Discovery and maintenance of the many-to-one counting strategy in 4-year-olds: A microgenetic study

83 British Journal of Developmental Psychology (2004), 22, 83–102 2004 The British Psychological Society www.bps.org.uk Discovery and maintenance of the many-to-one counting strategy in 4-year-olds: A microgenetic study Anke W. Blöte1 *, Sandra G. Van Otterloo2 , Claire E. Stevenson3 and Marcel V. J. Veenman 1 1 Leiden University, The Netherlands University of Amsterdam, The Netherlands 3 Leiden, The Netherlands 2 This study investigated the development of the many-to-one counting strategy in 4year-old children. In the first experiment, 52 children participated. Their development with respect to two kinds of tasks, a hidden-items task and a needed-items task, was studied over four sessions. Children (n = 28) who accurately used the many-to-one strategy in Session 4 also participated in the second experiment. These children were presented with more difficult hidden- and needed-items tasks. It was found that children often produced the strategy for the first time on tasks with relatively few items. Most children then kept producing it, even if they initially did not obtain much profit from its use because of counting errors. Increasing task difficulty resulted in children making more counting errors or reverting to invalid strategies depending on the nature of the new task. The purpose of this study was to learn more about the conditions under which new strategies are adopted. We paid attention to the discovery of the strategy, particularly the effect of task complexity on the first use of a correct strategy. The main focus of the research, however, was on strategy maintenance in relation to both strategy outcome and task demands. Two issues addressed were: (1) Will children maintain using a correct strategy that does not bring them immediate gains, or will they discontinue using it and revert to other strategies? (2) Once children have learned the correct strategy for solving a certain problem, what strategy will they use when presented with * Correspondence should be addressed to Anke W. Blöte, Department of Psychology, Section of Developmental and Educational Psychology, Leiden University, Wassenaarseweg 52, 2333 AK Leiden, The Netherlands (e-mail: bloete@fsw. leidenuniv.nl). 84 Anke W. Blöte et al. a more difficult version of the problem? Again, will they persist in using the correct strategy or will they regress to invalid strategies? When children try to solve a certain problem which is new to them, it may be that they do not know the correct strategy for solving the problem. If they then produce an invalid strategy which results in an incorrect answer and receive negative feedback, they will be in an impasse. This will cause children to try to solve the problem in new ways (e.g. Newell, 1990). When in the process of trying new strategies, a correct strategy is produced that results in a positive outcome, one would expect this strategy to be repeated in subsequent tasks. However, what would happen if the children made a small mistake in the application of the correct strategy? Would they go on producing the new strategy often enough to make it work correctly, or would they, at least for some time, discontinue using it? This issue is an important one because of the theoretical implications it has for the mechanisms that guide the production of new strategies in novices. Earlier studies have shown that children sometimes keep using new strategies that do not help them to improve their performance. This phenomenon has been coined ‘utilization deficiency’ (e.g. Miller, 1990; Miller & Harris, 1988). However, research on utilization deficiencies has been concerned with new strategies that are just relatively better, more effective or more efficient, than earlier used strategies, and not with a switch from invalid to valid strategies. Thus, it is not clear if the children in those studies realized that they did not profit from the use of the new strategies. The present study is concerned with a situation where the answers given by the children are either correct or incorrect. Feedback on these answers should therefore make it clear to them what the result of their actions is. Furthermore, the development is from invalid toward valid strategies, where the valid strategy is necessary (but not sufficient) to produce the correct answer. The children might blame the strategy for the wrong answer if an error occurs in the application of the newly discovered strategy. One would expect this to decrease the probability that the strategy would be used again in the next trials. However, if they were to continue producing this correct strategy while receiving negative feedback, this would suggest that metastrategic knowledge is guiding their strategy production (Kuhn, 2000; Kuhn & Pearsall, 1998). They could have a goal sketch (Siegler, 1996) related to the problem situation helping them to select the correct strategy. When studying the effect of strategy outcome on strategy development, one needs to pay attention to the gradual character of strategy development. It has been shown that children who have discovered a new strategy that either is less effortful (e.g. Siegler & Jenkins, 1989) or helps them to solve problems they could not solve before (e.g. Kuhn & Phelps, 1982) usually do not show a complete change in their approach to the task. If no direct instruction about a correct strategy is given, most children show a gradual transition from less advanced to advanced strategies. In the process of acquiring new strategies, variability in strategy use is the rule (Alibali, 1999; Kuhn & Phelps, 1982; Siegler, 1996). This gradual character of strategy development notwithstanding, one would expect that the effectiveness of a newly discovered strategy would have a noticeable effect on children’s subsequent use of it. A task that is suited to assess the effect of strategy effectiveness is not easily found. In addition to other more common requirements, the problem presented in the task has to meet the following two criteria: (1) The problem should, initially, elicit incorrect strategies and later on, in the restricted period in which the study takes place, stimulate children to develop a correct strategy. (2) The production of this correct strategy Many-to-one counting 85 should not automatically result in a correct answer; novel users of the strategy should make errors in applying it. Analysis of the tasks used in previous studies on strategy development revealed that most tasks do not meet the two criteria and are therefore not suited to the purpose of the present study. An exception are the many-to-one counting tasks described in the following paragraph. A problem such as ‘I want to give each of these three dolls two apples; how many apples do I need?’ can be solved by many-to-one counting, that is, by counting 1,2 at the first doll, 3, 4 at the second doll and 5, 6 at the third one resulting in the answer ‘I need six apples’. An essential characteristic of many-to-one counting tasks is that the number of rows and columns of the distribution are given but the items to be counted are not perceptually present. The 4- and 5-year-old children in a study by Becker (1993) used the many-to-one counting strategy in tasks such as ‘Give each of these four dolls three cookies; now we cover the cookies with this screen, how many cookies are under the screen?’ (Hidden-items task) and ‘How many cookies are needed in order to give each of these three dolls two cookies?’ (Needed-items task). Studies by Becker and Chung et al. (1999) made it clear that: (a) at the age of four, children start to use the many-to-one counting strategy in the hidden- and needed-items tasks; and (b) novel users of the strategy make counting errors while applying the strategy. This means that the many-toone counting tasks meet the two criteria stated above. Many-to-one counting, being an early stage in the development of multiplication skills (e.g. Anghileri, 1989; Becker, 1993), is also of interest in the context of children’s learning of arithmetic. First, young children acquire an understanding of one-to-one correspondence. When they encounter many-to-one correspondence problems, these children initially will not know how to correctly solve this kind of problem. They then may apply a strategy based on their knowledge of one-to-one correspondence such as counting the objects rather than the distribution of items over the objects or counting the number of items per object. Later on, children develop an understanding of multiplicative relations and learn to solve quantitative many-to-one problems (Nunes & Bryant, 1996). 3 array of coins Anghileri (1989) presented 4- to 12-year-old children with a 6 attached to a card. The number of rows and columns were discussed with the child, and then the card was turned face down on the table. The children were asked to figure out the total number of coins attached to the backside of the card. The youngest group of children counted each coin separately in the product set. Somewhat older children counted the perceptually not-present items groupwise, giving emphasis on the last item in each group. In this way, they could count by one and at the same time keep track of the number of items in each group. Children aged eight and older have been found to use repeated addition or counting in units larger than 1 as a strategy, e.g. 3, 6, 9, 12, 18 (Steffe, 1994). Finally, when children have learned the tables of multiplication, they retrieve the answer from memory. However, they continue to use counting and repeated addition as backup strategies (Lemaire & Siegler, 1995). It is well known now that task demands influence the level of cognitive competence children will show (Gelman & Meck, 1983; Siegler & Crowley, 1991; Sophian, 1998). If the task situation is relatively complex, children could either be hindered in their production of a new strategy or get confused in the application of that strategy. Both consequences could be explained by the restricted mental capacity that is available to execute the task (Miller & Seier, 1994). Coming up with a new strategy and then applying it may require more capacity than is available. Shannon (1978) reported an experiment in which the production of a counting strategy was indeed affected by task 86 Anke W. Blöte et al. demands. Increasing the number of items in the task caused some children to revert to a less organized count, that is, to a less advanced strategy. Questions posed in the present study In Experiment 1, three questions were asked about discovery and maintenance of the many-to-one strategy in young children. The first question addressed the effect of task complexity on the discovery of the many-to-one strategy. We hypothesized that the number of items in the task would influence the first production of the strategy. It was expected that the children would produce the strategy for the first time on the tasks with relatively few items per object. A second question was how effective children’s use of the many-to-one counting strategy would be. It was hypothesized that producing the strategy for the first time would use up so much of children’s mental capacity that there would not be enough left for the correct application of this strategy. It was therefore expected that novel users of the strategy would make counting errors. The third question addressed the role of strategy effectiveness in the subsequent production of the many-to-one strategy. We hypothesized that the outcome of the newly acquired target strategy would influence the subsequent production of the strategy. The expectation was that a poor outcome of the target strategy (because of counting errors) would result in fewer many-to-one counts in the following tasks. The role of task demands in the generalization of newly learned strategies was further studied in Experiment 2. The main question of the second experiment was if and how young children who have discovered the many-to-one counting strategy and have experienced its effectiveness change their strategies in more difficult many-to-one counting situations. The participants in this experiment were children who had mastered the many-to-one counting strategy in the first experiment. The study used a microgenetic design (Siegler & Crowley, 1991) to examine the course of strategy development. It was evident that only by following the children in the phase of developing the many-to-one strategy could we see both their first use of it and their persistence in using it. In this way, the present research expanded on the cross-sectional study of many-to-one counting carried out by Becker (1993). EXPERIMENT 1 Method We used the results of a pilot study (Chung et al., 1999) in adapting Becker’s (1993) method with respect to the age of the participants, the task procedures, and the scoring system. Participants This study initially started with 71 4-year-old children coming from four different schools in the Netherlands. (In the Netherlands, most children start school at age four.) Fifty-five of these children were able to count 15 objects. They were selected for participation in the study. Three of them were lost because of non-attendance on one of the test dates. Twenty of the remaining 52 children (10 girls and 10 boys) were Many-to-one counting 87 between 4 years and 0 months and 4 years and 6 months old at the start of the experiment, and 32 (15 girls and 17 boys) were between 4 years 7 months and 5 years 1 month. Materials A set of small zoo animals each about 4-cm tall were used in the hidden- and needed3 cm coloured items tasks. The items to be distributed to the zoo animals were 3 cardboard squares with stickers or prints on the face. There were four types of items: smiley faces, stars, footprints, and hearts. A screen approximately 75 cm wide and 40 cm high was used to hide and then cover the items from the subject during the hiddenitems tasks. Tasks In the basic counting task, a small bowl, containing 30 buttons, and an empty bowl were placed in front of the child on a table. The child was asked to count the buttons while putting them one by one in the empty bowl. The children were allowed to make one counting error. After making an error, they were corrected and asked to continue. The introduction task was given to prepare the children for the tasks in the following sessions. Five small toy bears were placed in a row on the table. The child received 10 cards and was asked to give each bear two cards. Children who had problems with this task were helped with the distribution of the cards. After the cards had been distributed, the child was asked to count these cards out loud. In the one-to-one counting task, a number of 10, 11, 13, or 15 cards were arranged in half a circle on the table. The children were asked to point to the cards while counting them. By doing this, we hoped to stimulate the children to point to cards perceptually not present in the following tasks. This would make the scoring of children’s strategy usage more reliable. During the hidden-items tasks, the child was presented with four identical zoo animals. The experimenter told the child that she would give each animal two (or three) cards. The cards were dealt behind a screen to prevent the child from counting the cards ahead of time. The experimenter then removed the screen and showed the child that each animal had received the number of cards mentioned earlier. Two seconds later, the cards were covered by the screen. The cards were therefore hidden, but the animals could still be seen. The child was asked to count how many cards were hidden under the screen. The child was asked to count out loud and show the experimenter how they solved the problem. If the child did not react or answered incorrectly, the question was repeated. It was emphasized that the number of all cards taken together was requested. After the second attempt, the child was told whether the answer was correct or not. During the needed-items tasks, the child was presented with two identical animals on the left-hand side of the table and three animals of another sort on the right. The child was then told that each animal sort was having its own party. The experimenter asked the child to give the two animals on the left-hand side two (or three) cards each (see Fig. 1). Then, the child was asked how many cards they needed in total to give each of the three other animals two (or three) cards as well. The child was asked to count out loud and show the experimenter how they solved the problem. If the child did not react or answered incorrectly, the question was repeated. After the second attempt, the child was told whether the answer was correct or not. 88 Anke W. Blöte et al. Figure 1. Placement of animals and cards in the needed-items task. Design and procedure The study used a microgenetic design (Siegler & Crowley, 1991). Five sessions took place during a period of 5 weeks. During the first week, there was a preliminary session that presented the children with the counting task and the introduction task. The testing sessions then took place in the following 4 weeks with a frequency of one session per week. Each child was presented with five tasks. First, the one-to-one counting task was administered, then the two hidden-items tasks, and finally the needed-items tasks. The hidden-items tasks were given before the needed-items task because they were easier and could warm up the children for the needed-items task (Becker, 1993). Because the present study did not specifically address task effects, the fixed order of the two experimental tasks was not considered a problem. The order of tasks with set sizes of two and three was counterbalanced for the two age groups. Furthermore, this order was reversed each week. Different animals and card types were used for each task to keep the children from getting bored with one sort. The children were tested individually in a quiet room at the child’s school by two female experimenters. One of them tested the child, and the other filmed and observed the child’s on-task behaviour. The children were about equally divided for the condition of experimenter and did not change experimenters over time. Scoring The same scoring system was used for the hidden- and needed-items tasks. This scoring system was adapted from Becker (1993). The children’s strategic behaviour and counting behaviour were scored. Strategies could be categorized in four main categories: (1) Correct strategy producing a correct answer: Explicit many-to-one counting. The children count the first animal’s cards and then continue counting further at the second and third animals, or the children count the first row of cards and then continue counting the other rows. The answer is correct. Implicit many-to-one counting. The children do not display a counting strategy but do come up with the correct answer. (2) Correct strategy producing an incorrect answer: Many-to-one counting with counting errors. There are three kinds of errors: (a) The children use the many-toone counting strategy but make one or two non-systematic counting errors and Many-to-one counting 89 therefore do not come up with the correct answer. (b) The children use the manyto-one counting strategy, but one row is added or not counted. (c) The children use the many-to-one counting strategy, but one column is added or not counted. (3) Not clear if strategy is correct, incorrect answer: Rudimentary many-to-one counting. The children count more than one card per animal and do count across animals, but one or more rows and columns are (partially) added or not counted. There are at least two row/column errors. (4) Incorrect strategies: Separate-groups counting. The children count the cards per set. They do not count across sets and give the number per set as an answer. Objects counting. The children count the animals. One-extra counting. The children count the animals plus one extra. Counting of example group. Only in the needed-items task do the children count the number of cards that the animals of the example group on the left-hand side have received. Repeating the number of cards. The children just repeat the number of cards given to each animal without counting them and without pointing to different animals. Ambiguous. Strategies that cannot be identified. No response. The children show no strategic behaviour and do not give an answer. Because implicit many-to-one counting (that is, a correct answer without any noticeable pointing or verbal counting before answering the ‘how many cards’ question) occurred in only about 1% of the cases, this category was combined with the explicit many-to-one counting category. For the same reason, the categories ‘one-extra counting’, ‘counting of example group’, ‘repeating the number of cards’, ‘ambiguous response’, and ‘no response’ have been coded as ‘other’. So, six categories remained to describe children’s strategic behaviour. (1) Many-to-one counting (M–1) implicit or explicit; (2) Many-to-one counting with errors (M–1 er.); (3) Rudimentary many-toone counting (Rudim.); (4) Separate-groups counting (Sep.Gr.); (5) Objects counting (Objects); and (6) Other. The second scoring was for the children’s counting behaviour. The categories were pointing (to the perceptually absent items) and counting out loud, pointing with internalized counting, counting out loud, internalized counting (lip movements or other signs of counting), and other (this included guessing). Sometimes, the children spontaneously corrected themselves, for example in a false start. When this occurred, only the last response that was given was coded. The scoring was done from the videotaped material by two raters. Slightly more than 10% of the children’s on-task behaviour was scored by both raters. The interrater agreement was 91% for the strategy categories and 99% for the counting method. Results Most children, while counting, pointed to the perceptually not present items, in the hidden-items task in 95%, and in the needed-items tasks in 92% of the counts. This high frequency of pointing behaviour greatly facilitated the scoring of the counting strategies used by the children. The children’s strategies produced in their first and second attempts of each task have been combined into the variable ‘best response’. The reason for analysing children’s best attempts instead of their second attempts is that in some cases (less than 4%), children reverted to a less advanced strategy in their second 90 Anke W. Blöte et al. attempt. All statistical tests will be reported on a .01-level of significance unless indicated otherwise. Discovery of the strategy During Week 1, six children correctly used the many-to-one strategy on all four tasks. They had apparently learned to use many-to-one counting before the experiment started. They were therefore excluded from the following analysis. Thirty-eight of the remaining 46 children produced the strategy at least once in the first week. Nearly all of these children started to produce the target strategy on a hidden-items task. They then persisted in using it in the next hidden-items task, but not on the following neededitems task (see Table 1). Only one child started to use the target strategy on a neededitems task. Eight children discovered the strategy after the first week. Five of them used it for the first time on a hidden-items task and three on a needed-items task. Throughout the 4 weeks, a total of 42 children used the many-to-one strategy for the first time on a hidden-items task. Of the 21 children who were presented with the twoitem hidden-items task as their first task, 20 started to use the target strategy on this two-item task. Only one child started it on the following three-item task. Of the 21 children who were first presented with the three-item task, 13 began to use the strategy on this three-item task. Eight children started on the following two-item task. Both order of task presentation and the number of items in the task had a significant effect on strategy production. The first task more often elicited the first use of the target strategy, 2 = 13.71. Apart from that, the children’s first production of the strategy more often occurred on the task with fewer items, 2 = 5.09, p <. 05. Only four children discovered the target strategy on the needed-items task, two did so on the two-item task, and two did so on the three-item task. Table 1. Frequencies of first use and maintenance of the many-to-one strategy in the first week for children who started with a two-items task (N = 19) or a three-items task (N = 19) Hidden-items task 1 Hidden-items task 2 Needed-items task 1 Needed-items task 2 2-items tasks first First use Maintenance 18 – 1 17 – 7 – 6 + 4a 3-items tasks first First use Maintenance 13 – 5 13 1 4 – 4 + 6b a 6 children had used the strategy before in the two-items needed-items task, and four children had used it before but only in the hidden-items tasks. b 4 children had used the strategy before in the three-items needed-items task, and 6 children had used it before, but only in the hidden-items tasks. Strategy development The progress in development of the advanced strategy during the 4 weeks of the experiment is presented in Table 2. In all sessions and on all tasks, the accurate Many-to-one counting 91 performance of the many-to-one counting strategy was prevalent. This not withstanding, it is clear that the occurrence of both inaccurate use of the many-to-one strategy and the use of less advanced strategies were not negligible. We conducted a MANOVA with three within factors, session (1–4), task (hidden, needed), and number of items (2, 3), one between factor, age (4, 4½), and with many-to-one counting (irrespective of any counting errors) as the dependent variable. Strategy scores 1 (manyto-one counting) and 2 (many-to-one counting with errors) had been recoded as 1 and the other scores as 0. Three main effects and one interaction effect were significant. First, the production of the many-to-one strategy increased over time, F(3, 48) = 16.50. Second, the use of the strategy was more frequent on the hidden- than on the neededitems tasks, F(1, 50) = 12.54. Third, the many-to-one counting strategy was used more often by the 4½-year-olds (M = 0.85) than by the 4-year olds (M = 0.64), F(1, 50) = 8.53. The number of items did not have a significant effect on the production of the advanced strategy. The Week Task effect was significant, F(3, 48) = 9.09. Initially, the children used the advanced strategy more often on the hidden- than on the needed-items tasks. However, this difference disappeared over time as the frequency of the strategy increased relatively more on the needed tasks than the hidden tasks. The initial difference in frequency of the target strategy between the hidden- and neededitems tasks is also visible in Table 1. Relatively speaking, many children who had used many-to-one counting in the hidden-items tasks regressed to more primitive strategies in the following needed-items tasks. Table 2. Frequencies (in percentages) of the different strategies for the two hidden- and two needed-items tasks over 4 weeks (N = 52) Week 1 Week 2 Week 3 Week 4 HiddenNeededHiddenNeededHiddenNeededHiddenNeededitems task items task items task items task items task items task items task items task M-1 M-1 er. Rudim. M-1 Separate groups Objects Other Total 45 31 3 7 3 11 100 37 6 3 12 24 18 100 58 22 5 8 0 7 100 63 9 3 5 12 8 100 71 15 2 4 0 8 100 62 20 2 5 8 3 100 76 12 5 4 0 3 100 72 15 3 2 8 0 100 Role of strategy effectiveness In studying the immediate effect of strategy effectiveness on strategy development, the following approach has been taken. The outcome of applying the many-to-one counting strategy on a certain task was linked to strategy production on the subsequent task. In each week, children’s strategy outcome and production were studied in relation to three pairs of subsequent tasks, (a) the first hidden-items task and the second hiddenitems task, (b) the second hidden-items task and the first needed-items task, and (c) the first needed-items task and the second needed-items task. In each analysis, those children were selected who had used many-to-one counting on the first task of the pair. 92 Anke W. Blöte et al. A cross-tabulation was then made of strategy outcome (answer correct or incorrect) on this first task by strategy production (many-to-one counting or an invalid strategy) on the second task. In Week 1, children did not often switch from many-to-one counting on one task to an invalid strategy on the following task. The transition from the first hidden-items task (with 17 children using the target strategy correctly and 19 making counting errors) to the next hidden-items task showed only one child reverting to an invalid strategy (see Table 1). From the first needed-items task (with 16 children using the target strategy correctly and 2 making counting errors) to the next one also, one child reverted. In both cases, the invalid strategy followed after many-to-one counting with counting error. The transition from the second hidden-items task to the first needed-items task more often resulted in regression to invalid strategies. Among the 36 children who produced the target strategy in the second hidden-items task 25 regressed to invalid strategies in the needed-items task, 16 out of 24 did so after correct many-toone counting and 9 out of 12 after many-to-one counting with an error. This difference as tested with chi-square was not significant. With regard to the following weeks, the same pattern as in the first week appeared for the transitions within pairs of hidden- and needed-items tasks. The transition from hidden- to needed-items tasks, however, was different as the number of invalid strategies used in the needed-items task decreased (see Table 3). From Week 2 to Week 4, in only 13 out of 132 times did children regress from many-to-one counting to an invalid strategy, 5 times after correct and 8 times after incorrect counting. In conclusion, no evidence was found that the outcome of the many-to-one strategy had an immediate effect on children’s strategy production be it in the first week of the study or in later weeks. To investigate the role of the target strategy’s effectiveness in the strategy’s long range development, we discriminated between children who, in Week 1, experienced negative outcomes on all their many-to-one counts and children who received positive or mixed outcomes with the advanced strategy. All children were grouped on the basis of their strategy usage in the first week taking their best attempts on all four tasks into account. There was a group of nine children who in Week 1 produced many-to-one counting with mixed outcomes and no invalid strategies (the ‘M–1 only’ group). As these children did not regress to invalid strategies in the first week, it was expected that they would not do so in the following weeks either. A second group of 11 children (the Correct M–1 group) used correct many-to-one counting and invalid strategies only. These children received positive feedback after using many-to-one counting and negative feedback after using the invalid strategies. One would expect therefore that they would abandon the use of the invalid strategies relatively quickly. A third group of 11 children (the Mixed M–1 group) applied many-to-one counting, on some trials making counting errors and on other trials counting correctly, along with invalid strategies. This group experienced success as well as failure after many-to-one counting. This could make them less certain about the value of the many-to-one strategy. It was expected that they therefore would not turn to the full use of many-to-one counting as quickly as the first group. A fourth group of 7 children (the Incorrect M–1 group) only used incorrect many-to-one counting and invalid strategies. As these children received negative feedback on their use of many-to-one counting as well as their other strategies, there would be no reason for them to favour the advanced strategy over invalid strategies. Figure 2 shows the production of the many-to-one counting strategy (both with and without counting errors) in three of the groups over the 4 weeks. The ‘M–1 only’ group has been left out because all but one of the children in this group had full use of the Many-to-one counting 93 strategy in all 4 weeks. We conducted a MANOVA with repeated-measures design on the sum scores (per task type and per week) of the use of the many-to-one strategy, with week (1–4) and task (hidden-, needed-items) as within factors and group (2, 3, 4) as between factors. No significant interaction effects relating to task were found. Therefore, a second MANOVA was performed without this task variable. The effect of group was significant, F(2, 26) = 10.14, M = 3.32, 3.48, and 2.07 for the respective groups. The children who showed at least one correct many-to-one count in Week 1 had a relatively high production of this advanced strategy over the 4 weeks, and the children who only performed incorrect many-to-one counts showed a relatively low production. Surprisingly, no difference was found between children who had only some of their counts correct and those who had all of them correct. The usage of the advanced strategy significantly increased over the 4 weeks, F(3, 24) = 36.32; M = 2.17, 3.20, 3.51, and 3.41 for Weeks 1–4, respectively. Regarding the role of strategy effectiveness in producing the many-to-one strategy the Time Group interaction was of particular interest. This interaction was nearly significant, F(3, 24) = 2.24, p = .054. Figure 2 shows that the children who only produced incorrect many-to-one counts along with invalid strategies in Week 1 did not change their strategy repertoire much in Week 2. In contrast, the other two groups did produce significantly more many-to-one counts in Week 2 than in Week 1, t(10) = 9.82 and t(10) = 5.24 for the Correct M-1 group and the Mixed M-1 group, respectively. In the next 2 weeks, the children of the Incorrect M–1 group caught up with the other groups to some extent. During the last 2 weeks, the difference between the three groups was no longer significant (however, Figure 2. Development of many-to-one counting in three groups of children with different strategy outcome in the first week. 94 Anke W. Blöte et al. this could be the result of ceiling effects in the Correct and Mixed M–1 groups). From Week 3 on, the Incorrect M–1 group started to produce some correct many-to-one counts. In Week 4, three out of the seven children in this group produced many-to-one counting on all four tasks. It should be noted that from the first week on, the Incorrect M–1 group produced fewer many-to-one strategies than the other two groups. The children in this group used the target strategy only once or twice in the first week. The slow development of their many-to-one counting could be related to the limited number of times the strategy was produced in the first week. We took the following steps to rule out initial production frequency as an explanation. We selected those children in the other two groups who, in the first week, had also used the strategy only once or twice. In the Mixed M–1 group, no children had Score 1, and only four children had Score 2. It was therefore decided not to use this group in the comparison. The Correct M-1 group counted seven children with Scores 1 or 2. The mean score in the first week of this group (M = 1.71) was not significantly different from that of the seven children in the Incorrect M-1 group (M = 1.43). A repeated-measures MANOVA then showed that the Week interaction effect, F(3, group effect, F(1, 14) = 5.22, p <.05, and the Group 10) = 7.09, were still significant. The mean scores of the Correct M-1 group were 3.57, 3.71, and 3.29 in Weeks 2–4, respectively. The mean scores of the Incorrect M-1 group were 1.71, 2.57, and 2.57, respectively.1 Thus, these results suggest that the outcome of children’s strategies and not their production rate played a role in later strategy production. The above-mentioned results were as expected, except that there were no differences between the Correct and Mixed M-1 groups. Evidently, making counting errors did not negatively influence strategy maintenance, providing children produced at least one correct many-to-one count. The children in the Mixed M-1 group, like those in the Correct M-1 group, very quickly disposed of the invalid strategies in their repertoire, whereas the children who only produced many-to-one counts with errors were slower in their adoption of the target strategy. Discussion An important finding of this study was that the negative feedback after an incorrect application of the many-to-one strategy did not keep the children from producing the strategy again on the next task. The results showed that in many cases, children who counted incorrectly in their first application of the target strategy still repeated the same strategy on the following task. This suggests that these children knew that the strategy was suited to solve the problem. Another finding was that the children who, in the first session, used many-to-one counting without success, along with invalid strategies, were slow in adopting many-to-one counting on all tasks. These results may seem contradictory: no immediate effect of outcome on strategy production yet longer-term effects in a selected group of children who initially made errors in all their counts. However, on closer examination, there may be no contradiction after all. It all could depend on a difference between the short-term effect of a negative outcome and the longer-term effect of a complete lack of success. The lack of success kept the children in the Incorrect M-1 group from turning to a full 1 The mean scores of the four children in the mixed M-1 group were 3.0, 3.5, and 3.5 in Weeks 2, 3, and 4, respectively. Many-to-one counting 95 use of the target strategy. (It did not keep them from using the strategy again.) For these children, it was just a strategy like the other strategies they used. None of their strategies helped them to find the correct answer. Other children who had successfully applied the target strategy at least once changed very quickly to a full use of the strategy, even if many of their counts were incorrect. It appears that the positive feedback they received on some of their many-to-one counts presented them with enough evidence about the relative value of this strategy. There is, however, an alternative explanation. It may be that the children in the Incorrect M-1 group were different from those in the other two groups as far as the level of their cognitive development was concerned. Their counting errors in Week 1 may have indicated that producing the target strategy was relatively difficult for them. The slow rate of their strategy development then may have had the same cause. The present study cannot exclude this latter interpretation. Another finding of the study was that the first production of the many-to-one counting strategy occurred relatively often in the hidden-items task with 2 items per object. This finding is in line with the Becker (1993) study. Thus, children produced the target strategy for the first time on the task with the lowest task demands. This suggests that the children either discovered the strategy on these tasks or realized that many-toone was the correct strategy to use on the task. They then very quickly generalized their strategic knowledge to the more difficult three-item task and less quickly to the neededitems tasks. The children initially were less likely to produce the advanced strategy on the needed-items tasks than on the hidden-items tasks, even with fewer items to count in this task and even after having had some experience with the many-to-one counting problems in the hidden-items tasks. Conclusions regarding differences between the hidden- and needed-items tasks are tentative because the order of presentation is a confounding factor. Becker (1993) found the same effect and also mentioned the problems in interpreting it. We will come back to the issue in the General Discussion, where we pay attention to the role of task difficulty in the generalization of the strategy. EXPERIMENT 2 In Experiment 2, the generalization of the many-to-one strategy to more complex tasks was studied. Children who had mastered the use of many-to-one counting in the first experiment were confronted with some new tasks that differed from those in the first experiment in one of two ways: (1) the number of objects and the number of items per object were larger; (2) there were no objects on the table to help the children keep count of the sets of items. We hypothesized that the increased task difficulty would result in mental-capacity problems. This would cause children to make counting errors in the application of the target strategy or to revert to invalid strategies. Evidently, when the number of items in the task increases or the indicators of the sets of items are removed, there is a higher chance of counting errors. The most interesting question therefore was whether children would regress to invalid strategies. 96 Anke W. Blöte et al. Method Participants The 33 children who, in Week 4 of Experiment 1, produced the many-to-one counting strategy on all 4 tasks without making counting errors were selected for participation in Experiment 2. Two of these children had to be removed because they were unable to count to 20 during a one-to-one counting task. Twenty is the highest number the children needed to count to complete the tasks for the current experiment. Two more children were lost due to non-attendance on the test date, and one child was excluded because he failed to produce the many-to-one strategy on the reminder task (see below). Therefore, 28 children participated, of whom 8 were 4 years old, and 20 were 4½ years old. These children had used the advanced strategy on most tasks during the 4 weeks of Experiment 1. More precisely, they had produced the strategy on 94% of the trials and did so without making counting errors on 86% of the trials, demonstrating that the selected children had truly mastered the use of the strategy. Materials In addition to the materials described in Experiment 1, a piece of cardboard with 2 4 attached cards (the same type of cards as used in the other tasks) and a 20-cm-wide and 30-cm-high screen were used in the non-visual hidden-items task. Tasks The first task was the basic counting task. Twenty-four cards were placed in a 4 6 arrangement in front of the child. The child was asked to count these cards out loud. This was done to ensure that each child could count to 20, which was the maximum number the children were required to count to during this experiment. If the child made more than one counting error or was unable to count to 20, the session was terminated. In the reminder task, the child was presented with the needed-items task with three cards from Experiment 1. This was done to remind the child of how the previous many-to-one counting tasks had been solved because the last session had 3 arrangement was chosen occurred 2 weeks earlier before a school vacation. A 3 because the answer ‘nine’ did not occur in the subsequent tasks in the session. The extended hidden-items task was similar to the hidden-items task used in Experiment 1, except for the number of animals and the number of cards per animal, which were larger. The child was presented with five zoo animals that had received four cards each. The extended needed-items task was similar to the needed-items task used in Experiment 1, except for the number of animals and the number of cards per animal, which again were larger. The child was asked to give four animals four cards each. The non-visual hidden-items task is an adaptation of one of the tasks used by Anghileri (1989). The child was shown a card and told that on the covered cardboard in front of them, a few of those same cards were attached. Subsequently, the top row of the covered 2 4 array on the cardboard was shown. The child was asked to count how many cards were next to each other (the other row was hidden by a screen). Next, the child was asked to count how many cards were underneath each other in the left-hand column (all other columns were covered by the screen). The child was then shown the entire array for 2 s. In the following step, the array was placed face down on the table with the screen on top, so that the cards could not be seen. The researcher checked if Many-to-one counting 97 the child remembered how many cards were next to each other and how many cards were underneath each other. The child was then asked to figure out the total number of cards under the screen. The child was asked to count out loud and show the experimenter how they solved the problem. If the child did not react or answered incorrectly, the question was repeated. After the second attempt, the child was told whether the answer was correct or not. In the non-visual needed-items task, the child was asked to imagine that there were three dogs. The child was told that each dog should receive two cards. The experimenter asked the child how many cards they would need all together to give each of the three dogs two cards. The child was asked to count out loud and show the experimenter how they solved the problem. If the child did not react or answered incorrectly, the question was repeated. After the second attempt, the child was told whether the answer was correct or not. Procedure The testing session took place 2 weeks after the last session of the first experiment. Each child was presented with six tasks. First, the basic counting task was administered. Second, the reminder task was presented. Third, the extended and non-visual tasks were administered. In both cases, the hidden-items tasks were given before the neededitems tasks, as was the case in Experiment 1. The order of task type (extended tasks and non-visual tasks) was counterbalanced. Different animals and card types were used for each task. The children’s counting strategies were scored according to the coding criteria described under Experiment 1. One category was added, Counting of seen cards. Only in the non-visual hidden-items task did the children add up the number of cards in a row to the number of cards in a column, both of which they had been asked to count. The inter-rater agreement based on 12.5% of the material was 95% for the categories of the strategy variable and 97% for counting method. Results In more than 90% of the counts, children pointed to the perceptually-not-present items while counting. Changes in strategy use All 28 children used the many-to-one counting strategy without errors in Week 4 of Experiment 1 and in the reminder task. Thus, all children had learned how to use the target strategy. To study the transfer to the two extended and two non-visual tasks, we first combined the strategies children used in their first and second attempts in a variable ‘best response’. Over the four tasks, children’s best responses counted 86% use of the advanced strategy, 50% accurate use, and 36% with counting errors. Table 3 shows that the production of the many-to-one strategy, irrespective of counting errors, did not change when children solved the extended tasks. The frequency of occurrence of this strategy was still 100%. There was, however, a considerable drop in correct outcome of this strategy. Table 3 further suggests that the children made more errors in the extended hidden- than in the extended needed-items task, but a paired t test yielded no significant difference between the two tasks. 98 Anke W. Blöte et al. Table 3. Frequencies (in percentages) of correct and incorrect many-to-one counting for the two extended and two non-visual tasks (N = 28) M-1 M-1 er. Invalid strategies Total Extended hidden-items task Extended needed-items task Non-visual hidden-items task Non-visual needed-items task 43 57 0 100 64 36 0 100 32 25 43 100 61 28 11 100 In the non-visual tasks, the errors were not just counting errors (see Table 3). The children also started to produce invalid strategies. (They did this more often on the hidden- than on the needed-items task, paired t = 2.8, p = .01.) Thus, changing the tasks by removing the objects the items had to be allotted to made children regress to invalid strategies as well as to inaccurate use of the advanced strategy. Discussion All children had used the many-to-one strategy without any errors on all tasks in Week 4 of Experiment 1 as well as on the reminder task of Experiment 2. The new tasks then resulted in children (a) reverting to invalid strategies and (b) making counting errors in applying the valid strategy. The question posed in the Introduction addressed the effect of task demands on children’s strategy use. Higher task demands require the children to spend more effort to solve the problem. Capacity problems then can influence strategy production and application. The expectation was that children would make counting errors in the new tasks but also would revert to less advanced strategies. The data show that it depended on the way in which the new task was different from the former tasks. Only if the perceptible objects the items had to be allotted to were removed did children start to produce less advanced strategies. Set size was not important to the production of the strategy. This finding is not in accordance with the results of the Shannon (1978) study. One likely explanation for this difference between the studies’ outcome is the nature of the tasks used. The items to be counted in our study were linked to a small number of objects lying in a straight line on the table. These objects were indicators of the subsets of items and therefore could have helped the children to organize their count. In contrast, in the Shannon study, the children had to organize their counting themselves, and no structure was provided. In this way, transfer from one task to a next one that had more items was more difficult in that study. The children more often regressed on the non-visual hidden- than the needed-items task. It should be noted that the order of presentation of the two tasks was fixed and therefore was a confounding factor. Possibly, the children regressed on the first nonvisual task (the hidden-items task), because it was a new task to them, and then recovered on the second, needed-items task. An alternative explanation is that the difference in strategy production had to do with the fact that the non-visual hiddenitems task conceptually was more different from the original task than the non-visual Many-to-one counting 99 needed-items task. The non-visual needed-items task referred to a task situation the children already knew from the former experiment, the only difference being that they now had to visualize the situation instead of seeing the reference set and the three animals to which the items had to be allotted. The non-visual hidden-items task, in contrast, did not relate the items to objects. In the instruction, only the array of items was mentioned and briefly shown. The children then had to find out how many cards 4 array. The task was relatively easy in the sense that the complete were in this 2 array was, like in the original hidden-items tasks, briefly shown to the children. However, the problem was more abstract than the original task in not relating the items to objects. GENERAL DISCUSSION The present study added to previous research on strategy development in two ways. First, it has shown that both initial use and maintenance of the many-to-one strategy were related to task characteristics. Second, it was found that after an incorrect manyto-one count, children would continue to produce many-to-one counting in the next task. However, there was also a group of children who took a relatively long time in improving their counting. Initially, all their many-to-one counts were incorrect. These children were relatively slow in their transition from less advanced strategies to complete many-to-one counting. At first, the many-to-one strategy was to these children just a possible strategy like the other (invalid) strategies. Task complexity Relatively often, children used the advanced strategy for the first time in tasks with two items per object. This suggests that set size is important to the initial use of the advanced strategy. Counting across subsets of two obviously is a less complex task than counting across subsets of three or four. Moreover, children will have more experience with sets of two and therefore have more insight in how to handle such sets (Becker, 1993). As a consequence, more processing capacity remains (Miller & Seier, 1994) to think about a strategy and subsequently produce the many-to-one strategy. Generalization of the target strategy to tasks with a larger set size per object and more objects did not pose a problem to the children. In the first experiment, after using the strategy in two-item tasks, they quickly transferred this knowledge to three-item tasks. Furthermore, all children who had learned the strategy in Experiment 1 also produced it in the more extended tasks of Experiment 2. This finding suggests that the number of items in the many-to-one counting task is not important to strategy generalization. The study also showed that discovery of the many-to-one strategy nearly always occurred on the hidden-items tasks. The knowledge was then generalized to solve the needed-items tasks (and later on the non-visual tasks). Here, like in Becker’s (1993) study, the hidden-items task appears the easiest of the two. With the hidden-items tasks, it is not completely clear whether children count the items in relation to the objects on the table and therefore do many-to-one counts or that they just count the items one-byone using their memory image of the array of items, without paying attention to the objects. Probably, the children rely on both their memory image of the perceptually- 100 Anke W. Blöte et al. not-present items and the objects lying on the table. A strong indication that the objects help them to perform a many-to-one count is presented by their responses in the nonvisual hidden-items task. If the children rely completely on their memory image of the items to do a one-by-one count instead of a many-to-one count, why then is the nonvisual hidden-items task so difficult for them? In this non-visual task, they have to 2 array shown to them. They can count these remember only the rather simple 4 then one by one based on their memory image of the array of items. The answer might be that the children interpret the task as a many-to-one counting task and therefore try to find a many-to-one strategy. The non-visual hidden-items task requires—by only referring to items and not to objects—an understanding of multiplication. It can therefore be considered the last stage in the development of many-to-one counting. In this development, children would start with counting items in a many-to-one way helped by an image of the array of items distributed over objects. They then learn to anticipate the quantity in a many-to-one distribution when answering questions about how many items are needed in order to give x objects y items. The example set on the table reminds them what the many-toone distribution should look like. Subsequently, the perceptual clues of the example set and the objects are no longer needed. Children can answer questions like ‘How many cards are needed to give three dogs two cards each?’ This task still refers to a concrete situation of distributing items over objects. On the highest level, in the non-visual hidden-items task, children no longer need this concrete distribution. They have learned to solve basic multiplication tasks without referring to objects. Strategy effectiveness The children often continued the production of the many-to-one strategy, although the occurrence of counting errors in the application of the strategy prevented them from finding the correct answer. A similar phenomenon has been described in the literature about utilization deficiency (Miller, 1990; Miller & Harris, 1988). Novice users have been found to persist in using a new and valid strategy, although it does not help them to improve their performance. An explanation that has been offered for this behaviour in previous studies is that the children in those studies may not have been aware of the outcome of their strategies because they did not receive feedback on their answers (Miller & Seier, 1994). However, in the present study, the children did receive negative feedback and still continued the use of the many-to-one strategy. An alternative explanation for children’s perseverance in using the advanced strategy might be that, even before they have mastered the use of the advanced strategy, they have a goal sketch (Siegler, 1996) about the requirements of a valid strategy. In the present study, it was emphasized in the instruction to the children that we wanted to know the number of ‘all cards taken together’. This may have triggered children’s understanding that they had to count on over sets of items and kept them from using Counting objects or Separate-groups counting as a strategy. In that case, their developing metastrategic knowledge guided their strategy production (Kuhn, 2000; Kuhn & Pearsall, 1998). In the same vein, in a study using children that were a little older, Kuhn and Phelps (1982) stressed the importance of metastrategic knowledge not only for the selection of effective strategies but also for the abandonment of less effective strategies. The children who initially have all their counts wrong keep trying the target strategy, be it at a lower rate. Eventually, they too increase their production of Many-to-one counting 101 the strategy. This suggests that they also develop metastrategic knowledge about the validity of their strategies. Educational implications The present study yielded two findings that might be important to the teaching of multiplication. First, children begin to understand the principle of multiplication in its most simple form, that is, many-to-one counting, at a very early age. The behaviour of Lisa, one of the girls in the study, illustrates what 4-year-olds are capable of. Lisa, after counting the perceptually not present items in the hidden-items task with 3 items concluded: ‘three times four equals twelve’! Second, most children do not need explicit instruction about how to solve these many-to-one counting tasks. Just by repeatedly working on these tasks and receiving feedback on the correctness of their answers, children spontaneously develop the understanding that is needed to find the right strategy. This offers the possibility of presenting children with many-to-one counting at an early age and build on their understanding as a basis for higher-level multiplication strategies. A girl in the study, Anne, vertically counted three cards (instead of four) at each of the four lions of the extended needed-items task and said that she would need 12 cards all together in order to give each lion the same number of cards as the elephants. The experimenter then told her that this was not correct and repeated the question: ‘How many cards would you need all together in order to give each lion four cards, just like the elephants?’ Realizing her mistake, Anne then horizontally counted on from 12 and pointed below each animal ‘giving’ (there were no cards there) each of them one more card, ‘13, 14, 15, 16 . . . I’d need 16 cards!’ This 4-year-old girl not only corrected her mistake but also used the most efficient strategy, the ‘min strategy’ (Siegler & Jenkins, 1989), to do this. With the two examples mentioned here, we are not just presenting the performance of gifted children. The finding that, in the second experiment, nearly two-thirds of the children produced the right strategy on a multiplication problem in which no objects were present to allot items to also suggests that children as young as 4 years of age can develop both metastrategic and strategic understanding of many-to-one counting. Acknowledgement We thank Patricia Miller for constructive feedback on an earlier version of the manuscript. References Alibali, M. (1999). How children change their minds: Strategy change can be gradual or abrupt. Developmental Psychology, 35, 127–145. Anghileri, J. (1989). An investigation of young children’s understanding of multiplication. Educational Studies in Mathematics, 20, 367–385. Becker, J. (1993). Young children’s numerical use of number words: Counting in many-to-one situations. Developmental Psychology, 29, 458–465. 102 Anke W. Blöte et al. Chung, P. C., Dijksman, T., Kilian, Y., Mead, G., Van der Hoeven, V., Van der Plaat, M., Van der Werf, K., Van Wijngaarden, M., & Blöte, A. W. (1999). Ontwikkeling in strategiegebruik bij kleuters. [Preschoolers’ strategy development]. Unpublished manuscript, Leiden University. Gelman, R., & Meck, E. (1983). Preschoolers’ counting: Principles before skill. Cognition, 13, 343–359. Kuhn, D. (2000). Metacognitive development. Current Directions in Psychological Science, 9, 178–181. Kuhn, D., & Pearsall, S. (1998). Relations between metastrategic knowledge and strategic performance. Cognitive Development, 13, 227–247. Kuhn, D., & Phelps, E. (1982). The development of problem-solving strategies. In H. Reese (Ed.), Advances in child development and behavior (Vol. 17, pp. 2–44). New York: Academic Press. Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change: Contributions to children’s learning of multiplication. Journal of Experimental Psychology: General, 124, 83–97. Miller, P. H. (1990). The development of strategies of selective attention. In D.F. Bjorklund (Ed.), Children’s strategies: Contemporary views of cognitive development (pp. 157–184). Hillsdale, NJ: Erlbaum. Miller, P. H., & Harris, Y. R. (1988). Preschoolers’ strategies of attention on same-different tasks. Developmental Psychology, 24, 628–633. Miller, P. H., & Seier, W. L. (1994). Strategy utilization deficiencies in children: When, where, and why. In H. W. Reese (Ed.), Advances in child development and behavior, (Vol. 25, pp. 107– 156). New York: Academic Press. Newell, A. (1990). Unified theories of cognition. Cambridge, MA: Harvard University Press. Nunes, T., & Bryant, P. (1996). Children doing mathematics. Oxford: Blackwell. Shannon, L. (1978). Spatial strategies in the counting of young children. Child Development, 49, 1212–1215. Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. New York: Oxford University Press. Siegler, R. S., & Crowley, K. (1991). The microgenetic method: A direct means for studying cognitive development. American Psychologist, 46, 606–620. Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ: Erlbaum. Sophian, C. (1998). A developmental perspective on children’s counting. In C. Donlan, (Ed.), The development of mathematical skills (pp. 27–41). Hove, UK: Psychology Press. Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–40). Albany: State University of New York Press. Received 15 August 2002; revised version received 11 March 2003