Children's Strategy Use in Computational Estimation PATRICK LEMAIRE, MIREILLE LECACHEUR, and FERNAND FARIOLI, CREPCO-CNRS and Universite de Provence (France)
Abstract This study reports an investigation of ten-year-old children's strategy use in computational estimation (i.e., give an approximate answer like 400 to an arithmetic problem like 224 + 213). Children used four strategies: rounding with decomposition, rounding without decomposition, truncation, and compensation. Strategies appeared to differ in frequency and effectiveness. Finally, children chose strategies in an adaptive way so as to obtain fast and accurate performance. Implications of these findings for understanding children's computational estimation performance and strategies in numerical cognition in general are discussed.
One of the most fundamental goals of research on early numeracy is to determine how children solve numerical problems. This goal has been pursued in domains of mathematical cognition as varied as acquisition of number concept and number names, subitizing and counting, as well as arithmetical problem solving; In all of these domains, researchers have found that children use several strategies to solve numerical problems (see Dehaene, 1997; Geary, 1994, for recent overviews). A much less-documented numerical activity is computational estimation. Computational estimation is defined as finding an approximate answer to arithmetic problems without actually (or before) computing the exact answer (e.g., 23 + 69 - 90). It is an important component of mathematical cognition, as it provides information about people's general understanding of mathematical concepts, relationships, and strategies, and about children's cognitive development in the domain of mathematics (e.g., Bestgen, Keys, Rybolt, & Wyatt, 1980; Carpenter, Coburn, Keys, & Wislon, 1976; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999; Sowder, 1988, 1992; Sowder & Wheeler, 1989). Moreover, computational estimation skills are useful in everyday situations in which a rough answer provides a contextually appropriate degree of precision. Examples of such situations include when we want to determine the number of people in a crowd, how many people can sit in a cathedral, or when we want to know how much we will spend on groceries in a supermar-
ket (Carpenter, Coburn, Keys, & Wislon, 1976; Rubenstein, 1985; Skvarcius, 1973; Trafton, 1978). The goal of the present work was to further investigate children's computational estimation. More specifically, we wanted to understand which strategies children use to do computational estimation, on which problems do they use each strategy, and how they choose and execute computational estimation strategies. PREVIOUS RESEARCH ON COMPUTATIONAL ESTIMATION Computational estimation has been studied in adults of varying levels of computational proficiency (Dowker, 1997; Dowker, Flood, Griffiths, Harriss, & Hook, 1996; Levine, 1982; Pelham, Sumarta, & Myakovsky, 1994) as well as in children and adolescents (Case & Sowder, 1990; Dowker, 1997; LeFevre, Greenham, & Waheed, 1993; Newman & Berger, 1984; Keys, Rybolt, Bestgen, & Wyatt, 1982; Sowder & Markovits, 1990). In most of the studies, participants were asked to give an approximate solution to arithmetic problems. Accuracy (as measured by the absolute or relative distance to the correct answer) and verbal protocols (i.e., people are asked to say how they found the solution) are the two main investigated measures. A few facts emerged from these studies. In all the previously cited research, people have been found to use several computational estimation strategies. A strategy is defined as the method to find a solution. For example, to estimate the solution of 46x58, people can round both operands (e.g., 50x60 — 3,000) or round only one operand (e.g., 46x60 — 2,760). People can also truncate one or two operands (e.g., 40 x 50 - 2,000). It is also possible to decompose a problem into a series of sub-problems, such as 40 + 70 + 10 to give an estimate for 43 + 76. The type of strategies vary for each type of problem and the number of strategies vary from one study to another, depending on the level of fine-grain analysis on which researchers have focused. In all studies, however, each participant has been found to use several strategies on different types of problems. Computational estimation strategies vary in frequency and efficiency. Some strategies have been found to be used
Canadian Journal of Experimental Psychology, 2000, 54:2, 141-148
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LIMITS OF PREVIOUS RESEARCH ON CHILDREN'S COMPUTATIONAL ESTIMATION STRATEGIES
answer the strategy gets you, but also how quickly it yields an answer. Two strategies with equal level of accuracy may still vary in effectiveness if one strategy yields an answer twice as fast as the other strategy. Therefore, complementing accuracy with speed measures would provide a more accurate picture of the effectiveness of each computational estimation strategy. Finally, previous studies established that people are flexible in their strategy use, in the sense that they use several strategies and choose among strategies on a problemby-problem basis. However, it is unknown how adaptive strategy choices are in computational estimation, or what are the predictors of children's strategy choices. That is, we do not know whether, as in other cognitive domains (e.g., Siegler, 1996, for a recent overview), children choose strategies to increase speed and accuracy. As past research on arithmetic problem-solving has shown (e.g., Lemaire, Barrett, Fayol, & Abdi, 1994; Lemaire, Fayol, & Abdi, 1991; Lemaire & Siegler, 1995), knowing whether children are adaptive in their strategy use is critical for understanding how children choose among strategies. In the present study, we tested the prediction that 10-year-old children would use computational strategies in an adaptive way, as shown by significant correlations between problem characteristics and strategy use.
Previous research on computational strategies was limited in several respects. First, in most previous studies, people had to give estimates of problems involving different arithmetical operations or types of numbers (see Dowker, 1997; LeFevre et al., 1993, for exceptions). In many studies, people were given addition and multiplication of whole and decimal estimation problems (e.g., 76x89; .46x.26; 546/33.5; .767.89). It is possible that, in such a situation, people use different strategies because they have to solve different problem types. That is, they use a different strategy for different types of problems (i.e., addition versus multiplication problems). Furthermore, they could use several strategies for each type of problem, but the diversity of strategies may result from the problem set including very different kinds of problems (e.g., multiplication of whole and decimal numbers). In sum, differences in strategy repertoire or accuracy may result from people using different strategies on different kinds of problems. In the present research, following LeFevre et al.'s (1993) and Dowker's (1997) studies, we gave children only three-digit by three-digit addition problems in order to determine whether multiple strategy use and strategy execution are critical dimensions in children's computational strategies. A second limit of previous research stems from the conclusion that strategies vary in effectiveness. This conclusion has been drawn only from measures of inaccuracy. No study timed people as they did computational estimation. To know whether two strategies are equally effective, it is important to determine not only how close to the exact
OVERVIEW OF THE PRESENT STUDY The goal of the present study was to systematically examine strategic aspects of children's performance in computational estimation. To do this, we used Lemaire and Siegler's (1995) general conceptual framework on strategic aspects of children's cognition. This framework makes a distinction between (a) strategy repertoire, (b) strategy distribution, (c) strategy execution, and (d) strategy selection. Strategy repertoire means the strategies that children use. Strategy is generally defined here as "a procedure or set of procedures to achieve a higher-level goal or task" (Lemaire & Reder, 1999, p. 365). As in previous studies, strategy repertoire refers to the variety of methods that children use to provide an estimate. Strategy distribution refers to when each strategy is used. It involves both the relative frequencies of each strategy and the types of problems on which the strategy is used. Are some strategies used more often than others, more often with some problems than with other problems? Strategy execution means how fast and accurately each strategy is executed, as well as the different variants of strategies. Finally, strategy selection refers to how strategies are chosen, that is, the decisions about which strategy to use on each problem. Issues of strategy selection concern how participants adjust their strategy use to problem characteristics (e.g., problem size). This adjustment of strategy use is also called strategy adaptivity and is measured by correlations between children's percent use of each strategy and problem characteristics (e.g., problem size). This framework
more often than others. For example, both children and adults tend to round two numbers more often than to round one number when they estimate the sum or product of two numbers (e.g., Dowker, 1997; Dowker et al., 1996; LeFevre et al., 1993). Moreover, strategies are not equally efficient: They vary in accuracy (Baroody, 1989; Dowker, 1997; LeFevre et al., 1993; Sowder & Berger, 1984). Each strategy yields an answer which is more or less close to the correct answer. For example, when people round both operands to estimate 46 + 57, they obtain a more accurate answer (100) than when they round the first (107) or second operand (106). Finally, use of computational estimation strategies is affected by characteristics of problems, such as problem size or distance to the adjacent decade (LeFevre et al., 1993). People provide better estimates for problems of small magnitude (i.e., 18 + 23) than for problems of large magnitude (e.g., 78 + 83). Also, people tend to provide better estimates to so-called small adjustment problems (i.e., problems with unit digits close to the adjacent decade like in 42 + 51) than to large adjustment problems (i.e., problems with unit digits far from the adjacent decade like in 47 + 56).
Estimation Strategies
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TABLE 1 Problem Set Used in This Experiment
228 243 247 259 294 317 325 328 354 364 378 389 164 186 192 252 271 274 295 313 347 351 361 371
iiH K h Hh Hh Hh ih Hh HK H H H H -( -( -( -t -( -i -t 4 -t
723 548 748 524 559 629 429 497 257 287 487 472 673 552 457 461 436 542 384 535 581 614 633 493
415 459 493 496 512 536 548 564 568 649 659 687 397 418 421 423 481 486 525 641 646 673 757 762
H H th h HH Hh Hh -ih Hh HiH H H H 4 -I -t -i -t -t -t -t -t
538 346 349 215 469 256 354 357 294 292 298 245 352 581 327 341 348 353 264 317 213 296 172 234
proved useful to analyze children's strategy use in the acquisition of multiplication problems (Lemaire & Siegler, 1995) as well as younger and older adults' strategies in multiplication problem-solving (Siegler & Lemaire, 1997). In the present study, fifth-grade children (10 years old) were asked to provide estimates of 48 three- by three-digit addition problems (e.g., 459 + 346), because previous studies have tested either older or younger participants (e.g., Baroody, 1989; Dowker., 1997; LeFevre et al., 1993). After each problem, children were asked to say how they found the solution. Verbal protocols, speed, and accuracy were collected for each problem so as to identify which computational estimation strategy is used on each trial, and how each strategy is used, selected, and executed. Based on previous work, the main predictions of this study were: (a) children use several strategies, (b) the strategies are not used equally often, (c) some strategies are more efficient than others, and (d) children choose strategies in an adaptive way. Findings consistent with these predictions were of interest to us for two general reasons. First, they extend the relevance of Lemaire and Siegler's (1995) framework concerning strategic dimensions of children's cognition in a new domain. Second, they deepen our understanding of determinants of children's computational estimation performance. Method PARTICIPANTS
Twenty-three fifth-grade children (12 girls and 11 boys) from a French urban public school in Aix-en-Provence
(France) participated in the study. At the time of testing, the mean age of the children was 128 months (range - 120-132 months). STIMULI
The stimuli were 48 addition problems (i.e., a + b) in which a and b were three-digit numbers (see Table 1). Given certain experimental effects that are known in the domain of mental arithmetic, we controlled the following factors across the three types of problems: (a) zero effects: no operand had a 0 unit digit (e.g., 390); (b) tie effects: no decade or unit digit was repeated across operands (e.g., we avoided 154 + 474 or 241 + 347); (c) side of the larger operand: the first operand was larger than the second in half of the problems; (d) problem size: in each category, the mean problem size was similar; (e) distance to the adjacent decade: 16 problems had one operand close to the adjacent decade (e.g., 421 + 327), 16 had both operands close to the adjacent decade (e.g., 252 + 461), and 16 problems had zero operands close to the adjacent decade (e.g., 164 + 673). An operand was considered as close to the adjacent decade if the unit digit of that operand was 1 or 2 away from the adjacent decade (e.g., 321, 548) and far from the adjacent decade otherwise (e.g., 324,146). PROCEDURE Before encountering the experimental problems, children were told that they were going to do estimation. As computational estimation is not a practised skill in French schools, computational estimation was explained to each child as giving an approximate answer that was as close as possible to the correct answer without actually calculating the correct answer. An example was worked out with the children. Children were told, "for example, if I have to estimate 123 + 256, I can do 120 + 260 - 380, and give 380 as an approximate solution to the problem. I can also do 125 + 255 or 130 + 260, or anything else that yields an approximate answer." The experimental problems were presented in 60-point bold Palatine font in the centre of a 14-inch computer screen controlled by a Power Macintosh PC 1400. Each trial began when a 750-ms ready signal (the wordpret? French for "ready?") appeared in the centre of the screen. The equations were then displayed horizontally in the centre of the screen. The equations were in the form a + b. The symbol and numbers were separated by spaces equal to the width of one character. Problems remained on the screen until the participant answered. Timing of each trial began when the problem appeared on the screen and ended when the experimenter pressed a button on a button box, the latter event occurring as soon as possible after the participant's response. The software, PsyScope (Cohen, MacWhinney, Flatt, & Provost, 1995), collected data with 1-ms accuracy. In order to stress the estimation nature of the expected
Lemaire, Lecacheur, and Farioli
144 response, we asked children to be as fast and accurate as possible. After each response, children were asked "How did you find that solution?" On each trial, the experimenter recorded children's responses and verbal protocols. Problems remained on the screen during verbal protocols because pilot testing revealed that this made it easier for children to describe their strategy. The experimenter wrote verbal protocols down verbatim. Those protocols were subsequently coded by an independent coder for each problembased on the information recorded by the experimenter. Two raters who independently classified strategy use on 100 randomly chosen trials agreed on 97% of them. The stimuli were presented in two 24-problem trial blocks. Each child was permitted a 5-10 minute rest between blocks. Before the experimental trials, children were given 10 practice problems to familiarize themselves with the apparatus, procedure, and task. Results and Discussion Results are reported in four sections. The first describes children's strategy repertoire, the second, strategy distribution. How strategies were executed is analyzed in the third section, and the final section examines how children chose strategies. WHICH STRATEGIES WERE USED?
Children's verbal protocols indicated that four strategies were used to estimate: (a) rounding with decomposition, (b) rounding without decomposition, (c) compensation, and (d) truncation. Rounding with decomposition consisted of sequentially adding hundreds, tens, and units of both operands. At any point in the addition, children rounded one number. For the problem 459 + 356, a child might do this: 400 + 300 + 60 + 60. Rounding without decomposition consisted of rounding one of the two operands or both operands to the closest decade or hundred. As an example, for 536 + 256, children may round 536 to 540 and 256 to 250 and then add the two rounded numbers, 540 + 250 790. In the compensation strategy, children added or subtracted a small quantity to the sum of the two rounded operands. For the problem 378 + 487, children might round up to 400 + 500, and then subtract 20 or 30 to compensate for rounding both operands to larger values. Finally, the truncation strategy was used when children neglected unit or ten digits. For example, some children estimated 641 + 317 as 600 + 300. On some rare occasions, in which a strategy could be coded as either rounding or truncating, we based our decision on children's verbal protocols. So, for example, in a problem like 228 + 723, a child solved 200 + 700; this was coded as "truncation," as the child said that she had dropped the 28 and 23. The diverse strategy use reflected individual as well as group-level behaviour. Of the children, 95% used two or more strategies, 71% used three or more, and 38% used all
TABLE 2 Children's Performance on Computational Estimation Strategies Strategies
Rounding with decomposition Rounding without decomposition Compensation Truncation
% of trials in which strategy used
Mean RT (in seconds)
% deviation from correct solution
30
7.6
4.8
34 2 27
8.5 9.6 6.6
5.2 4.4 5.2
four. Rounding with decomposition was used at least once by 22 children, rounding without decomposition by 21 children, compensation by 7 children, and truncation by 14 children. WHEN WAS EACH STRATEGY USED?
Table 2 indicates the frequency of use of each strategy. Overall, rounding with or without decomposition were the two most frequently used strategies, followed by truncation. Compensation was used on only 2% of the problems.1 HOW WERE STRATEGIES EXECUTED? As shown in Table 2, strategies varied in both latencies and accuracy. The truncation strategy was faster than all other strategies. Rounding with or without decomposition were faster than compensation, ft (5) > 5.35, ps < 01 but were of equal speed, t(19) < 1. Accuracy here was measured in mean percent deviation from the correct response, after eliminating 1% of the problems on which estimates were 50% or more away from the correct response, so as to control for extreme scores. The only significant differences were between compensation and both rounding without decomposition and truncation, rs (12, 5) < 5.12, ps < .01. Strategies differed in speed and accuracy as well as in how they were executed. For each of the three most often-used strategies, several variants were noted. Children used two variants of rounding with decomposition. In the first variant, children first rounded hundreds, second rounded tens, third added hundreds, fourth added tens, and finally added these two final quantities. For 459 + 356, they did: (1) 459 - 400; 356 - 300; (2) 49 - 40; 56 - 50; (3) 400 + 300 700; (4) 40 + 50 - 90; and (5) 700 + 90 - 790. In the second variant, they did exactly the same thing but concatenated operations 2 and 4 into a single operation and executed operation 3 after operation 1. For 459 + 356, they did: (1) 459 - 400; 356 - 300; (2) 400 + 300 - 700; (3) 49 + 56 90; and (4) 700 + 90 - 790. In the third variant, children first rounded and added the hundreds, then rounded and added the decades together, and finally added these two Children calculated the correct solution on 5% of the problems and strategy could not be clearly and unambiguously identified from children's verbal protocol on 2% of the problems.
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145 TABLE 3 Summary of Problem-Based Correlation Analyses (N - 48)
Correct Sum Percent use of rounding with decomposition Percent use of rounding without decomposition Percent use of truncation
.09
-.28 -.11
Operand 1
Operand 2
Rounding difficulty
.14
.14
.58**
.06 -.51**
-.06 -.02
-.61** -.11
Note. Rounding difficulty was the sum of the absolute values of distances to the closest decades for both operands (e.g., for problem 228 + 7,234, it was [(230-228) + (7,234-720) - 5]. **p < .01.
quantities. For 459 + 356, they did: (1) 459 - 400; 356 300; (2) 400 + 300 - 700; (3) 49 - 40; 56 - 50; (4) 40 + 50 - 90; and (5) 700 + 90 - 790. The second variant was used most often (55%) followed by the first (28%), then by the third (17%) variant, X2(2, N - 22) - 12.74,/> < .01. Rounding without decomposition was used by children with two variants. In both variants, children added the two rounded operands. Either they rounded the two operands in a similar way, as when they rounded both operands to the nearest hundred (e.g., 418 + 581 - 400 + 500) or they rounded the two operands in a different way like when they rounded one operand to the nearest hundred and the other to the nearest decade (e.g., 418 + 581 - 410 + 500). Children used the first variant more often than the second (80% versus 20%, x2(l, N - 21) = 16.13,/> < .01). Truncation was also executed in two different ways. Children truncated either unit digits only (e.g., 450 + 350) or unit and decade digits (e.g., 400 + 300). They used the first variant much more often than the second variant (93% versus 7%, x2(l, W - 14) = 58.76,p < .01). HOW DID CHILDREN CHOOSE AMONG STRATEGIES? To understand how children chose among strategies, mean percent use of each of the three mostly used strategies were correlated with four variables: (a) size of the correct sum (e.g., for the problem 228 + 723, this was 951), (b) size of the first operand (e.g., for the problem 228 + 723, this was 228), (c) size of the second operand (e.g., for the problem 228 + 723, this was 723), and (d) rounding difficulty as measured by the sum of the absolute values of the distances to the closest decades for both operands (e.g., for the problem 228 + 723, rounding difficulty was calculated as [(230-228) + (723-720) - 2 + 3-5]. Table 3 summarizes correlation results and clearly shows that (a) the rounding difficulty was the only variable that significantly correlated with mean percent use of rounding with or without decomposition, and (b) the size of the first operand was the only variable that significantly correlated with mean percent use of truncation. Stepwise regression analyses of mean percent use of each strategy confirmed this finding. For each of rounding with or without decomposi-
tion, rounding difficulty accounted for 34% and 37% of variance, respectively. For truncation, the size of the first operand accounted for 26% of the variance. This suggests that to decide whether or not to use rounding, with or without decomposition, children considered the difficulty of the rounding process, and that the size of the first operand was critical in children's use of truncation. One possible reason for truncation correlating more with the size of the first operand than with the rounding difficulty variable is that, as children saw a large first number, they viewed the problem as harder to solve via rounding because problems with larger operands are harder to solve. General Discussion The goal of the present study was to investigate aspects of children's strategy use in computational estimation. Tenyear-old children had to provide estimates to 48 three-digit by three-digit addition problems. Verbal protocols revealed that four strategies were used and that these strategies varied in speed and accuracy. The data also revealed that children were very adaptive in their strategy choices. The present findings have implications for understanding children's computational estimation and for studying strategy choices in estimation. IMPLICATIONS FOR UNDERSTANDING COMPUTATIONAL ESTIMATION
A number of previous studies had found that children use several strategies when they do computational estimation and that strategies are not used equally often (e.g., Dowker, 1997; Levine, 1982; LeFevre et al., 1993; Keys et al., 1982). The present results are consistent with this conclusion. Four strategies appeared in this study: rounding with decomposition, rounding without decomposition, compensation, and truncation. Rounding with or without decomposition were the most frequently used strategies, followed by truncation and by compensation. Another result from this study that is consistent with previous findings is that strategies vary in effectiveness. In the present study, the fastest strategy was truncation and the slowest was compensation. Compensation yielded the most
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accurate estimates and rounding without decomposition and truncation yielded the least accurate estimates. This study was the first to combine latency and accuracy measures, and this was interesting for two reasons. First, differences between strategy effectiveness were revealed in both of these measures. Second, some strategies appeared faster but less accurate, and other strategies, slower but more accurate. To give an example, the most accurate strategy, compensation, was also the slowest strategy, and the fastest strategy, truncation, was one of the least accurate strategies. Unknown is whether speed and accuracy are really independent measures of strategy effectiveness in computational estimation. Further studies will have to determine if, at a comparable level of speed, two estimation strategies can yield equally accurate estimates, or, if two strategies are equally fast given a comparable level of accuracy. One original contribution of the present study regarding strategy execution concerns the use of variants of each strategy. For each strategy, an unequal number of variants was used (from 2 to 3), and each variant was not used equally often. Further studies of children's estimation strategies may investigate these variants in greater detail so as to learn why some variants are used more often than others and how each variant varies in effectiveness. This point could not be documented here due to the small number of observations of each variant. Another original contribution of this work is the finding that children's estimation strategies are chosen in an adaptive way. This was revealed by significant correlations between problem characteristics (i.e., rounding difficulty) and strategy use. Children chose a given strategy when that strategy was the most appropriate for individual problems. Specifically, when rounding was more difficult, they used rounding with decomposition and when rounding was easiest they used either rounding without decomposition or truncation. This was interesting because it showed that estimation is like other arithmetic or cognitive domains in which the use of adaptive strategy choices has been found in both children and adults (e.g., Geary & Brown, 1991; Geary & Burlingham-Dubree, 1989; LeFevre, Bisanz, Daley, Buffone, & Sadesky, 1996; LeFevre, Sadesky, & Bisanz, 1996; Lemaire & Siegler, 1995; Siegler, 1987, 1988; Siegler & Lemaire, 1997). Finding that children are adaptive in their estimation strategy use suggests that children choose their estimation strategies with a strategy selection process similar to the strategy selection process used in mental arithmetic. Empirical research and computational models in mental arithmetic established that, during mental calculation, features of problems affect children's strategy choices (Geary, 1991; Lemaire & Fayol, 1995; Lemaire & Siegler, 1995; Siegler, 1988; Siegler & Jenkins, 1989; Siegler & Shipley, 1995). The present study showed that computational estimation strategy choices are also affected by problem characteristics.
Lemaire, Lecacheur, and Farioli IMPLICATIONS FOR INVESTIGATING STRATEGY CHOICES IN COMPUTATIONAL ESTIMATION
Even though this work documented strategic aspects of children's computational estimation, there were several limitations that may result from the manner in which strategic aspects were assessed. Consistent with previous studies, strategies have been found here to vary in effectiveness. For example, truncation was the fastest strategy. However, truncation was also the least-often used strategy. It is therefore possible that truncation was used only when it was fast. More generally, one issue regarding strategy effectiveness in this and previous studies of computational estimation strategies stems from an inherent confounding of strategy effectiveness and strategy choices. In all studies (the present one included), strategies are not only found to vary in speed and/or accuracy but also in how frequently they are used. Similarly, it was surprising here to observe that rounding with decomposition yielded shorter latencies than rounding without decomposition. One possible reason for this is that each strategy was used on different sets of problems, and was used with unequal frequency. Future studies will have to use methods to disentangle strategy choices from other strategic parameters. Another set of limitations to the present study concerns the impossibility of unambiguously assessing the role of strategy effectiveness on computational estimation strategy choices. This can be achieved via problem-based regression analyses in which percent use of a given strategy is regressed against strategy effectiveness variables (such as strategy speed or accuracy). Here it was not possible to run such analyses because strategy speed or accuracy was inherently confounded with strategy choice. Assessing the role of strategy effectiveness may prove important and interesting in future research. For example, Siegler and Lemaire's (1997) study of mental calculation showed that strategy effectiveness was the best predictor of calculation strategy use. Strategy effectiveness was an even better predictor than problem characteristic. Given that children practise computational estimation skills less frequently than calculation skills, finding out whether strategy use is mainly determined by different factors than strategy effectiveness would provide fruitful insight to the differences between calculation strategy choice and estimation strategy choice processes. More generally, it would help delineate further conditions (e.g., task, participant, or activity characteristics) under which strategy effectiveness and problem features best predict strategy choices. One potential control for strategy frequency in assessing strategy effectiveness, as well as the role of strategy effectiveness on computational strategy use, is the choice/no-choice method proposed by Siegler and Lemaire (1997). In this method, children are asked to solve problems under two conditions. In one condition, they are free to choose among strategies. In the other condition, they are required to use a
Estimation Strategies given strategy on all problems. The no-choice condition reveals strategy characteristics that are independent of strategy choice; this finding sheds light on which strategy is really the fastest and/or most accurate. Such a method also provides the type of strategy speed and accuracy information that is necessary to understand how speed and/or accuracy affect strategy choices. Finally, as demonstrated by Lemaire and Siegler (1995), the choice/no-choice method is appropriate to investigate the change in strategy use with age. It can be used in further studies to document changes in children's and adults' computational estimation strategies. In sum, consistent with previous findings on computational estimation, the present results showed that computational estimation is a domain of mathematical cognition in which children use several strategies, in which strategies vary in frequency and execution, and in which children choose their strategies in an adaptive way. The present study also suggests that computational estimation is a specific domain in mathematical cognition, with its own strategies and its own variables affecting strategy use. Future studies will help to further understand the specificities of computational estimation performance as well as developmental changes in children's and adults' strategic aspects of computational estimation. This research was supported by the CNRS (French NSF). We thank Morgane Cay-Maubuisson and Marie-Caroline Mini for their help in data collection and analyses, and Jo-Anne LeFevre and two anonymous reviewers for their helpful comments on a previous version of this article. Correspondence should be addressed to P. Lemaire, CNRS & Universite de Provence, 29 av. R. Schuman, 13621 Aix-en-Provence, France (E-mail:
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Sommaire Dans cet article, nous rapportons une etude visant a mieux comprendre les performances cognitives d'enfants de 10 ans a une tache d'estimation calculatoire, tache relativemem peu etudiee dans le domaine de la cognition numerique et pourtant souvent accomplie dans notre vie quotidienne. Nous avons analyse comment les enfants estiment (i.e., donner un resultat approximatif sans calculer la reponse exacte) la somme a des problemes d'addition de deux nombres a trois chiffres, comme 224+213. Nous avons egalement caracterise les aspects strategiques sous-jacents a leurs performances, d'apres le cadre conceptual d'etude des strategies cognitives propose par Lemaire et Siegler (1995). Les resultats montrent que les enfants utilisent quatre strategies, que ces strategies different du point de vue de leur efficacite (i.e., temps de resolution et deviation de la reponse estimee par rapport a la reponse correcte) et que les enfants choisissent des strategies differentes a chaque probleme en fonction des caracteristiques (e.g., taille) des problemes. Ainsi, les enfants ont utilse la strategic d'arrondissement
avec decomposition, d'arrondissement sans decomposition, la troncature et la compensation. Chaque strategic pouvait etre executee sous differentes variantes. La strategic la plus rapide etait la troncature, suivie de 1'arrondissement avec decomposition, puis de 1'arrondissement sans decomposition et de la compensation. La strategic permettant d'obtenir la reponse la plus proche de la reponse correcte etait la strategic de compensation, suivie de 1'arrondissement avec decomposition et de 1'arrondissement sans decomposition a egalite avec la troncature. Au niveau des determinants des choix strategiques, la difficulte a effectuer un arrondissement etait le meilleur predicteur de 1'utilisation des strategies d'arrondissement avec et sans decomposition, tandis que la taille du premier operande etait le meilleur predicteur de 1'utilisation de la troncature. L'analyse de ces predicteurs indique que les choix strategiques des enfants de 10 ans sont adaptatifs dans le sens ou Us s'ajustent au mieux aux types de problemes a resoudre.
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