6
DEVELOPING PROFICIENCY WITH WHOLE NUMBERS

Whole numbers are the easiest numbers to understand and use. As we described in the previous chapter, most children learn to count at a young age and understand many of the principles of number on which counting is based. Even if children begin school with an unusually limited facility with number, intensive instructional activities can be designed to help them reach similar levels as their peers.1 Children’s facility with counting provides a basis for them to solve simple addition, subtraction, multiplication, and division problems with whole numbers. Although there still is much for them to work out during the first few years of school, children begin with substantial knowledge on which they can build.

In this chapter, we examine the development of proficiency with whole numbers. We show that students move from methods of solving numerical problems that are intuitive, concrete, and based on modeling the problem situation directly to methods that are more problem independent, mathematically sophisticated, and reliant on standard symbolic notation. Some form of this progression is seen in each operation for both single-digit and multidigit numbers.

We focus on computation with whole numbers because learning to compute can provide young children the opportunity to work through many number concepts and to integrate the five strands of mathematical proficiency. This learning can provide the foundation for their later mathematical development. Computation with whole numbers occupies much of the curriculum in the early grades, and appropriate learning experiences in these grades improve children’s chances for later success.



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Adding + It Up: Helping Children Learn Mathematics 6 DEVELOPING PROFICIENCY WITH WHOLE NUMBERS Whole numbers are the easiest numbers to understand and use. As we described in the previous chapter, most children learn to count at a young age and understand many of the principles of number on which counting is based. Even if children begin school with an unusually limited facility with number, intensive instructional activities can be designed to help them reach similar levels as their peers.1 Children’s facility with counting provides a basis for them to solve simple addition, subtraction, multiplication, and division problems with whole numbers. Although there still is much for them to work out during the first few years of school, children begin with substantial knowledge on which they can build. In this chapter, we examine the development of proficiency with whole numbers. We show that students move from methods of solving numerical problems that are intuitive, concrete, and based on modeling the problem situation directly to methods that are more problem independent, mathematically sophisticated, and reliant on standard symbolic notation. Some form of this progression is seen in each operation for both single-digit and multidigit numbers. We focus on computation with whole numbers because learning to compute can provide young children the opportunity to work through many number concepts and to integrate the five strands of mathematical proficiency. This learning can provide the foundation for their later mathematical development. Computation with whole numbers occupies much of the curriculum in the early grades, and appropriate learning experiences in these grades improve children’s chances for later success.

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Adding + It Up: Helping Children Learn Mathematics Whole number computation also provides an instructive example of how routine-appearing procedural skills can be intertwined with the other strands of proficiency to increase the fluency with which the skills are used. For years, learning to compute has been viewed as a matter of following the teacher’s directions and practicing until speedy execution is achieved. Changes in career demands and the tasks of daily life, as well as the availability of new computing tools, mean that more is now demanded from the study of computation. More than just a means to produce answers, computation is increasingly seen as a window on the deep structure of the number system. Fortunately, research is demonstrating that both skilled performance and conceptual understanding are generated by the same kinds of activities. No tradeoffs are needed. As we detail below, the activities that provide this powerful result are those that integrate the strands of proficiency. Operations with Single-Digit Whole Numbers As students begin school, much of their number activity is designed to help them become proficient with single-digit arithmetic. By single-digit arithmetic, we mean the sums and products of single-digit numbers and their companion differences and quotients (e.g., 5+7=12, 12–5=7, 12–7=5 and 5×7=35, 35÷5=7, 35÷7=5). For most of a century, learning single-digit arithmetic has been characterized in the United States as “learning basic facts,” and the emphasis has been on memorizing those facts. We use the term basic number combinations to emphasize that the knowledge is relational and need not be memorized mechanically. Adults and “expert” children use a variety of strategies, including automatic or semiautomatic rules and reasoning processes to efficiently produce the basic number combinations.2 Relational knowledge, such as knowledge of commutativity, not only promotes learning the basic number combinations but also may underlie or affect the mental representation of this basic knowledge.3 The domain of early number, including children’s initial learning of single-digit arithmetic, is undoubtedly the most thoroughly investigated area of school mathematics. A large body of research now exists about how children in many countries actually learn single-digit operations with whole numbers. Although some educators once believed that children memorize their “basic facts” as conditioned responses, research shows that children do not move from knowing nothing about the sums and differences of numbers to having the basic number combinations memorized. Instead, they move through a series of progressively more advanced and abstract methods for working out the answers

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Adding + It Up: Helping Children Learn Mathematics to simple arithmetic problems. Furthermore, as children get older, they use the procedures more and more efficiently.4 Recent evidence indicates children can use such procedures quite quickly.5 Not all children follow the same path, but all children develop some intermediate and temporary procedures. Most children continue to use those procedures occasionally and for some computations. Recall eventually becomes the predominant method for some children, but current research methods cannot adequately distinguish between answers produced by recall and those generated by fast (nonrecall) procedures. This chapter describes the complex processes by which children learn to compute with whole numbers. Because the research on whole numbers reveals how much can be understood about children’s mathematical development through sustained and interdisciplinary inquiry, we give more details in this chapter than in subsequent chapters. Word Problems: A Meaningful Context One of the most meaningful contexts in which young children begin to develop proficiency with whole numbers is provided by so-called word problems. This assertion probably comes as a surprise to many, especially mathematics teachers in middle and secondary school whose students have special difficulties with such problems. But extensive research shows that if children can count, they can begin to use their counting skills to solve simple word problems. Furthermore, they can advance those counting skills as they solve more problems.6 In fact, it is in solving word problems that young children have opportunities to display their most advanced levels of counting performance and to build a repertoire of procedures for computation. Most children entering school can count to solve word problems that involve adding, subtracting, multiplying, and dividing.7 Their performance increases if the problems are phrased simply, use small numbers, and are accompanied by physical counters for the children to use. The exact procedures children are likely to use have been well documented. Consider the following problems: Sally had 6 toy cars. She gave 4 to Bill. How many did she have left? Sally had 4 toy cars. How many more does she need to have 6? Most young children solve the first problem by counting a set of 6, removing 4, and counting the remaining cars to find the answer. In contrast,

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Adding + It Up: Helping Children Learn Mathematics they solve the second problem by counting a set of 4, adding in more as they count “five, six,” and then counting those added in to find the answer. Children solve these problems by “acting out” the situation—that is, by modeling it. They invent a procedure that mirrors the actions or relationships described in the problem. This simple but powerful approach keeps procedural fluency closely connected to conceptual understanding and strategic competence. Children initially solve only those problems that they understand, that they can represent or model using physical objects, and that involve numbers within their counting range. Although this approach limits the kinds of problems with which children are successful, it also enables them to solve a remarkable range of problems, including those that involve multiplying and dividing. Since children intuitively solve word problems by modeling the actions and relations described in them, it is important to distinguish among the different types of problems that can be represented by adding or subtracting, and among those represented by multiplying or dividing. One useful way of classifying problems is to heed the children’s approach and examine the actions and relations described. This examination produces a taxonomy of problem types distinguished by the solution method children use and provides a framework to explain the relative difficulty of problems. Four basic classes of addition and subtraction problems can be identified: problems involving (a) joining, (b) separating, (c) part-part-whole relations, and (d) comparison relations. Problems within a class involve the same type of action or relation, but within each class several distinct types of problems can be identified depending on which quantity is the unknown (see Table 6–1). Students’ procedures for solving the entire array of addition and subtraction problems and the relative difficulty of the problems have been well documented.8 For multiplication and division, the simplest kinds of problems are grouping situations that involve three components: the number of sets, the number in each set, and the total number. For example: Jose made 4 piles of marbles with 3 marbles in each pile. How many marbles did Jose have? In this problem, the number and size of the sets is known and the total is unknown. There are two types of corresponding division situations depending on whether one must find the number of sets or the number in each set. For example:

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Adding + It Up: Helping Children Learn Mathematics Table 6–1 Addition and Subtraction Problem Types Problem Type   Join (Result Unknown) (Change Unknown) (Start Unknown)   Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether? Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether? Connie had some marbles. Juan gave her 5 more. Now she has 13 marbles. How many marbles did Connie have to start with? Separate (Result Unknown) (Change Unknown) (Start Unknown)   Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left? Connie had 13 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did Connie give to Juan? Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many marbles did Connie have to start with? Part- Part- (Whole Unknown)   (Part Unknown) Whole Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether?   Connie has 13 marbles: 5 are red and the rest are blue. How many blue marbles does Connie have? Compare (Difference Unknown) (Compare Quantity Unknown) (Referent Unknown)   Connie has 13 marbles. Juan has 5 marbles. How many more marbles does Connie have than Juan? Juan has 5 marbles. Connie has 8 more than Juan. How many marbles does Connie have? Connie has 13 marbles. She has 5 more marbles than Juan. How many marbles does Juan have?   SOURCE: Carpenter, Fennema, Franke, Levi, and Empson, 1999, p. 12. Used by permission of Heinemann. All rights reserved.

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Adding + It Up: Helping Children Learn Mathematics Jose has 12 marbles and puts them into piles of 3. How many piles does he have? Jose has 12 marbles and divides them equally into 3 piles. How many marbles are in each pile? Additional types of multiplication and division problems are introduced later in the curriculum. These include rate problems, multiplicative comparison problems, array and area problems, and Cartesian products.9 As with addition and subtraction problems, children initially solve multiplication and division problems by modeling directly the action and relations in the problems.10 For the above multiplication problem with marbles, they form four piles with three in each and count the total to find the answer. For the first division problem, they make groups of the specified size of three and count the number of groups to find the answer. For the other problem, they make the three groups by dealing out (as in cards) and count the number in one of the groups. Although adults may recognize both problems as 12 divided by 3, children initially think of them in terms of the actions or relations portrayed. Over time, these direct modeling procedures are replaced by more efficient methods based on counting, repeated adding or subtracting, or deriving an answer from a known number combination.11 The observation that children use different methods to solve problems that describe different situations has important implications. On the one hand, directly modeling the action in the problem is a highly sensible approach. On the other hand, as numbers in problems get larger, it becomes inefficient to carry out direct modeling procedures that involve counting all of the objects. Children’s proficiency gradually develops in two significant directions. One is from having a different solution method for each type of problem to developing a single general method that can be used for classes of problems with a similar mathematical structure. Another direction is toward more efficient calculation procedures. Direct-modeling procedures evolve into the more advanced counting procedures described in the next section. For word problems, these procedures are essentially abstractions of direct modeling that continue to reflect the actions and relations in the problems. The method children might use to solve a class of problems is not necessarily the method traditionally taught. For example, many children come to solve the “subtraction” problems described above by counting, adding up, or thinking of a related addition combination because any of these methods is easier and more accurate than counting backwards. The method traditionally presented in textbooks, however, is to solve both of these problems by

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Adding + It Up: Helping Children Learn Mathematics subtracting, which moves students toward the more difficult and error-prone procedure of counting down. Ultimately, most children begin to use recall or a rapid mental procedure to solve these problems, and they come to recognize that the same general method can be used to solve a variety of problems. Single-Digit Addition Children come to understand the meaning of addition in the context of word problems. As we noted in the previous section, children move from counting to more general methods to solve different classes of problems. As they do, they also develop greater fluency with each specific method. We call these specific counting methods procedures. Although educators have long recognized that children use a variety of procedures to solve single-digit addition problems,12 substantial research from all over the world now indicates that children move through a progression of different procedures to find the sum of single-digit numbers.13 This progression is depicted in Box 6–1. First, children count out objects for the first addend, count out objects for the second addend, and count all of the objects (count all). This general counting-all procedure then becomes abbreviated, internalized, and abstracted as children become more experienced with it. Next, they notice that they do not have to count the objects for the first addend but can start with the number in the first or the larger addend and count on the objects in the other addend (count on). As children count Box 6–1 Learning Progression for Single-Digit Addition

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Adding + It Up: Helping Children Learn Mathematics on with objects, they begin to use the counting words themselves as countable objects and keep track of how many words have been counted on by using fingers or auditory patterns. The counting list has become a representational tool. With time, children recompose numbers into other numbers (4 is recomposed into 3+1) and use thinking strategies in which they turn an addition combination they do not know into one they do know (3+4 becomes 3+3+1). In the United States, these strategies for derived number combinations often use a so-called double (2+2, 3+3, etc.). These doubles are learned very quickly. As Box 6–1 shows, throughout this learning progression, specific sums move into the category of being rapidly recalled rather than solved in one of the other ways described above. Children vary in the sums they first recall readily, though doubles, adding one (the sum is the next counting word), and small totals are the most readily recalled. Several procedures for single-digit addition typically coexist for several years; they are used for different numbers and in different problem situations. Experience with figuring out the answer to addition problems provides the basis both for understanding what it means to say “5+3=8” and for eventually recalling that sum without the use of any conscious strategy. Children in many countries often follow this progression of procedures, a natural progression of embedding and abbreviating. Some of these procedures can be taught, which accelerates their use,14 although direct teaching of these strategies must be done conceptually rather than simply by using imitation and repetition.15 In some countries, children learn a general procedure known as “make a 10” (see Box 6–2).16 In this procedure the solver makes a 10 out of one addend by taking a number from the other addend. Educators in some countries that use this approach believe this first instance of regrouping by making a 10 provides a crucial foundation for later multidigit arithmetic. In some Asian countries this procedure is presumably facilitated by the number words.17 It has also been taught in some European countries in which the number names are more similar to those of English, suggesting that the procedure can be used with a variety of number-naming systems. The procedure is now beginning to appear in U.S. textbooks,18 although so little space may be devoted to it that some children may not have adequate time and opportunity to understand and learn it well. There is notable variation in the procedures children use to solve simple addition problems.19 Confronted with that variation, teachers can take various steps to support children’s movement toward more advanced procedures. One technique is to talk about slightly more advanced procedures and why

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Adding + It Up: Helping Children Learn Mathematics Box 6–2 Make a Ten: B+6=? they work.20 The teacher can stimulate class discussion about the procedures that various students are using. Students can be given opportunities to present their procedures and discuss them. Others can then be encouraged to try the procedure. Drawings or concrete materials can be used to reveal how the procedures work. The advantages and disadvantages of different procedures can also be examined. For a particular procedure, problems can be created for which it might work well or for which it is inefficient. Other techniques that encourage students to use more efficient procedures are using large numbers in problems so that inefficient counting procedures cannot easily be used and hiding one of the sets to stimulate a new way of thinking about the problem. Intervention studies indicate that teaching counting-on procedures in a conceptual way makes all single-digit sums accessible to U.S. first graders, including children who are learning disabled and those who do not speak English as their first language.21 Providing support for children to improve their own procedures does not mean, however, that every child is taught to use all the procedures that other children develop. Nor does it mean that the teacher needs to provide every child in a class with

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Adding + It Up: Helping Children Learn Mathematics support and justification for different procedures. Rather, the research provides evidence that, at any one time, most children use a small number of procedures and that teachers can learn to identify them and help children learn procedures that are conceptually more efficient (such as counting on from the larger addend rather than counting all).22 Mathematical proficiency with respect to single-digit addition encompasses not only the fluent performance of the operation but also conceptual understanding and the ability to identify and accurately represent situations in which addition is required. Providing word problems as contexts for adding and discussing the advantages and disadvantages of different addition procedures are ways of facilitating students’ adaptive reasoning and improving their understanding of addition processes. Single-Digit Subtraction Subtraction follows a progression that generally parallels that for addition (see Box 6–3). Some U.S. children also invent counting-down methods that model the taking away of numbers by counting back from the total. But counting down and counting backward are difficult for many children.23 Box 6–3 Learning Progression for Single-Digit Subtraction

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Adding + It Up: Helping Children Learn Mathematics A considerable number of children invent counting-up procedures for situations in which an unknown quantity is added to a known quantity.24 Many of these children later count up in taking-away subtraction situations (13–8=? becomes 8+?=13). When counting up is not introduced, many children may not invent it until the second or third grade, if at all. Intervention studies with U.S. first graders that helped them see subtraction situations as taking away the first x objects enabled them to learn and understand counting-up-to procedures for subtraction. Their subtraction accuracy became as high as that for addition.25 Experiences that focus on part-part-whole relations have also been shown to help students develop more efficient thinking strategies, especially for subtraction.26 Students examine a join or separate situation and identify which number represents the whole quantity and which numbers represent the parts. These experiences help students see how addition and subtraction are related and help them recognize when to add and when to subtract. For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplishments in arithmetic.27 For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplish-ments in arithmetic. Examining the relationships between addition and subtraction and seeing subtraction as involving a known and an unknown addend are examples of adaptive reasoning. By providing experiences for young students to develop adaptive reasoning in addition and subtraction situations, teachers are also anticipating algebra as students begin to appreciate the inverse relationships between the two operations.28 Single-Digit Multiplication Much less research is available on single-digit multiplication and division than on single-digit addition and subtraction. U.S. children progress through a sequence of multiplication procedures that are somewhat similar to those for addition.29 They make equal groups and count them all. They learn skip-count lists for different multipliers (e.g., they count 4, 8, 12, 16, 20,…to multiply by four). They then count on and count down these lists using their fingers to keep track of different products. They invent thinking strategies in which they derive related products from products they know. As with addition and subtraction, children invent many of the procedures they use for multiplication. They find patterns and use skip counting (e.g., multiplying 4×3 by counting “3, 6, 9, 12”). Finding and using patterns and other thinking strategies greatly simplifies the task of learning multiplication tables (see Box 6–4 for some examples).30 Moreover, finding and describing

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Adding + It Up: Helping Children Learn Mathematics 51.   On the basis of these results for multidigit addition and subtraction, Siegler, in press, suggests, as a broad principle, that conceptual instruction should occur prior to teaching of procedures. Rittle-Johnson and Alibali, 1999, reported similar results with respect to mathematical equivalence. 52.   Beishuizen, 1993; Beishuizen, Van Putten, and Van Mulken, 1997; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, Human, 1997; Hiebert and Wearne, 1993, 1996. 53.   Fuson, 1992a, 1992b. 54.   Carpenter, Franke, Jacobs, Fennema, Empson, 1998; Carraher, Carraher, and Schliemann, 1987; Cobb and Wheatley, 1988; Fuson and Burghardt, 1993; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997; Hiebert and Wearne, 1996; Kamii, 1989; Labinowicz, 1985; Nunes, 1992; Olivier, Murray, and Human, 1990; Saxe, 1988; Ambrose, Baek, and Carpenter, in press. 55.   Kamii and Dominick, 1998. 56.   Hiebert and Wearne, 1993, 1996. 57.   Carpenter, Franke, Jacobs, Fennema, and Empson, 1998; Fuson and Briars, 1990; Hiebert and Wearne, 1993, 1996. 58.   Cobb and Bauersfeld, 1995; Fuson, 1992a, 1992b; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997; Kamii, 1989. 59.   Carpenter, Franke, Jacobs, Fennema, and Empson, 1998; Fuson and Briars, 1990; Hiebert and Wearne, 1993, 1996. See Carroll and Porter, 1998, for some alternative algorithms. For a discussion of principles for creating classroom environments that incorporate these features of effective teaching, see Fuson, De La Cruz, Smith, Lo Cicero, Hudson, Ron, and Steeby, 2000. 60.   Uttal, Scudder, DeLoache, 1997. 61.   Hiebert and Wearne, 1996. 62.   Beishuizen, Gravemeijer, and van Lieshout, 1997; Bowers, Cobb, and McClain, 1999; Fuson and Burghardt, 1993, in press; Carpenter, Franke, Jacobs, Fennema, and Empson, 1998; McClain, Cobb, and Bowers, 1998; Fuson and Briars, 1990. 63.   Fuson, 1990. 64.   Cauley, 1988; Fuson and Burghardt, 1993, 1997, in press; Hiebert and Wearne, 1996. 65.   VanLehn, 1986. 66.   Fuson and Briars, 1990; Hiebert and Wearne, 1993, 1996. For examples of student difficulties with numeration and the base-10 system, see Bednarz and Janvier, 1982. 67.   Student-invented procedures are sometimes not really algorithms because the steps are not precisely specified but instead follow a path that emerges through the process—and that path may be slightly different if the same problem is posed again. Because such procedures can often be made into algorithms by deliberate specification of the steps, the distinction between algorithms and ad hoc procedures is seldom maintained in the literature. (See, e.g., the articles in Morrow and Kenney, 1998.) 68.   Carpenter, Franke, Jacobs, Fennema, Empson, 1998; Fuson and Burghardt, 1993; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997. 69.   Fuson, Wearne, Hiebert, Murray, Human, Olivier, Carpenter, and Fennema, 1997. 70.   Fuson and Burghardt, 1993, in press.

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Adding + It Up: Helping Children Learn Mathematics 71.   Bowers, Cobb, and McClain, 1999; Hiebert and Wearne, 1996; Kamii, 1989. 72.   Carpenter, Franke, Jacobs, Fennema, and Empson, 1998; Fuson, 1986a; Fuson and Briars, 1990; Hiebert and Wearne, 1993, 1996; Kamii, 1989. 73.   Fuson, 1986a; Fuson and Briars, 1990. 74.   For example, Carroll and Porter, 1998; Kamii, 1994; Lampert, 1986a, 1986b. 75.   Baek, 1998; Ambrose, Baek, and Carpenter, in press. 76.   See, for example, the 1999 edition of Scott Foresman-Addison Wesley Math, Grade 4. 77.   Lampert, 1992; Murray, Olivier, and Human, 1992. 78.   See, for example, Scott Foresman’s Seeing Through Arithmetic, Grade 4 (Hartung, Van Engen, and Knowles, 1955). 79.   For example, Bowers, Cobb, and McClain, 1999, and Carpenter, Franke, Jacobs, Fennema, and Empson, 1998; Fuson and Burghardt, in press; Hiebert and Wearne, 1996; Kamii, 1994. In a comparison study, Hiebert and Wearne, 1993, showed that students who spent more time on fewer problems and were asked to explain their procedures outperformed their more traditionally taught peers. 80.   Fuson, 1986a; Fuson, Smith, Lo Cicero, 1997. 81.   For example, VanLehn, 1986, and Fuson and Burghardt, in press. 82.   For example, Bowers, Cobb, and McClain, 1999; Carpenter, Franke, Jacobs, Fennema, and Empson, 1998; Fuson, 1986a; Fuson and Burghardt, in press; Hiebert and Wearne, 1993, 1996; Kamii, 1994. 83.   Beberman, 1959; Rathmell and Trafton, 1990. For a similar discussion about estimation, see Buchanan, 1978. Beishuizen, 1993, discusses students connecting mental arithmetic procedures to using base-10 blocks and hundreds squares. 84.   Cohen, 1982. 85.   Stigler, 1984; Hatano, 1988. 86.   Sowder, 1992. 87.   Hope and Sherrill, 1987. 88.   Davis, 1984. 89.   Markovits and Sowder, 1988. 90.   Reys, Rybolt, Bestgen, and Wyatt, 1982. 91.   Markovits and Sowder, 1994; Rubenstein, 1985. 92.   Sowder and Wheeler, 1989. References Ambrose, R., Baek, J., & Carpenter, T.P. (in press). Children’s construction of multiplication and division algorithms. In A.J.Baroody & A.Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ: Erlbaum. Armstrong, G.A. (1991). Use of the part-whole concept for teaching word problems to grade three children (Doctoral dissertation, National College of Education, 1990). Dissertation Abstracts International, 52(03), 833A.

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