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## 6DEVELOPING 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 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 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 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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Adding + It Up: Helping Children Learn Mathematics Baek, J.-M. (1998). Children’s invented algorithms for multidigit multiplication problems. In L.J.Morrow & M.J.Kenney (Eds.), The teaching and learning of algorithms in school mathematics (1998 Yearbook of the National Council of Teachers of Mathematics, pp. 151–160). Reston, VA: NCTM. Baroody, A.J. (1984a). Children’s difficulties in subtraction: Some causes and cures. Arithmetic Teacher, 32(3), 14–19. Baroody, A.J. (1984b). The case of Felicia: A young child’s strategies for reducing memory demands during mental addition. Cognition and Instruction, 1, 109–116. Baroody, A.J. (1985). Mastery of the basic number combinations: Internalization of relationships or facts? Journal of Research in Mathematics Education, 16, 83–98. Baroody, A.J. (1987a). Children’s mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York: Teachers College Press. Baroody, A.J. (1987b). The development of counting strategies for single-digit addition. Journal for Research in Mathematics Education, 18, 141–157. Baroody, A.J. (1992). The development of kindergartners’ mental-addition strategies. Learning and Individual Differences, 4, 215–235. Baroody, A.J. (1994). An evaluation of evidence supporting fact-retrieval models. Learning and Individual Differences, 6, 1–36. Baroody, A.J. (1996). Self-invented addition strategies by children classified as mentally handicapped. American Journal of Mental Retardation, 101, 72–89. Baroody, A.J. (1999a). Children’s relational knowledge of addition and subtraction. Cognition and Instruction, 17, 137–175. Baroody, A.J. (1999b). The roles of estimation and the commutativity principle in the development of third-graders’ mental multiplication. Journal of Experimental Child Psychology, 74 [Special issue on mathematical cognition], 157–193. Beberman, M. (1959). Introduction. In C.H.Shutter & R.L.Spreckelmeyer (Eds.), Teaching the third R. Washington, DC: Council for Basic Education. Bednarz, N., & Janvier, B. (1982). The understanding of numeration in primary school. Educational Studies in Mathematics, 13, 33–57. Beishuizen, M. (1993). Mental procedures and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in Mathematics Education, 24, 294–323. Beishuizen, M., Gravemeijer, K.P.E., & van Lieshout, E.C.D.M. (Eds.). (1997). The role of contexts and models in the development of mathematical strategies and procedures (pp. 163–198). Utrecht: CD-B Press/Freudenthal Institute. Beishuizen, M., Van Putten, C.M., & Van Mulken, F. (1997). Mental arithmetic and strategy use with indirect number problems up to one hundred. Learning and Instruction, 7, 87–106. Bergeron, J.C., & Herscovics, N. (1990). Psychological aspects of learning early arithmetic. In P.Nesher & J.Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education. ICMI study series (pp. 31–52). Cambridge, UK: Cambridge University Press. Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. Cognition and Instruction, 17, 25–64.

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Adding + It Up: Helping Children Learn Mathematics Bransford, J.D., Brown, A.L., & Cocking, R.R. (Eds.). (1999). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press. Available: http://books.nap.edu/catalog/6160.html. [July 10, 2001]. Brown, J.S., & Van Lehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4, 379–426. Brownell, W.A. (1944). Rate accuracy and process in learning. Journal of Educational Psychology, 35, 321–337. Brownell, W.A. (1987). AT classic: Meaning and skill—maintaining the balance. Arithmetic Teacher, 34(8), 18–25. (Original work published 1956) Brownell, W.A., & Chazal, C.B. (1935). The effects of premature drill in third-grade arithmetic. Journal of Educational Research, 29, 17–28. Buchanan, A.D. (1978). Estimation as an essential mathematical skill (Professional Paper No. 39, SWRL-PP-39). Los Alamitos, CA: Southwest Regional Laboratory for Educational Research and Development. (ERIC Document Reproduction Service No. ED 167 385) Carnine, D.W., & Stein, M. (1981). Organizational strategies and practice procedures for teaching basic facts. Journal for Research in Mathematics Education, 12, 65–69. Carpenter, T.P. (1985). Learning to add and subtract: An exercise in problem solving. In E.A.Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 17–40). Hillsdale, NJ: Erlbaum. Carpenter, T.P., Ansell, E., Franke, M.L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children’s problem-solving processes. Journal for Research in Mathematics Education, 24, 428–441. Carpenter, T.P., Fennema, E., & Franke, M.L. (1996). Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction. Elementary School Journal, 97, 3–20. Carpenter, T.P., Fennema, E., Franke, M.L., Empson, S.B., & Levi, L.W. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Carpenter, T.P., Fennema, E., Peterson, P.L., Chiang, C.P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26, 499–531. Carpenter, T.P., Franke, M.L., Jacobs, V.R., Fennema, E., & Empson, S.B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3–20. Carpenter, T.P., & Moser, J.M. (1984). The acquisition of addition and subtraction concepts in grades one through three . Journal for Research in Mathematics Education, 15, 179–202. Carpenter, T.P., Moser, M.J., & Romberg, T.A. (Eds.). (1982). Addition and subtraction: A cognitive perspective. Hillsdale, NJ: Erlbaum. Carraher, T.N., Carraher, D.W., & Schliemann, A.D. (1987). Written and oral mathematics. Journal for Research in Mathematics Education, 18, 83–97. Carroll, W.M., & Porter, D. (1997). Invented procedures can develop meaningful mathematical procedures. Teaching Children Mathematics, 3, 370–74. Carroll, W.M., & Porter, D. (1998). Alternative algorithms for whole-number operations. In L.J.Morrow & M.J.Kenney (Eds.), The teaching and learning of algorithms in school mathematics (1998 Yearbook of the National Council of Teachers of Mathematics, pp.

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Adding + It Up: Helping Children Learn Mathematics Griffin, S.A., Case, R., & Siegler, R.S. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K.McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 25–49). Cambridge, MA: MIT Press. Harel, G., & Confrey, J. (1994). The development of multiplicative reasoning in the learning of mathematics. Albany: State University of New York Press. Hartung, M.L., Van Engen, H., & Knowles, L. (1955). Seeing through arithmetic. Chicago: Scott Foresman. Hatano, G. (1988, Fall). Social and motivational bases for mathematical understanding. New Directions for Child Development, 41, 55–70. Hiebert, J., Carpenter, T., Fennema, E., Fuson, K.C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann. Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and student learning in second grade. American Educational Research Journal, 30, 393–425. Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14, 251–83. Hope, J.A., & Sherrill, J.M. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education, 18(2), 98–111. Huinker, D.M. (1991). Effects of instruction using part-whole concepts with one-step and two-step word problems in grade four (Doctoral dissertation University of Michigan, 1990). Dissertation Abstracts International, 52(01), 103 A. Jerman, M. (1970). Some strategies for solving simple multiplication combinations. Journal for Research in Mathematics Education, 1, 95–128. Kamii, C. (1989). Young children continue to reinvent arithmetic—2nd grade: Implications of Piaget’s theory. New York: Teachers College Press. Kamii, C. (1994). Young children continue to reinvent arithmetic—3rd grade: Implications of Piaget’s theory. New York: Teachers College Press. Kamii, C. & Dominick, A. (1998). The harmful effects of algorithms in grades 1–4. In L. J.Morrow & M.J.Kenney (Eds.), The teaching and learning of algorithms in school mathematics (1998 Yearbook of the National Council of Teachers of Mathematics, pp. 130–140). Reston VA: NCTM. Kouba, V. (1989). Children’s solution procedures for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20, 147–158. Labinowicz, E. (1985). Learning from children: New beginnings for teaching numerical thinking. Menlo Park, CA: Addison-Wesley. Lampert, M. (1986a). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3, 305–342. Lampert, M. (1986b). Teaching multiplication. Journal of Mathematical Behavior, 5, 241– 280. Lampert, M. (1992). Teaching and learning long division for understanding in school. In G.Leinhardt, R.T.Putnam, & R.A.Hattrup (Eds.), The analysis of arithmetic for mathematics teaching (pp. 221–282). Hillsdale, NJ: Erlbaum. LeFevre, J., Bisanz, J., Daley, K.E., Buffone, L., Greenham, S.L., & Sadesky, G.S. (1996). Multiple routes to solution of single-digit multiplication problems. Journal of Experimental Psychology: General, 125, 284–306.

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