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Strategy choice and strategy discovery
LeamIng and In.Wmchon Vol. 1, pp. 8%1M, Printed III Great Bntain. AU rights reserved
1991
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0959-4752/91$0.00 + so 1991 Pergamcm Press plc
STRATEGY CHOICE AND STRATEGY DISCOVERY ROBERT
S. SIEGLER
Carnegie Mellon University,
Pittsburgh, U.S.A.
Abstract This article summarizes a number of recent findings and ideas concerning how children choose among existing strategies and how they discover new ones. Across a wide range of domains, individual children use diverse strategies. In all of these domains, they choose adaptively among the alternative approaches. Consistent individual differences are present within the strategy choices of both middle and lower class populations, and these predict performance on standardized achievement tests. The ideas about strategy choices have been formalized within running computer simulations that produce performance much like children’s both in terms of performance at any one time and in terms of changes over time. The article concludes with a description of recent work on how children discover new strategies and with a discussion of the relation of the present perspective on cognitive development to that embodied in traditional Piagetian and neo-Piagetian theories.
Strategy Choice and Strategy Discovery A huge body of research documents the strategies that children use to memorize, conceptualize, reason, and solve problems under various conditions. Much less is known, however, about an issue of equal importance: the. processes through which they come to use one strategy rather than another. I believe that understanding how children choose among strategies is essential to understanding both how they think at any one time and how their thinking changes over time. In short, it is basic to understanding cognitive development. For strategy choice to be meaningful, there must be multiple strategies to choose among. Although many studies have depicted development in terms of a 1 : 1 correspondence between children’s age and the strategy that they use, recent trialby-trial analyses have found that children of a single age often use a wide variety of strategies. The finding has emerged in domains as diverse as arithmetic (Cooney, Swanson, & Ladd, 1988; Gear-y & Burlingham-Dubree, 1989; Goldman, Pellegrino, & Mertz, 1988; Siegler & Robinson, 1982), causal reasoning (Shultz, Fisher, Pratt, & Rulf, 1986), spatial reasoning (Ohlsson, 1984), referential communications (Kahan & Richards, 1986), serial recall (McGilly & Siegler, 1989), reading and spelling (Jorm & Address for correspondence: R. S. Siegler, Department Pittsburgh, PA 15213-3890, U.S.A. 89
of Psychology,
Carnegie Mellon University,
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Share, 1983; Siegler, 1986), and judgments of plausibility (Kuhara-Kojima & Hatano, 1989). The variability in strategy use is evident within as well as between individuals; even the same child presented the same problem on two successive days often uses different strategies on the two occasions (Siegler, 1987a; Siegler & McGilly, 1989). The fact that children use diverse strategies is not a mere idiosyncracy of human cognition. Good reasons exist for people to know and use multiple strategies. Strategies differ in their accuracy, in the amounts of time needed to execute them, in their processing demands, and in the range of problems to which they apply. Strategy choices involve tradeoffs among these properties. The broader the range of strategies that children know, the more precisely they can shape their approaches to the demands of particular circumstances. Even young children often capitalize on the strengths of different strategies and use each most often on problems where its advantages are greatest. Studies of children’s strategy choices have revealed some striking unities across domains, both in empirical data and in the cognitive mechanisms that seem to underlie the data. For example, on many tasks, children use backup strategies. These are strategies other than retrieval that can be used to solve problems that can also potentially be solved by retrieval. Thus, counting from one or from the larger addend are backup strategies for simple addition; sounding out words or looking them up in a dictionary are backup strategies for spelling; writing down a phone number or rehearsing it are backup strategies for serial recall; and so on. A variety of studies have demonstrated that within each domain, children most often use backup strategies on the most difficult problems, with problem difficulty measured either in terms of percentage of errors or length of solution times. Correlations among percentage of use of backup strategies on each problem, percentage of errors on each problem, and mean solution time on each problem have ranged from I = 0.83 to r = 0.92 in 4- and 5-year-olds’ addition, 6- and 7-year-olds’ subtraction and word identification, 7- and 8-year-olds’ spelling and time telling, and 8- and 9-year-olds’ multiplication (Siegler, 1986, 1987a, 1988a, b; Siegler & Robinson, 1982; Siegler & Shrager, 1984). Such choices are adaptive because they mean that children use the more time consuming backup strategies primarily on problems where such strategies are needed for accurate performance. The adaptive strategy choices are evident among lower income minority children and students who are not doing well in school as well as among middle income students and those who are doing well (Kerkman & Siegler, submitted for publication). Once it is recognized that children know multiple strategies and choose effectively among them, the question arises: How do they construct such strategies in the first place? Are new strategies constructed primarily after other strategies fail? When children first use strategies, do they understand why they are effective? Do they generalize them broadly from the beginning, or do they at first use them only in limited circumstances? Microgenetic studies, in which individual children who do not yet know a strategy are given prolonged experience in the domain, with their performance then being subjected to trial-by-trial scrutiny, seem especially useful for studying strategy construction. They allow identification of the trial on which a new strategy is first used, which in turn allows examination of what the experience of discovery was like, what led up to the discovery, and how the discovery was generalized beyond its initial use. Studies of this type have yielded a variety of nonintuitive findings. Among them are results indicating
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that discoveries are often made not in response to impasses or failure but rather in the context of successful performance; that short-lived transition strategies often precede more enduring approaches; and that generalization of new approaches often occurs very slowly, even when children can provide compelling rationales for their usefulness (Siegler & Jenkins, 1989). Below, I summarize in greater detail my recent research on strategy choice and strategy discovery.
The Ubiquity of Multiple Strategy Use The first context in which I became aware that children were using multiple strategies was 4- and 5-year-olds’ addition of small numbers (Siegler & Robinson, 1982). Videotapes of the children’s performance made it clear that they sometimes put up their fingers and counted them, sometimes put up their fingers and recognized how many fingers were up without counting, sometimes counted without any obvious external referent, and sometimes retrieved the answer from memory. Since then, my colleagues and I have documented children’s use of diverse strategies in many domains. To multiply, 8 to lo-year-olds sometimes repeatedly add one of the multiplicands, sometimes write the problem and then recognize the answer, sometimes write and then count groups of hatch marks that represent the problem, and sometimes retrieve the answer from memory (Siegler, 1988a). To tell time, 7- to 9-year-olds sometimes count forward from the hour by ones and/or fives, sometimes count backward from the hour by ones and/or fives, sometimes count from reference points such as the half hour, and sometimes retrieve the time that corresponds to the clock hands’ configuration (Siegler & McGilly, 1989). To spell, 7- and 8-year-olds sometimes sound out words, sometimes look them up in dictionaries, sometimes write out alternative forms and try to recognize which is correct, and sometimes recall the spelling from memory (Siegler, 1986). To serially recall lists, 5- to 8-year-olds sometimes repeatedly recite the lists during the delay period, sometimes recite the list once and stop, and sometimes just wait (McGilly & Siegler, 1989, 1990). These diverse strategies are not artifacts of one child using one strategy and a different child another one. Most individuals of all ages in all of these domains have been found to use at least 2 strategies; in most of the domains, the norm has been to use 3 or more. This multiple strategy use is apparent even within classes of similar problems and, in the limiting case, on the same problem presented to the same child twice on consecutive days (Siegler, 1987a; Siegler & McGilly, 1989). In both subtraction and time telling, fully one-third of children used different strategies on the identical problem on two successive days. The change was not in general from less advanced to more advanced strategies. For example, almost as many children retrieved the answer on the first day and used a backup strategy on the second as did the reverse. These experiments have called into serious question previous models in these areas that postulated that children of a given age consistently use a particular strategy. To illustrate, a number of models of first and second graders’ addition (Ashcraft, 1982, 1987; Groen & Parkman, 1972; Svenson, 1975) postulate that 6- and 7-year-olds consistently add by using the min strategy of counting-on from the larger addend (e.g.,
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they would solve 3 + 5 by starting with 5 and then thinking “6 is 1, 7 is 2, 8 is 3”). However, Siegler (1987b) found that children of these ages used the min strategy on only a minority of trials; on other trials, they counted from 1, retrieved answers from memory, decomposed problems into simpler form, or guessed. Similarly, the McGilly and Siegler (1989) finding that the majority of 5-year-olds use repeated rehearsal on some trials, as well as using single rehearsal and no rehearsal on others, contradicted claims that preschoolers generally do not rehearse (Hagen, Hargrave, & Ross, 1973; Schneider & Pressley, 1989).
Validity of the Strategy Assessments The validity of all of the above findings depends critically on the validity of the method by which strategy use on each trial was assessed. In all of my studies published prior to 1987, the assessments of strategy use were based entirely on videotapes of ongoing overt behavior. Since then, I have relied on immediately-retrospective verbal reports as well as observations of overt behavior. Specifically, strategy assessments in the more recent studies have been based on overt behavior when that behavior unambiguously indicated the strategy, and on immediately retrospective verbal reports when ongoing overt behavior was absent or ambiguous. Results of the recent studies have yielded evidence for the validity of the strategy assessments and have also indicated that the assessment method allows accurate detection of many instances of covert strategy use without creating more use of the strategies than would occur if the immediately-retrospective reports were not obtained. Siegler (1987b) provided evidence for the validity of the strategy assessments. As mentioned above, a number of investigators had hypothesized that 6- and 7-year-olds add by using the min strategy. A considerable body of evidence was consistent with this hypothesis. The size of the smaller addend is consistently the best predictor of first and second graders’ solution times on different problems (Ashcraft, 1982, 1987; Groen & Parkman, 1972; Kaye, Post, Hall, & Dineen, 1986; Svenson, 1975). It is a good predictor in absolute as well as relative terms, accounting for 60% to 75% of the variance in solution times in a number of studies. The min model, based on_ the assumption that young children invariably use the min strategy, fits the performance of individual children as well as group averages, children in Europe as well as in North America, and children in special education classes as well as standard ones (Groen & Resnick, 1977; Kaye et al., 1986; Svenson, 1975). Despite all this support, the model is wrong. Siegler (1987b) videotaped 5- to 7-year-olds’ ongoing behavior as they solved simple addition problems and asked them immediately after the problem how they solved it. When solution time data were averaged over trials (and over strategies), the results closely replicated the previous finding that solution times were a linear function of the smaller addend. Smaller addend size accounted for 76% of variance in solution times on the 45 problems that were examined. The trial-by-trial assessments of strategy use, however, indicated that children used 5 different strategies and that the min strategy was used on only 36% of trials. Dividing the solution time data according to what strategy children were classified as using
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lent considerable credence to the strategy assessments. On trials where children were classified as using the min strategy, the min model was an even better predictor of solution times than in the data set as a whole. It accounted for 86% of the variance in solution times. In contrast, on trials where children reported using one of the other strategies, the min model was never a good predic&or of performance, either in absolute or relative terms. It never accounted for as much as 40% of variance, and was never either the best or the second best predictor. Thus, the trial-by-trial strategy assessments appeared to yield a valid picture of addition performance, and one quite at odds with that derived from analyses of data averaged over trials. (See Siegler, 1987b, for a statistical analysis of why the min model fit the averaged data despite the min strategy not being used on most trials. Also see Siegler, 1989, for a similar analysis and conclusion about the Woods, Resnick, and Groen [1975] model of 2nd through 4th graders’ subtraction.) Not surprisingly, obtaining verbal reports as well as videotaped records of ongoing overt behavior results in higher estimates of strategy use than does reliance on overt behavior alone. This could be due either to the requests for verbal reports artificially inflating the amount of strategy use, or to the verbal reports leading to detection of strategies that children did use but without any accompanying overt behavior. To distinguish between these interpretations, McGilly and Siegler (1990) randomly assigned 5, 7-, and 9-year-olds in a serial recall experiment to self-report and no-selfreport conditions. Those in the self-report condition were asked immediately after each trial what, if anything, they had done in the delay period between when the list was presented and when they needed to recall it. Those in the no-self-report condition received an identical procedure, except that they were not asked any questions about their strategies. When verbal reports as well as overt behavior were considered in assessing strategies within the self-report condition, children in that condition were classified as rehearsing on a much higher percentage of trials, 74% vs. 47%. However, when the strategy use of children in both groups was scored solely on the basis of overt behavior, children in the self-report condition were classified as rehearsing slightly less often, 39% vs 47%. There was no obvious reason to think that overt strategic behavior was unaffected by the questions but covert strategic behavior was affected. Thus, the similar amounts of overt behavior obtained from parallel strategy assessment methods indicated that the higher estimates of rehearsal in the self-report condition were not due to the requests for self reports creating greater use of strategies. Instead, the higher estimates apparently stemmed from the verbal reports allowing detection of covert rehearsal activities that would otherwise have gone undetected.
Three Ways in Which Strategy Choices Are Adaptive These studies have revealed three distinct senses in which children’s strategy choices are adaptive. One concerns the choice of whether to use retrieval or a backup strategy (an approach other than retrieval). As noted earlier, the more difficult the problem, the more often children use backup strategies to solve it. This has been found in the above-cited studies of addition, subtraction, multiplication, spelling, word identification, and time telling. Such choices are useful, because they enable children to use the
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fast retrieval strategy on problems where that approach yields correct answers and to use slower backup strategies on problems where they are necessary to produce accurate performance. Consistent with this analysis, preventing children from using backup strategies by imposing a very short time limit (4 set) produces sharp falloffs in accuracy, with the dropoffs being largest on precisely the problems on which children are most likely to use backup strategies when they are allowed to choose freely (Siegler & Robinson, 1982). Children also choose adaptively among alternative backup strategies. For example, in choosing whether to add by counting from one or by counting from the larger addend (the min strategy), children are especially likely to use the min strategy on problems such as 9 + 2, that have both a large difference between the addends and a relatively small minimum addend. Such problems also are the ones where the efficiency of the min strategy is greatest relative to counting from one (Siegler, 1987b). Studies of serial recall have led to appreciation of a third sense in which children’s strategy choices are adaptive: trial-to-trial changes in strategy use. McGilly and Siegler (1989) found that children changed strategies most often when they had recalled the list incorrectly on the previous trial and had not rehearsed. When they succeeded despite not rehearsing or failed after rehearsing, they were significantly less likely to switch strategies. Thus, children switched to the more effortful approach of rehearsing when the less effortful approach of just waiting failed to yield accurate recall, but not when it allowed successful performance. To summarize, the generalizations that individual children often use diverse strategies and that they choose among them in adaptive ways have proved to hold true for many types of domains beyond the areas of simple addition and subtraction in which they were first formulated. The work on time telling indicates that the generalizations apply to numerical tasks other than arithmetic. The work on reading and spelling indicates that they hold true for tasks that do not involve numbers. The work on serial recall indicates that they hold true for memory as well as problem solving strategies, and for tasks on which children lack extensive problem-specific experience as well as ones in which they have such experience. Thus, a view of development that depicts children as actively choosing among multiple competing strategies may turn out to be more generally accurate than the traditional view in which a child of a given age is depicted as invariably using a particular approach.
Individual Differences
and Cross-Domain
Consistencies
A major goal of my research has been to reveal cross-domain commonalities in strategy choices. One way in which this goal has been pursued has been described already: through establishing that use of multiple strategies and adaptive patterns of strategy choices are present in diverse domains. Another approach to the same goal has been to attempt to identify stable individual. difference patterns across domains. To examine individual differences in strategy choices, Siegler (1988b) had each of a group of first graders perform three tasks: addition, subtraction, and reading (word identification). A cluster analysis of the children’s percent correct, mean RT, and frequency of use of backup strategies on each of the three tasks indicated considerable
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consistency in performance. The clustering program divided children into 3 groups, which we labeled the “good students”, the “not-so-good students”, and the “perfectionists”. The contrast between good and not-so-good students was evident along all of the dimensions that might be expected from the names. Good students more often used retrieval in both addition and subtraction. They were correct more often on both retrieval and nonretrieval trials on all 3 tasks. They also were faster in executing backup strategies on all 3 tasks and faster in using retrieval on both addition and subtraction problems. The relation of the performance of the perfectionists to that of children in the other 2 groups was more complex. Despite their being as fast and as accurate as the good students on both retrieval and backup strategy trials, the perfectionists used retrieval less often even than the not-so-good students. Convergent validation for these findings was obtained in a second experiment with a somewhat different strategy assessment method, different problems, and different children (Siegler, 1988b; Experiment 2). Four months after the experiment, all children were given the Metropolitan Achievement Test. Differences between perfectionists and good students on the one hand and not-so-good students on the other echoed those in the experimental setting. The perfectionists’ average mathematics scores were at the 81st percentile, the good students’ at the 80th percentile, and the not-so-good students’ at the 43rd percentile. Thus, differences between not-so-good students and the other 2 groups would have been evident on standardized achievement tests. However, differences between good students and perfectionists would not have been. Instead, these seem to be alternative ways of doing well in early mathematics and reading. Kerkman and Siegler (submitted for publication) extended the individual differences research to lower-income African-American children. The sample came from quite poor back~ounds; 80% qualified for the federal free school lunch program. The children were examined twice, once in first grade and once in second, to examine stability of the individual different classifications over time. The same three groups emerged in the lower income sample as in the previouslystudied middle income ones. Proportions of children in the three groups also were quite similar. Stability over time was evident in that most children were classified as being in the same group on the two testings. Probably the most striking finding, though, was that the lower income children’s strategy choices were just as adaptive as those of middle income children. Percent backup strategy use and percent errors on each problem correlated r = 0.91 for addition, r = 0.92 for subtraction, and r = 0.86 for word identification. The findings demonstrate that in domains in which children have substanti~ experience with individual problems, lower-income as well as middle-income children produce reasonable and systematic strategy choices.
Computer
Simulations of Strategy Choice
To test whether their ideas about strategy choice mechanisms were sufficient to account for the patterns of speed, accuracy, and strategy use observed in preschoolers’ addition, Siegler and Shrager (1984) formulated a computer simulation of how children choose among competing strategies, how the choices contribute to their knowledge of
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addition, and how previous choices shape future patterns of answers and strategy choices. The same framework was later used to simulate the development of simple subtraction (Siegler, 1987a) and multiplication (Siegler, 1988a). Results of these simulations indicated a close fit along a number of dimensions between children’s behavior and the output of the simulations. For example, in multiplication, the correlation between the children’s and the simulation’s percent correct, percent backup strategy use, and mean solution time on each problem were r = 0.90, T = 0.90, and r = 0.95 for the 3 measures respectively. Thus, the same problems were the most difficult for the simulation as for the children, and the simulation, like the children, chose to use backup strategies most often on those problems. Also like children, the simulation showed substantial learning from its experience with problems. Percent correct increased from 61% during the first 10% of problems to 89% on the last 10%. Over the same period, percent use of retrieval increased from 24% to 77% of trials. The simulation also allowed an explicit demonstration of a nonintuitive but accurate prediction of the underlying model: that the relative difficulty of problems for 8- and 9-year-olds, just learning to multiply, would more accurately predict the problems’ difficulty for adults than would any structural variable, such as the product or the sum of the multiplicands. Despite these successes, several limitations produced by the simulations’ basic structure also became evident. One problem was that the model was well worked out only for the choice between retrieving and using a backup strategy. If the choice process is general, the same basic procedure should govern choices among alternative backup strategies. A second problem was that the simulation was inflexible, in the sense that it always tried to retrieve first. Reder’s (1987) evidence from question answering indicated that if retrieval was unsuccessful on previous trials, adults were more likely to try alternative strategies first. I suspected that the same would be true of children’s arithmetic, since the retrieval mechanism is likely to be the same for question answering and arithmetic. A third problem was the models’ inability to generalize. The simulations approached each problem as a separate entity. Again, it seemed unreasonable that knowing that the min strategy worked well on 9 + 1 and 9 + 3 would not lead children to suspect that it also worked well on 9 + 2. The new strategy choice model was developed to overcome these limits while preserving the strengths of the original model. The basic approach was to generalize the principles on which the model was based along two dimensions: the types of choices that were made and the types of data that were considered in making the choices. With regard to the types of choices that were made, the Siegler and Shrager model’s procedure for choosing among answers was generalized to choosing among strategies as well as answers. With regard to the types of data that were considered, the database was extended to include not only the answers that the model had generated but also the speeds and accuracies that each strategy produced. The new model’s organization centers on the process of a strategy being used to solve a problem. The process generates information about the speed, accuracy, and particular answers that accompany the problem and the strategy. From this information, the model builds data bases that include information about the speeds and accuracies that each strategy produces on the total set of problems, on problems with particular features (e.g., a smaller addend of l), and on particular problems (e.g., 7 + 1). These three types
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of information are used to compute the activation of each strategy for a given problem, which in turn determines the probability of using that strategy on the problem. Another innovation is the model’s treatment of how new strategies ever come to be used on tasks where existing strategies work relatively well. The model accomplishes this through assigning novelty points to new strategies. These novelty points act in the same way as the activations of other strategies, with one exception. With each use of the new strategy, some of its novelty points are lost. If the new strategy leads to quick and accurate responding, the activation it gains through generating effective performance more than compensates for the decrease in novelty points. If it generates inaccurate and/or slow responding, novelty points are gradually used up without compensating gains in activation. Thus, useful new strategies become stronger, and ineffective ones weaker. As in the earlier strategy choice models, answers, both correct and incorrect, become associated with problems through being stated on those problems. Also as in the earlier models, a retrieved answer is stated only when its strength exceeds the confidence criterion on the trial. These features allow the model to generate the effects it did previously, while becoming more flexible and more general. Siegler and Shipley (in preparation) tested this model’s ability to account for learning and performance on the 81 basic addition facts with addends l-9. It proved to generate a wide range of basic characteristics of 5- to S-year-olds’ addition. Its accuracy progressed from 55% correct early in its run to 99% at the end. Its percent use of retrieval progressed from 16% at the early point to 98% at the end. Its percent correct, mean solution time, and percent use of retrieval on each problem all correlated at least r = 0.80 with children’s. The best predictors of its performance also mimicked those of children: at first the size of the sum was the best predictor of errors and solution times, then the size of the smaller addend, and finally the product of the two addends. The simulation also revealed why the product eventually became the best predictor of the simulation’s performance, as it had been found to be with adults (Miller, Keating, & Perlmutter, 1984; Widaman, Geary, Little, & Cormier, 1989) - an averaging of the effects of previous use of counting from 1 and using the min strategy. Further, varying two parameter values - accuracy of execution of backup strategies and stringency of confidence criteria - and leaving the rest of the simulation constant, led to patterns of performance like those of the good students, not-so-good students, and perfectionists in the individual difference studies. The simulation also generalized from its experience to choose strategies adaptively on problems it had not encountered before. In sum, the new simulation maintained the strong points of the earlier ones and produced a number of desired new outcomes as well.
Discovery of New Strategies Probably the most important gap in the strategy choice model is a lack of an account of how new strategies are discovered. To provide a data base relevant to how this process occurs, Siegler and Jenkins (1989) studied 4- and 5-year-olds’ construction of the min strategy for adding numbers. First, a pretest was given to identify children who could add by counting from 1 but who did not yet know the min strategy. Eight children
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who met these criteria then participated in an 11 week experiment in which they were presented addition problems approximately 3 times per week. The first 7 weeks of this period were spent on problems with addends l-5. A number of children discovered the min strategy during this period, but none generalized it very widely. Therefore, in week 8, we presented them challenge problems, such as 22 + 3, on which counting from 1 and retrieval would work badly but on which the min strategy would work well. In weeks 9-11, children were presented a mixed set of problems; some were small number problems, some were challenge problems, and some were in-between (e.g., 8 + 3). Throughout the experiment, children’s strategies were classified on a trial-by-trial basis. This allowed identification of the first trial on which they used the new strategy, what led up to the discovery, and how the new strategy was generalized beyond its initial use. During the 11 weeks, 7 of the 8 children discovered the min strategy. Many of the discoveries were accompanied by an impressive amount of insight into the new strategy’s advantages. For example, “Ruth” first used the new strategy on 4 + 3. When asked “Can you tell me why you started (counting) from 4”, she said “I don’t have to count a very long ways if I start from 4”. Despite this understanding, Ruth, like the other children, used the min strategy only occasionally in the sessions between the discovery and presentation of the challenge problems. Encountering the challenge problems, however, led her and the other children who had made the discovery to greatly increase their use of the new strategy. This pattern held true not only for the data set as a whole (as shown in Fig. 1) but when the analysis was limited to strategy use on the types of problems presented at the beginning of the study, that is problems with addends of 5 or less. On these problems, use of the min strategy increased from 11% to 45% of trials. The microgenetic experiment also yielded several other findings that ran counter to typical depictions of discovery. Most discoveries were not responses to children encountering impasses; the discoveries were made on ordinary problems, usually ones
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Trial Black (O=Challenge Problems) Figure 1. Percent use of min strategy relative to all counting strategies before, during, and after challenge problems. Each trial block represents five experimental sessions, except for the challenge problems block, which represents two sessions. Thus, the -3 trial block represents min strategy use 11-15 sessions before the challenge problems among children who had discovered the strategy by that time.
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children had solved previously using other strategies. Further, children’s answers were generally correct both on the problem just before the .discovery and on most prior problems in that session. The only unusual characteristic of performance on problems just before the discovery was solution times more than twice as long as usual. This suggested that some type of cognitive ferment was occurring, though it eventually was resolved through correct solutions generated by typical strategies. Another striking finding of the study was the fact that none of the children ever attempted strategies inconsistent with the principles underlying addition. The situation was reminiscent from the classic conversation in The Memoirs of Sherlock Holmes. Holmes: But there was the curious incident of the dog in the nightime. Watson: The dog did nothing in the nightime. Holmes: That was the curious incident. In Siegler and Jenkins (1989), the curious incident was that not one child adopted a strategy that violated the principles underlying addition. Several potential illegitimate strategies were procedurally similar to legitimate approaches and therefore might have been expected to be tried. For example, children might have tried counting the first addend twice, or counting on from the second addend the number of times indicated by the larger addend, regardless of whether it was first or second. In fact, not one child used these or other illegitimate strategies on even one trial. This raised the question: “Why not?” One possibility is that the strategy generation process was constrained by what Siegler and Jenkins termed a goal sketch. Such a goal sketch specifies the hierarchy of objectives that a satisfactory strategy must meet. The hierarchical structure directs searches of existing knowledge toward procedures that can meet the goals. In so doing, it directs searches away from illegitimate procedures. When legitimate procedures for meeting each goal have been identified, the goal sketch provides a schematic outline of how the components can be organized into a new strategy. In the case of addition, such a goal sketch would include the information that each set being added must be represented, that a quantitative representation of the combined sets must be generated, and that a number corresponding to this quantitative representation must be advanced as the answer. Such goal sketches seem likely to be both widely useful and widely used. For example, in Gelman and Gallistel’s (1978) order-invariance experiment, they may have provided the framework within which 5year-olds constructed unusual counting procedures that met specific experimental demands without violating counting principles. Such goal sketches would also help children from applying in a new and inappropriate domain a strategy that was entirely appropriate in its original domain. In this way, they would function much like the goal information within the condition side of the productions in a production system. Even flawless and well-elaborated goal sketches do not guarantee that successful strategies will be generated. They cannot even guarantee that flawed strategies will not be generated. Flawed strategies can emerge despite such goal sketches if the learner must generate an answer and knows no procedure for doing so within the constraints of the goal sketch. This appears to be what happened in the “buggy” subtraction algorithms described by Brown and VanLehn (1980). In these studies, children often knew how to solve long subtraction problems unless the problems demanded borrowing across a zero. In this case, they produced buggy algorithms. Protocol analyses of children’s reactions
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to such problems have indicated that the children often explicitly stated that they did not know how to solve the problem. Only the experimenter’s insistence that they try to solve the problems anyway led to the flawed approaches (VanLehn, in preparation). This view suggests that children can sometimes generate sketches of the goals that successful strategies in a domain must meet, even when they do not know any strategies that meet all of the goals. At a general level of analysis, they probably do this through considering the causal relations among entities within the domain and the way in which existing strategies within the domain map onto this causal structure. Formulating more detailed descriptions of ‘how such goal sketches are generated, how they constrain searches for components of new strategies, and how they contribute to assembling new strategies from the component parts promise to advance considerably our understanding of an important part of learning.
Implications for Theories of Cognitive Development I would like to conclude this article by considering the relation between my own current approach to cognitive development and that of Piagetian and neo-Piagetian approaches. The research program described in this article has pursued many of the same goals as these more traditional approaches. For example, one of the central goals of each approach has been to reveal unities across domains in children’s thinking. Piagetian and neo-Piagetian approaches have found such unities primarily in structural similarities in reasoning across tasks. The present approach has identified a number of other types of cross-task unities: unities in children’s use of multiple strategies on varied tasks; unities in the adaptive quality of their strategy choices from quite young ages through adulthood; unities in individual difference patterns across domains; and most important, unities in the processes that produce strategy choices and strategy discoveries in different domains. The sources of these unities, however, is quite different than those typically posited by Piagetian and neo-Piagetian theories. The sources that are emphasized in the present theory are commonalities in the strategy choice process, rather than structurally-imposed constraints on the type of reasoning that is possible, the capacity of working memory, or the type of representation that can be formed. That is, within the present approach, parallels across diverse domains are attributed to basic properties of the cognitive system: the processes that give rise to strategy choices and strategy discoveries. With development, new strategies are constructed, choices among existing strategies shift toward the more advanced strategies, and efficiency of execution of all strategies improves. However, the processes that give rise to the strategy choices and discoveries remain the same. A substantial gulf at first appears to separate the present theory from Piagetian and neo-Piagetian alternatives. The approaches are in all likelihood more compatible than they initially appear, though. Both structural and processing constraints contribute to the types of changes that are evident in children’s thinking. Focusing on either alone gives rise to an imbalanced depiction of cognitive development. Consistent with this view, several prominent neo-Piagetian theorists have attempted recently to incorporate both types of unities within a single theory. For example, in Halford’s (1990) most
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recent statement of his theory, choices among alternative strategies are identified as one of three critical factors determining children’s reasoning. Similarly, the most recent statements of Case’s theory (e.g., Case & Griffin, 1990) emphasize central conceptual structures, which play the same role within that theory as goal sketches play in the present one. Formulating a single theory that clearly articulates how structural and processing features of the cognitive system work together to produce children’s thinking at any one time and changes in that thinking over time remains perhaps the largest goal of the field of cognitive development. Acknowledgements-The research was supported in part by National Institutes of Health Grant HD-19011, in part by a grant from the Spencer Foundation, and in part by a grant from the McDonnell Foundation. Thanks are due to Dr. Ann Taylor and the teachers and children at the Carnegie Mellon Children’s School. Correspondence may be sent to Robert S. Siegler, Psychology Department, Carnegie Mellon University, Pittsburgh, PA 15213.
References Ashcraft, M. H. (1982). The development
of mental arithmetic: A chronometric
approach. Developmennrl
Review, 2, 213-236.
Ashcraft, M. H. (1987). Children’s knowledge of simple arithmetic: A developmental model and simulation. In C. J. Brainerd, R. Kail, & J. Bisanz (Eds.), Formal methods in developmentalpsychology (pp. 302-338). New York: Springer-Verlag. Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4, 379-426.
Case, R., & Griffin, S. (1990). Child cognitive development: The role of central conceptual structures in the development of scientific and social thought. In C. A. Hauert (Ed.), Developmental psychology: Cognitive, perceptuo-motor and neuropsychological perspectives. Amsterdam: North Holland. Cooney, J. B., Swanson, H. L., & Ladd, S. F. (1988). Acquisition of mental multiplication skill: Evidence for the transition between counting and retrieval strategies. Cognition and Instruction, 5, 323-345. Geary, D. C., & Burlingham-Dubree, M. (1989). External validation of the strategy choice model for addition. Journal of Experimental Child Psychology, 47, 175-192. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Goldman, S. R., Pellegrino, J. W., & Mertz, D. L. (1988). Extended practice on basic addition facts: Strategy changes in learning disabled students. Cognition and Instruction, 5, 223-265. Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79, 329-343.
Groen, G. J., & Resnick, L. B. (1977). Can preschool children invent addition algorithms? Journal of Educational Psychology, 69, 645-652.
Hagen, J. W., Hargrove,
S., & Ross, W. (1973). Prompting and rehearsal in short-term
memory. Child
Development,
44,201-204. G. S. (1990). Children’s
Halford, understanding: The development of mental models. Hillsdale, NJ: Erlbaum. Jorm, A. F., & Share, D. L. (1983). Phonological recoding and reading acquisition. Applied Psycholinguistics, 4, 103-147.
Kahan, L. D., & Richards, D. D. (1986). The effects of context on children’s referential communication strategies. Child Development, 57, 1130-1141. Kaye, D. B., Post, T. A., HalI, V. C., & Dineen, J. T. (1986). The emergence of information retrieval strategies in numerical cognition: A development study. Cognition and Zn.st&tion, 3, 137-166. Kerkman, D., & Siegler, R. S. (submitted for publication). Individual differences and adaptive flexibility in lower-income children’s strategy choices.
Kuhara-Kojima, K., & Hatano, G. (1989). Strategies for recognizing sentences among high and low critical thinkers. Japanese Psychological Research, 31, l-19. McGilly, K., & Siegler, R. S. (1989). How children choose among serial recall strategies. Child Development, 60, 172-182.
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McGilly, K., & Siegler, R. S. (1990). The influence of encoding and strategic knowledge on children’s choices among serial recall strategies. Developmental Psychology, 26, 931-941. Miller, K., Perhnutter, M., & Keating, D. (1984). Cognitive arithmetic: Comparison of operations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 10,46-t%. Ohlsson, S. (1984). Induced strategy shifts in spatial reasoning. Acta Psychologica, 57,4747. Reder, L. (1987). Strategy selection in question answering. Cognitive Psychology, 19,90-138. Schneider, W., & Pressley, M. (1989). Memory development between 2 and 20. New York: SpringerVerlag. Shultx, T. R., Fisher, G. W., Pratt, C. C., & Rulf, S. (1986). Selection of causal rules. Child Development, 57, 143-152. Siegler, R. S. (1986). Unities in strategy choices across domains. In M. Perlmutter (Ed.), Minnesota symposium on child development, (Vol. 19, pp. l-48). Hillsdale, NJ: Erlbaum. Siegler, R. S. (1987a). Strategy choices in subtraction. In J. Sloboda and D. Rogers (Eds.), Cognitive process in mathematics (pp. 81-106). Oxford: Oxford University Press. Siegler, R. S. (198%). The perils of averaging data over strategies: An example from children’s addition. Journal of Experimental Psychology: General, 116, 250-264. Siegler, R. S. (1988a). Strategy choice procedures and the development of multiplication skill. Journal of Experimental Psychology: General, 117, 258-275. Siegler, R. S. (1988b). Individual differences in strategy choices: Good students, not-so-good students, and perfectionists. Child Development, 59, 833-851. Siegler, R. S. (1989). Hazards of mental chronometry: An example from children’s subtraction. Journal of Educational Psychology, 81, 497-506. Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ: Erlbaum. Siegler, R. S., & McGilly, K. (1989). Strategy choices in children’s time-telling. In I. Levin & D. Zakay (Eds.), Psychological time: A life span perspective. Elsevier: The Netherlands. Siegler, R. S., & Robinson, M. (1982). The development of numerical understandings. In H. Reese & L. P. Lipsitt (Eds.), Advances in child development and behavior, (Vol. 16, pp. 241-312). New York: Academic Press. Siegler, R. S., & Shrager, J. (1984). A model of strategy choice. In C. Sophian (Ed.), Origins of cognitive skills (pp. 229-293). Hillsdale, NJ: Erlbaum. Svenson; 0. (1975). Analysis of time required by children for simple additions. Acta Psychologica, 39, 289-302. Widaman, K. F., Geary, D. C., Cormier, P., & Little, T. D. (1989). A componential model for mental addition. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15, 898-919. Woods, S. S., Resnick, L. B., & Groen, G. J. (1975). Experimental test of five process models for subtraction. Journal of Educational Psychology, 67, 17-21.
1991
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0959-4752/91$0.00 + so 1991 Pergamcm Press plc
STRATEGY CHOICE AND STRATEGY DISCOVERY ROBERT
S. SIEGLER
Carnegie Mellon University,
Pittsburgh, U.S.A.
Abstract This article summarizes a number of recent findings and ideas concerning how children choose among existing strategies and how they discover new ones. Across a wide range of domains, individual children use diverse strategies. In all of these domains, they choose adaptively among the alternative approaches. Consistent individual differences are present within the strategy choices of both middle and lower class populations, and these predict performance on standardized achievement tests. The ideas about strategy choices have been formalized within running computer simulations that produce performance much like children’s both in terms of performance at any one time and in terms of changes over time. The article concludes with a description of recent work on how children discover new strategies and with a discussion of the relation of the present perspective on cognitive development to that embodied in traditional Piagetian and neo-Piagetian theories.
Strategy Choice and Strategy Discovery A huge body of research documents the strategies that children use to memorize, conceptualize, reason, and solve problems under various conditions. Much less is known, however, about an issue of equal importance: the. processes through which they come to use one strategy rather than another. I believe that understanding how children choose among strategies is essential to understanding both how they think at any one time and how their thinking changes over time. In short, it is basic to understanding cognitive development. For strategy choice to be meaningful, there must be multiple strategies to choose among. Although many studies have depicted development in terms of a 1 : 1 correspondence between children’s age and the strategy that they use, recent trialby-trial analyses have found that children of a single age often use a wide variety of strategies. The finding has emerged in domains as diverse as arithmetic (Cooney, Swanson, & Ladd, 1988; Gear-y & Burlingham-Dubree, 1989; Goldman, Pellegrino, & Mertz, 1988; Siegler & Robinson, 1982), causal reasoning (Shultz, Fisher, Pratt, & Rulf, 1986), spatial reasoning (Ohlsson, 1984), referential communications (Kahan & Richards, 1986), serial recall (McGilly & Siegler, 1989), reading and spelling (Jorm & Address for correspondence: R. S. Siegler, Department Pittsburgh, PA 15213-3890, U.S.A. 89
of Psychology,
Carnegie Mellon University,
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Share, 1983; Siegler, 1986), and judgments of plausibility (Kuhara-Kojima & Hatano, 1989). The variability in strategy use is evident within as well as between individuals; even the same child presented the same problem on two successive days often uses different strategies on the two occasions (Siegler, 1987a; Siegler & McGilly, 1989). The fact that children use diverse strategies is not a mere idiosyncracy of human cognition. Good reasons exist for people to know and use multiple strategies. Strategies differ in their accuracy, in the amounts of time needed to execute them, in their processing demands, and in the range of problems to which they apply. Strategy choices involve tradeoffs among these properties. The broader the range of strategies that children know, the more precisely they can shape their approaches to the demands of particular circumstances. Even young children often capitalize on the strengths of different strategies and use each most often on problems where its advantages are greatest. Studies of children’s strategy choices have revealed some striking unities across domains, both in empirical data and in the cognitive mechanisms that seem to underlie the data. For example, on many tasks, children use backup strategies. These are strategies other than retrieval that can be used to solve problems that can also potentially be solved by retrieval. Thus, counting from one or from the larger addend are backup strategies for simple addition; sounding out words or looking them up in a dictionary are backup strategies for spelling; writing down a phone number or rehearsing it are backup strategies for serial recall; and so on. A variety of studies have demonstrated that within each domain, children most often use backup strategies on the most difficult problems, with problem difficulty measured either in terms of percentage of errors or length of solution times. Correlations among percentage of use of backup strategies on each problem, percentage of errors on each problem, and mean solution time on each problem have ranged from I = 0.83 to r = 0.92 in 4- and 5-year-olds’ addition, 6- and 7-year-olds’ subtraction and word identification, 7- and 8-year-olds’ spelling and time telling, and 8- and 9-year-olds’ multiplication (Siegler, 1986, 1987a, 1988a, b; Siegler & Robinson, 1982; Siegler & Shrager, 1984). Such choices are adaptive because they mean that children use the more time consuming backup strategies primarily on problems where such strategies are needed for accurate performance. The adaptive strategy choices are evident among lower income minority children and students who are not doing well in school as well as among middle income students and those who are doing well (Kerkman & Siegler, submitted for publication). Once it is recognized that children know multiple strategies and choose effectively among them, the question arises: How do they construct such strategies in the first place? Are new strategies constructed primarily after other strategies fail? When children first use strategies, do they understand why they are effective? Do they generalize them broadly from the beginning, or do they at first use them only in limited circumstances? Microgenetic studies, in which individual children who do not yet know a strategy are given prolonged experience in the domain, with their performance then being subjected to trial-by-trial scrutiny, seem especially useful for studying strategy construction. They allow identification of the trial on which a new strategy is first used, which in turn allows examination of what the experience of discovery was like, what led up to the discovery, and how the discovery was generalized beyond its initial use. Studies of this type have yielded a variety of nonintuitive findings. Among them are results indicating
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that discoveries are often made not in response to impasses or failure but rather in the context of successful performance; that short-lived transition strategies often precede more enduring approaches; and that generalization of new approaches often occurs very slowly, even when children can provide compelling rationales for their usefulness (Siegler & Jenkins, 1989). Below, I summarize in greater detail my recent research on strategy choice and strategy discovery.
The Ubiquity of Multiple Strategy Use The first context in which I became aware that children were using multiple strategies was 4- and 5-year-olds’ addition of small numbers (Siegler & Robinson, 1982). Videotapes of the children’s performance made it clear that they sometimes put up their fingers and counted them, sometimes put up their fingers and recognized how many fingers were up without counting, sometimes counted without any obvious external referent, and sometimes retrieved the answer from memory. Since then, my colleagues and I have documented children’s use of diverse strategies in many domains. To multiply, 8 to lo-year-olds sometimes repeatedly add one of the multiplicands, sometimes write the problem and then recognize the answer, sometimes write and then count groups of hatch marks that represent the problem, and sometimes retrieve the answer from memory (Siegler, 1988a). To tell time, 7- to 9-year-olds sometimes count forward from the hour by ones and/or fives, sometimes count backward from the hour by ones and/or fives, sometimes count from reference points such as the half hour, and sometimes retrieve the time that corresponds to the clock hands’ configuration (Siegler & McGilly, 1989). To spell, 7- and 8-year-olds sometimes sound out words, sometimes look them up in dictionaries, sometimes write out alternative forms and try to recognize which is correct, and sometimes recall the spelling from memory (Siegler, 1986). To serially recall lists, 5- to 8-year-olds sometimes repeatedly recite the lists during the delay period, sometimes recite the list once and stop, and sometimes just wait (McGilly & Siegler, 1989, 1990). These diverse strategies are not artifacts of one child using one strategy and a different child another one. Most individuals of all ages in all of these domains have been found to use at least 2 strategies; in most of the domains, the norm has been to use 3 or more. This multiple strategy use is apparent even within classes of similar problems and, in the limiting case, on the same problem presented to the same child twice on consecutive days (Siegler, 1987a; Siegler & McGilly, 1989). In both subtraction and time telling, fully one-third of children used different strategies on the identical problem on two successive days. The change was not in general from less advanced to more advanced strategies. For example, almost as many children retrieved the answer on the first day and used a backup strategy on the second as did the reverse. These experiments have called into serious question previous models in these areas that postulated that children of a given age consistently use a particular strategy. To illustrate, a number of models of first and second graders’ addition (Ashcraft, 1982, 1987; Groen & Parkman, 1972; Svenson, 1975) postulate that 6- and 7-year-olds consistently add by using the min strategy of counting-on from the larger addend (e.g.,
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they would solve 3 + 5 by starting with 5 and then thinking “6 is 1, 7 is 2, 8 is 3”). However, Siegler (1987b) found that children of these ages used the min strategy on only a minority of trials; on other trials, they counted from 1, retrieved answers from memory, decomposed problems into simpler form, or guessed. Similarly, the McGilly and Siegler (1989) finding that the majority of 5-year-olds use repeated rehearsal on some trials, as well as using single rehearsal and no rehearsal on others, contradicted claims that preschoolers generally do not rehearse (Hagen, Hargrave, & Ross, 1973; Schneider & Pressley, 1989).
Validity of the Strategy Assessments The validity of all of the above findings depends critically on the validity of the method by which strategy use on each trial was assessed. In all of my studies published prior to 1987, the assessments of strategy use were based entirely on videotapes of ongoing overt behavior. Since then, I have relied on immediately-retrospective verbal reports as well as observations of overt behavior. Specifically, strategy assessments in the more recent studies have been based on overt behavior when that behavior unambiguously indicated the strategy, and on immediately retrospective verbal reports when ongoing overt behavior was absent or ambiguous. Results of the recent studies have yielded evidence for the validity of the strategy assessments and have also indicated that the assessment method allows accurate detection of many instances of covert strategy use without creating more use of the strategies than would occur if the immediately-retrospective reports were not obtained. Siegler (1987b) provided evidence for the validity of the strategy assessments. As mentioned above, a number of investigators had hypothesized that 6- and 7-year-olds add by using the min strategy. A considerable body of evidence was consistent with this hypothesis. The size of the smaller addend is consistently the best predictor of first and second graders’ solution times on different problems (Ashcraft, 1982, 1987; Groen & Parkman, 1972; Kaye, Post, Hall, & Dineen, 1986; Svenson, 1975). It is a good predictor in absolute as well as relative terms, accounting for 60% to 75% of the variance in solution times in a number of studies. The min model, based on_ the assumption that young children invariably use the min strategy, fits the performance of individual children as well as group averages, children in Europe as well as in North America, and children in special education classes as well as standard ones (Groen & Resnick, 1977; Kaye et al., 1986; Svenson, 1975). Despite all this support, the model is wrong. Siegler (1987b) videotaped 5- to 7-year-olds’ ongoing behavior as they solved simple addition problems and asked them immediately after the problem how they solved it. When solution time data were averaged over trials (and over strategies), the results closely replicated the previous finding that solution times were a linear function of the smaller addend. Smaller addend size accounted for 76% of variance in solution times on the 45 problems that were examined. The trial-by-trial assessments of strategy use, however, indicated that children used 5 different strategies and that the min strategy was used on only 36% of trials. Dividing the solution time data according to what strategy children were classified as using
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lent considerable credence to the strategy assessments. On trials where children were classified as using the min strategy, the min model was an even better predictor of solution times than in the data set as a whole. It accounted for 86% of the variance in solution times. In contrast, on trials where children reported using one of the other strategies, the min model was never a good predic&or of performance, either in absolute or relative terms. It never accounted for as much as 40% of variance, and was never either the best or the second best predictor. Thus, the trial-by-trial strategy assessments appeared to yield a valid picture of addition performance, and one quite at odds with that derived from analyses of data averaged over trials. (See Siegler, 1987b, for a statistical analysis of why the min model fit the averaged data despite the min strategy not being used on most trials. Also see Siegler, 1989, for a similar analysis and conclusion about the Woods, Resnick, and Groen [1975] model of 2nd through 4th graders’ subtraction.) Not surprisingly, obtaining verbal reports as well as videotaped records of ongoing overt behavior results in higher estimates of strategy use than does reliance on overt behavior alone. This could be due either to the requests for verbal reports artificially inflating the amount of strategy use, or to the verbal reports leading to detection of strategies that children did use but without any accompanying overt behavior. To distinguish between these interpretations, McGilly and Siegler (1990) randomly assigned 5, 7-, and 9-year-olds in a serial recall experiment to self-report and no-selfreport conditions. Those in the self-report condition were asked immediately after each trial what, if anything, they had done in the delay period between when the list was presented and when they needed to recall it. Those in the no-self-report condition received an identical procedure, except that they were not asked any questions about their strategies. When verbal reports as well as overt behavior were considered in assessing strategies within the self-report condition, children in that condition were classified as rehearsing on a much higher percentage of trials, 74% vs. 47%. However, when the strategy use of children in both groups was scored solely on the basis of overt behavior, children in the self-report condition were classified as rehearsing slightly less often, 39% vs 47%. There was no obvious reason to think that overt strategic behavior was unaffected by the questions but covert strategic behavior was affected. Thus, the similar amounts of overt behavior obtained from parallel strategy assessment methods indicated that the higher estimates of rehearsal in the self-report condition were not due to the requests for self reports creating greater use of strategies. Instead, the higher estimates apparently stemmed from the verbal reports allowing detection of covert rehearsal activities that would otherwise have gone undetected.
Three Ways in Which Strategy Choices Are Adaptive These studies have revealed three distinct senses in which children’s strategy choices are adaptive. One concerns the choice of whether to use retrieval or a backup strategy (an approach other than retrieval). As noted earlier, the more difficult the problem, the more often children use backup strategies to solve it. This has been found in the above-cited studies of addition, subtraction, multiplication, spelling, word identification, and time telling. Such choices are useful, because they enable children to use the
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fast retrieval strategy on problems where that approach yields correct answers and to use slower backup strategies on problems where they are necessary to produce accurate performance. Consistent with this analysis, preventing children from using backup strategies by imposing a very short time limit (4 set) produces sharp falloffs in accuracy, with the dropoffs being largest on precisely the problems on which children are most likely to use backup strategies when they are allowed to choose freely (Siegler & Robinson, 1982). Children also choose adaptively among alternative backup strategies. For example, in choosing whether to add by counting from one or by counting from the larger addend (the min strategy), children are especially likely to use the min strategy on problems such as 9 + 2, that have both a large difference between the addends and a relatively small minimum addend. Such problems also are the ones where the efficiency of the min strategy is greatest relative to counting from one (Siegler, 1987b). Studies of serial recall have led to appreciation of a third sense in which children’s strategy choices are adaptive: trial-to-trial changes in strategy use. McGilly and Siegler (1989) found that children changed strategies most often when they had recalled the list incorrectly on the previous trial and had not rehearsed. When they succeeded despite not rehearsing or failed after rehearsing, they were significantly less likely to switch strategies. Thus, children switched to the more effortful approach of rehearsing when the less effortful approach of just waiting failed to yield accurate recall, but not when it allowed successful performance. To summarize, the generalizations that individual children often use diverse strategies and that they choose among them in adaptive ways have proved to hold true for many types of domains beyond the areas of simple addition and subtraction in which they were first formulated. The work on time telling indicates that the generalizations apply to numerical tasks other than arithmetic. The work on reading and spelling indicates that they hold true for tasks that do not involve numbers. The work on serial recall indicates that they hold true for memory as well as problem solving strategies, and for tasks on which children lack extensive problem-specific experience as well as ones in which they have such experience. Thus, a view of development that depicts children as actively choosing among multiple competing strategies may turn out to be more generally accurate than the traditional view in which a child of a given age is depicted as invariably using a particular approach.
Individual Differences
and Cross-Domain
Consistencies
A major goal of my research has been to reveal cross-domain commonalities in strategy choices. One way in which this goal has been pursued has been described already: through establishing that use of multiple strategies and adaptive patterns of strategy choices are present in diverse domains. Another approach to the same goal has been to attempt to identify stable individual. difference patterns across domains. To examine individual differences in strategy choices, Siegler (1988b) had each of a group of first graders perform three tasks: addition, subtraction, and reading (word identification). A cluster analysis of the children’s percent correct, mean RT, and frequency of use of backup strategies on each of the three tasks indicated considerable
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consistency in performance. The clustering program divided children into 3 groups, which we labeled the “good students”, the “not-so-good students”, and the “perfectionists”. The contrast between good and not-so-good students was evident along all of the dimensions that might be expected from the names. Good students more often used retrieval in both addition and subtraction. They were correct more often on both retrieval and nonretrieval trials on all 3 tasks. They also were faster in executing backup strategies on all 3 tasks and faster in using retrieval on both addition and subtraction problems. The relation of the performance of the perfectionists to that of children in the other 2 groups was more complex. Despite their being as fast and as accurate as the good students on both retrieval and backup strategy trials, the perfectionists used retrieval less often even than the not-so-good students. Convergent validation for these findings was obtained in a second experiment with a somewhat different strategy assessment method, different problems, and different children (Siegler, 1988b; Experiment 2). Four months after the experiment, all children were given the Metropolitan Achievement Test. Differences between perfectionists and good students on the one hand and not-so-good students on the other echoed those in the experimental setting. The perfectionists’ average mathematics scores were at the 81st percentile, the good students’ at the 80th percentile, and the not-so-good students’ at the 43rd percentile. Thus, differences between not-so-good students and the other 2 groups would have been evident on standardized achievement tests. However, differences between good students and perfectionists would not have been. Instead, these seem to be alternative ways of doing well in early mathematics and reading. Kerkman and Siegler (submitted for publication) extended the individual differences research to lower-income African-American children. The sample came from quite poor back~ounds; 80% qualified for the federal free school lunch program. The children were examined twice, once in first grade and once in second, to examine stability of the individual different classifications over time. The same three groups emerged in the lower income sample as in the previouslystudied middle income ones. Proportions of children in the three groups also were quite similar. Stability over time was evident in that most children were classified as being in the same group on the two testings. Probably the most striking finding, though, was that the lower income children’s strategy choices were just as adaptive as those of middle income children. Percent backup strategy use and percent errors on each problem correlated r = 0.91 for addition, r = 0.92 for subtraction, and r = 0.86 for word identification. The findings demonstrate that in domains in which children have substanti~ experience with individual problems, lower-income as well as middle-income children produce reasonable and systematic strategy choices.
Computer
Simulations of Strategy Choice
To test whether their ideas about strategy choice mechanisms were sufficient to account for the patterns of speed, accuracy, and strategy use observed in preschoolers’ addition, Siegler and Shrager (1984) formulated a computer simulation of how children choose among competing strategies, how the choices contribute to their knowledge of
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addition, and how previous choices shape future patterns of answers and strategy choices. The same framework was later used to simulate the development of simple subtraction (Siegler, 1987a) and multiplication (Siegler, 1988a). Results of these simulations indicated a close fit along a number of dimensions between children’s behavior and the output of the simulations. For example, in multiplication, the correlation between the children’s and the simulation’s percent correct, percent backup strategy use, and mean solution time on each problem were r = 0.90, T = 0.90, and r = 0.95 for the 3 measures respectively. Thus, the same problems were the most difficult for the simulation as for the children, and the simulation, like the children, chose to use backup strategies most often on those problems. Also like children, the simulation showed substantial learning from its experience with problems. Percent correct increased from 61% during the first 10% of problems to 89% on the last 10%. Over the same period, percent use of retrieval increased from 24% to 77% of trials. The simulation also allowed an explicit demonstration of a nonintuitive but accurate prediction of the underlying model: that the relative difficulty of problems for 8- and 9-year-olds, just learning to multiply, would more accurately predict the problems’ difficulty for adults than would any structural variable, such as the product or the sum of the multiplicands. Despite these successes, several limitations produced by the simulations’ basic structure also became evident. One problem was that the model was well worked out only for the choice between retrieving and using a backup strategy. If the choice process is general, the same basic procedure should govern choices among alternative backup strategies. A second problem was that the simulation was inflexible, in the sense that it always tried to retrieve first. Reder’s (1987) evidence from question answering indicated that if retrieval was unsuccessful on previous trials, adults were more likely to try alternative strategies first. I suspected that the same would be true of children’s arithmetic, since the retrieval mechanism is likely to be the same for question answering and arithmetic. A third problem was the models’ inability to generalize. The simulations approached each problem as a separate entity. Again, it seemed unreasonable that knowing that the min strategy worked well on 9 + 1 and 9 + 3 would not lead children to suspect that it also worked well on 9 + 2. The new strategy choice model was developed to overcome these limits while preserving the strengths of the original model. The basic approach was to generalize the principles on which the model was based along two dimensions: the types of choices that were made and the types of data that were considered in making the choices. With regard to the types of choices that were made, the Siegler and Shrager model’s procedure for choosing among answers was generalized to choosing among strategies as well as answers. With regard to the types of data that were considered, the database was extended to include not only the answers that the model had generated but also the speeds and accuracies that each strategy produced. The new model’s organization centers on the process of a strategy being used to solve a problem. The process generates information about the speed, accuracy, and particular answers that accompany the problem and the strategy. From this information, the model builds data bases that include information about the speeds and accuracies that each strategy produces on the total set of problems, on problems with particular features (e.g., a smaller addend of l), and on particular problems (e.g., 7 + 1). These three types
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of information are used to compute the activation of each strategy for a given problem, which in turn determines the probability of using that strategy on the problem. Another innovation is the model’s treatment of how new strategies ever come to be used on tasks where existing strategies work relatively well. The model accomplishes this through assigning novelty points to new strategies. These novelty points act in the same way as the activations of other strategies, with one exception. With each use of the new strategy, some of its novelty points are lost. If the new strategy leads to quick and accurate responding, the activation it gains through generating effective performance more than compensates for the decrease in novelty points. If it generates inaccurate and/or slow responding, novelty points are gradually used up without compensating gains in activation. Thus, useful new strategies become stronger, and ineffective ones weaker. As in the earlier strategy choice models, answers, both correct and incorrect, become associated with problems through being stated on those problems. Also as in the earlier models, a retrieved answer is stated only when its strength exceeds the confidence criterion on the trial. These features allow the model to generate the effects it did previously, while becoming more flexible and more general. Siegler and Shipley (in preparation) tested this model’s ability to account for learning and performance on the 81 basic addition facts with addends l-9. It proved to generate a wide range of basic characteristics of 5- to S-year-olds’ addition. Its accuracy progressed from 55% correct early in its run to 99% at the end. Its percent use of retrieval progressed from 16% at the early point to 98% at the end. Its percent correct, mean solution time, and percent use of retrieval on each problem all correlated at least r = 0.80 with children’s. The best predictors of its performance also mimicked those of children: at first the size of the sum was the best predictor of errors and solution times, then the size of the smaller addend, and finally the product of the two addends. The simulation also revealed why the product eventually became the best predictor of the simulation’s performance, as it had been found to be with adults (Miller, Keating, & Perlmutter, 1984; Widaman, Geary, Little, & Cormier, 1989) - an averaging of the effects of previous use of counting from 1 and using the min strategy. Further, varying two parameter values - accuracy of execution of backup strategies and stringency of confidence criteria - and leaving the rest of the simulation constant, led to patterns of performance like those of the good students, not-so-good students, and perfectionists in the individual difference studies. The simulation also generalized from its experience to choose strategies adaptively on problems it had not encountered before. In sum, the new simulation maintained the strong points of the earlier ones and produced a number of desired new outcomes as well.
Discovery of New Strategies Probably the most important gap in the strategy choice model is a lack of an account of how new strategies are discovered. To provide a data base relevant to how this process occurs, Siegler and Jenkins (1989) studied 4- and 5-year-olds’ construction of the min strategy for adding numbers. First, a pretest was given to identify children who could add by counting from 1 but who did not yet know the min strategy. Eight children
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who met these criteria then participated in an 11 week experiment in which they were presented addition problems approximately 3 times per week. The first 7 weeks of this period were spent on problems with addends l-5. A number of children discovered the min strategy during this period, but none generalized it very widely. Therefore, in week 8, we presented them challenge problems, such as 22 + 3, on which counting from 1 and retrieval would work badly but on which the min strategy would work well. In weeks 9-11, children were presented a mixed set of problems; some were small number problems, some were challenge problems, and some were in-between (e.g., 8 + 3). Throughout the experiment, children’s strategies were classified on a trial-by-trial basis. This allowed identification of the first trial on which they used the new strategy, what led up to the discovery, and how the new strategy was generalized beyond its initial use. During the 11 weeks, 7 of the 8 children discovered the min strategy. Many of the discoveries were accompanied by an impressive amount of insight into the new strategy’s advantages. For example, “Ruth” first used the new strategy on 4 + 3. When asked “Can you tell me why you started (counting) from 4”, she said “I don’t have to count a very long ways if I start from 4”. Despite this understanding, Ruth, like the other children, used the min strategy only occasionally in the sessions between the discovery and presentation of the challenge problems. Encountering the challenge problems, however, led her and the other children who had made the discovery to greatly increase their use of the new strategy. This pattern held true not only for the data set as a whole (as shown in Fig. 1) but when the analysis was limited to strategy use on the types of problems presented at the beginning of the study, that is problems with addends of 5 or less. On these problems, use of the min strategy increased from 11% to 45% of trials. The microgenetic experiment also yielded several other findings that ran counter to typical depictions of discovery. Most discoveries were not responses to children encountering impasses; the discoveries were made on ordinary problems, usually ones
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Trial Black (O=Challenge Problems) Figure 1. Percent use of min strategy relative to all counting strategies before, during, and after challenge problems. Each trial block represents five experimental sessions, except for the challenge problems block, which represents two sessions. Thus, the -3 trial block represents min strategy use 11-15 sessions before the challenge problems among children who had discovered the strategy by that time.
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children had solved previously using other strategies. Further, children’s answers were generally correct both on the problem just before the .discovery and on most prior problems in that session. The only unusual characteristic of performance on problems just before the discovery was solution times more than twice as long as usual. This suggested that some type of cognitive ferment was occurring, though it eventually was resolved through correct solutions generated by typical strategies. Another striking finding of the study was the fact that none of the children ever attempted strategies inconsistent with the principles underlying addition. The situation was reminiscent from the classic conversation in The Memoirs of Sherlock Holmes. Holmes: But there was the curious incident of the dog in the nightime. Watson: The dog did nothing in the nightime. Holmes: That was the curious incident. In Siegler and Jenkins (1989), the curious incident was that not one child adopted a strategy that violated the principles underlying addition. Several potential illegitimate strategies were procedurally similar to legitimate approaches and therefore might have been expected to be tried. For example, children might have tried counting the first addend twice, or counting on from the second addend the number of times indicated by the larger addend, regardless of whether it was first or second. In fact, not one child used these or other illegitimate strategies on even one trial. This raised the question: “Why not?” One possibility is that the strategy generation process was constrained by what Siegler and Jenkins termed a goal sketch. Such a goal sketch specifies the hierarchy of objectives that a satisfactory strategy must meet. The hierarchical structure directs searches of existing knowledge toward procedures that can meet the goals. In so doing, it directs searches away from illegitimate procedures. When legitimate procedures for meeting each goal have been identified, the goal sketch provides a schematic outline of how the components can be organized into a new strategy. In the case of addition, such a goal sketch would include the information that each set being added must be represented, that a quantitative representation of the combined sets must be generated, and that a number corresponding to this quantitative representation must be advanced as the answer. Such goal sketches seem likely to be both widely useful and widely used. For example, in Gelman and Gallistel’s (1978) order-invariance experiment, they may have provided the framework within which 5year-olds constructed unusual counting procedures that met specific experimental demands without violating counting principles. Such goal sketches would also help children from applying in a new and inappropriate domain a strategy that was entirely appropriate in its original domain. In this way, they would function much like the goal information within the condition side of the productions in a production system. Even flawless and well-elaborated goal sketches do not guarantee that successful strategies will be generated. They cannot even guarantee that flawed strategies will not be generated. Flawed strategies can emerge despite such goal sketches if the learner must generate an answer and knows no procedure for doing so within the constraints of the goal sketch. This appears to be what happened in the “buggy” subtraction algorithms described by Brown and VanLehn (1980). In these studies, children often knew how to solve long subtraction problems unless the problems demanded borrowing across a zero. In this case, they produced buggy algorithms. Protocol analyses of children’s reactions
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to such problems have indicated that the children often explicitly stated that they did not know how to solve the problem. Only the experimenter’s insistence that they try to solve the problems anyway led to the flawed approaches (VanLehn, in preparation). This view suggests that children can sometimes generate sketches of the goals that successful strategies in a domain must meet, even when they do not know any strategies that meet all of the goals. At a general level of analysis, they probably do this through considering the causal relations among entities within the domain and the way in which existing strategies within the domain map onto this causal structure. Formulating more detailed descriptions of ‘how such goal sketches are generated, how they constrain searches for components of new strategies, and how they contribute to assembling new strategies from the component parts promise to advance considerably our understanding of an important part of learning.
Implications for Theories of Cognitive Development I would like to conclude this article by considering the relation between my own current approach to cognitive development and that of Piagetian and neo-Piagetian approaches. The research program described in this article has pursued many of the same goals as these more traditional approaches. For example, one of the central goals of each approach has been to reveal unities across domains in children’s thinking. Piagetian and neo-Piagetian approaches have found such unities primarily in structural similarities in reasoning across tasks. The present approach has identified a number of other types of cross-task unities: unities in children’s use of multiple strategies on varied tasks; unities in the adaptive quality of their strategy choices from quite young ages through adulthood; unities in individual difference patterns across domains; and most important, unities in the processes that produce strategy choices and strategy discoveries in different domains. The sources of these unities, however, is quite different than those typically posited by Piagetian and neo-Piagetian theories. The sources that are emphasized in the present theory are commonalities in the strategy choice process, rather than structurally-imposed constraints on the type of reasoning that is possible, the capacity of working memory, or the type of representation that can be formed. That is, within the present approach, parallels across diverse domains are attributed to basic properties of the cognitive system: the processes that give rise to strategy choices and strategy discoveries. With development, new strategies are constructed, choices among existing strategies shift toward the more advanced strategies, and efficiency of execution of all strategies improves. However, the processes that give rise to the strategy choices and discoveries remain the same. A substantial gulf at first appears to separate the present theory from Piagetian and neo-Piagetian alternatives. The approaches are in all likelihood more compatible than they initially appear, though. Both structural and processing constraints contribute to the types of changes that are evident in children’s thinking. Focusing on either alone gives rise to an imbalanced depiction of cognitive development. Consistent with this view, several prominent neo-Piagetian theorists have attempted recently to incorporate both types of unities within a single theory. For example, in Halford’s (1990) most
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recent statement of his theory, choices among alternative strategies are identified as one of three critical factors determining children’s reasoning. Similarly, the most recent statements of Case’s theory (e.g., Case & Griffin, 1990) emphasize central conceptual structures, which play the same role within that theory as goal sketches play in the present one. Formulating a single theory that clearly articulates how structural and processing features of the cognitive system work together to produce children’s thinking at any one time and changes in that thinking over time remains perhaps the largest goal of the field of cognitive development. Acknowledgements-The research was supported in part by National Institutes of Health Grant HD-19011, in part by a grant from the Spencer Foundation, and in part by a grant from the McDonnell Foundation. Thanks are due to Dr. Ann Taylor and the teachers and children at the Carnegie Mellon Children’s School. Correspondence may be sent to Robert S. Siegler, Psychology Department, Carnegie Mellon University, Pittsburgh, PA 15213.
References Ashcraft, M. H. (1982). The development
of mental arithmetic: A chronometric
approach. Developmennrl
Review, 2, 213-236.
Ashcraft, M. H. (1987). Children’s knowledge of simple arithmetic: A developmental model and simulation. In C. J. Brainerd, R. Kail, & J. Bisanz (Eds.), Formal methods in developmentalpsychology (pp. 302-338). New York: Springer-Verlag. Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4, 379-426.
Case, R., & Griffin, S. (1990). Child cognitive development: The role of central conceptual structures in the development of scientific and social thought. In C. A. Hauert (Ed.), Developmental psychology: Cognitive, perceptuo-motor and neuropsychological perspectives. Amsterdam: North Holland. Cooney, J. B., Swanson, H. L., & Ladd, S. F. (1988). Acquisition of mental multiplication skill: Evidence for the transition between counting and retrieval strategies. Cognition and Instruction, 5, 323-345. Geary, D. C., & Burlingham-Dubree, M. (1989). External validation of the strategy choice model for addition. Journal of Experimental Child Psychology, 47, 175-192. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Goldman, S. R., Pellegrino, J. W., & Mertz, D. L. (1988). Extended practice on basic addition facts: Strategy changes in learning disabled students. Cognition and Instruction, 5, 223-265. Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79, 329-343.
Groen, G. J., & Resnick, L. B. (1977). Can preschool children invent addition algorithms? Journal of Educational Psychology, 69, 645-652.
Hagen, J. W., Hargrove,
S., & Ross, W. (1973). Prompting and rehearsal in short-term
memory. Child
Development,
44,201-204. G. S. (1990). Children’s
Halford, understanding: The development of mental models. Hillsdale, NJ: Erlbaum. Jorm, A. F., & Share, D. L. (1983). Phonological recoding and reading acquisition. Applied Psycholinguistics, 4, 103-147.
Kahan, L. D., & Richards, D. D. (1986). The effects of context on children’s referential communication strategies. Child Development, 57, 1130-1141. Kaye, D. B., Post, T. A., HalI, V. C., & Dineen, J. T. (1986). The emergence of information retrieval strategies in numerical cognition: A development study. Cognition and Zn.st&tion, 3, 137-166. Kerkman, D., & Siegler, R. S. (submitted for publication). Individual differences and adaptive flexibility in lower-income children’s strategy choices.
Kuhara-Kojima, K., & Hatano, G. (1989). Strategies for recognizing sentences among high and low critical thinkers. Japanese Psychological Research, 31, l-19. McGilly, K., & Siegler, R. S. (1989). How children choose among serial recall strategies. Child Development, 60, 172-182.
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McGilly, K., & Siegler, R. S. (1990). The influence of encoding and strategic knowledge on children’s choices among serial recall strategies. Developmental Psychology, 26, 931-941. Miller, K., Perhnutter, M., & Keating, D. (1984). Cognitive arithmetic: Comparison of operations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 10,46-t%. Ohlsson, S. (1984). Induced strategy shifts in spatial reasoning. Acta Psychologica, 57,4747. Reder, L. (1987). Strategy selection in question answering. Cognitive Psychology, 19,90-138. Schneider, W., & Pressley, M. (1989). Memory development between 2 and 20. New York: SpringerVerlag. Shultx, T. R., Fisher, G. W., Pratt, C. C., & Rulf, S. (1986). Selection of causal rules. Child Development, 57, 143-152. Siegler, R. S. (1986). Unities in strategy choices across domains. In M. Perlmutter (Ed.), Minnesota symposium on child development, (Vol. 19, pp. l-48). Hillsdale, NJ: Erlbaum. Siegler, R. S. (1987a). Strategy choices in subtraction. In J. Sloboda and D. Rogers (Eds.), Cognitive process in mathematics (pp. 81-106). Oxford: Oxford University Press. Siegler, R. S. (198%). The perils of averaging data over strategies: An example from children’s addition. Journal of Experimental Psychology: General, 116, 250-264. Siegler, R. S. (1988a). Strategy choice procedures and the development of multiplication skill. Journal of Experimental Psychology: General, 117, 258-275. Siegler, R. S. (1988b). Individual differences in strategy choices: Good students, not-so-good students, and perfectionists. Child Development, 59, 833-851. Siegler, R. S. (1989). Hazards of mental chronometry: An example from children’s subtraction. Journal of Educational Psychology, 81, 497-506. Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ: Erlbaum. Siegler, R. S., & McGilly, K. (1989). Strategy choices in children’s time-telling. In I. Levin & D. Zakay (Eds.), Psychological time: A life span perspective. Elsevier: The Netherlands. Siegler, R. S., & Robinson, M. (1982). The development of numerical understandings. In H. Reese & L. P. Lipsitt (Eds.), Advances in child development and behavior, (Vol. 16, pp. 241-312). New York: Academic Press. Siegler, R. S., & Shrager, J. (1984). A model of strategy choice. In C. Sophian (Ed.), Origins of cognitive skills (pp. 229-293). Hillsdale, NJ: Erlbaum. Svenson; 0. (1975). Analysis of time required by children for simple additions. Acta Psychologica, 39, 289-302. Widaman, K. F., Geary, D. C., Cormier, P., & Little, T. D. (1989). A componential model for mental addition. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15, 898-919. Woods, S. S., Resnick, L. B., & Groen, G. J. (1975). Experimental test of five process models for subtraction. Journal of Educational Psychology, 67, 17-21.