How Children Change Their Minds: Strategy Change Can Be Gradual Or Abrupt

Developmental Psychology Copyright 1999 by the American Psychological Association, Inc. 1999, Vol. 35, No. 1, 127-145 0012-1649/99/$3.00 How Children Change Their Minds: Strategy Change Can Be Gradual or Abrupt Martha Wagner Alibali Carnegie Mellon University This study investigated patterns of change in children's strategies for solving mathematical equivalence problems. The strategies children expressed in speech and in gesture were assessed both before and after an instructional intervention. In the intervention, children received either no input, accuracy feedback, or feedback plus instruction about a principle, an analogy, or a procedure. From pretest to posttest, many children changed both the variability of their strategy use and the content of their strategy repertoires. Patterns of change depended on type of instruction and on children's initial level of variability. Children who received instruction were especially likely to generate new strategies, and children with high variability were especially likely to abandon prior strategies. Gradual change was most common; however, many children modified their repertoires abruptly. Abrupt strategy change was especially prevalent among children who received procedure-based instruction and among children with low initial variability. Developmental researchers typically seek to identify common- Nevertheless, many theoretical accounts of developmental alities in children's behavior. The implicit goal is to identify change accord variability a central explanatory role. Indeed, a consistent patterns that hold across children and across contexts, in variety of theoretical formulations, including Piagetian, neo- an effort to characterize children's predominant way of thinking or Piagetian, information-processing, connectionist, and dynamic behaving at a given age. From this perspective, the "signal" of systems frameworks, posit variability as either a prerequisite for or interest in developmental data must be discerned from the "noise" an indicator of developmental change. Within some theoretical caused by variability across or within children. Consequently, frameworks, variability is causally related to change. For example, much empirical work about development emphasizes consistency in the Piagetian view, the impetus for change comes from disequi- in children's thinking. librium, which can be caused by "internal conflict" between com- peting approaches to a problem (Piaget, 1975/1985). Within other frameworks, variability is a concomitant of change but is not the Portions of the data were presented at the annual meeting of the Jean causal force driving change. For example, in the dynamic systems Piaget Society, Chicago, Illinois, June 1994; the biennial meeting of the framework, behavioral variability is manifested at transition points Society for Research in Child Development, Indianapolis, Indiana, March (Thelen, 1989), but variability appears to be an indicator of ongo- 1995; and the Seventh European Conference on Developmental Psychol- ing change rather than a cause of change. ogy, Krakow, Poland, August 1995. This research was supported by a National Science Foundation Graduate Despite its theoretical importance, until recently, surprisingly Fellowship, a William Rainey Harper Dissertation Fellowship, and a little empirical research has examined variability per se. However, P.E.O. Scholar Award to Martha Wagner Alibali and by Grant R01 in the past several years, along with the shift toward more process- HD31185 from the National Institute of Child Health and Human Devel- oriented accounts of developmental change, researchers have be- opment to Susan Goldin-Meadow. gun to take variability seriously (Siegler, 1996; Thelen & Smith, I thank Susan Goldin-Meadow for her contributions at all stages of this 1994). When the potential importance of variability is recognized, project. I also thank Karen Adolph, Erin Alexander, Pete Alibali, Larry what was previously considered noise becomes data of central Barsalou, Miriam Bassok, Jeff Bisanz, R. Breckinridge Church, Philip interest. Accordingly, developmentalists using many different Garber, Lisa Haverty, Janellen Huttenlocher, Spencer Kelly, James Mc- methodologies have begun to document and quantify variability in Clelland, David McNeill, Jill Morford, Bethany Rittle-Johnson, and Robert Siegler for helpful discussions and for constructive feedback on previous children's behavior (e.g., Acredolo & O'Connor, 1991; Acredolo, versions of the article. I am also grateful to Lucia Flevares, Michael Kang, O'Connor, & Horobin, 1989; Crowley & Siegler, 1993; Siegler & Spencer Kelly, Martha Scott, and Sharon Syc for assistance with data Jenkins, 1989). Variability is found not only in strategic behaviors collection, transcription, and coding, and to Brian Junker, Aidan Palmer, but also at many other levels, such as problem representation and Stella Salvatierra, and Zdenek Valenta for assistance with statistical anal- synaptogenesis (see Siegler, 1988). At all levels, variability allows ysis and interpretation. Finally, I am especially grateful to the students, organisms to adapt to local environmental conditions. As such, it teachers, and administrators at St. Ambrose, St. Ann, St. Bede, St. Gall, St. makes sense that children's behavior is variable. Michael, and St. Rita elementary schools in Chicago, Illinois, who made Across several different theoretical frameworks and content this work possible. Correspondence concerning this article should be addressed to Martha domains, one consistent empirical finding has been that periods of Wagner Alibali, Department of Psychology, Carnegie Mellon University, conceptual transition are marked by high variability. Knowledge Pittsburgh, Pennsylvania 15213. Electronic mail may be sent to change has been associated with variability in patterns of motor alibali @ andrew.cmu.edu. performance (Adolph, 1997; Thelen, 1989; Thelen & Ulrich, 127 128 ALIBALI 1991), variability in the methods children use to arrive at problem change they will make. Children who start with low levels of solutions (Strauss & Rimalt, 1974), variability across problems in variability should generate new strategies, thereby increasing vari- children's verbally reported strategies (Siegler, 1995), and variability, and children who start with high levels of variability should ability within individual problems in the strategies children ex- abandon strategies, thereby decreasing variability. In this frame- press in speech and in the accompanying gestures (Church & work, strategy generation can be viewed as a process of variation, Goldin-Meadow, 1986; Perry, Church, & Goldin-Meadow, 1988). and strategy abandoning can be viewed as a process of selection Thus, there is strong empirical support for the view that variability operating on variability. is heightened during periods of knowledge transition. It also seems likely that the surrounding environment could In general, the available data suggest that conceptual change is induce changes in children's strategy mix. In particular, environ- a cyclical process that involves moving from periods of lesser to mental contexts that involve instruction (such as school) are a greater and then back to lesser variability. This view suggests that, highly salient source of new problem-solving strategies. Instruc- at a minimum, the process of change must involve adding new tion may serve to introduce strategies into children's repertoires, structures (e.g., representations, productions, or strategies) to the either directly or by setting up conditions that facilitate strategy repertoire, thereby increasing variability, and then removing or discovery. Instruction may also serve to eliminate strategies from integrating old structures, thereby decreasing variability. For ex- children's repertoires, either directly or by implication. Thus, ample, within a strategy choice framework, the process of change instruction could serve both to heighten and to diminish variability. can be viewed as a process of expansion and then contraction of The present study examines instruction as a source of strategy the strategy repertoire (see, e.g., Siegler, 1995). change, by assessing how children modify their strategy reper- Thus, there is growing evidence that there are systematic and toires in response to several different types of instruction. Specif- cyclical changes in the variability of children's behavior over the ically, the study examines how children change their strategies in course of learning. However, relatively little is known about the response to (a) feedback about whether solutions are correct, (b) sources of these changes or about the overall shape of change. The instruction about a specific problem-solving procedure, (c) instruc- present study addressed two specific questions about the process of tion about a principle presented in verbal form, and (d) instruction strategy change in children learning a mathematical concept. First, about a principle presented through an analogy. These particular what leads children to generate or abandon strategies? Two poten- types of instruction were chosen on the basis of prior findings tial sources of change are considered: the child's initial level of about knowledge change and instruction. First, feedback about variability and the environmental input (in this study, instruction) correctness has been shown to promote strategy change (Ellis, provided to the child. Second, is strategy change gradual or Siegler, & Klahr, 1993). Second, analogies have been shown

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