr. Angelo, and he writes the number on the overhead transparency. “And what about the last one?” he asks. “Forty-five thousand,” call out several students.

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Adding + It Up: Helping Children Learn Mathematics Writing “45,000,” Mr. Angelo says, “Good, you are all seeing the trick. What is it? Who can say it?” Several hands shoot into the air. Ethel says, “You just add the same number of zeros as are all together in the number and in the number you are multiplying by. Easy.” “Right,” says Mr. Angelo. “Let’s try some more and see if you are getting it.” He writes three more examples: 30×70= 40×600= 45×6000= “So who can do these?” he asks, looking over the students. “What’s the first one?” “Three hundred!” announces Robert, confidently. Mr. Angelo pauses and looks at the other students. “Who can tell Robert what he did wrong?” There is a moment of silence and then Susan raises her hand, a bit hesitantly. “I think it should be twenty-one hundred,” she says. “You have to multiply both the 3 and the 7, too, in ones like this. So 3 times 7 is 21, and then add two zeros—one from the 30 and one from the 70.” “Good!” replies Mr. Angelo. “Susan reminded us of something important for our trick. It’s not just about adding the right number of zeros. You also have to look to see whether the number you are multiplying by begins with something other than a 1, and if it does, you have to multiply by that number first and then add the zeros.” He writes 2100 after the equals sign and continues with the remaining examples. Mr. Angelo writes another three examples on the overhead: 4.5×0.1= 4.5×0.01= 4.5×0.001= “I wonder whether I can fool you. Now we are going to multiply by decimals that are also powers of 10: one tenth, one hundredth, one thousandth, and so on. We’ll do easy ones to start.” Who knows the first one?” he asks. “Luis?” “Point four five,” replies Luis. Nodding, Mr. Angelo rephrases Luis’s answer: “Forty-five hundredths.” He then points to the second, “How about this one?” Nadya responds, “Point zero four five,” almost inaudibly. “That’s right. Forty-five thousandths,” Mr. Angelo affirms, and he writes the number on the overhead. “And what about the last one?” “Point zero zero forty-five,” responds the girl near the front again. Mr. Angelo writes “0.0045” and says, “Good, does anyone see the rule. Who can say it?” After a long pause, one hand in the back goes up. “You just move the decimal point.”

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Adding + It Up: Helping Children Learn Mathematics “Right,” says Mr. Angelo. “You move the decimal point to the left as many places as there are in the multiplier.* But think now. What did we decide happens to the product when we multiply a decimal by 10, 100, or 1,000? These are the powers of 10 that are greater than one, right?” This time several hands go up. “You just add the same number of zeros to the end of the number as are in the number you are multiplying by.” “Okay, that is what we said. But now we are ready for a better rule now that we have looked at some powers of 10 that are less than one. They are numbers like one tenth, one hundredth, one thousandth, and so on. Instead of having two completely different rules, it is better to have one good rule. And here it is. Listen carefully: “When you multiply by a power of 10 that is greater than one, you move the decimal point to the right as many places as the number of zeros in the multiplier. When you multiply by a power of 10 that is less than one, you move the decimal point to the left as many places as there are in the multiplier.” Mr. Angelo illustrates the movement of the decimal point with a colored pen. He explains, “You can remember which way to move the decimal point if you remember that multiplying by a number greater than one makes the product bigger and multiplying by a number less than one makes the product smaller. Right makes bigger, left makes smaller.” “Let’s practice this a bit now and get it under our belts.” Mr. Angelo passes out a worksheet with 40 exercises that resemble what was done in class. He goes over the first exercise to make sure his students remember what to do. While the students work, Mr. Angelo circulates around the room, answering questions and giving hints. The students make a variety of computational errors, but most seem able to use the rule correctly. Mr. Angelo is pleased with the outcome of his lesson. *   Mr. Angelo is referring to the number of places between the decimal point and the last nonzero digit in the multiplier. Strictly speaking the first factor in a product is the multiplier. But because of the commutative property, Mr. Angelo uses the term for whichever factor he wishes to focus on.

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Adding + It Up: Helping Children Learn Mathematics This lesson focuses on mathematical procedures for multiplying by powers of 10. 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. Mr. Angelo asks brief questions to engage students in the steps he is taking. By giving the students a rule, he simplifies their learning, heading off frustration and making getting the right answer the point—and likely to be attained. Concerned about the spring testing, he attempts to ensure that his students develop a solid grasp of the procedure and can use it reliably. 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. Although Mr. Angelo integrates these two “rules,” he does not work in the underlying conceptual territory. He does not, for example, explain why, for problems such as 30×70=?, students multiply the 3 and the 7. He might have shown them that 30×70=3×10×7×10 and that, using associativity and commutativity, one can multiply 3 by 7 and then multiply that product by 10 times 10, or 100. Instead, he skips this opportunity to help the procedure make sense and instead adds an extra twist to the rule. He also does not show his students what they are doing when they “move the decimal point.” In fact, of course, one does not “move” the decimal point. 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. For example, since .05 is one tenth times .5, in .5×10–1=?, the 5 can be thought of as shifting one place to the right—to the hundredths place, which is one tenth of one tenth. If a 5 is in the tens place, then multiplying by 10 shifts it to the left one place, to the hundreds place: What was 50 is now 500. 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. Students miss an opportunity to see and use the power of place-value notation: that the placement of digits in a numeral determines their value. A 5 in the tens place equals 50; in the hundredths place, 0.05; and in the ones place, 5. Mr. Angelo offers his students an effective and mathematically justifiable rule, but he does so without exploring its conceptual underpinnings.

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Adding + It Up: Helping Children Learn Mathematics In lessons such as Mr. Angelo’s, mathematics entails following rules and practicing procedures, often with little attention to the underlying concepts.5 Procedural fluency is given central attention. Adaptive reasoning is not Mr. Angelo’s goal: He does not offer a justification for the rule he is teaching, nor does he engage students in reasoning about the structure of the place-value notation system that is its foundation. He focuses instead on ensuring that they can use it correctly. Other aspects of mathematical proficiency are also not on his agenda. Instead, Mr. Angelo has a clear purpose for the lesson, and to accomplish that purpose he controls its pace and content. Students speak only in response to closed questions calling for a short answer, and students do not interact with one another. When a student gets an answer wrong, Mr. Angelo signals that immediately and asks someone else to provide the correct answer. The lesson is paced quickly. We turn now to our second teacher, Ms. Lawrence, who is working with her fifth graders on adding fractions (Box 9–3). Ms. Lawrence’s goals are different from Mr. Angelo’s. Although she also structures the lesson to accomplish her goals, unlike Mr. Angelo, she emphasizes explanation and reasoning along with procedures. The pace of the lesson is carefully controlled to allow students time to think but with enough momentum to engage and maintain their interest. Box 9–3 Ms. Lawrence— Teaching Fifth Graders About Adding Fractions After a few minutes in which the class does mental computation to warm up, Ms. Lawrence reviews equivalent fractions by asking the students to provide other names for She asks the class what fractions are called that “name the same number.” On the chalkboard she writes a problem involving the addition of fractions with like denominators: She asks the students how to find the sum. One student, Betsy, volunteers that you just add the numerators and write the sum over the denominator. “Why does this work?” Ms. Lawrence asks. She asks Betsy to go to the board and explain. Confidently, Betsy draws two pie diagrams, one for each fraction, and explains that the denominator tells the size of the pieces and the numerators how many pieces all together:

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Adding + It Up: Helping Children Learn Mathematics In response, Ms. Lawrence poses another problem, this time involving unlike denominators: “How would we find the sum of these two?” she asks. Stepping back, she gives the students a chance to think. She then asks whether the sum would be less than or greater than 1. Several students raised their hands, eager to respond. Ms. Lawrence calls on Susan, who explains that the sum would be less than 1 because is less than and equals exactly 1. Ms. Lawrence then asks how you could find the exact sum. Jim raises his hand and offers and as equivalent fractions with a common denominator. Ms. Lawrence writes on the chalkboard as Jim dictates: She asks Jim why he chose 12 as the common denominator. “Twelve is the smallest number that both 3 and 4 go into,” replies Jim. “How did you come up with that?” Ms. Lawrence asks. “By multiplying 3 and 4,” he answers. Ms. Lawrence turns to the class. “Let’s take a closer look. Jim got the equivalent fractions by multiplying the numerator and denominator of each fraction by the denominator of the other fraction. So if we show all the steps, it looks like this.” She then reworks the problem to make her point, justifying each step by giving a property of the rational numbers: Ms. Lawrence stops and looks at the students. “How do we know that what Jim did makes sense? How do we know that he is adding the same fractions as in the original problem: and This is really important. Maybe he has just added two other fractions.”

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Adding + It Up: Helping Children Learn Mathematics “Oh!” exclaims Lucia. “I know! Two thirds is equivalent to eight twelfths. We could show that with a picture like what Betsy drew for three eighths and four eighths. If we draw two thirds on a pie that has three pieces, those two pieces will actually make eight pieces on that same pie if it’s divided into 12. But the eight pieces, eight twelfths, will equal the same total amount of pie as two pieces that are each one third of the pie.” She pauses, and beams, looking at Ms. Lawrence expectantly. “Is that right?” “Yes, you explained it well,” says Ms. Lawrence. “Can someone come up and make pictures to show what Lucia just said?” Several hands go up, and Ms. Lawrence picks Nicole, who comes to the board and represents accurately what Lucia said. Ms. Lawrence makes a few additional remarks to make sure that all the students understand. Ms. Lawrence continues with three more examples, showing all the steps in each. She then asks the students to generalize the process by writing “a rule that would work for any two fractions.” Several students volunteer a verbal rule. “Let’s try this out on a couple of less obvious examples,” she says, writing on the overhead projector: Ms. Lawrence asks the students to work on these problems in pairs. As the students work, she walks around, listening, observing, and answering questions. Satisfied that the students seem to understand and are able to carry out the procedure, she assigns a page from their textbook for practice. The assignment contains a mixture of problems in adding fractions, including some fractions that already have like denominators and many that do not, and in adding whole numbers as well as several word problems. Ms. Lawrence wants the practice that she provides to require the students to think and not merely follow the algorithm blindly. She believes that this way of working will equip them well for the standardized test her district administers in April and the basic skills test they have to take at the beginning of sixth grade. She expects the students to remember the procedure because they have had opportunities to learn why it makes sense. She knows that this approach is understandable to her students’ parents, while at the same time she is stretching them beyond what some have been demanding—a solid focus on basic skills. She feels comfortable with the balance she has struck on these issues. SOURCE: This vignette was constructed to embody the principles from Good, Grouws, and Ebmeier, 1983.

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Adding + It Up: Helping Children Learn Mathematics In this lesson, Ms. Lawrence is trying to develop her students’ ability to add fractions with like or unlike denominators. She wants them to understand how to convert fractions to fractions with the same denominator and add them, and to have a reliable procedure for doing so. She also wants them to understand why the procedure works. Her lesson is designed to engage the students actively in the conceptual and procedural development of the topic. She begins by reviewing equivalent fractions, a concept both familiar and necessary for the new work. She poses a variety of questions and expects the students to explain their reasoning. She does not stop with well-articulated statements of the procedure but demands explanation and connection to the underlying meaning. She seeks to make the procedure make sense by asking for and providing explanations. In this lesson, time is spent in a variety of ways to address Ms. Lawrence’s goals: The students spend time practicing mental computation, developing a general rule for adding fractions, explaining and making sense of others’ explanations, and working with a partner to practice on more complex examples of what they were learning. The lesson proceeds at a steady pace, but one that affords time for developing the ideas. Ms. Lawrence checks to see whether the students are understanding before she assigns them independent work, and the assignment mixes familiar and extension problems to help strengthen students’ proficient command of the content. 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. Our third teacher, Mr. Hernandez, is working on making and linking different representations of rational numbers (Box 9–4). He works hard to engage all his students in active work on the mathematics. Toward that end, he asks challenging questions that allow for a variety of solutions, and he expects the students to push themselves. He is conscious of the district and state basic skills assessments, but he has concluded that if he invests in this sort of work with his students, it pays off in their preparedness for the test. Occasionally, he finds that the approach is not working for some of his students, and he seeks ways to build their skills more solidly. He worries a bit, since the parents have been quite vocal in his school, with much pressure about getting students to algebra in eighth grade. He takes a strong stand on the importance of developing a solid foundation with number and representation, particularly with rational numbers. This lesson is different from either Mr. Angelo’s or Ms. Lawrence’s. Mr. Hernandez has selected a task that draws on students’ past experience

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Adding + It Up: Helping Children Learn Mathematics it, not just how much time is allocated for mathematics but how that time is spent. 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. A second set of issues concerns instruction over time. Although learning is fundamentally temporal, too little research has addressed the ways in which instruction develops over time. Many studies are restricted to isolated fragments of teaching and learning, providing little understanding of how the interactions of teachers, students, and content emerge over time, and how earlier interactions shape later ones. How do ideas developed in class affect later work, and what affects teachers’ and students’ ability and inclination to make such links, as well as their use of such connections over time? How is time used, and how does its use by teachers and students affect the quality of instruction? A third arena concerns students and how their diversity affects instruction. Too little research offers insight into the experience of students and how the instruction offered, together with their responses to it, affects their learning. Still more important, there are too few well-designed studies that would offer insight into how instruction might be developed to work effectively for all students. Too often, research on classroom teaching and learning either studies faceless, colorless students and teachers out of context, or it is situated in particular contexts but lacks a design that permits analyses that could provide the knowledge needed for effective instruction in mathematics. Fourth, too little research has addressed what it takes for students to learn mathematics in class. What do students need to do, and know how to do, in order to profit from the instruction offered by each of our four teachers? A cursory glance at any mathematics class makes plain that the skills, abilities, knowledge, and dispositions displayed by students are not the same, and yet teachers and researchers rarely attend to what students need to know and be able to do in order to use instruction effectively. People seem to assume implicitly that instruction acts on students and that opportunities to learn are actually moments of learning. Research that examined both what students have to know and do in mathematics instruction and what teachers can do to enable all students to make use of that instruction would add significantly to the knowledge base on teaching and learning mathematics. A fifth set of issues has to do with reconnecting research on teacher knowledge with instructional effectiveness. Although most people believe that teachers’ knowledge of mathematics and of students makes a difference for

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Adding + It Up: Helping Children Learn Mathematics the quality of teaching, little empirical confirmation of this belief can be found. Moreover, too little is known about the mathematical knowledge that teachers need and how it is used in instruction. We discuss this point more in chapter 10, but it is important to the discussion in this chapter, too. Every time we reiterate that how teachers use texts, manipulatives, and calculators makes the difference, we are hovering around questions concerning what teachers know and how they make use of that knowledge in teaching. Finally, too little of the extant research probes the work of teaching at a sufficiently fine grain to contribute to the development of a conceptual and practical language of practice. Much of the interactive work in instruction remains unexamined, which leaves to teachers the unnecessary challenge of reinventing their practice from scratch, armed with only general advice. Suggestions that a class “discuss the solutions to a problem” provides little specificity about what constitutes a productive discussion and runs the risk of a free-for-all session that resembles sharing more than instruction. Research needs to be designed to illuminate what is entailed in a “discussion” and to probe the specific moves that teachers and students engage in that lead to productive rather than an unproductive discussions. Instruction that develops mathematical proficiency is neither simple, common, nor well understood. It comes in many forms and can follow a variety of paths. As this chapter demonstrates, such instruction offers numerous fertile sites for research that could make a profound difference in teachers’ practice and their students’ learning. Notes 1.   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. The particular version employed here is based on the work of Cohen and Ball, 1999, 2000, in press. 2.   Cohen and Ball, 1999, 2000, in press. 3.   This lesson is typical of lessons observed in many U.S. classrooms during the past half-century. See, for example, the report by Fey, 1979, or the more recent TIMSS video study (Stigler and Hiebert, 1999). 4.   Note that Mr. Angelo has avoided 10°, partly because the rule is stated in terms of moving the decimal point, and multiplying by 10°=1 leaves the number unchanged. 5.   U.S. eighth-grade lessons from the TIMSS video study were characterized the same way. See Stigler and Hiebert, 1999. 6.   Cohen and Ball, 2000.

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Adding + It Up: Helping Children Learn Mathematics 7.   Berliner and Biddle, 1995. Opportunity to learn was also studied in what is now called the First International Mathematics Study (Husén, 1967), although there it was based on teachers’ perceptions of students’ opportunity to learn. 8.   McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, and Cooney, 1987. 9.   Knapp, Shields, and Turnbull, 1995; Mason, Schroeter, Combs, and Washington, 1992; Steele, 1992. 10.   Berliner, 1979. 11.   Stevenson and Stigler, 1992, p. 150. 12.   Freeman and Porter, 1989; Porter, 1993. 13.   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. 14.   Doyle, 1983, 1988; Stein, Grover, and Henningsen, 1996. 15.   Henningsen and Stein, 1997; Stein, Grover, and Henningsen, 1996. 16.   Clark and Yinger, 1979. 17.   Shavelson and Stern, 1981. 18.   Boaler, 1997. 19.   Good and Brophy, 2000. 20.   Good and Brophy, 2000. 21.   Smith, 1996. 22.   For example, Hatano, 1988, suggests that students are motivated to learn with understanding when they encounter novel problems regularly, are encouraged to seek comprehension over efficiency, and engage in dialogue. 23.   National Research Council, 1999b, pp. 29–38. 24.   Feather, 1982. 25.   Bandura, 1997; Bandura and Schunk, 1981; Dweck and Elliott, 1983. 26.   Good and Brophy, 2000. 27.   Brophy, 1998, Brophy and Kher, 1986; Good and Brophy, 2000. 28.   These principles and the discussion that follows are based largely on a synthesis by Baroody, 1999. For related research and syntheses, see also Baroody, 1987, 1996; Cawley, 1985; and Geary, 1993. For practical advice for teaching, see Thornton and Bley, 1994. 29.   Baroody, 1999. 30.   See Donlan, 1998, for example, for a discussion of students with speech deficiencies. See Nunes and Moreno, 1998, for a discussion of hearing impairment. 31.   Becker, 1981; Leder, 1987. See also Leder, 1992. 32.   Ladson-Billings, 1999. 33.   Foster, 1995. 34.   Steele, 1992. 35.   Knapp, 1995. 36.   Good and Brophy, 2000. 37.   See, for example, Ball and Bass, 2000; Cobb, Boufi, McClain, and Whitenack, 1997; Hiebert and Wearne, 1993; Lampert, 1990; Wood, 1999. 38.   Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997.

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Adding + It Up: Helping Children Learn Mathematics 39.   Oakes, 1985: Oakes, Gamoran, and Page, 1992. 40.   Kulik, 1992; Linchevski and Kutsher, 1998; Mason and Good, 1993; Mosteller, Light, and Sachs, 1996; Slavin, 1987, 1993. 41.   Loveless, 1998. 42.   Linchevski and Kutscher, 1998. 43.   Argys, Rees, and Brewer, 1996. 44.   Druckman and Bjork, 1994, pp. 83–111; Johnson, Johnson, and Maruyama, 1983; Sharan, 1980; Slavin, 1980, 1983, 1995. 45.   Ellis and Gauvain, 1992. 46.   Fennema, Carpenter, Franke, Levi, Jacobs, and Empson, 1996; Thompson and Briars, 1989. 47.   Hiebert, 1990. 48.   Case, 1985. 49.   Flanders, 1987; McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, and Cooney, 1987; Schmidt, McKnight, and Raizen, 1997. 50.   Siegler and Stern, in press; Sophian, 1997. 51.   Carpenter, Fennema, Peterson, Chiang, and Loef, 1989; Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz, 1991; Fennema, Carpenter, Franke, Levi, Jacobs, and Empson, 1996; Hiebert and Wearne, 1993. 52.   Cooper, 1989; Epstein, 1988; Miller and Kelley, 1991. 53.   Epstein, 1998; Good and Brophy, 2000. 54.   Good and Brophy, 2000. 55.   Fuson, 1986; Fuson and Briars, 1990; Wearne and Hiebert, 1988. 56.   Cohen, 1990; Hart, 1996; Resnick and Omanson, 1987. 57.   Ball, 1992a, 1992b. 58.   Thompson and Lambden, 1994. 59.   Fuson, Wearne, Hiebert, Murray, Human, Olivier, Carpenter, and Fennema, 1997; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997. 60.   Fuson, 1986. 61.   Fey, 1989; NCTM, 1974. 62.   Brolin and Björk, 1992; Groves 1993, 1994a, 1994b; Hembree and Dessart, 1986, 1992; Ruthven, 1996, 1998; Shuard, 1992. 63.   Hembree and Dessart, 1986, 1992. 64.   Ruthven, 1996. 65.   Brolin and Björk, 1992. 66.   Groves, 1993, 1994a, 1994b. 67.   Shuard, 1992. 68.   Mitchell, Hawkins, Jakwerth, Stancavage, and Dossey, 1999. 69.   National Research Council, 1999a, p. 48. 70.   Stigler and Hiebert, 1999.

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