W hat Leads Children to Adopt New Strategies 3

Learning of class inclusion by 5-year-olds in response to empirical and logical explanations of an adult’s answers was examined. Contrary to the view that young children possess an empirical bias, 5-year-olds learned more, and continued learning for longer, when given logical explanations of correct answers than when given empirical explanations. Once children discovered how to solve the problems, they showed few regressions. Many children in the microgenetic experiment followed the path of change anticipated from previous cross-sectional studies, but children in the cross-sectional part of the study seemed to follow a different path. Reasons for the superior effectiveness of the logical explanations were discussed. What Leads Children to Adopt New Strategies 3 What Leads Children to Adopt New Strategies? A Microgenetic / Cross Sectional Study of Class Inclusion The primary goal of the present study was to compare young children’s learning from logical and empirical explanations of reasoning. A second main goal was to examine the relation between short-term microgenetic changes and longterm age-related changes. These goals were pursued in the context of the classic Piagetian class inclusion task, a task that can be solved either logically or empirically and that therefore allows direct comparison of the effects of encountering the two types of explanations. This introductory section includes four parts. In the first part, we describe current understanding and unresolved issues regarding children’s adoption of new strategies in general. In the second, we examine current knowledge and unresolved issues regarding acquisition of class inclusion. In the third, we introduce the microgenetic / cross-sectional design as a way of addressing the main issues in the present study. In the fourth, we describe what we did in the study and our hypotheses regarding the results. Acquiring New Strategies A number of recent studies have examined how children discover new strategies (e.g., Alibali, 1999; Chen & Klahr, 1999; Goldin-Meadow & Alibali, 2002; Granott, 2002; Kuhn et al., 1995). These studies have shown that discovering a strategy is often only the first step in strategic change; prior strategies often persist, even when the newly discovered strategy has clear advantages and even when children can describe those advantages (Siegler & Jenkins, 1989). Thus, the adoption of new strategies is often quite slow. It is also the case, however, that the rate of adoption of new strategies varies considerably across studies. Two factors that seem to be related to this rate have What Leads Children to Adopt New Strategies 4 been identified: accuracy and efficiency. A recent literature review (Siegler, in press) concluded that the largest factor in determining the rate of adoption of newly discovered strategies is the difference between their accuracy and that of previously available strategies. In the relatively few cases in which newly discovered strategies quickly replace previous approaches, the new strategies have entailed large improvements in accuracy, generally from consistently incorrect to consistently correct. Differences between the efficiency of new and previous strategies also appear to influence the rate of adoption of new strategies, though to a lesser degree than differences in accuracy do. When both new and old strategies generate correct performance, new strategies that confer large advantages in speed or substantial reductions in processing steps are adopted more rapidly than new strategies that confer less dramatic advantages (Siegler, in press). In the present study, we examined a third factor that may influence the rate and completeness of adoption of new strategies: whether there are logical as well as empirical reasons for adopting them. Some strategies are based on the fundamental logic of the problem. For example, on conservation of weight problems, one correct logical strategy is to note that nothing was added or subtracted when the clay was remodeled, so the weight must be the same. Other strategies are empirical; for example, rather than solving the conservation of weight problem logically, a child could take the empirical approach of weighing the clay before and after its shape was transformed, and noting that the weight remained the same. Many other problems also can be solved either empirically or logically. For example, number conservation tasks can be solved either by counting the objects in each row or by noting that nothing was added or subtracted; transitivity of length problems can be solved either by measuring the length of the sticks being compared or by drawing the logical inference “If A > B and B > C, then A > C”; arithmetic problems of the form A + B – B = ___ can be solved either by adding the first two numbers and What Leads Children to Adopt New Strategies 5 subtracting the third or by deducing that the answer must be “A,” because adding and subtracting the same number cannot have an effect; and so on. The specific issue addressed in the present study was whether young children learn more rapidly and completely when they are encouraged to use logical or empirical strategies for solving a problem. In particular, we compared 5-year-olds’ learning in response to logical and empirical explanations of correct answers to class inclusion problems. Contrasting acquisition of two strategies for solving the same problem was a more direct way of establishing factors that influence acquisition of new strategies than previous techniques, which involved reviewing findings from different tasks and populations, and then attempting to isolate the factors associated with more and less rapid and complete acquisition (Siegler, in press). There were reasonable arguments for expecting either logical or empirical strategies to be acquired more effectively. First consider arguments that suggest that young children should acquire empirical strategies more easily. Age peers of the children whose learning was examined in the present study, 5-year-olds, have been found to rely on empirical strategies in a number of situations in which older children and adults use logical reasoning strategies. When presented tautologies (e.g., “I either am holding a green chip in my hand or I’m not”) or contradictions (“I am holding and not holding a green chip in my hand”), and asked whether they needed the experimenter to open her hands to tell if the statement was true or false, 5-year-olds usually chose the empirical strategy; in both cases, most children said they needed the experimenter to open her hands to tell (Osherson & Markman, 1975). Most 5-year-olds also confuse logically determinate and logically indeterminate problems and are subject to “positive capture,” in which they think that a case consistent with a conclusion implies that the conclusion must be true (Fay & Klahr, 1996; Klahr & Chen, 2003). On class inclusion tasks in which What Leads Children to Adopt New Strategies 6 children are asked whether a set or the larger of two subsets has more objects (e.g., “I have five dogs and three cats; do I have more dogs or more animals”), most 5year-olds compare the number of objects in the two subsets (e.g., five dogs versus three cats) rather than reasoning that there must be more in a set than in any of its subsets (Piaget, 1952). Young children also have been found to use empirical strategies on other tasks somewhat similar to class inclusion. When asked if they could place more spoons than silverware on the table (Markman, 1978) or whether they could draw a picture that has more cats than animals (S. Miller, 1986), most 5year-olds said “yes,” rather than reasoning that this was logically impossible. Also favoring the prediction that children should learn more from empirical explanations, 5-year-olds seem to have a particular fondness for the type of empirical strategy relevant to class inclusion, counting. They use counting even on tasks where such a strategy is inappropriate. For example, K. Miller (1984) found that 5-year-olds created “fair” divisions of food between two birds by counting out equal numbers of pieces of food, regardless of the sizes of the pieces. Similarly, Levin (1989) found that 5-year-olds often counted the passage of two temporal durations and judged the higher number to indicate the longer time, regardless of whether the counting had proceeded at the same speed. This reliance on counting strategies, and on empirical rather than logical strategies in general, suggested that 5-year-olds would learn empirical strategies more readily than logical ones. On the other hand, Siegler and Svetina (2002) suggested that logical strategies might be learned more quickly than empirical ones. The above-cited findings of young children using empirical strategies in situations where older children and adults use logical ones may reflect the young children not understanding the logical strategies. If the logic was made clear to them, they might prefer the logical strategies (or have no preference between the two). In addition, several of the microgenetic studies in which newly discovered strategies have been What Leads Children to Adopt New Strategies 7 adopted most rapidly have involved acquisition of logical reasoning strategies. These include strategies for solving matrix completion, A + B B, and 20 Questions problems (Siegler & Stern, 1998; Siegler & Svetina, 2002; Thornton, 1999). Thus, the fact that young children use empirical strategies to solve problems that adults and older children solve via logical strategies does not imply that children would choose empirical strategies if they understood that the logical strategies could be used to solve the problems. Moreover, there were reasons to think that young children might prefer logical reasoning strategies if they understood how to execute and apply them to particular problems. Logical strategies often have large advantages in speed and accuracy. In the context of class inclusion, solving the problem logically both requires less time and avoids the possibility of errors in counting and numerical comparison. Anticipating these benefits, or learning about them through experience with the strategies, might lead children to consistently use new logical strategies more rapidly than new empirical ones. In addition, to the extent that children are searching for basic logical understanding of domains, as theorists such as R. Gelman & Williams (1998) and Keil (1998) have proposed, logical strategies might be adopted faster and more completely than empirical strategies precisely because they are logically compelling. The Development of Class Inclusion Understanding The development of class inclusion was first studied by Piaget (1952). Piaget defined understanding of class inclusion as the ability to compare sets of objects that are at different levels of a hierarchical organization. He viewed it as a crucial type of reasoning because it brings together understanding of classes and relations, requires understanding of hierarchies, and requires reliance on logical rather than intuitive reasoning under conditions that somewhat favor intuitive reasoning What Leads Children to Adopt New Strategies 8 (Inhelder & Piaget, 1964). These features also have made the task of continuing interest. To investigate the development of class inclusion, Piaget (1952) and Inhelder and Piaget (1964) presented children with drawings of two types of entities from a common set, for example five dogs and two cats, and asked whether there were more entities in the set or in the larger subset (more animals or dogs). Children younger than age 7 or 8 tended to say that there were more objects in the larger subset, which Inhelder and Piaget interpreted as supporting their theory that preoperational children are unable to simultaneously view a single object as being a member of both a class and a subclass (e.g., as being both an animal and a dog.) Inhelder and Piaget’s findings triggered a flurry of research on the development of understanding of class inclusion. These studies replicated the original finding that children younger than 7 or 8 years rarely solve the Piagetian class inclusion task consistently correctly. Reviews of the class inclusion literature (Halford, 1993; Winer, 1980) indicate that not until around 8 years of age do 50% of children in the U. S. correctly answer the standard Piagetian version of the task, and that not until age 9 or 10 do 75% of children do so. Subsequent research also extended Inhelder and Piaget’s findings in several directions. Success on class inclusion tasks was found to correlate positively with success on other measures of logical reasoning, including conservation of number, liquid quantity, and solid quantity (Tomlinson-Keasey, Eisert, Kahle, Hard-Brown, & Keasey, 1979). Many task variables were found to influence class inclusion performance (Trabasso, et al., 1978; Wilkinson, 1976). One involved the number of objects in the subsets: Performance was best when the subsets were equally numerous and worst when one subset had far more objects than the other (Ahr & Youniss, 1970). Linguistic variables also proved highly influential: Describing the set via collection nouns (e. g. forest, army, team), which call attention to part-whole What Leads Children to Adopt New Strategies 9 relations, greatly facilitated performance relative to describing the set via class nouns (trees, soldiers, children), as in the original Piagetian studies (Markman & Seibert, 1976). Another influential variable was the typicality of subordinate class members; using typical category exemplars (e. g. dogs and horses) was associated with better performance than using atypical ones (flies and bees) (Carlson & Abrahamson, 1976). Children also were found to use class inclusion reasoning more often on tasks that did not include misleading properties; for example, after being told “A pug is a kind of dog,” 4-year-olds respond “yes” more often than chance when asked, “Is a pug a kind of animal?” (Smith, 1979). Much less is known about how children learn about class inclusion than about their performance at different ages. However, some informative data have been obtained. These data can be described within Siegler’s (1996) five dimensions of change framework, which characterizes changes in terms of their path, rate, breadth, source, and variability. The path of change concerns the sequence of knowledge states or problem solving approaches that children use while gaining competence. The rate of change involves the amount of time/experience before children’s first use of a new approach and the amount of time/experience that separates initial use of a new approach from consistent use of it. The breadth of change involves changes in performance on related tasks. The source of change concerns the causes that set the change in motion. The variability of change concerns betweenand within-child variability on the other dimensions of change. Analyzing the acquisition of class inclusion understanding in terms of these five dimensions indicates that current understanding is largely limited to two areas. The greatest amount of previous research has examined the path of change. Several investigators (Chapman & McBride, 1992; Hodkin, 1987; McCabe, Siegel, Spence, & Wilkinson, 1982) have suggested a three-phase progression. In the first phase, between ages 4 and 6 years, children are said to solve class inclusion problems by What Leads Children to Adopt New Strategies 10 comparing subclasses, which leads to below-chance performance. In the second phase, around 6 to 7 or 8 years, children answer correctly at about a chance level (50%). This has been hypothesized to come about through guessing (e.g., Hodkin, 1987), though it also could come about through some children answering consistently correctly and other children answering consistently incorrectly (or through all children using a strategy other than guessing that generates a chance level of performance.) In the third phase, beginning at around age 8 or 9 years, children are said to compare the set to the larger subset, and therefore to generate consistently correct answers and explanations. Some information also is available regarding the source of change. In particular, explicitly telling children about the relation between superordinate and subordinate classes has been shown to help children learn about class inclusion (Judd & Mervis, 1979). Growth of working memory and acquisition of inclusion schema also appear to be sources of improved understanding of class inclusion (Halford, 1993; Rabinowitz, Howe, & Lawrence, 1989). Little is known about the other three dimensions of cognitive change. With regard to the breadth of change, success at class inclusion correlates positively with success on other logical reasoning tasks, but whether gaining understanding of class inclusion has broader effects is unknown. With regard to the rate of change, the fact that 5-year-olds solve some variants of the problem but that many 8-year-olds do not solve the classic Piagetian task might be taken to imply a slow rate. However, it is unclear that the classic task and the variants are solved via the same cognitive processes, and very little is known about the rate of acquisition of any particular task in response to relevant experience. With regard to the variability of change, essentially nothing is known. To advance understanding of these five dimensions of cognitive change, we employed a microgenetic/cross-sectional design in the present experiment. What Leads Children to Adopt New Strategies 11 Microgenetic/Cross-sectional Designs Over the past two decades, an increasing number of investigators have adopted the microgenetic approach as a means for studying cognitive development (Kuhn, 1995; Miller & Coyle, 1999; Siegler, in press). The main reasons are the precise descriptions of changing competence that the approach can yield and the implications of those precise descriptions for understanding change processes. The micogenetic approach is defined by three primary characteristics: 1) observations span the whole period of rapidly changing competence; 2) the density of observation within this period is high, relative to the rate of change; 3) observations of changing performance are analyzed intensively to indicate the processes that give rise to them. The second characteristic is especially important. Dense sampling of performance while the performance is changing provides the temporal resolution needed to adequately describe the process of change. Often, microgenetic designs track strategic change on a trial-by-trial basis (e.g., Adolph, 1997; Alibali, 1999). The data yielded by such dense sampling of strategy use allow identification of the precise trial on which a child discovered a new strategy, which in turn allows analyses of the events that led up to the discovery and of how the strategy was generalized beyond its initial use. An especially encouraging characteristic of microgenetic studies is that despite the diversity of content areas and populations to which they have been applied, findings from them have been surprisingly consistent (Kuhn, 1995; Miller & Coyle, 1999; Siegler, in press). A common finding regarding the path of change is that just before discovery of a new approach, performance becomes more variable (Goldin-Meadow, Alibali & Church, 1993; Graham & Perry, 1993; Siegler, 1995; Siegler & Svetina, 2002). The rate of change tends to be gradual, with less adequate, initial approaches continuing to be used well after more advanced approaches also have emerged (Bjorklund, Miller, Coyle & Slawinski, 1997; Kuhn et What Leads Children to Adopt New Strategies 12 al., 1995; Siegler & Jenkins, 1989). The breadth of change usually is relatively narrow (Kuhn et al, 1995; Schauble, 1990, 1996; Siegler, in press). Variability tends to be high: children learn via different paths, at different rates, and with differing amounts of generalization. Finally, certain sources of cognitive growth, such as encouragement to explain observations, operate over a wide age range and in diverse content domains (Bielaczyc, Pirolli & Brown, 1995; Chi, de Leeuw, Chiu & La Vancher, 1994; Pine & Messer, 2000; Siegler, in press). Thus, examining cognitive growth along the five dimensions has proven useful for identifying regularities in how change occurs. One question that has been raised about microgenetic studies concerns the similarity of changes within them to changes with age in traditional cross-sectional and longitudinal designs. There is widespread agreement that the changes are similar at a general level but much less agreement as to the extent of the similarity (Granott, 1998; Kuhn, 1995; Miller & Coyle, 1999). For example, after reviewing the microgenetic literature, Miller and Coyle (p. 212) concluded, “Although the microgenetic method reveals how behavior can change, it is less clear whether behavior typically does change in this way in the natural environment” (italics in original). To address this issue, Siegler and Svetina (2002) proposed the microgenetic / cross-sectional design. The basic strategy was to combine cross-sectional and microgenetic components within the same experimental design, using the same population, tasks, instructions, and measures of performance. In the first use of this design, Siegler and Svetina found that microgenetic and age-related changes in acquisition of matrix completion were highly similar. The question investigated here was whether microgenetic and cross-sectional analyses of acquisition of class inclusion would also show extensive parallels. The Present Study What Leads Children to Adopt New Strategies 13 In the microgenetic experiment in the present study 5-year-olds were presented 30 class inclusion problems, 10 problems in each of three sessions. The children were randomly divided into four groups; the groups varied in the explanations provided to children regarding why the correct answer was correct. Consider the explanation that children in each group received following incorrect answers to a problem that involved six dogs and three cats. Children in the empirical explanations group were told that there were more animals than dogs because there were nine animals but only six dogs. To illustrate the counting procedure that yielded this outcome, the experimenter pointed to each animal in turn before saying there were nine, and to each dog in turn before saying there were six. Children in the logical explanations group were told that there were more animals than dogs because dogs are just a type of animal, so if there were dogs and some other animals, there must be more animals than dogs. Children in the logical and empirical explanations group were provided both explanations. Children in the no explanations group were given feedback on the correctness of their answer (as were children in the other three groups) but were not provided any explanation. On trials on which children answered incorrectly, they were asked to judge how smart the experimenter’s explanation of the correct answer was. This was intended to provide a measure of conceptual understanding of the logical and empirical explanations. Previous studies have shown that conceptual understanding, as indexed by ability to answer questions about the intelligence of strategies, often diverges from procedural knowledge, as indexed by use of correct strategies. Sometimes conceptual understanding precedes use of correct strategies (Siegler & Crowley, 1994), sometimes the reverse is true (Briars & Siegler, 1984), and sometimes the two types of knowledge are highly similar (Cauley, 1988). This microgenetic design allowed us to characterize the source, rate, path, breadth, and variability of change. The source of change was examined by What Leads Children to Adopt New Strategies 14 comparing changes in performance among children who received the four types of explanations. The rate of change was examined by analyzing the number of trials that elapsed before children first used the correct strategy and the number of trials before they did so consistently. The path of change was examined by analyzing performance in the period of rapid learning—the trials immediately before and immediately after children discovered the correct solution to the problem. The breadth of change was examined by analyzing the relation between changes in strategy use and changes in conceptual understanding, as measured by children’s evaluations of the smartness of logical and empirical explanations as well as their answers to problems. The variability of change was examined by analyzing individual differences in the other dimensions of change. In the cross-sectional part of the study, 5-, 6-, 7-, 8-, 9-, and 10-year-olds were presented 10 class inclusion problems without any feedback. The problems were randomly sampled from the set of 30 presented to 5-year-olds who participated in the microgenetic part of the study. Because children in the cross-sectional and microgenetic parts of the study were chosen from the same population, and because the identical problems, procedures, and measures were presented in both parts, it was possible to evaluate in an unusually direct way the similarity between microgenetic and age-related change. We tested four hypotheses. The first hypothesis was that accuracy in all four groups in the microgenetic segment of the study would improve over trials and sessions. If 5-year-olds’ predominant strategy at the outset was to compare the two subsets, as the literature suggested, the feedback that children in all groups received would disconfirm their initial approach and therefore move them from below chance to chance or above chance performance. This was expected to be due to the feedback, rather than to simple exposure to class inclusion problems or What Leads Children to Adopt New Strategies 15 experience answering them. Thus, no comparable improvements in accuracy were expected over trials for 5-year-olds in the cross-sectional part of the study. The second hypothesis was that both logical and empirical explanations would increase learning. Both types of explanation illustrated strategies that would yield accurate performance. In contrast, feedback would disconfirm incorrect approaches but would not indicate a correct one. The third hypothesis was that logical explanations would elicit greater learning than would empirical explanations. Previous findings in which 5-year-olds used empirical rather than logical strategies may have been due to the children not realizing that the problems could be solved logically. That is, the preschoolers might reflexively assume that questions asked by an adult must be solved empirically, because that almost always is the case. How often are young children (other than those of academics) presented questions where the answer can be deduced from the question itself? This view, that preschoolers may not consider the possibility that the information needed to answer a question is contained in the question itself, is different from saying that preschoolers cannot employ such strategies. Indeed, we believe that if it is clear to young children that a problem can be solved deductively, they will find such strategies preferable to empirical approaches. The sources of appeal of the logical strategies for young children would presumably be the same as for older individuals: perfect accuracy, rapid execution, and intellectual elegance. Although this final characteristic is difficult to define, we hoped to measure it through obtaining children’s and adults’ evaluations of the smartness of empirical and logical strategies. A final hypothesis involved the relation between microgenetic and age-related change. Evidence from microgenetic studies suggests that after using an incorrect strategy for a substantial period of time, children often begin to guess or oscillate among alternative approaches before generating a stable, advanced strategy. What Leads Children to Adopt New Strategies 16 (Siegler & Svetina, 2002; Spencer et al., 2000; Thelen & Corbetta, 2002; van Geert, 2002.) It seemed likely that receiving feedback on an unfamiliar task such as class inclusion would lead after a short time to guessing or inconsistency in the present task as well. Thus, we expected to see the same three state path of change in the present microgenetic context as has been proposed in the literature on age-related change in class inclusion performance – from reliance on subset-subset comparisons, which produce systematically incorrect performance; to guessing or oscillation among approaches, which produces approximately chance performance; to set-subset comparisons, which produce consistently correctly performance. Method Participants The children were 200 Caucasian 5to 10-year-olds of Slovenian ethnic background who were recruited from kindergartens and elementary classrooms of several schools in a predominantly middle class area in Ljubljana, Slovenia: 100 5year-olds, M = 66.7 months, range = 60-71 months; 20 6-year-olds, M = 79.4 months, range = 75-83 months; 20 7-year olds, M = 87.4 months, range = 84-91 months; 20 8year-olds, M = 100.5 months, range = 96-107 months; 20 9-year-olds, M = 111.8 months, range = 108-119 months; and 20 10-year-olds, M = 124.4 months, range = 120-130 months. Most children in each school returned the consent forms; the 200 children were randomly selected from among those children. Children seemed to enjoy the study; the only children who did not complete the study were five who either were ill for an extended period or moved. A 35-year-old male researcher served as the experimenter. Task and Procedure Class Inclusion Task Children in the microgenetic part of the study received 30 trials, 10 trials in each of 3 sessions, with the sessions presented over a period of 2 weeks, with an What Leads Children to Adopt New Strategies 17 average of 5.4 days between successive sessions for each child. Children in the cross-sectional part of the study received a randomly chosen 10 of these 30 problems in a single session. On each trial, children in both the microgenetic and cross sectional segments were presented a piece of paper with a black and white drawing. Each drawing included 8-10 objects (e. g. animals), of which 2 or 3 belonged to one sub-class (e. g. cats) and 6 or 7 to a different subclass (e. g. dogs). The drawings included diverse categories of objects from children’s everyday experience (Figure 1). The categories were trees (apple and pines), food (pieces of bread and sausage), toys (bears and balls), musical instruments (violins and trumpets), adults (men and women), children (boys and girls), clothing (jackets and pants; skirts and pants), footwear (shoes and boots), garden tools (rakes and spades), tools (hammers and saws), furniture (chairs and chests; tables and chairs), headwear (caps and hats), silverware (forks and spoons), fruit (apples and pears; bananas and strawberries), vehicles (trucks and buses), vegetable (beets and carrots), birds (crows and owls), kitchen tools (pots and pans), flowers (tulips and daisies), letters (A and G; B and E), numbers (1’s and 2’s), and animals (geese and chickens; horses and cows; rabbits and turtles; dogs and cats; cats and dogs). (Dogs and cats are listed twice, because on one problem there were more dogs and on another problem there were more cats.) Ratios of subclasses to classes were 2:6, 2:7, 3:6, or 3:7. Problems were presented in random order, with the randomization done separately for each child. On every item, children were asked to count the number of objects in one subset, then the number in the other subset, and then the number in the set. For example, on one problem, the experimenter said (pointing to the corresponding objects), “Look at these pictures; first count all the dogs, then all the cats, and then all the animals." After each answer, the experimenter said, “OK”. Next, the experimenter asked the child either (e.g.,) “Are there more dogs or more What Leads Children to Adopt New Strategies 18 animals in the picture?” or “Are there more animals or more dogs in the picture?” followed by the question “How did you know?” The set was mentioned first on half of the questions within each session; the larger subset was mentioned first on the other half. Experimental Conditions The 80 5-year-olds who were in the microgenetic experiment were randomly assigned to the four experimental conditions. Children in all four groups were told, “Yes, that’s right” when they answered correctly and were told “No, that’s wrong” when they answered incorrectly. Where the groups differed was in the explanation that the experimenter provided on trials where a child answered incorrectly. In the empirical explanations condition, the experimenter said, “A couple of days ago, I asked a teacher about that problem. She said there were more animals than dogs, because there were nine animals (pointing to each of the animals) and only six dogs (pointing to each of the dogs).” In the logical explanations condition, the experimenter said, “A couple of days ago, I asked a teacher about that problem. She said there were more animals than dogs, because dogs are a type of animal, so if there are some dogs and some other animals, there must be more animals than there are dogs.” In the both explanations condition, the experimenter provided both empirical and logical explanations by saying that she had asked two teachers and one had provided one explanation and the other had provided the other (each type of explanation was stated first on half of the trials.) In the no explanations condition, the experimenter said, “A couple of days ago, I asked a teacher about that problem. She said that there were more animals than dogs because he/she just knew it.” In all four groups, children were asked after hearing the explanation, “Do you think this answer was very smart, not smart, or kind of smart?" The 120 5to 10-year-old participants in the cross-sectional segment of the study were presented 10 class inclusion problems, as in Session 1 of the What Leads Children to Adopt New Strategies 19 microgenetic experiment. However, they were not provided explanations or feedback concerning the correctness of their answers. Adults’ Evaluations of the Explanations To test the assumption that adults would see logical explanations as superior to empirical ones, we asked 60 Caucasian college students of Slovenian ethnic background, aged 18-21, in their first semester in the School of Arts, Ljubljana, Slovenia, to rank how smart different types of explanations to class inclusion problems were. Specifically, we asked them to rank four potential responses to class inclusion problems: correct answer and logical explanation; correct answer and empirical explanation; correct answer and no explanation; and incorrect answer and no explanation. For example, the college students were presented an overhead showing eight animals and the following story: “There were three horses and five cows in the yard. A group of children came by. They stopped and a teacher asked the children whether there were more cows or more animals in the yard. John said there were more animals in the yard because cows were just a type of animal. Ann said there were more animals in the yard because there were eight animals and only five cows. Brigit said there were more animals in the yard because she just knew it. Dennis said there were more cows in the yard because he just knew it.” The college students were asked to rank the responses on each problem from smartest (1) to least smart (4). The overhead displaying a given problem and the four responses to it remained visible until the college students completed their ranking. Each student was presented four such problems, each including the same four types of answers and explanations. Cronbach's alpha coefficients indicated that students were consistent in their judgments across the four stories—coefficient alphas for the logical and empirical explanations were .84 and .91, respectively. What Leads Children to Adopt New Strategies 20 Results To test our assumption that adults viewed the logical explanations as more intelligent than the empirical ones, we first analyzed the college students’ ratings of the four answers and explanations. An ANOVA yielded differences among their evaluations of the four responses, F(3, 177) = 310.92, p < .001, h = .84. The logical explanations (mean ranking of 1.07 on the 1-4 scale) were ranked as smarter than the empirical explanations (mean 2.08), which were ranked as smarter than correct answers without explanations (mean 2.88), which were ranked as smarter than incorrect answers without explanations (mean 3.94). These data supported our expectation that adults would find logical explanations smarter than empirical ones, and empirical explanations smarter than no explanation. The remaining analyses examined children’s performance in the main experiment. On each trial, we coded children’s answer and their explanation for it. However, the explanations proved uninformative. The large majority of children said, “I don’t know,” or failed to respond when asked why they answered as they did. Efforts to motivate children to provide more informative explanations were unsuccessful. For this reason, we relied exclusively on the pattern of answers to infer strategy use. The results are presented in two sections. In the first section, we describe findings from the microgenetic experiment; these results are organized in terms of the source, path, rate, breadth, and variability of change. In the second section, we describe changes with age in class inclusion performance and compare microgenetic and age related changes. If not otherwise stated, post-hoc comparisons are Newman-Keuls tests with p < .05. Microgenetic Experiment Source of Change What Leads Children to Adopt New Strategies 21 To examine the effects of the logical and empirical explanations on children’s class inclusion performance, we first conducted a 2 (logical explanations: present or absent) X 2 (empirical explanations: present or absent) X 3 (session) ANOVA on each child’s number of correct answers. The analysis revealed main effects of logical explanations, F(1, 76) = 4.94, p < .05, h = .06, and session, F(2, 152) = 35.20, p < .001, h = .32. Children who received logical explanations generated more correct answers than children who did not, and number of correct answers increased over sessions. A logical explanations by session interaction was also present, F(2, 152) = 5.85, p < .01, h = .07. To better understand the source of the interaction, separate logical explanations X empirical explanations analyses were undertaken for each session. In Session 1, performance did not differ as a function of the type of explanation that children received. In contrast, in both Sessions 2 and 3, children who received logical explanations answered the class inclusion problems more accurately than did those who did not, F(1, 76) = 6.23, p < .05, h = .08, and F(1, 76) = 6.10, p < .05, h = .07. As shown in Figure 2, children who received logical explanations already answered somewhat more accurately than those who did not in Session 1, but the size of the difference was greater in Sessions 2 and 3. No main effect of empirical explanations and no interaction involving that variable were found. Children only were exposed to the logical and empirical explanations when they answered incorrectly; therefore, the session variable only roughly corresponded to children’s experience with the experimental manipulations, because different children in a given group received different numbers of explanations. To more directly examine the effect of the history of explanations on children’s accuracy, and also to circumvent the problem of autocorrelated residuals, we performed a What Leads Children to Adopt New Strategies 22 transition analysis (Diggle, Liang, & Zeger, 1994). This involved a logit model of the following form: 4 LogitP[Yij = 1 ̇Hij] = b0 + gi + S akf(Hij) k=1 Where Yij is the jth response of the ith child, Hij is the history of responses of childi up to trialj, b0 is the intercept, gi is a random effect for childi, ak is a coefficient for each experimental group indicating the effect of the experience provided to that group, and f(Hij) is the number of incorrect answers (and therefore the number of explanations in the groups that received explanations) encountered by childi up to trialj. Results of this transition analysis indicated significant effects for number of trials with explanations in the group that received both logical and empirical explanations (odds ratio = 1.23, standard error = .0393, p < .001) and in the group that received logical explanations alone (odds ratio = 1.15, standard error = .0469, p < .001.) There were no effects for the empirical explanations group or for the no explanations group. The effect of logical explanations and the lack of effect of empirical ones were not due to pre-existing differences among the children who were presented the two experimental manipulations. Performance on the first trial of the study, the only trial before children in the four groups received different treatment, was closely similar; percent correct of children who did and did not receive logical explanations was identical, 40% correct, and percent correct among children who did and did not receive empirical explanations was almost identical, 42% and 38% respectively. The data also allowed us to examine whether feedback in the absence of explanations was sufficient to improve children’s accuracy. Surprisingly, feedback was insufficient for learning to occur. The evidence came from a comparison of the performance of 5-year-olds in the cross-sectional comparison, who did not receive What Leads Children to Adopt New Strategies 23 feedback, with that of peers in the no explanations condition of the microgenetic experiment, who did receive feedback. The 5-year-olds in the cross-sectional comparison received 10 class inclusion problems, selected from the same set as the 10 problems received by peers in the no explanations condition in Session 1. The problems were not identical but did not differ in any systematic way. The number of correct answers on the 10 problems of 5-year-olds in the crosssectional comparison did not differ from the number of correct answers of peers in the microgenetic experiment who received feedback without explanations, t(38) = 1.01, p > .05, d = .32. In addition, accuracy of both groups declined slightly from the first five to the second five problems that they received, suggesting that neither exposure to the problems nor receiving feedback elicited learning. Moreover, in the no explanations condition in the microgenetic experiment, percent correct did not increase over the three sessions, F(2, 38) = 2.55, p > .05, h = .12. Thus, feedback on whether responses were correct was insufficient for 5-year-olds to improve their class inclusion performance. Rate of Change The trial of discovery was operationally defined as the first trial within the first set of five consecutive correct answers that the child generated. Thus, if a child answered incorrectly on trials 1 and 4 and then correctly on trials 5-9, the child would be said to have made the discovery on trial 5. Given that one of the two alternative answers on each problem was correct, the chance probability of generating five consecutive correct answers was .5 or approximately 3%. This criterion for discovery was chosen to minimize the probability of both false positives and false negatives. The results did not vary substantially with the particular criterion; for example, 70% of children made the discovery at some time during the three sessions by the five consecutive correct answers criterion, and 74% would have made it if a three consecutive correct answers criterion had been used. What Leads Children to Adopt New Strategies 24 Most children who ever met the criterion met it quite early in the experiment. In particular, of the children who met the criterion in any of the three sessions, 82% met it in Session 1. To examine the rate of discovery more precisely, we graphed the cumulative percentage of discoveries that had occurred by each of the 30 trials within the microgenetic experiment (Figure 3). In addition to illustrating that more children in the two groups that received logical explanations made the discovery, Figure 3 also illustrates that the difference between those who did and did not receive logical explanations arose through the logical explanations condition stimulating discoveries over a longer period. The difference is particularly clear in comparing the rates of discovery of children who only received the logical explanations with that of children who only received the empirical ones. Through the first 5 trials, more children who only received empirical explanations than who only received logical ones made the discovery (35% versus 20%.) In contrast, many more children who received logical explanations made the discovery after the fifth trial (45% versus 15%). Viewed from a different perspective, 69% of children who received logical explanations who had not yet made the discovery by Trial 5 made it afterward, whereas only 30% of children who received empirical explanations did, c(df = 1, N = 23) = 3.49, p < .10, V = .39. Once children met the discovery criterion, they used the correct approach with striking consistency. This can be seen by examining performance in the next session, which was removed by a mean of 5.4 days from the session in which the discovery occurred. Children in previous multi-session microgenetic studies have often shown a great deal of regression to previous strategies from one session to the next (e.g., Siegler & Jenkins, 1989; Siegler & Stern, 1998), but this was not the case in the present study. Children who discovered the correct strategy in Session N answered correctly on 97% of the first five questions in Session N+1. The findings What Leads Children to Adopt New Strategies 25 did not depend on the original discovery criteria requiring a relatively long string of consecutive correct answers. When a looser discovery criterion of three consecutive correct answers was used, children who met the criterion in one session answered 91% of problems correctly on the first three trials of the following session. To examine the effect of the instructional variables on the consistency of use of the newly discovered correct strategy, we performed a 2 (logical explanations: present or absent) X 2 (empirical explanations: present or absent) ANOVA on the number of incorrect answers after the child met the discovery criterion. Consistency of use proved to be unaffected by these variables. Once children discovered the correct solution, they used it consistently, regardless of the type of explanation that they were given; in none of the four conditions did children average more than 1.3 errors in the more than 20 trials (on average) following their discovery. Path of Change To examine where the path leading to the discovery began, we compared to chance children’s percent correct on all trials before the discovery. The results showed that the percentage correct on trials before the discovery (34%) was below chance, t(40) = 3.85, p < .001, d = .60. However, the data appeared to include two sub-periods, an earlier period in which accuracy was far below chance and a later period in which it was at a chance level. It also appeared that the five consecutive correct trials that operationally defined the discovery marked the transition to a third period, one of consistently correct performance. Thus, acquisition of understanding of class inclusion appeared to include three periods: a period of consistently incorrect responding, a period of chance responding, and a period of consistently correct responding. Several sources of evidence were consistent with the conclusion that most children began the study with a systematically incorrect approach to the problems. On the first item, the only item presented before children received feedback, What Leads Children to Adopt New Strategies 26 accuracy was below chance. Excluding the 15 children who came to the experiment already knowing how to solve the problems (as defined by their generating five correct responses on the first five items), only 26% of children answered the first problem correctly, which was well below chance, t(64) = 4.34, p < .001, d = .55. Another source of evidence was that among the 30% of children who never met the criterion of five consecutive correct answers, percent correct was far below chance, 6% correct, t(23) = 23.03, p < .001, d = 4.65. Lending further support to the conclusion of an initial state that produced below chance responding, accuracy of age peers in the cross-sectional sample, who were not presented feedback on any of the 10 problems they encountered, was also well below chance, 30%, t(19) = 3.57, p < . 01, d = .56. More than half of the 5-year-olds in the cross-sectional sample (55%) were correct on 10% or fewer of the problems. Evidence for a period of chance levels of accuracy and a period of consistently correct performance came from a comparison of the percentage of correct answers on the five trials before the discovery and the five trials immediately after it. This comparison was limited to the 28 children who made the discovery, performed at least three trials prior to it, and performed at least three trials after it; the reason for the last two criteria was to allow reasonably stable estimation of performance both before and after the discovery. Performance on the trials immediately prior to discovery was at a chance level, 46% correct, t(27) = .85, p > .05, d = .16. Performance on the trials immediately after the discovery was very accurate and far above chance, 90% correct, t(27) = 7.45, p < .001, d = 1.99. Again, the particular criteria for inclusion in the analyses were not critical; the same pattern emerged with a criterion of at least two trials before the discovery, 43% correct, t(32) = 1.44, p > .05, d = .27; and after it, 90% correct, t(32) = 8.38, p < .001, d = 2.13.