Cognitive Strategies in Mathematics, Part I: on

Scandinavian Journal of Educational Research

ISSN: 0031-3831 (Print) 1470-1170 (Online) Journal homepage: http://www.tandfonline.com/loi/csje20

Cognitive Strategies in Mathematics, Part I: on children's strategies for solving simple addition problems Ivar Bråten To cite this article: Ivar Bråten (1998) Cognitive Strategies in Mathematics, Part I: on children's strategies for solving simple addition problems, Scandinavian Journal of Educational Research, 42:1, 5-24, DOI: 10.1080/0031383980420101 To link to this article: http://dx.doi.org/10.1080/0031383980420101

Published online: 19 May 2010.

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Scandinavian Journal of Educational Research, Vol. 42, No. 1, 1998

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Cognitive Strategies in Mathematics, Part I: on children's strategies for solving simple addition problems Downloaded by [University of Oslo] at 23:38 09 November 2015

IVAR BRÅTEN Institute for Educational Research, University of Oslo, PO Box 1092, Blindem, N-0317 Oslo 3, Norway

ABSTRACT This article first argues that successful students are generally strategic and self-regulate their learning and problem solving in school. The domain of addition is then focused and alternative strategies for solving simple addition problems are discussed. It is also shown of an addition strategy use is embedded within a functional system and influenced by metacognitive, motivational and social factors, as well as by the knowledge available within the domain. In conclusion it is indicated that addition-strategy instruction with poor learners should be comprehensive—in the sense that it should include many system components.

INTRODUCTION In two articles the aim is to demonstrate the exciting consequences of merging the fields of cognitive psychology, the psychology of mathematics and special education. Cognitive theory is providing increasingly powerful tools for breaking down tasks, such as mathematical problem solving, into component processes and strategies (Mayer, 1992). This may contribute to the understanding of how humans learn and think within the domain of mathematics, which is the critical question asked by those psychologists who are concerned specially with this subject matter (Resnick & Ford, 1981). When applied to mathematics, cognitive theory of the processes and strategies required to perform intellectual tasks may also lead to improved teaching and assessment of mathematical problem solving. In particular, improved teaching and assessment of problem solving may be profitable to students who have learning difficulties in mathematics. The field of special education has long focused on identifying individual differences in students' processes and strategies, and it has been suggested that students with learning difficulties use different cognitive strategies in solving problems than other students (for example, Swanson, 1988). Advances in the cognitive analysis of mathematical problem solving may allow special educators to describe more precisely the processes and strategies that distinguish students with learning difficulties from other students and to design better training programmes that ameliorate the problems of such students. 0031-3831/98/010005-20 © 1998 Carfax Publishing Ltd

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6 /. Brdten The remainder of this article is divided into two main sections. The first section discusses some hallmarks of skilled academic self-regulation in general terms, while the second section deals with the particular subject-matter area of addition from the same perspective. Alternative cognitive strategies that children use when trying to solve simple addition problems will be described, and the most economical and advanced counting-based backup strategy, the so-called counting-on-from-the-largeraddend strategy (COL strategy), will be discussed in detail. It will also be argued that, in accordance with the general idea of strategic processing within cognitive psychology, children's addition-strategy use is embedded within a complex strategy system where multiple cognitive, metacognitive, motivational and social components interact. The second of these two articles (Braten & Throndsen, 1998) presents a detailed single case study of a girl being taught an advanced addition strategy. This single case study illustrates a multicomponential approach based on the notion of the complex addition-strategy system discussed in the current article. SELF-REGULATED LEARNING AND PROBLEM SOLVING IN SCHOOL Contemporary research has yielded the important finding that successful students are strategic. They actively engage in strategy use and spontaneously invent increasingly advanced strategies to improve their performance in various subjects (see, for example, Resnick & Ford, 1981; Weinstein & Mayer, 1986; Siegler & Jenkins, 1989). While the issue of strategy definition is still somewhat controversial (Schneider & Weinert, 1990), it seems unquestionable that strategies are goal directed and non-obligatory actions, that is, they are not the only way strategy users can achieve their goals (Siegler & Jenkins, 1989). Moreover, it can be argued that even though strategies may not always be consciously chosen or used they are potentially conscious and potentially controllable procedures (Pressley et al., 1987). Another important research finding is that good students possess and utilize metacognitive knowledge to control their learning and problem solving. Specifically, they have knowledge about how, when and where to use specific strategies, and they actively monitor their ongoing strategy use in ways that allow them to appropriately continue, modify, terminate or shift strategies (Pressley et al, 1987; Brown & Pressley, 1994). In addition to strategies and metacognition, successful students have wellorganized content knowledge that sometimes enables or prompts strategy use and sometimes diminishes the need for strategic activity (Pressley et al., 1987). As regards motivational characteristics, it has been emphasized that successful students often have an incremental conception of ability, that is, they believe that they can improve their ability by investing greater effort (Borkowski & Muthukrishna, 1992; Meece, 1994). Correspondingly, they attribute successful learning and problem solving to effort and the effective use of strategies, and failure is attributed to lack of effort and inappropriate strategy use (Pressley et al., 1987; Borkowski & Muthukrishna, 1992). They recognize that success often comes only after some frustration and that strategic actions should be shielded from competing behaviours, distractions and emotions (Pressley et al., 1987; Brown & Pressley, 1994). Good students also possess high self-efficacy for accomplishing the academic tasks that

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confront them: they believe that they have the capacity to learn and solve problems at designated levels (Bandura, 1986; Schunk, 1994), often by exerting task-appropriate strategic effort (Brown & Pressley, 1994). As regards goal orientation, successful students pursue learning—and task-oriented rather than performance—or ego-oriented goals (Meece, 1994). This means that they are intrinsically motivated—in the sense that enhancing competence and understanding by mastering challenging tasks is more important to them than demonstrating ability by doing better than others. Finally, successful students are motivated by concrete, multiple images of possible selves, that is, visions of selves in the near and distant future that have goal properties (Borkowski & Muthukrishna, 1992). In terms of academic self-regulation, skilled self-regulation may be said to occur only when the components described above are fully integrated and function properly, that is, when strategies, metacognition and domain-specific knowledge are actively co-ordinated to attain academic goals, motivated in part by long-standing success in gaining knowledge through strategy use. Academic self-regulation thus refers to the complex process whereby students can activate and persistently use strategies, metacognition, domain-specific knowledge and underlying motives in a systematic, goal-directed way. This view of self-regulation fits well with the notion that, rather than being passive recepients of information, capable academic thinkers are mentally active participants in their own learning and problem solving and exert a large degree of control over the attainment of their own goals. The concept of self-regulation focuses on students learning and solving problems on their own. It is clear, however, that social factors can help develop self-regulation. This means that the major components of the self-regulatory system are developed initially by early home experiences and then reshaped by later classroom experiences. The ideal student probably has a history of being supported in all the components by parents, schools and society at large (Borkowski & Muthukrishna, 1992). The view that self-regulation develops in social contexts also accords with Vygotsky's (1978) theory of cognitive development. Initially, children experience mature problem solving in interpersonal situations where more knowledgeable people model good information processing and guide and assist them when they cannot manage on their own (Pressley et al, 1992). Eventually, children adopt as their own the processes that adults have externalized for them and encouraged them to use. They thus internalize the co-ordination of strategies, metacognition, knowledge and motivation that they have both observed and participated in. As the explicit verbal processing that characterizes adult-child interactions becomes intrapsychological functioning (Vygotsky, 1978), children's self-directed or private speech becomes the primary mechanism of self-regulation (Rohrkemper, 1989). SELF-REGULATION IN ADDITION: THE DISCOVERY AND USE OF EFFICIENT ADDITION STRATEGIES Children's Addition Strategies

Simple, single-digit addition is a key aspect of school mathematics, and learning it is fundamental for continued studies in mathematics. Early attempts at

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8 /. Brdten understanding children's addition strategies (for example, Groen & Parkman, 1972; Svenson, 1975; Svenson & Broquist, 1975) primarily utilized chronometric data, that is, they analysed and interpreted the mean times required for solving different problems in terms of their component cognitive processes. These early studies tended to conclude that children of a given age consistently use a single strategy. It was postulated that first and second graders consistently used the min strategy to solve simple addition problems. This strategy involves counting up from the larger addend the number of times indicated by the smaller addend. For example, a student solving '2 + 7' would start at 7 and count upwards for 2 counts: '7 [pause]; 8 [ + 1], 9 [ + 2]—9.' In accordance with the terminology employed by Baroody (1994a, b), this strategy will be termed the counting-on-from-the-larger-addend (COL) strategy in this paper. Even though a variety of findings based on chronometric data seemed to support the view that young students consistently add by means of the COL strategy, students' own verbal reports of what they did when they solved addition problems were at variance with this result. Students often said that they used a variety of strategies (for example, Svenson et ah, 1976; Baroody, 1984). Siegler (1987) combined the use of solution-time and verbal-report measures to determine whether 5-7-year-olds consistently used the COL strategy or a variety of strategies. His analyses showed that the COL strategy was but one of four different strategies that the children used. In addition, they also counted all starting with 1, decomposed problems into easier ones (for example, 12 + 3 = 10 + (2 + 3)) and retrieved answers to problems. The use of various strategies characterized individual as well as group performance. It has become increasingly apparent that children of any one age use diverse strategies with simple addition problems and that normal development in this area primarily involves changes in the mix of existing strategies as well as the construction of new strategies and the abandonment of old ones (Siegler & Jenkins, 1989; Siegler, 1990). The advantage of having various addition strategies at one's disposal is clearly seen in children's choices between retrieval and a backup strategy. In accordance with the definition used by Siegler & Jenkins (1989), a backup strategy is defined as any strategy other than retrieval, and it typically involves a form of counting of one or both addends. Retrieval is a very fast strategy, but a backup strategy can help children solve addition problems that retrieval cannot. It has been shown that children primarily choose retrieval on easy problems where a fast strategy can be executed accurately, and they use a backup strategy on difficult problems where it is necessary for accurate performance (for example, Siegler, 1990). Moreover, children have been shown to make comparable adaptive choices among alternative backup strategies. For example, children most frequently use the COL strategy, and least frequently count all starting from one, on problems where the difference between the two addends is large and/or the size of the smaller addend is small (for example, 7 + 2) (Siegler, 1987). Siegler & Jenkins (1989) used a microgenetic approach to investigate how fourand five-year-old children add new strategies to their repertoires. Their findings

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supported the view that children use multiple addition strategies rather than relying sequentially on different strategies one at a time. However, while different addition strategies coexisted and continued to compete over an extended period of time, it was also clear that, over the course of the study, children increasingly used more advanced strategies. In what follows, alternative counting-based backup strategies for computing sums are listed and denned in order of sophistication. Our developmental description accords with the ordering of strategies from the least to the most advanced implied by Siegler & Jenkins' (1989) study. It also draws heavily on Baroody's (1994a) recent outline of counting-based addition strategies. Developmentally, the concrete counting-all (CC) strategy is the most basic addition strategy. It involves counting out fingers or other objects (such as blocks) one by one to represent each addend. Then, all the objects are recounted to compute the sum. On a problem such as 3 + 5, a child using the CC strategy would typically count ' 1 , 2, 3' while successively putting up three fingers and then ' 1 , 2, 3, 4, 5' while successively putting up five other fingers. Finally, the child would start with 1 again and recount all eight fingers displayed ('1, 2, 3, 4, 5, 6, 7, 8'). (This strategy is called the sum strategy by Siegler & Jenkins, 1989.) Children can short cut the CC strategy in different ways by using pattern recognition. For example, when solving 3 + 5 the child might use finger patterns to short cut the process of representing both addends by simultaneously putting up three fingers on the left hand and then simultaneously putting up five fingers on the right hand without counting. Finally, the child begins with 1 and counts all the fingers displayed to compute the sum. Another possibility is that the child creates finger patterns to represent both addends and then uses visual or kinaesthetic pattern recognition to determine the sum. This 'finger-recognition strategy' (Siegler & Jenkins, 1989) completely eliminates the counting process. (See Baroody, 1994a, for other CC short cuts (CCS).) Counting entities (CE) strategies are somewhat more sophisticated than CC and CCS strategies because they involve the concrete representation of only one addend (Baroody, 1994a). On solving 3 + 5, a child may count and successively put up five fingers to represent the second addend, and then verbally count up to the cardinal value of the first addend ('1, 2, 3') and continue this count while pointing to the fingers previously put up ('4, 5, 6, 7, 8'). Compared to the concrete counting strategies mentioned above, abstract counting strategies are more advanced because the child executes a keeping-track process simultaneously with the sum count (Baroody, 1994a, b). On solving 3 + 5, for example, this keeping-track process ensures that the child only counts five numbers beyond the cardinal value of the first addend. With abstract strategies, concrete objects may still be used, but only to help the child keep track of how far to count beyond the first addend, not to represent the second addend per se (Carpenter & Moser, 1982). According to Baroody (1994a), the most basic of the abstract counting strategies is counting all starting with the first addend (CAF). When using CAF, a child starts with 1 and counts up to the cardinal value of the first addend and then continues the count for a number of steps corresponding to the cardinal value of the second

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10 /. Brdten addend, perhaps simultaneously putting up one finger on each count. Thus, on solving 3 + 5, the child would count ' 1 , 2, 3, ... 4, 5, 6, 7, 8'. Counting on from the first addend (COF) short cuts the CAF strategy by starting with the cardinal value of the first addend. On solving 3 + 5, the child counts '3 [pause]; 4 [ + 1], 5 [ + 2], 6 [ + 3], 7 [ + 4], 8 [ + 5]—8', perhaps simultaneously putting up one finger on each count beyond the first addend. Counting all starting with the larger addend (CAL) differs from CAF in that the child now starts with the larger addend and counts ' 1 , 2, 3, 4, 5, ... 6, 7, 8', thus minimizing the cognitively demanding keeping-track process. Counting on from the larger addend (COL) short cuts the CAL strategy by starting with the cardinal value of the larger addend irrespective of the order of the addends. On solving 3 + 5, this involves reversing the order of the addends and counting '5 [pause]; 6 [ + 1], 7 [ + 2], 8 [ + 3]—8'. With both CAL and COL, children may or may not use concrete objects to support the keeping-track process. According to Baroody (1994a), modified versions of the four abstract counting strategies described above also exist. With these strategies, concrete objects are used to represent the first addend as well as to keep track. They are thus intermediate in the sense that they are more advanced than concrete strategies, but less advanced than CAF, COF, CAL and COL. It should be noted in this context that the 'shortcut-sum' strategy described by Siegler & Jenkins (1989) as an important transitional strategy to COL, denotes a broad category encompassing both CAL and CAF as well as the modified versions of these strategies (MCAL and MCAF, respectively). Capturing COL The COL strategy is the most economical and advanced counting-based backup strategy in addition. Therefore, an analysis of children's movement into this 'mode of thought' may have not only theoretical but also practical importance. However, there has been considerable debate about how and why children come to discover and use the COL strategy. Specifically, there has been controversy over the role that other, less advanced, strategies play as a transition to COL. Moreover, there have been marked differences in opinion about the developmental mechanism underlying children's construction of the COL strategy. Transitional strategies. Groen & Resnick (1977) suggested that the COF strategy plays a critical role in the transition to COL. Despite the fact that this hypothesis was not directly supported by their chronometric analyses, Resnick & Neches (1984) and Neches (1987) later developed a computer simulation in which COF was the key transitional strategy to COL. This model indicates that when children have learned to count on from the first addend, they, in turn, compare the effort involved in counting inverse problems such as 2 + 3 and 3 + 2 and, consequentially, shift from always counting on from the first addend to counting on from whichever addend is larger. According to Siegler & Jenkins (1989), one problem with Neches' model is that the comparision of inverse problems seems to imply an amount of prescience that children are unlikely to share.

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Using a microgenetic approach, Siegler & Jenkins (1989) were able to rule out the hypothesis that COF is the key transitional step to COL. Instead, their analyses seemed to identify the short cut-sum strategy as the crucial mediator of change. On a problem such as 3 + 5, this strategy involved counting ' 1 , 2, 3, 4, 5, 6, 7, 8' and answering '8', perhaps putting up one finger on each count. Siegler & Jenkins (1989) noted that the short cut-sum strategy resembled the CC strategy, in that the child counted all the numbers between one and the sum of the addends, and the COL strategy, in that the child began the counting of the numbers that corresponded to the second addend from a point greater than one. At times, it also resembled the COL strategy in that the child reversed the order of the addends and counted the larger addend first (for example, ' 1 , 2, 3, 4, 5, ... 6, 7, 8'). Experience with the short-cut-sum strategy could thus give opportunity for learning the advantages of reversing the order of the addends. In Siegler & Jenkins' (1989) microgenetic study, almost all of the children used the short cut-sum strategy, and most of them first used it shortly before discovering the COL strategy. In other words, discovering and using this strategy made it likely that discovery of the COL strategy would follow soon after. It should also be noted that only one of the children in Siegler & Jenkins' study ever used COF, and that this use probably occurred after this child had already discovered the COL strategy. Empirical considerations thus join up with conceptual ones in pointing to the short cut-sum approach as a key transitional strategy to COL. In accordance with the findings of Siegler & Jenkins (1989), Baroody (1987) found that abstract counting-all strategies such as CAF and CAL typically serve as the transitional step between CC and COL. (As mentioned earlier, the short cut-sum strategy described by Siegler & Jenkins seems to span both CAF and CAL, as well as modified versions of these two strategies.) However, Baroody also showed that children may capture the COL strategy by different routes, with one possible route leading from CAL to COF to COL. The COF strategy may thus serve as a possible but rather uncommon transitional strategy to COL (Baroody, 1994b). Developmental mechanism. In considering the developmental mechanism, Secada et al. (1983) have argued that a counting-on strategy is the product of conceptual development. In their component-skills analysis of the counting-on procedure, they identified as Subskill 1 the ability to produce the correct sequence of counting words beginning from an arbitrary point in the sequence, for example, to count-up from 5: 'five, six, seven, eight, ...'. Secada et al. (1983) claimed that two more subskills could be identified on the basis of four meanings of the first word said in countingon. Subskill 2 requires a connection between a cardinal (how many?) and a counting meaning. On solving 5 + 3, the child must shift from the cardinal meaning of 5 given in the problem to the meaning of 5 within a counting act. This cardinal-count transition requires that the child understands that the number word denoting the cardinality of the first addend is also the final count word produced when the items in that set are counted in the sum counting of both addends together. Subskill 3 involves going from the first to the second addend within the act of counting the sum. The child shifts from the meaning of the first word said (for example, 'five' in

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12 /. Brdten 5 + 3) as an abbreviation of the first addend count to its meaning as the beginning of the count of the second addend, and then continues the sum count by saying the next counting word for the first item in the second addend set. This shift seems to require the understanding that the items in each addend set are not only in separate sets but are also embedded within the same sum set. While the component skills and the related conceptual bases that Secada et al. (1983) proposed as prerequisites for counting-on were supported by their data from 73 first graders, their study seems to leave room for increased clarity. Most notable is the likelihood that the hypothesized prerequisites for counting-on also apply to both concrete and abstract counting-all strategies. This implies that the prerequisites identified may be necessary conditions for the discovery of counting-on, but they are not sufficient conditions for it (Baroody, 1994b). In Siegler & Jenkins' (1989) study, performance just before discovery of the COL strategy was characterized by a relatively long solution time. This may have resulted from some type of cognitive conflict or interference among competing strategies. Likewise, the solution time tended to be much longer than average in those trials where children first used the COL strategy, also indicating some type of heightened cognitive activity. Siegler & Jenkins (1989) have speculated that the generation of new strategies is governed by a goal sketch. This mechanism 'specifies the hierarchy of objectives that a satisfactory strategy must meet. The hierarchical structure directs searches of existing knowledge for procedures that can meet the goals. When such procedures have been identified, they are assembled into a new strategy' (Siegler & Jenkins, 1989, p. 115). Specifically, discovery of the COL strategy is hypothesized to depend on activation and integration of the following five component skills: (i) identifying the larger addend, (ii) reversing the addend order (if necessary), (iii) quantitatively representing the larger addend by repeating its number, (iv) counting on from an arbitrary number word, and (v) simultaneously keeping track of the running total of counts and the count of the smaller addend. According to Siegler & Jenkins (1989), sources of these components may reside in already existing strategies within the domain of addition and information about related numerical domains, such as counting and numerical-magnitude comparisions. Unfortunately, Siegler & Jenkins' analyses do not seem to go very far towards explaining the exact developmental mechanism underlying the construction of COL. In fact, their component-skills analysis does not seem to add much information to Secada et al.'s (1983) account, and the cause and nature of the heightened cognitive activity suggested by the relatively long solution times immediately before and on discovery trials were not clarified by their study. Within the domain of addition, the commutativity principle states that the order of addends may be changed without altering the value of the sum. Understanding this principle has been considered a necessary conceptual basis for discovering the COL strategy (Groen & Resnick, 1977; Resnick & Ford, 1981). However, as Baroody (1994b) has pointed out, some children seem to adopt the COL strategy even without understanding commutativity, as part of their striving for cognitive economy and trying to minimize computational effort. As a matter of fact, how

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children may discover the efficient approach of counting on and disregarding the addend order even without understanding the commutativity principle in addition remains somewhat unclear. In what follows, Baroody's (1994b) promising attempt to show that 'number-after rules' are developmental mechanisms which underlie the discovery of COL will be summarized. According to Baroody (1994b), children come to master n + 1 combinations such as 4 + 1 and 5 + 1 by discovering the number-after rule: 'The sum of an n + 1 combination is the number after n in the count sequence' (p. 6). When children recognize that this principle also applies to 1 + n combinations, they are said to operate with a generalized number-after rule, which may or may not involve an understanding of commutativity (Baroody, 1994b). It seems to be a good hunch that number-after rules could play a key role in the development of counting-on strategies. Logically, the number-after rule for n + 1 combinations may form a cognitive platform for adopting counting-on with other larger-plus-smaller addend (L + S) combinations. Using counting-all strategies to solve addition problems involving one should make salient the relationship between children's ability to count-up from a number greater than one and addition. Children may then utilize knowledge of this relationship on n + 2 combinations and identify the answer as being two numbers after n in the count sequence. While the number-after rule for n + 1 combinations may provide a conceptual basis for discovering that it is possible to start with the cardinal value of the first addend and count on, a generalized number-after rule also encompassing 1 + n combinations may lead children to discover the COL strategy, which also disregards addend order. As they work with 1 + n combinations, children may gain the insight that addend order is unimportant in solving the problem (protocommutativity), or that the answers to 1 + n and n + 1 problems are the same (commutativity). It should be noted in this context that the construction and application of a generalized number-after rule is thought to occur gradually. Children first tend to apply a number-after rule for n + 1 combinations indiscriminately and respond not only to n + 1 combinations with the number after n but also to other L + S combinations (such as 5 + 3) with the number after L. This 'protonumber-after rule' (Baroody, 1994b) may thus develop before a discriminate use of an n + 1 rule. A discriminate^ applied n + 1 rule may, in turn, precede the development of a generalized rule for both n + 1 and 1 + n combinations. Moreover, number-after rules may gradually undergo automatization and only automatic rules may ensure efficient responding to all combinations involving one and general use of counting on. Baroody (1994b) has summarized the hypothesized relationship between number-after rules and counting-on strategies in this way: In this view, counting on should appear after children make the transition from using a protonumber-after rule to using a (discriminately-applied) number-after rule for n + 1 combinations. It would also predict that the invention of COL often follows the mastery of the 1 + n combinations. However, even before the number-after rules become fully automatic, they could serve as a basis for experimenting with counting on (COF) and disregarding addend order (COL), (p. 9)

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14 /. Brdten Baroody's (1994b) longitudinal single-case study of a trainable mentally handicapped boy showed that an automatic n + 1 rule was used just prior to COF. It also showed that when the boy eventually responded automatically to 1 + n combinations, he soon after began using COL. Thus, this study seemed to confirm the hypothesis that discovery of a number-after rule for n + 1 combinations underlies the development of counting-on strategies, and the hypothesis that discovery of a generalized number-after rule encompassing 1 + n combinations enables progression to a strategy which also disregards addend order. Baroody (1994b) supplemented this interesting case study with longitudinal data on 10 other mentally handicapped children. The data for all these children were consistent with the hypothesis that a number-after rule for n + 1 combinations is discovered before and may provide a basis for the invention of counting-on strategies. However, the relationship between a generalized number-after rule and strategies that disregard addend order was not entirely clear in this follow-up study. Finally, Baroody (1994b) reported on a longitudinal study of seventeen kindergarden children with normal intelligence quotients (IQs) which was also conducted to confirm or disprove the findings of the first case study. Five of these children invented and used the COL strategy on a regular basis during the study. All five cases seemed to confirm that a generalized number-after rule for both n + 1 and 1 + n combinations may serve as a basis for disregarding addend order and inventing and using the COL strategy. It should be acknowledged, however, that the idea that number-after rules often promote the development of advanced addition strategies needs more empirical backing. More specifically, it seems pertinent to test this hypothesis by means of a manipulative approach that teaches the n + 1 rule to children and evaluates its effect on counting on. Moreover, teaching directed towards the discovery of a generalized number-after rule should be evaluated for its effect on COL. The Addition Strategy System As indicated earlier, contemporary research holds that strategic processing is embedded within a functional system where multiple cognitive and non-cognitive components interact. In essence, this means that purposeful, goal-directed learning activities are thought to function within an interactive system also encompassing: (i) metacognitive knowledge and monitoring of strategies, (ii) an organized, non-strategic knowledge base, and (iii) a variety of motivational factors. Moreover, the development of this strategy system is typically seen as socially mediated (see also, Braten, 1993). In what follows, it will be argued that, in accordance with this general idea of strategic processing, children's addition strategy use is also influenced by metacognitive, motivational and social factors, as well as by their available knowledge within the domain of addition. Metacognition and addition strategies. It seems plausible to assume that children's metacognitive knowledge about addition strategies will affect their addition-strategy use. A child might know, for example, that a specific addition strategy is very costly

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in terms of effort and, at the same time, gives little benefit in terms of reliability when applied to certain problems. It then stands to reason that the child will use this strategy very infrequently for these problems in comparison with other strategies that are known to be both more efficient and more reliable. Even though metacognitive knowledge has been shown to predict children's strategy use in various domains, only a few studies have investigated the impact of metacognition on mathematical performance, and even fewer have focused on young elementary-school children. However, several studies indicate that teaching metacognitive knowledge about strategies to children can improve their mathematical performance (for example, Swing & Peterson, 1988), and it has been found that even high-school students often fail to monitor their progress in mathematical problem solving (Schoenfield, 1987). It has also been shown that middle-school students with learning disabilities and average-achieving students are less metacognitively active than intellectually gifted students when solving mathematical word problems (Montague & Applegate, 1993). Moreover, Martha Carr and her associates have shown that children have knowledge about the mathematical strategies that they use, and that this knowledge, for example, about the speediness of a strategy, predicts correct strategy use (Carr & Jessup, 1993; Carr et ah, 1993). Despite some evidence that metacognitive knowledge is necessary for good strategy use in mathematics, then, this evidence is not conclusive. Fortunately, new evidence has recently come to light that directly bears on the relationship between metacognition and addition-strategy use in young children. Carr & Jessup (1994) studied metacognitive influences on the development and use of addition strategies in 60 children, half girls and half boys, during their first school year. The children were interviewed individually in October, January and May. At each of the three interviews, the children solved 20 simple addition and subtraction problems with solutions ranging from 4 to 32. Sufficient counters were available to solve all of the problems, as external aids. Following each solution, the children were asked about how they solved each problem, and the children's responses to this question were checked against the strategies observed. Strategies were categorized as being external, internal or retrieval. A strategy was considered to be external if there was observable movement of either fingers or counters, and to be retrieval if the child described pulling the information from memory or that the information just 'popped' into the head. Metacognitive knowledge about strategies was assessed by asking the children for their rationales for the use of different strategies while they completed the problems. The children were also asked about different situations in which they would use each strategy. After they had completed all of the problems, they were asked about possible strategies that were not used. In a separate session after the individual interviews had been completed, the children were observed while they solved addition and subtraction problems within a group setting. As regards children's correct strategy use, the results of Carr & Jessup's (1994) study indicated that girls develop a mastery, of external strategies during the autumn term of the first school year and maintain their use of correct external strategies during the spring term. Boys, on the other hand, tend to shift to correct retrieval

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16 /. Brdten during the second half of the school year. Both boys and girls were more likely to use correct retrieval in the group sessions than in the individual sessions, and for boys the shift to correct retrieval occurred earlier in the group sessions than in the individual sessions. As regards metacognition, it was found that girls began the first grade with poorer metacognitive knowledge about strategies than boys but finished it with significantly better metacognition. (Both boys and girls acquired more metacognitive knowledge across the first school year.) An examination of correlational data showed that the children's metacognitive knowledge was significantly correlated at each time point (that is, in October, January and May) with correct retrieval, external and internal strategy use. Metacognition at prior time points was also found to predict later correct strategy use. Carr & Jessup (1994) concluded that a clear and consistent relationship between metacognition and correct strategy use was found across the first school year. Metacognition was related to correct strategy use at each time point, and it also predicted correct strategy use at later time points. Thus, this study seems to confirm that metacognition may be an important contributor to young elementary-school children's development of mathematics strategies.

Motivation and addition strategies. Motivational factors are also likely to affect children's addition-strategy use. Specifically, the attributional pattern of children who predominantly possess an 'incremental theory of intelligence' (Dweck, 1986), informing them that intellectual competence is a direct consequence of their own effort, may be hypothesized to induce experimentation with different addition strategies and acquisition of a repertoire of strategies. Concurrently, incremental theorists may develop metacognitive knowledge about how, when and why these strategies can be used to improve performance. In contrast, the attributional pattern of children who predominantly possess an 'entitist theory of intelligence' (Dweck, 1986) and believe that intellectual competence is inversely related to effort may restrict addition-strategy development to strategies that indicate little effort and high intelligence (such as retrieval). This lack of experimentation with different addition strategies may not only hamper strategy and skill development in mathematics, but also lead to less metacognitive knowledge about strategies in entitist than in incrementalist children. Unfortunately, there is a dearth of studies that examine the influence of motivational factors on young elementary-school children's mathematics-strategy use and metacognitive development in mathematics. However, Schunk (1983) has provided early evidence that the performance of elementary-school children with poor subtraction skills could be improved by giving them attributional feedback. Subtraction and long-division skills of older students have also been shown to be influenced by both attributions and self-efficacy (Relich et al, 1986; Schunk & Cox, 1986; Schunk & Gunn, 1986). Moreover, self-efficacy and expectancies for success have been found to be related to students' strategy use in high-school mathematics (Pokay & Blumenfeld, 1990; Zimmerman & MartinezPons, 1990). Recently, Fusco (1994) has shown that attributions and metacognitive activities may interplay in high-school students' solving of mathematical word

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problems. In this study, effort attribution seemed to go hand in hand with metacognitive regulation during actual problem solving. In Carr & Jessup's (1994) recent study, the effects of attributional patterns on addition-strategy use were examined across the first school year. How children's attributional patterns affect and are affected by metacognitive knowledge about strategies was also examined. At each individual interview in October, January and May, the children were asked about their beliefs about the roles of effort and ability as contributors to mathematics outcomes. In addition, children were asked about their attributions for the successes and failures in the addition and subtraction problems they had just solved. Specifically, the children had to choose between an incrementalist and an entitist explanation for the causes of successes and failures in these problems. Two of the questions from this (nine-item) attributional scale were also given to the children in the group sessions, and an individual attributional subset score was created by summing the answers to these two questions from the individual interview. Thus, three attributional scores were created in this study: (i) a total attributional score for the individual interview, (ii) a subset score of the individual interview, and (iii) a group attributional score which included the same two questions as the individual subset. As regards total attributional scores, the children made more ability attributions (that is, they chose more entitist explanations) over the school year, but no significant gender differences were found. However, when the individual subset scores were examined, it was found that the girls shifted more towards an entitist position than the boys. This gender difference resulted from the girls' increasing tendency to report ability attributions for failure. Thus, a tendency by the girls to attribute failure outcomes to low ability appeared already in the first grade. While the girls were more likely to make ability attributions in the individual sessions, they were much more likely to make effort attributions in the group sessions. The boys were also more likely to make effort attributions in the group sessions, but these attributions reflected their individual attributions. An examination of correlational data showed that the relationship between attributions and strategy use did not coalesce until the end of the year, when effort attributions became positively related to internal strategy use and retrieval. It should be remembered in this context that the girls tended to shift from being relatively incrementalist to being more entitist during the first school year. In light of the finding that effort attributions were positively correlated with correct strategy use by the end of the year, this transition from incrementalist beliefs to entitist beliefs does not bode well for girls. The girls, in fact, seemed to be acquiring attributional beliefs that are counterproductive to good strategy use. Evidence for the hypothesis that attributional beliefs in effort are related to metacognitive knowledge about strategies was only found in October. Social factors and addition strategies. Because children learn mathematics within the social context of the home and classroom, their addition-strategy use is likely to be influenced by other people. But even though classrooms and classmates would be expected to influence the development and use of mathematics strategies, little is known about how and how much peers contribute to strategy development.

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18 /. Brdten However, as young children seem to infer that those who seek help have low abilities (Graham & Barker, 1990), they may also believe that external, effortful strategies are signs of low ability or intelligence (Carr & Jessup, 1994). In accordance with this, many of the second-grade children studied by Carr et al. (1993) reported that external strategies were not well thought of by their peers. Thus, young elementaryschool children seem to acquire an understanding of the desirability of certain strategies for social reasons. Carr & Jessup's (1994) study of first graders examined the degree to which social rationales for mathematics strategies affect children's strategy use. At the individual interviews, the children were asked about their own attitudes toward retrieval, internal and external addition and subtraction strategies. They were also asked to provide a rationale for their strategy rankings, and these rationales were scored as social if the children referred to other children, teachers, or parents as part of the rationale. Next, the children were asked about their peers' attitudes toward the same strategies, that is, they were asked how they thought that their classmates would rank the strategies and justify their rankings. The impact of social influences was also assessed by observing the children working within a group setting and comparing their strategy use within this social context with their strategy use during individual problem solving. Finally, children's attributional beliefs were assessed both in group sessions and in individual interviews. This made it possible to determine whether attributions were affected by the presence of classmates. An analysis of the children's rankings of strategies showed that boys were more likely to rank retrieval and internal strategies consistently above external strategies, while girls made the opposite response and ranked external strategies as the best. The gender difference found in children's strategy use was thus reflected in their ranking of different strategies. At the beginning and end of the school year both boys and girls made similar rankings of strategies on behalf of their classmates as they did themselves. In the middle of the school year both boys and girls were more likely to prefer more internal strategies for themselves than for their classmates. But by the end of the first grade, the children's own strategy preferences seem to be comparable to their perceptions of their classmates' preferences and to reflect actual strategy use. This indicates that first graders' mathematics strategy use is, in part, socially driven. In January and May, when boys shifted over to retrieval as their preferred strategy, they also reported more social rationales for their strategy ranking than girls. By the end of the school year, boys were also more likely than girls to give social rationales on behalf of their classmates. If children's social rationales affect their strategy use then social rationales should correlate with strategy use. In brief, the correlational data reported by Carr & Jessup (1994) are in line with the hypotheses that during the first grade children become aware of the social utility of addition strategies, show preferences for such strategies and attempt to use the preferred strategies. As children acquired social rationales for their strategy use they were more likely to use retrieval and internal strategies and to suppress external strategy use. Moreover, the children's social rationales became increasingly related to their reported preference for retrieval and internal strategies during the school year. The fact that the children's strategy use

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and strategy preferences seemed in part to be based on their perceptions of the social utility of strategies suggests that children's addition-strategy development is not a purely cognitive but also a socially influenced process. When comparing children's strategy use in an individual setting with their strategy use in a group setting, Carr & Jessup (1994) found that both boys and girls were more likely to use retrieval correctly in the group setting than in the individual setting and that boys shifted earlier to retrieval in the group setting than in the individual setting. This indicates that correct use of retrieval is driven by the peer influences of the group. Both boys and girls were also more likely to attempt to use retrieval in the group setting than in the individual setting. As regards attributional beliefs, Carr & Jessup (1994) found that the children altered their attributions when in the company of others. Specifically, both boys and girls tended to be more effort oriented in the group setting than in the individual setting. For boys, the presence of peers seemed to strengthen their individual effort attributions, while girls seemed to shift from being more ability oriented in the individual setting to being more effort oriented in the group setting. Thus, when asked to make attributions in groups, the boys' attributional pattern seemed to predominate with both boys and girls making effort attributions. Likewise, the boys' strategy pattern seemed to predominate in the group setting, with both boys and girls using retrieval more often in groups. Carr & Jessup's study (1994) suggested that boys' strategy development during the first school year may be especially susceptible to social factors. As mentioned earlier, boys made social rationales by spring that reflected their shift to retrieval, and they were much more likely to use retrieval in the group setting in May. Typically , boys commented that retrieval was a competitive strategy that allowed them to 'finish first' or to 'beat the other children'. Moreover, the relationships found between children's reports of social rationales and strategy use, and between social rationales and strategy rankings, seem to support the idea that, for boys at least, mathematics strategy development during the first grade is guided by the social factor of competitiveness. Finally, it should be noted that social influences seem to go hand in hand with entitist attributions to suppress external strategy use and promote retrieval and internal strategy use. (However, children's social rationales and their attributional beliefs were never correlated as hypothesized by Carr & Jessup, 1994.) Domain-specific knowledge and addition strategies. Finally, children's addition strat-

egy use is probably influenced by their knowledge base within the domain. For example, if the answer to an addition problem is stored in the knowledge base, a child does not need to compute an answer with the help of a backup strategy, but can simply rely on what that child already knows and retrieve the answer immediately. If the presented problem is not entirely congruous with prior knowledge, however, the child may have to analyse the task and select a task-appropriate backup strategy on the basis of metacognitive knowledge about strategies. It seems obvious that relying on the knowledge base and simply retrieving the answer is an important developmental step that serves adaptive purposes. Increasing demands are made on

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20 /. Brdten children's arithmetic skills with increasing age. When addition must be rapid, fact retrieval from long-term memory takes less time than backup computation, and without the knowledge of basic addition facts it would be almost impossible to solve such complex problems as 3614 + 1987. Knowing basic single-digit addition facts is an enabling condition for the application of strategies required to solve such complex addition problems (Pressley, 1986). Siegler & Shrager (1984) tried to describe the complex relationship between children's knowledge bases and strategy use in addition by means of the distributions of associations model. This model consists of a representation of knowledge about particular problems and a process that operates on this representation to produce performance and learning. Children are thought to associate both correct and incorrect answers with a specific problem in their representation of knowledge. The representation of knowledge of each problem can be classified along a dimension of the peakedness of its distribution of associations, with a peaked distribution having the most associative strength concentrated in the correct answer. The process that operates on the representation involves the sequential phases of retrieval, elaboration of the representation and the application of an algorithm. This means that a child presented with an addition problem will first try to retrieve an answer. But if that child is not sufficiently confident of any answer, he or she will elaborate the representation of the problem, for example, by putting up fingers corresponding to the two addends and trying to recognize their number. If the child still does not know the answer he or she might count the fingers and report the last count as the answer. In Siegler & Shrager's (1984) model, the probability of retrieving any given answer to an addition problem is proportional to that answer's associative strength relative to the total associative strengths of all the answers to the problem. And, once retrieved, an answer is only stated if its associative strength exceeds a response threshold known as the confidence criterion. If the associative strength of the answer that is retrieved does not exceed the confidence criterion, the child can again retrieve an answer from the problem's distribution of associations and state it if its associative strength exceeds the confidence criterion. However, if the retrieval process fails to yield a statable answer, the child computes an answer by using a backup addition strategy. Within this model, backup strategies are used primarily on the most difficult problems, because the peakedness of the distribution of associations determines both the problem difficulty and the frequency of use of backup strategies. It should be noted that the distributions of associations model implies a bidirectionality of influences between domain-specific knowledge and addition strategies. Knowledge of answers to specific addition problems influences which strategy is chosen, with accurate and highly differentiated knowledge allowing for the fast, efficient strategy of retrieval. Conversely, the structure of the knowledge representation is influenced by children's addition-strategy use, with the accuracy of execution of backup strategies influencing knowledge of particular problems. Specifically, more accurate execution of backup strategies contributes to the building of more peaked distributions of associations, which ultimately allow retrieval to be executed more accurately as well.

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Siegler and his colleagues later developed a strategy-choice model that has the strengths of the original distributions of associations model and a number of others as well (for an outline of this revised strategy-choice model, see, for example, Siegler & Jenkins, 1989; Siegler, 1990). Like the original model, the revised strategy-choice model includes a representation and a process. However, children are hypothesized to not only know about problems but also about strategies, and about the interaction between problems and strategies. The process that operates on the revised model's representation involves two phases, strategy choice and strategy execution. A strategy is chosen with a probability proportional to its strength relative to the strength of all strategies, with the strength of a strategy on a problem being a joint function of how fast and accurate it has been for that problem in the past and how fast and accurate it has been across all problems in the domain. Once a strategy is chosen, the model attempts to execute it to produce an answer. But if the strategy cannot be executed, as when retrieval is chosen but no retrieved answer exceeds the confidence criterion, the process returns to the strategy-choice phase. The process continues until a strategy is chosen that can be executed and which produces an answer. Unlike the original model, the revised strategy-choice model does not always choose retrieval first and a backup strategy only after retrieval has failed. It should be noted that the revised model of Siegler and associates sticks to the original view of a bilateral influence between knowledge and strategies. Thus, knowledge of answers to specific problems and knowledge of how different strategies have worked in the past influence which strategy is chosen for each problem, while strategy use, in turn, is critical to the acquisition of knowledge of problems and strategies. In particular, children's use of backup strategies early in learning seems to play a critical role in shaping their knowledge into an organization that allows retrieval to take over as the dominant strategy later on. Siegler's original strategy-choice model focuses on the choice between stating a retrieved answer and using a backup strategy in addition. In the revised version of the model, the problem of choices between alternative backup strategies is also addressed. Still, neither version of the model specifies the prior knowledge necessary for mastering different backup strategies to any great extent. However, as discussed in connection with the COL strategy, certain forms of conceptual understanding and related component skills function as prerequisites for efficient backup strategy use. The point is that children's knowledge possession enables good strategy use. This means that spontaneous use of a backup strategy (such as COL) depends on the relationship that certain qualifications in the addition problem have to the knowledge base, with good strategy use occurring when problems activate (are consistent with) the knowledge base, and with production deficiencies occurring when problems do not link up with the knowledge base. In memory research, enabling effects have been shown to occur even when subjects are under strong instructional control to produce strategies (Pressley et al., 1987). Within the domain of addition, an important area of future research on knowledgestrategy interactions concerns differential instructional effects as a function of prior knowledge.

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CONCLUSION Contemporary cognitive research has not only ascertained that successful students are strategic, but also that strategies function within a system characterized by complex interactions between strategies, knowledge, metacognition, motivation and social factors. Skilled self-regulation may be said to occur when this system is fully integrated and functions properly. This view of self-regulation also seems to hold good for the domain of simple, single-digit addition, where successful students both use advanced strategies and self-regulate their addition by coordinating multiple aspects of problem solving. More specifically, the general idea that strategic processing is embedded within a strategy system encompassing cognitive, metacognitive, motivational and social components also seems to hold good as far as strategies for solving simple, single-digit addition problems are concerned. However, since most of the research discussed in the preceding section was done with the relatively broad categories of external strategies, internal strategies, and retrieval, the idea of a functional additionstrategy system still needs empirical backing from research using a more fine-grained differentiation of addition strategies. It might be the case, for example, that the various backup strategies outlined in this paper to different degrees are influenced by (and influence) the various cognitive, metacognitive, motivational and social system components under discussion. But even though it should be acknowledged that the interactions between critical components of the addition-strategy system are not yet fully understood, a reasonable instructional strategy that can be derived from our, albeit incomplete, insight is that strategy training with poor learners in addition should profitably be comprehensive, in the sense that it includes many, if not all, of the components assumed necessary for efficient addition-strategy use.

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