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Adding + It Up: Helping Children Learn Mathematics 4 THE STRANDS OF MATHEMATICAL PROFICIENCY During the twentieth century, the meaning of successful mathematics learning underwent several shifts in response to changes in both society and schooling. For roughly the first half of the century, success in learning the mathematics of pre-kindergarten to eighth grade usually meant facility in using the computational procedures of arithmetic, with many educators emphasizing the need for skilled performance and others emphasizing the need for students to learn procedures with understanding.1 In the 1950s and 1960s, the new math movement defined successful mathematics learning primarily in terms of understanding the structure of mathematics together with its unifying ideas, and not just as computational skill. This emphasis was followed by a “back to basics” movement that proposed returning to the view that success in mathematics meant being able to compute accurately and quickly. The reform movement of the 1980s and 1990s pushed the emphasis toward what was called the development of “mathematical power,” which involved reasoning, solving problems, connecting mathematical ideas, and communicating mathematics to others. Reactions to reform proposals stressed such features of mathematics learning as the importance of memorization, of facility in computation, and of being able to prove mathematical assertions. These various emphases have reflected different goals for school mathematics held by different groups of people at different times. Our analyses of the mathematics to be learned, our reading of the research in cognitive psychology and mathematics education, our experience as learners and teachers of mathematics, and our judgment as to the mathematical knowledge, understanding, and skill people need today have led us to adopt a
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Adding + It Up: Helping Children Learn Mathematics composite, comprehensive view of successful mathematics learning. This view, admittedly, represents no more than a single committee’s consensus. Yet our various backgrounds have led us to formulate, in a way that we hope others can and will accept, the goals toward which mathematics learning should be aimed. In this chapter, we describe the kinds of cognitive changes that we want to promote in children so that they can be successful in learning mathematics. Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency to capture what we believe is necessary for anyone to learn mathematics successfully. Mathematical proficiency, as we see it, has five components, or strands: 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 productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. These strands are not independent; they represent different aspects of a complex whole. Each is discussed in more detail below. The most important observation we make here, one stressed throughout this report, is that the five strands are interwoven and interdependent in the development of profi ciency in mathematics (see Box 4–1). Mathematical proficiency is not a one-dimensional trait, and it cannot be achieved by focusing on just one or two of these strands. In later chapters, we argue that helping children acquire mathematical proficiency calls for instructional programs that address all its strands. As they go from pre-kindergarten to eighth grade, all students should become increasingly proficient in mathematics. 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. The five strands are interwoven and interdependent in the development of proficiency in mathematics. The five strands provide a framework for discussing the knowledge, skills, abilities, and beliefs that constitute mathematical proficiency. This frame-
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Adding + It Up: Helping Children Learn Mathematics Box 4–1 Intertwined Strands of Proficiency work has some similarities with the one used in recent mathematics assessments by the National Assessment of Educational Progress (NAEP), which features three mathematical abilities (conceptual understanding, procedural knowledge, and problem solving) and includes additional specifications for reasoning, connections, and communication.2 The strands also echo components of mathematics learning that have been identified in materials for teachers. At the same time, research and theory in cognitive science provide general support for the ideas contributing to these five strands. Fundamental in that work has been the central role of mental representations. How learners represent and connect pieces of knowledge is a key factor in whether they will understand it deeply and can use it in problem solving. Cognitive
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Adding + It Up: Helping Children Learn Mathematics scientists have concluded that competence in an area of inquiry depends upon knowledge that is not merely stored but represented mentally and organized (connected and structured) in ways that facilitate appropriate retrieval and application. Thus, learning with understanding is more powerful than simply memorizing because the organization improves retention, promotes fluency, and facilitates learning related material. The central notion that strands of competence must be interwoven to be useful reflects the finding that having a deep understanding requires that learners connect pieces of knowledge, and that connection in turn is a key factor in whether they can use what they know productively in solving problems. Furthermore, cognitive science studies of problem solving have documented the importance of adaptive expertise and of what is called metacognition: knowledge about one’s own thinking and ability to monitor one’s own understanding and problem-solving activity. These ideas contribute to what we call strategic competence and adaptive reasoning. Finally, learning is also influenced by motivation, a component of productive disposition.3 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. Conceptual Understanding Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know.4 Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten.5 If students understand a method, they are unlikely to remember it incorrectly. They monitor what they remember and try to figure out whether it makes sense. They may attempt to explain the method to themselves and correct it if necessary. Although teachers often look for evidence of conceptual understanding in students’ ability to verbalize connections among concepts and representations, conceptual understanding need not be explicit. Students often understand before they can verbalize that understanding.6 Conceptual understanding refers to an integrated and functional grasp of mathematical ideas.
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Adding + It Up: Helping Children Learn Mathematics A significant indicator of conceptual understanding is being able to represent mathematical situations in different ways and knowing how different representations can be useful for different purposes. To find one’s way around the mathematical terrain, it is important to see how the various representations connect with each other, how they are similar, and how they are different. The degree of students’ conceptual understanding is related to the richness and extent of the connections they have made. For example, suppose students are adding fractional quantities of different sizes, say They might draw a picture or use concrete materials of various kinds to show the addition. They might also represent the number sentence as a story. They might turn to the number line, representing each fraction by a segment and adding the fractions by joining the segments. By renaming the fractions so that they have the same denominator, the students might arrive at a common measure for the fractions, determine the sum, and see its magnitude on the number line. By operating on these different representations, students are likely to use different solution methods. This variation allows students to discuss the similarities and differences of the representations, the advantages of each, and how they must be connected if they are to yield the same answer. Connections are most useful when they link related concepts and methods in appropriate ways. Mnemonic techniques learned by rote may provide connections among ideas that make it easier to perform mathematical operations, but they also may not lead to understanding.7 These are not the kinds of connections that best promote the acquisition of mathematical proficiency. Knowledge that has been learned with understanding provides the basis for generating new knowledge and for solving new and unfamiliar problems.8 When students have acquired conceptual understanding in an area of mathematics, they see the connections among concepts and procedures and can give arguments to explain why some facts are consequences of others. They gain confidence, which then provides a base from which they can move to another level of understanding. With respect to the learning of number, when students thoroughly understand concepts and procedures such as place value and operations with single-digit numbers, they can extend these concepts and procedures to new areas. 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
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Adding + It Up: Helping Children Learn Mathematics Thus, learning how to add and subtract multidigit numbers does not have to involve entirely new and unrelated ideas. The same observation can be made for multiplication and division. Conceptual understanding helps students avoid many critical errors in solving problems, particularly errors of magnitude. For example, if they are multiplying 9.83 and 7.65 and get 7519.95 for the answer, they can immediately decide that it cannot be right. They know that 10×8 is only 80, so multiplying two numbers less than 10 and 8 must give a product less than 80. They might then suspect that the decimal point is incorrectly placed and check that possibility. Conceptual understanding frequently results in students having less to learn because they can see the deeper similarities between superficially unrelated situations. Their understanding has been encapsulated into compact clusters of interrelated facts and principles. The contents of a given cluster may be summarized by a short sentence or phrase like “properties of multiplication,” which is sufficient for use in many situations. If necessary, however, the cluster can be unpacked if the student needs to explain a principle, wants to reflect on a concept, or is learning new ideas. Often, the structure of students’ understanding is hierarchical, with simpler clusters of ideas packed into larger, more complex ones. A good example of a knowledge cluster for mathematically proficient older students is the number line. In one easily visualized picture, the student can grasp relations between all the number systems described in chapter 3, along with geometric interpretations for the operations of arithmetic. It connects arithmetic to geometry and later in schooling serves as a link to more advanced mathematics. As an example of how a knowledge cluster can make learning easier, consider the cluster students might develop for adding whole numbers. If students understand that addition is commutative (e.g., 3+5=5+3), their learning of basic addition combinations is reduced by almost half. By exploiting their knowledge of other relationships such as that between the doubles (e.g., 5+5 and 6+6) and other sums, they can reduce still further the number of addition combinations they need to learn. Because young children tend to learn the doubles fairly early, they can use them to produce closely related sums.10 For example, they may see that 6+7 is just one more than 6+6. These relations make it easier for students to learn the new addition combinations because they are generating new knowledge rather than relying on rote memorization. Conceptual understanding, therefore, is a wise investment that pays off for students in many ways.
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Adding + It Up: Helping Children Learn Mathematics Procedural Fluency 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. In the domain of number, procedural fluency is especially needed to support conceptual understanding of place value and the meanings of rational numbers. It also supports the analysis of similarities and differences between methods of calculating. These methods include, in addition to written procedures, mental methods for finding certain sums, differences, products, or quotients, as well as methods that use calculators, computers, or manipulative materials such as blocks, counters, or beads. 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. Students need to be efficient and accurate in performing basic computations with whole numbers (6+7, 17–9, 8×4, and so on) without always having to refer to tables or other aids. 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. A good conceptual understanding of place value in the base-10 system supports the development of fluency in multidigit computation.11 Such understanding also supports simplified but accurate mental arithmetic and more flexible ways of dealing with numbers than many students ultimately achieve. Connected with procedural fluency is knowledge of ways to estimate the result of a procedure. It is not as critical as it once was, for example, that students develop speed or efficiency in calculating with large numbers by hand, and there appears to be little value in drilling students to achieve such a goal. But many tasks involving mathematics in everyday life require facility with algorithms for performing computations either mentally or in writing. 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. Students need to see that procedures can be developed that will solve entire classes of problems, not just individual problems. By studying algorithms as “general procedures,” students can gain insight into the fact that mathematics is well structured (highly organized, filled with patterns, predictable) and that a carefully developed procedure can be a powerful tool for completing routine tasks. It is important for computational procedures to be efficient, to be used accurately, and to result in correct answers. Both accuracy and efficiency can be improved with practice, which can also help students maintain fluency. Students also need to be able to apply procedures flexibly. Not all computational situations are alike. For example, applying a standard pencil-and-paper algorithm to find the result of every multiplication problem is neither neces-
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Adding + It Up: Helping Children Learn Mathematics sary nor efficient. Students should be able to use a variety of mental strategies to multiply by 10, 20, or 300 (or any power of 10 or multiple of 10). Also, students should be able to perform such operations as finding the sum of 199 and 67 or the product of 4 and 26 by using quick mental strategies rather than relying on paper and pencil. Further, situations vary in their need for exact answers. Sometimes an estimate is good enough, as in calculating a tip on a bill at a restaurant. Sometimes using a calculator or computer is more appropriate than using paper and pencil, as in completing a complicated tax form. Hence, students need facility with a variety of computational tools, and they need to know how to select the appropriate tool for a given situation. Procedural fluency and conceptual understanding are often seen as competing for attention in school mathematics. But pitting skill against understanding creates a false dichotomy.12 As we noted earlier, the two are interwoven. Understanding makes learning skills easier, less susceptible to common errors, and less prone to forgetting. 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. For example, it is difficult for students to understand multidigit calculations if they have not attained some reasonable level of skill in single-digit calculations. On the other hand, once students have learned procedures without understanding, it can be difficult to get them to engage in activities to help them understand the reasons underlying the procedure.13 In an experimental study, fifth-grade students who first received instruction on procedures for calculating area and perimeter followed by instruction on understanding those procedures did not perform as well as students who received instruction focused only on understanding.14 Without sufficient procedural fluency, students have trouble deepening their understanding of mathematical ideas or solving mathematics problems. The attention they devote to working out results they should recall or compute easily prevents them from seeing important relationships. Students need well-timed practice of the skills they are learning so that they are not handicapped in developing the other strands of proficiency. When students practice procedures they do not understand, there is a danger they will practice incorrect procedures, thereby making it more difficult to learn correct ones. For example, on one standardized test, the grade 2 national norms for two-digit subtraction problems requiring borrowing, such as 62–48=?, are 38% correct. 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). If students learn to subtract with understanding, they rarely make
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Adding + It Up: Helping Children Learn Mathematics Box 4–2 A common error in multidigit subtraction this error.15 Further, when students learn a procedure without understanding, they need extensive practice so as not to forget the steps. If students do understand, they are less likely to forget critical steps and are more likely to be able to reconstruct them when they do. 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. If students have been using incorrect procedures for several years, then instruction emphasizing understanding may be less effective.16 When children learn a new, correct procedure, they do not always drop the old one. Rather, they use either the old procedure or the new one depending on the situation. Only with time and practice do they stop using incorrect or inefficient methods.17 Hence initial learning with understanding can make learning more efficient. When skills are learned without understanding, they are learned as isolated bits of knowledge.18 Learning new topics then becomes harder since there is no network of previously learned concepts and skills to link a new topic to. This practice leads to a compartmentalization of procedures that can become quite extreme, so that students believe that even slightly different problems require different procedures. That belief can arise among children in the early grades when, for example, they learn one procedure for subtraction problems without regrouping and another for subtraction problems with regrouping. Another consequence when children learn without understanding is that they separate what happens in school from what happens outside.19 They have one set of procedures for solving problems outside of school and another they learned and use in school—without seeing the relation between the two. This separation limits children’s ability to apply what they learn in school to solve real problems. Also, students who learn procedures without understanding can typically do no more than apply the learned procedures, whereas students who learn
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Adding + It Up: Helping Children Learn Mathematics with understanding can modify or adapt procedures to make them easier to use. For example, students with limited understanding of addition would ordinarily need paper and pencil to add 598 and 647. Students with more understanding would recognize that 598 is only 2 less than 600, so they might add 600 and 647 and then subtract 2 from that sum.20 Strategic Competence Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them. This strand is similar to what has been called problem solving and problem formulation in the literature of mathematics education and cognitive science, and mathematical problem solving, in particular, has been studied extensively.21 Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them. Although in school, students are often presented with clearly specified problems to solve, outside of school they encounter situations in which part of the difficulty is to figure out exactly what the problem is. Then they need to formulate the problem so that they can use mathematics to solve it. Consequently, they are likely to need experience and practice in problem formulating as well as in problem solving. They should know a variety of solution strategies as well as which strategies might be useful for solving a specific problem. For example, sixth graders might be asked to pose a problem on the topic of the school cafeteria.22 Some might ask whether the lunches are too expensive or what the most and least favorite lunches are. Others might ask how many trays are used or how many cartons of milk are sold. Still others might ask how the layout of the cafeteria might be improved. With a formulated problem in hand, the student’s first step in solving it is to represent it mathematically in some fashion, whether numerically, symbolically, verbally, or graphically. Fifth graders solving problems about getting from home to school might describe verbally the route they take or draw a scale map of the neighborhood. Representing a problem situation requires, first, that the student build a mental image of its essential components. Becoming strategically competent involves an avoidance of “number grabbing” methods (in which the student selects numbers and prepares to perform arithmetic operations on them)23 in favor of methods that generate problem models (in which the student constructs a mental model of the variables and relations described in the problem). To represent a problem accurately, students must first understand the situation, including its key features. They then need to generate a mathematical representation of the problem that captures the core mathematical elements and ignores the irrelevant features. This
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Adding + It Up: Helping Children Learn Mathematics step may be facilitated by making a drawing, writing an equation, or creating some other tangible representation. Consider the following two-step problem: At ARCO, gas sells for $1.13 per gallon. This is 5 cents less per gallon than gas at Chevron. How much does 5 gallons of gas cost at Chevron? In a common superficial method for representing this problem, students focus on the numbers in the problem and use so-called keywords to cue appropriate arithmetic operations.24 For example, the quantities $1.83 and 5 cents are followed by the keyword less, suggesting that the student should subtract 5 cents from $1.13 to get $1.08. Then the keywords how much and 5 gallons suggest that 5 should be multiplied by the result, yielding $5.40. In contrast, a more proficient approach is to construct a problem model— that is, a mental model of the situation described in the problem. A problem model is not a visual picture per se; rather, it is any form of mental representation that maintains the structural relations among the variables in the problem. One way to understand the first two sentences, for example, might be for a student to envision a number line and locate each cost per gallon on it to solve the problem. In building a problem model, students need to be alert to the quantities in the problem. It is particularly important that students represent the quantities mentally, distinguishing what is known from what is to be found. Analyses of students’ eye fixations reveal that successful solvers of the two-step problem above are likely to focus on terms such as ARCO, Chevron, and this, the principal known and unknown quantities in the problem. Less successful problem solvers tend to focus on specific numbers and keywords such as $1.13, 5 cents, less, and 5 gallons rather than the relationships among the quantities.25 Not only do students need to be able to build representations of individual situations, but they also need to see that some representations share common mathematical structures. Novice problem solvers are inclined to notice similarities in surface features of problems, such as the characters or scenarios described in the problem. More expert problem solvers focus more on the structural relationships within problems, relationships that provide the clues for how problems might be solved.26 For example, one problem might ask students to determine how many different stacks of five blocks can be made using red and green blocks, and another might ask how many different ways hamburgers can be ordered with or without each of the following:
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Adding + It Up: Helping Children Learn Mathematics 17. Alibali, 1999; Lemaire and Siegler, 1995; Siegler and Jenkins, 1989. 18. Researchers have shown clear disconnections between students’ “street mathematics” and school mathematics, implying that skills learned without understanding are learned as isolated bits of knowledge. See, for example, Nunes, 1992a, 1992b; Saxe, 1990. It should be emphasized that, as discussed above, conceptual understanding requires that knowledge be connected. See Bransford, Brown, and Cocking, 1999; Hiebert and Carpenter, 1992. 19. Saxe, 1990. 20. Carpenter, Franke, Jacobs, Fennema, and Empson, 1998. 21. See Schoenfeld, 1992; and Mayer and Wittrock, 1996, for reviews. 22. Wiest, 2000. 23. Such methods are discussed by Schoenfeld, 1988. 24. Mayer and Hegarty, 1996. 25. Hagarty, Mayer, and Monk, 1995. 26. Bransford, Brown, and Cocking, 1999, pp. 19–38. See also Krutetskii, 1968/1976, ch. 13. 27. For each of the five levels in the stack of blocks, there are two options: red or green. Similarly, for each of the five toppings on the hamburger, there are two options: include the topping or exclude it. The connection might be made explicit as follows: Let each level in the stack of blocks denote a particular topping (e.g., 1, catsup; 2, onions; 3, pickles; 4, lettuce; 5, tomato) and let the color signify whether the topping is to be included (e.g., green, include; red, exclude). Such a scheme establishes a correspondence between the 2×2×2×2×2=32 stacks of blocks and the 32 kinds of hamburgers. 28. Pólya, 1945, defined such problems as follows: “In general, a problem is called a ‘routine problem’ if it can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some well-worn conspicuous example” (p. 171). 29. Siegler and Jenkins, 1989. 30. English, 1997a, p. 4. 31. English, 1997a, p. 4. See English, 1997b, for an extended discussion of these ideas. 32. For example, Inhelder and Piaget, 1958; Sternberg and Rifkin, 1979. 33. Alexander, White, and Daugherty, 1997, p. 122. 34. Davis and Maher, 1997, p. 94. 35. Davis and Maher, 1997, pp. 99–100. 36. Davis and Maher, 1997, pp. 101–102. 37. Alexander, White, and Daugherty, 1997, propose these three conditions for reasoning in young children. There is reason to believe that the conditions apply more generally. 38. Carpenter and Levi, 1999; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997; Schifter, 1999; Yaffee, 1999. 39. Maher and Martino, 1996. 40. There is a precedent for this term: “Students come to think of themselves as capable of engaging in independent thinking and of exercising control over their learning process [contributing] to what can best be called the disposition to higher order
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Adding + It Up: Helping Children Learn Mathematics thinking. The term disposition should not be taken to imply a biological or inherited trait. As used here, it is more akin to a habit of thought, one that can be learned and, therefore, taught” (Resnick, 1987, p. 41). 41. Schoenfeld, 1989. 42. Dweck, 1986. 43. See, for example, Stevenson and Stigler, 1992. Other researchers claim that Asian children are significantly more oriented toward ability than their U.S. peers and that in both groups attributing success to ability is connected with high achievement (Bempechat and Drago-Severson, 1999). 44. For evidence that U.S. students’ attitudes toward mathematics decline as they proceed through the grades, see Silver, Strutchens, and Zawojewski, 1997; Strutchens and Silver, 2000; Ansell and Doerr, 2000. 45. McLeod, 1992. 46. Thompson, 1992. 47. Cobb, Yackel, and Wood, 1989, 1995. For a more general discussion of classroom norms, see Cobb and Bauersfeld, 1995; and Fennema and Romberg, 1999. 48. National Research Council, 1989, p. 10. 49. 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. 50. Steele, 1997. 51. Fuson 1992a, 1992b; Hiebert, 1986; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997. A recent synthesis by Rittle-Johnson and Siegler, 1998, on the relationship between conceptual and procedural knowledge in mathematics concludes that they are highly correlated and that the order of development depends upon the mathematical content and upon the students and their instructional experiences, particularly for multidigit arithmetic. 52. Hiebert and Wearne, 1996. 53. Ball and Bass, 2000. 54. The NAEP data reported on the five strands are drawn from chapters in Silver and Kenney, 2000. 55. Kouba and Wearne, 2000. 56. Wearne and Kouba, 2000. 57. Kouba, Carpenter, and Swafford, 1989, p. 83. 58. The NAEP long-term trend mathematics assessment “is more heavily weighted [than the main NAEP] toward students’ knowledge of basic facts and the ability to carry out numerical algorithms using paper and pencil, exhibit knowledge of basic measurement formulas as they are applied in geometric settings, and complete questions reflecting the direct application of mathematics to daily-living skills (such as those related to time and money)” (Campbell, Voelkl, and Donahue, 2000, p. 50). 59. Kouba and Wearne, 2000, p. 150. 60. Kouba and Wearne, 2000, p. 155. 61. Silver, Alacaci, and Stylianou, 2000. 62. Shannon, 1999.
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Adding + It Up: Helping Children Learn Mathematics 63. Carpenter, Corbitt, Kepner, Lindquist, and Reys, 1981. 64. See Leder, 1992, and Fennema, 1995, for summaries of the research. In NAEP, gender differences may have increased slightly at grade 4 in the past decade, although they are still quite small; see Ansell and Doerr, 2000. 65. Ansell and Doerr, 2000. 66. For a review of the literature on race, ethnicity, social class, and language in mathematics, see Secada, 1992. Relevant findings from NAEP can be found in Silver, Strutchens, and Zawojewski, 1997; and Strutchens and Silver, 2000. 67. Beaton, Mullis, Martin, Gonzalez, Kelly, and Smith, 1996, pp. 124–125, 128; Mullis, Martin, Gonzalez, Gregory, Garden, O’Connor, Chrostowski, and Smith, 2000, pp. 137–144. 68. Mullis, Martin, Gonzalez, Gregory, Garden, O’Connor, Chrostowski, and Smith, 2000, pp. 132–136. 69. Swafford and Brown, 1989, p. 112. 70. Knapp, Shields, and Turnbull, 1995; Mason, Schroeter, Combs, and Washington, 1992; Steele, 1997. 71. Ladson-Billings, 1999, p. 1. 72. Reese, Miller, Mazzeo, and Dossey, 1997. 73. Tate, 1997. 74. Reese, Miller, Mazzeo, and Dossey, 1997, p. 31. 75. Reese, Miller, Mazzeo, and Dossey, 1997. 76. Zernike, 2000. 77. Reese, Miller, Mazzeo, and Dossey, 1997. 78. Secada, 1992. 79. Mullis, Jenkins, and Johnson, 1994. 80. Oakes, 1990. 81. Backer and Akin, 1993. 82. Committee for Economic Development, 1995; National Research Council, 1989, 1998; U.S. Department of Labor, Secretary’s Commission on Achieving Necessary Skills, 1991. References Alexander, P.A., White, C.S., & Daugherty, M. (1997). Analogical reasoning and early mathematics learning. In L.D.English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 117–147). Mahwah, NJ: Erlbaum. Alibali, M.W. (1999). How children change their minds: Strategy change can be gradual or abrupt. Developmental Psychology, 35, 127–145. Ansell, E., & Doerr, H.M. (2000). NAEP findings regarding gender: Achievement, affect, and instructional experiences. In E.A.Silver & P.A.Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 73– 106). Reston, VA, National Council of Teachers of Mathematics. Backer, A., & Akin, S. (Eds.). (1993). Every child can succeed: Reading for school improvement. Bloomington, IN: Agency for Instructional Television. Baddeley, A.D. (1976). The psychology of memory. New York: Basic Books.
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