As street vendors, these children solved the problems mostly orally by ... Seo and Ginsburg demonstrate the richness of insight that can arise when researchers.
European Journal of Psychology of Education 2009, Voi. XXIV, n'3, 335-359 © 2009,1.S.P.A.
Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education Li even Verschaffel Koen Luwel Katholieke Universiteit Leuven, Belgium Joke Torbeyns Leuven Education College / Katholieke Universiteit Leuven, Belgium Wim Van Dooren Katholieke Universiteit Leuven, Belgium
Some years ago, Hatano differentiated between routine and adaptive expertise and made a strong plea for the development and implementation of learning environments that aim at the latter type of expertise and not just the former. In this contribution we reflect on one aspect of adaptivity, namely the adaptive use of solution strategies in elementary school arithmetic. In theflrstpart of this article we give some conceptual and methodological reflections on the adaptivity issue. More specifically, we critically review definitions and operationalisations of strategy adaptivity that only take into account task and subject characteristics and we argue for a concept and an approach that also involve the sociocultural context. The second part comprises some educational considerations with respect to the questions why, when, for whom, and how to strive for adaptive expertise in elementary mathematics education.
Introduction A few years ago, the international research community of developmental and educational psychologists lost one of its principal scholars, namely Giyoo Hatano. In his foreword to the book The development of arithmetic concepts and skills: Constructing adaptive expertise edited by Baroody and Dowker (2003), the late Hatano (2003, p. xi) argues that one of the most important issues in the psychology of (mathematics) education is how students can be taught curricular subjects so that they develop adaptive expertise. He describes adaptive This research was partially supported by Grant GOA 2006/01 "Developing adaptive expertise in mathematics education" from the Research Fund K.U. Leuven, Belgium. We want to thank Brian Greer and the anonymous reviewers for their valuable comments and suggestions.
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expertise as "the ability to apply meaningfully leamed procedures flexibly and creatively" and opposes it to routine expertise, i.e., "simply being able to complete school mathematics exercises quickly and accurately without understanding". Although the constructs of adaptive and routine expertise were already introduced by Hatano more than two decades ago (Hatano, 1982), and although the terms 'adaptivity' and 'flexibility' have been used with increasing frequency by researchers and practitioners in the field of elementary mathematics education for a long time, few attempts have been made to rigorously and systematically study in the academic domain of elementary mathematics (a) adaptive expertise as a competence, (b) the acquisition of adaptive expertise, (c) its cultivation, and (d) its assessment. Of course, the older and recent histories of the research fields of (cognitive) psychology and (mathematics) education afford numerous valuable building blocks for such an endeavor. For instance, there is the classical work of Gestalt psychologists such as Duncker (1945) on 'functional fixedness' and Luchins and Luchins (1959) on 'problem-solving sets', which provides strong evidence for the inflexible nature of human thinking (at least under certain experimental conditions) as well as Wertheimer's (1945) richly documented analysis of the difference between blind following of procedures and detection and exploitation of (mathematical) structure. Flexibility (or its converse rigidity) also played a central role in Krutetskii's (1976) famous study of mathematical ability and in several other analyses of differences in ability (e.g., Cattell, 1946; Guilford, 1967; Spearman, 1927), creativity (e.g., Jausovec, 1994), and expertise (e.g., Adelson, 1984; Cañas, Quesada, Antoli, & Fajardo, 2003; Feltovich, Spiro, & Coulson, 1997; Frensch & Stemberg, 1989) in general and in mathematics in particular (e.g., Berar, 2004; Dunn, 1975). Furthermore, we can refer to studies of the thinking processes of eminent mathematicians (e.g., Cajori, 1917; Wertheimer, 1945) and skilled calculating prodigies (Heavey, 2003) demonstrating remarkable flexibility in their thinking. Finally, mathematics educators have for a long time emphasized the educational importance of recognizing and stimulating flexibility in children's self-constructed strategies as a major pillar of their innovative approaches of (elementary) mathematics education and have designed and implemented instructional materials and interventions aimed at the development of such flexibility (see e.g., Brownell, 1945; Freudenthal, 1991; Frobisher& Threlfall, 1998; Thompson, 1999; Wittmann & Müller, 1990-1992). The rise of cognitive psychology during the last decades of the 20th century led to a more flne-grained analysis of the processes and mechanisms underlying flexibility and adaptivity. As will be explained later in more detail, the series of (computer) models of how children choose adaptively among strategies for particular tasks in elementary arithmetic and of how they come to use new strategies developed by the cognitive psychologist Siegler and his associates (for an overview see: Shrager & Siegler, 1998; Siegler, 1998, 2000) arguably constitutes the most ambitious and explicit theoretical account of strategy adaptivity in (elementary) arithmetic. Also highly relevant are insights from cognitive psychology with respect to automated and controlled reasoning processes, for instance the dual-process theory, according to which our cognition and behavior operate in parallel in two different modes, called System 1 (SI) and System 2 (S2), roughly corresponding to our common-sense notions of intuitive and analytical thinking. The heuristic processes of SI are characterized as being fast, automatic, effortless, unconscious, and rather inflexible, because they are triggered by similarity of tasks to stored prototypes (i.e., tasks where a certain strategy has been repeatedly applied with success). S2 processes, on the other hand, are slow, conscious, effortful, and relatively flexible, because they are based on the underlying task structures, decoupled from superficial context (Kahneman, Slovic, & Tversky, 1982; Stanovich & West, 2000). Mostly SI-heuristics provide correct responses (that is also why they originate and develop), but sometimes S2 needs to override the response generated by SI to obtain a correct response. So, according to this theory, the key to flexibility lies in the activation of S2 processes when necessary. Although the rise of cognitive psychology has greatly improved our understanding of flexibility and its role in mathematical expertise, other, more recent, developments within our research field have suggested that the quintessence of adaptivity can only be grasped by going 'beyond the purely cognitive' - to use Bmner's (1986) famous phrase. Compared to the well-
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established cognitive perspective on the development and enhancement of expertise, Hatano and Oura (2003) present a broader view on (adaptive) expertise that also addresses affective and socioeultural aspects. This broader perspective is exemplified by their emphasis on the observation that the process of gaining adaptive expertise always occurs in particular sociocultural contexts and is accompanied by changes in interest, values, and identity (see also Cobb & Hodge, 2007; Leron & Hazzan, 1997), Such a change in emphasis is echoed in Boater's (2000, p, 118) plea for a greater consideration of the 'macro context' in which (math) educational research is conducted, and for "extending our focus beyond the concepts and procedures that pupils learn to the practices in which they engage as they are learning them and the mediation of cognitive forms by the environments in which they are produced". Similarly, reflecting upon their longstanding developmental research project on mathematical learning in the early grades of elementary school, Yackel and Cobb (1996, p, 459) write: (,,,) we began the project intending to focus on learning primarily from a cognitive perspective, with constructivism as our guiding framework. However, as we attempted to make sense of our experiences in the classroom, it was apparent that we needed to broaden our interpretative stance by developing a socio-cultural perspective on mathematical activity. The above pleas for considering (mathematics) learning and teaching in terms of situated learning in communities of practice are also in line with other theoretical approaches by activity theorists such as Engestrom (1987), who view school mathematics as an 'activity system' (Engestrom, 1987; Engestrom & Miettinen, 1999) and, in doing so, offer a powerñil theoretical lens for analysing the learning of elementary arithmetic from a perspective that "overcomes and transcends the dualism between individual subject and objective soeietal circumstances" (Engestrom & Miettinen, 1999, p, 3), Although the specific role of affective and sociocultural aspects in the development of adaptive expertise is still not very well understood, more and more scholars consider them as pivotal components of this development (see e,g,. Bereiter & Scardamalia, 1993; Bransford, 2001; Dai & Stemberg, 2004; Ellis, 1997; Perkins, 1995), Dai and Stemberg (2004, p, 5) even claim that it is exactly this evolution in our discipline that allows us to overcome the limitations of cognitivism and to restore "the adaptive nature of intellectual ñinctioning and development". In this contribution, we reflect on the adaptive choice and use of solution strategies in elementary school arithmetic. In the first part of this article we present and discuss different conceptual and methodological approaches to the adaptivity issue in this domain, while the second part addresses some educational considerations. However, some important preliminary comments must be made. First, we emphasize that the opposition between routine and adaptive expertise in mathematics (education) does not only apply to mathematical strategies or procedures' which will be the focus of this article - but also to other aspects of mathematical expertise. For instance, it can also be applied to mathematical modeling (see e,g,. Van Dooren, Verschaffel, Greer, & De Bock, 2006) and to representational forms (see e,g,, Elia, Gagatsis, & Deliyianni, 2005; Goldin, 2003; Kaput, 1992; Lesh & Doerr, 2003; Warner, 2005; Warner, Davis, Alcock, & Coppolo, 2002), Second, until now we have treated the terms 'flexibility' and 'adaptivity' as synonymous. Throughout our review of the literature we found some authors who use the terms as alternatives (e,g,, Baroody, 2003; Feltovich et a l , 1997), but mostly only one of them is used. Surveying the literature, it seems that the term 'flexibility' is primarily used to refer to switching (smoothly) between different strategies, whereas the term 'adaptivity' puts more emphasis on selecting the most appropriate strategy. However, as will become clear further on, this distinction in meaning certainly does not apply to all authors using (one of) these terms. Whereas nobody uses 'adaptivity' to refer (merely) to the use of multiple strategies, some authors (e,g,, Blöte, Van der Burg, & Klein, 2001) apply the term 'flexibility' also or only in cases where (based on the above-mentioned definition) the term 'adaptivity' would be more appropriate. In the present article, we will, henceforth, use the dual term 'flexibility/
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adaptity' as the overall term, 'flexibility' for the use of multiple strategies, and 'adaptivity' for making appropriate strategy choices. Third, the above definition of flexibility/adaptivity does not involve any reference to the level of consciousness and deliberateness of the strategy choice being made. Many researchers believe that metacognitive processes - here defined as conscious awareness and deliberate control - regulate strategy selection and, thus, strategy flexibility/adaptivity. Clearly, deliberate monitoring and control of strategy selection is present in many everyday situations. For instance, when an adult has to calculate a tip in a restaurant or when a pupil has to do the computational work to obtain the answer of a word problem, (s)he can either do mental calculation, apply a written algorithm, or use a calculator. It is commonly agreed that conscious monitoring and control are involved, at least to some extent, in such strategy choices. Especially when the strategy choice relates to a task that represents a genuine problem to the solver, the choice process will typically be affected by metacognitive knowledge and beliefs, and by metacognitive skills of monitoring and control (see e.g., Ishida, 2002). On the other hand, there is quite a lot of evidence suggesting that in quick and simple strategy selections, like doing simple additions or subtractions in the number domain up to 20, people's selection of a particular strategy does not result from deliberate consideration of the choices and conscious awareness of the factors that influenced that choice, but rather from more autonomous, implicit processes (Cary & Reder, 2002; Ellis, 1997; Siegler, 1996). In this respect, one can refer to the older Chronometrie studies of mental addition (see e.g.. Groen & Parkman, 1972) which provided empirical support for the so-called MIN model (a strategy choice model wherein the smaller addend is always added to the larger one), suggesting very strongly that there is some vestige of 'counting on' of which people are not consciously aware. So, although the term 'strategy selection' may suggest that the process involves deliberate choice and conscious awareness of factors influencing that choice, this process often lacks conscious deliberation and we argue, therefore, that strategy selection can also occur implicitly, without conscious consideration of alternative strategies. In other words, we share the position taken by Siegler and Jenkins (1989), who argue that a strategy must not necessarily be rationally chosen or consciously executed, but can also be selected and executed without involving any consciousness^. Finally, while this article focuses on procedural or strategic competence in elementary mathematics, it is clear that this competence is inextricably related to conceptual knowledge. There is a lot of evidence that conceptual knowledge plays a direct and/or indirect role both in the invention of new strategies and in the choice between strategies (Baroody, 2003; Baroody & Rosu, 2006; Hiebert, 1986; Rittle-Johnson & Siegler, 1998; Star, 2005; Verschaffel, Greer, & De Corte, 2007). The meaningfulness of the flexible/adaptive application of strategies is a principal element in Hatano's (2003) definition of adaptive expertise quoted at the beginning of this article. According to Baroody and Rosu (2006), a sharp distinction between conceptual knowledge, on the one hand, and computational flueney with basic and larger number combinations, on the other hand, "may be a by-product of traditional instructional practices that foster routine expertise" (p. 1). In their opinion, adaptive expertise can only be enhanced through instruction that simultaneously fosters conceptual development with computational fluency. As will be argued later on, the close relationship between conceptual and procedural knowledge does not only manifest itself by the important role that conceptual knowledge plays in the development of procedural flexibility, but also by how engendering procedural flexibility is used by mathematics educators in service of the realization of conceptual goals.
Conceptual and methodological considerations Strategy flexibility As explained before, strategy flexibility is sometimes defined and operationalized as using a variety of solution strategies, without any further qualification. For instance, at the
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beginning of their ascertaining study ofthe role of flexibility in accurate mental addition and subtraction in the number domain up to 100, Heirdsfield and Cooper (2002, p. 59) raise the following question: "Why are some students more accurate and flexible (=using a variety of efficient strategies) than others?" Further on, they distinguish between pupils who are 'accurate and flexible' and others who are 'accurate and inflexible' merely on the question whether they did or did not solve all items from a given problem set by means of the same procedure. While, as explained earlier, some authors would be reluctant to identify 'strategy flexibility' with 'strategy variety' (as Heirdsfield and Cooper do), probably nobody would be willing to give that kind of flexibility also the label 'adaptivity'. Although possessing a variety of strategies and being able to switch smoothly between these strategies can be considered an essential stepping-stone towards adaptivity, merely applying different solution strategies on a series of similar mathematical items or problems can hardly be considered as evidence of adaptivity. After all, on the one hand, one can switch fluently between different strategies without acting adaptively, and, on the other hand, using consistently one single strategy for a whole series of arithmetic tasks might sometimes be more adaptive than switching between a diversity of strategies available in one's repertoire. Adaptivity to task variables Strategy flexibility/adaptivity has been defined and operationalized frequently in relation to certain task characteristics. For instance, Van der Heijden (1993, p. 80) defines it as follows: "Flexibility in strategy use involves the flexible adaptation of one's solution procedures to task characteristics". He operationalized it by analyzing whether children systematically use the 1010-procedure and the G10-procedure for, respectively, additions and subtractions in the number domain up to lOO"*. Starting from exactly the same definition, Blote et al. (2001, p. 628) operationalize flexibility of strategy use in addition and subtraction with numbers up to 100 as follows: "A student is considered a flexible problem solver if he or she chooses the solution procedures in relation to the number characteristics of the problem, for example, NIOC for 62-29 and AlO for 62-24^". In a similar way, in his plea for developing flexibility in elementary school arithmetic, Thompson (1999, p. 147) concurs that "mental calculation places great emphasis on the need to select an appropriate computational strategy for the actual numbers in the problem". All these authors, first, distinguish different strategies for doing additions and subtractions up to 100, and, based on an analysis ofthe strengths and weaknesses of these different strategies vis-à-vis certain types of problems*, they then define certain 'problem type ' strategy type combinations' as flexible/adaptive and others as not. Although this operationalization of flexibility/adaptivity can already be considered as more sophisticated than the one wherein it is simply identified with the use of multiple strategies, it remains, in our view, highly problematic to define and operationalize strategy flexibility/adaptivity in terms of task characteristics alone. Indeed, it is possible that, for a particular subject and/or under particular circumstances, the strategy choice process that Van der Heijden (1993) and Blote et al. (2001) call 'fiexible' may become 'inflexible', and vice versa. A similar criticism on this conception of strategy flexibility/adaptivity in the domain of mental arithmetic is raised by Threlfall (2002, p. 39): As in a number of other decision contexts, it is difficult to determine what the criteria for choice of a mental calculation method might actually be. Even though some problems do seem to suit some 'strategies' more than others, 'choice' could not be just about the number characteristics ofthe problem. Hereafter, we consider two other groups of factors that need to be incorporated into a comprehensive concept of adaptivity, besides task variables, namely subject and context variables.
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Adaptivity to subject variables A first additional set of factors, which has been intensively and systematically investigated and modeled by cognitive psychologists such as Siegler and his associates (Shrager & Siegler, 1998; Siegler, 1998, 2000; see also Torbeyns, Arnaud, Lemaire, & Verschaffel, 2004), are subject variables. According to SCADS (Strategy Choice and Discovery Simulation), Siegler's computer model of how children's mastery of simple arithmetic sums develops, whether a particular strategy (e.g., retrieval or an extensive counting strategy or a shortcut counting strategy) is chosen to solve a particular item by a particular child depends basically on how accurate and how quick that strategy is for that particular item and for that particular child, in comparison to other concurrent strategies available in the child's repertoire. In other words, SCADS always tends to select and apply the strategy that produces the most beneficial combination of speed and accuracy for a particular sum (on the basis of the accumulated data regarding speed and accuracy available in the system's data base)^. Irreflitably, the adaptivity concept underlying this computer model - which claims to be an accurate simulation of how 'real' children make strategy choices and develop new strategies in the domain of elementary arithmetic - reflects a more complex and more subtle view on the strategy choice process, wherein affordances inherent in the task have to be seen in relation to, and balanced with, features of the individual who is solving the task, especially how well (s)he masters the distinct strategies. Of course, conceptualizations of strategy choice that aim to include both task and subject variables necessitate capturing this assumed complexity of the strategy choice process in the research methodology. A method that is being increasingly used for this purpose is the choice/no-choice method (Siegler & Lemaire, 1997; see also, Torbeyns, Arnaud, et al., 2004). This method requires testing each participant under two types of conditions. In the choice condition, participants can freely choose which strategy they use to solve each problem. In the no-choice condition, they must use one particular strategy to solve all problems. Ideally, the number of no-choice conditions equals the number of strategies available in the choice condition. The obligatory use of one particular strategy on all problems in the no-choice condition by each participant allows the researcher to obtain unbiased estimates of the speed and accuracy of the strategy. Comparison of the data about the accuracy and the speed of the different strategies as gathered in the no-choice conditions, with the strategy choices made in the choice condition, allows the researcher to assess the adaptivity of individual strategy choices in the choice condition in a scientifically appropriate way: Does the subject (in the choice condition) solve each problem by means of the strategy that yields the best performance - in terms of accuracy and speed - on this problem, as evidenced by the information obtained in the no-choice conditions? This method has been successfully applied to assess children's and adults' strategy choices in diverse mathematical domains, including solving one-step muliplications (Siegler & Lemaire, 1997) and one-step additions and subtractions (Torbeyns, Verschaffel, & Ghesquière, 2004, 2005), currency conversion (Lemaire & Lecacheur, 2001), computational estimation (Lemaire & Lecacheur, 2002) and numerosity judgment (Luwel, Verschaffel, & Lemaire, 2005).
Adaptivity to context variables As argued in the introduction, recent theoretical developments, and especially the rise of the sociocultural perspective, have revealed that the issue of flexibility/adaptivity is even more complicated than is suggested by cognitive (computer) models such as the one by Siegler and his colleagues (Shrager & Siegler, 1998; Siegler, 1998, 2000). More particularly, studies within that sociocultural view suggest that people switch adaptively between arithmetic strategies not only depending on task variables (i.e., the nature of the given numbers in the problem) and subject variables (i.e., how accurately and how quickly they can perform the competing strategies on the given problem, given their personal knowledge and skills for applying these strategies), but also on variables related to the (sociocultural) context. We acknowledge that
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the adaptation of strategy choices to contextual variations has also been considered and investigated by cognitive psychologists in numerous studies covering a wide array of task domains. For instance, in a discussion of (at least) four different ways in which children's strategy choices can be adaptive. Siegler (1996, p. 145) mentions that "they react appropriately to changing circumstances, such as situational emphases on speed or on accuracy". Other types of contextual variables that have been studied within this paradigm include: the demands on cognitive resources or working memory (Hecht, 2002; Klayman, 1985), time restrictions (Luwel & Verschaffet, 2003), the requirement to adapt to immediate or long-term goals (Crowley & Siegler, 1993), the base rate with which different types of problems are presented (Lemaire & Reder, 1999)^, the characteristics of the item(s) immediately preceding the actual item (Siegler & Araya, 2005). However, with the rise of the sociocultural perspective, alertness to these contextual factors and how they are perceived, interpreted, and valued by the individual problem solver or learner, as well as the nature of the contextual factors being considered, has drastically changed. Whereas the contextual factors addressed by these cognitive psychologists are typically factors that are located in the foreground of the situation and lend themselves rather easily to experimental manipulation and control, those that emerge in the sociocultural line of research are located more in the background of the situation and are more resistant to straightforward operationalization and experimental control. In an excellent review article entitled Strategy choice in sociocultural context, Ellis (1997) presents and discusses the evidence for the claim that sociocultural influences play a powertlil role, not only in shaping the repertoires of strategies individuals have available to solve problems, but also in the choices they make among those available strategies. Relying on the available theoretical and empirical sociocultural literature, she shows that, with age and experience, children not only construct a data base including information about strategy speed and accuracy; they also develop (implicit) knowledge regarding the sociocultural norms - or, to quote Ellis (1997, p. 492) regarding "what a given culture defines as appropriate, adaptive, and wise" - which will also guide their strategy choices. By focusing on only a subset of problem-solving situations (e.g., contexts assumed to be impervious of social influences) and only a subset of strategies (e.g., strategies that do not involve reliance on other people or on cultural artifacts), by largely neglecting the role of the cultural factors (both differences between cultures and within cultures) and by paying little or no attention to affective aspects, the cognitive-psychological view on strategy choice has been tested under very constrained conditions and has, consequently, resulted in a restricted set of parameters likely to influence strategy choice (i.e., the speed and accuracy with which an individual can apply the different strategies available in his/her strategy repertoire). However, according to Ellis (1997, p. 510), these (...) specific problems presented and the conditions under which they have been studied are particularly well suited for the parameters Siegler has emphasized in his model, i.e., speed and accuracy. Within Western cultures, and on school-like tasks, speed and accuracy are extremely salient features of performance. Clearly however, accuracy and speed are not the only variables that influence strategy choice. Observations of problem solving in non-Western cultures and during joint problem solving suggest a host of other variables that might fruitfully be examined within a framework of strategy choice. Like concems for speed and accuracy, these variables are predicted to influence strategy choice at an implicit level. Although, until now, few investigations have explicitly examined how sociocultural factors determine strategy choice in the particular domain of elementary arithmetic, Ellis' (1997) review cites quite a lot of indirectly relevant as well as a few directly relevant studies. With respect to research on how cultural values shape performance on cognitive tasks, she first reviews evidence for (sub)cultural differences in (a) the weights assigned to speed versus accuracy of performance, (b) the value placed on solutions constructed in the head versus by means of external aids, (c) the importance accorded to individual uniqueness versus conformity, and (d) the attitude towards independence versus help-seeking, which all may codetermine people's adaptive strategy
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choices, Ellis then documents how selections among available strategies are also shaped by the relative importance assigned to social goals versus task goals. Examples of such social goals are: (a) the desire to present oneself favorably to others and (b) the aim to continue valued social opportunities rather than showing efficient solution behavior. Conceiving of classroom situations and psychological experiments "as situations characterized by both social and task goals" (Ellis, 1997, p, 508) - or, to state it differently, as situations determined by, respectively, a 'didactical contract' (Brousseau, 1997) or an 'experimental contract' (Greer, 1997) - can contribute, according to Ellis, to our understanding of strategy choice, particularly those choices that seem less than optimal at first glance (in terms of accuracy and speed). Hereafter, we refer to a few illustrative studies wherein this sociocultural perspective has been applied to the topic of strategy choice in (elementary) arithmetic, A first example is the well-known pioneering study by Nunes, Schliemann, and Carraher (1993), which nicely documents how the sociocultural context wherein people are confronted with a particular problem (co-)detennines the kind of arithmetic strategy (oral versus written arithmetic) being used. More particularly, these scholars analyzed the calculation strategies of third-grade children from lower-income families who were involved in the informal economy in larger Brazilian cities (e,g,, candy selling, shoe shining, car washing) when solving mathematically isomorphic tasks in different contexts, such as when actually being engaged in their informal selling activities versus being given the same problems as a computational exercise in a school context. As street vendors, these children solved the problems mostly orally by means of mental calculation, whereas when acting as students they typically solved them using paper and pencil algorithms (leading also to much lower rates of correct responses and also errors that were quantitatively extreme,) As a second example, we refer to Carr and Jessup's (1997) investigation of how parents and teachers influence the development of gender differences in mathematics strategy use in the first grade. Children were interviewed about their strategy use, their metacognitive knowledge about specific strategies, and their perceptions of parents' and teachers' attitudes toward various strategies. Parents and teachers completed questionnaires about the types of strategy and metacognitive instruction they provided. Boys correctly used retrieval during the first grade more than girls and girls correctly used overt (eounting) strategies more than boys. Interestingly, these gender-related differences in strategy choice were related to their distinct perceptions of adults' beliefs and actions. The strategy choices of boys were influenced by their belief that adults like strategies indieating ability and by teacher instruction on retrieval strategies, whereas girls' strategy use was not related to perceived adult beliefs or actions. So, as exemplarily shown by the two above examples, children's strategy ehoiees in elementary arithmetic are not only determined by task and subject variables, but also by characteristics of the sociocultural context wherein they have to demonstrate their arithmetic skills, including (a) what representational and computational tools (such as currency, counting aids, calculators, rulers, pricing tables or charts, etc) are available and/or allowed, (b) what aspects of their strategic behavior - instead of, or as well as, the salient aspects of speed and correctness - seem (most) valued in the classroom and/or testing context, such as simplicity, elegance, formality, generality, intelligibility, certitude, originality, etc, of the solution strategy, and (c) the relative importance of the goal to do well on the school task or test versus other, social goals (see also: Ellis, 1997; Lave & Wenger, 1991; Rogoff, 1990; Star, 2005), Some possible ways of taking these complicating contextual factors (more) seriously into account when investigating (the development of) strategy flexibiiity/adaptivity from a sociocultural perspective, are: - assessing the effects of cultural knowledge on strategy choice by examining strategy choices in a variety of cultures that are known or hypothesized to embrace different values and norms; - varying conditions within a single eulture by means of the systematic manipulation of the 'experimental contract' (Greer, 1997) (e,g,, by explicitly or implicitly rewarding particular aspects of one's strategic behavior, like the correctness, speed, elegance, originality, formality, etc, of the solution strategy being chosen);
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- doing individual interviews to get deeper insight into subjects' perception and interpretation of the context wherein they have to make strategy choices and how this influences their strategy choices^; - applying video-based observations of how children (leam to) make strategy choices in real, daily classroom settings, using existing textbooks and being taught by regular teachers - if possible, complemented with (video-based stimulated) interviews afterwards. This recommendation to pay more attention to the contextual background variables - and particularly to students' educational histories and current instructional settings - is echoed in Bisanz' (2003) commentary at the end of Baroody and Dowker's (2003) book, wherein he complains that "too much psychological work on the development of mathematical thinking is amazingly mute with respect to educational issues" (p. 438) and wherein he makes a plea for integrating the analysis of instructional materials and of teacher behavior into the study of acquisition of (adaptive) expertise in elementary arithmetic. Commenting on one of the chapters in the book, Bisanz (2003, p. 447) writes: Seo and Ginsburg demonstrate the richness of insight that can arise when researchers analyze not only the responses or interpretations of children but also the environments in which those children develop. In this case the analysis of textbooks and teacher behavior were essential to gaining insights that otherwise would have been regarded only as speculative. We would like to see more of this sort of sensitivity and thoroughness in the mainstream of developmental research. While fully endorsing Bisanz' plea, we would like to take it even a step ñirther and add that integrated research on strategy adaptivity should not only investigate these instructional environments in which this adaptivity does or does not develop by means of 'ecologically valid' ascertaining studies, but also, and maybe even primarily, through design experiments, wherein researchers work at "the construction of 'artificial objects', namely teaching units, sets of coherent teaching units and curricula as well as the investigation of their possible effects in different educational 'ecologies'", as argued convincingly by Wittmann (1995, p. 363) as well as by many other others over the past 10-15 years (Freudenthal, 1991; Gravemeijer, 1994; Greeno, Collins, & Resnick, 1996; Schäuble & Glaser, 1996; Verschaffel et al., 2007). Given our previous pleas for taking into account students' educational histories over an extended period, a long-term perspective seems essential in these design experiments as well. Towards a definition of strategy adaptivity Based on the previous theoretical and methodological reflections, we propose the following working definition of what it means to be adaptive in one's strategy choices: By an adaptive choice ofa strategy we mean the conscious or unconscious selection and use of the most appropriate solution strategy on a given mathematical item or problem, for a given individual, in a given sociocultural context. Before moving to the second part of the article, wherein we will discuss some educational implications, we give some comments on this definition. First, as emphasized earlier, by the phrase 'the most appropriate strategy' we certainly do not simply mean 'the strategy that leads most quickly to the correct answer' (as in the cognitive-psychological sense of the term). Of course, we do not exclude that in a particular instructional or testing setting, the term 'most appropriate' might have that meaning, but in many cases the meaning will be more subtle and multidimensional (see also Ellis, 1997; Star, 2005). In this respect, we refer to a study with the choice/no-choice method by Torbeyns et al. (2005) wherein the predicted relation between first-grade children's strategy adaptivity and their arithmetic performance on near-tie sums over 10 (like 6+7= or 8+7=) was not found.
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implying that he high-achieving first graders did not apply the different strategies for solving these near-tie sums more adaptively than did their lower achieving peers. The authors' interpretation of that unexpected finding - based on a post-hoc analysis of the children's textbook and interviews with the teachers - was that the high-achieving pupils might have based their strategy selections more on (their interpretation of) the socio-mathematical norms and practices in their mathematics classroom (Yackel & Cobb, 1996), which strongly favored, or even imposed, one particular strategy for doing near-tie sums, namely the so-called 'tie strategy' (6+7=6+6+1=12+1=13), above another strategy, namely 'decomposition to 10' '(6+7=6+4+3=10+3=13), and not because this was cognitively the most efficient strategy (in terms of accuracy and speed) for them. Stated differently, according to Torbeyns et al. (2005), these high-achieving children demonstrated a level of adaptivity that was suboptimal for them (in the cognitive-psychological sense of the word), because they made strategy selections in line with (their interpretation of) the classroom norms and practices about what it means to behave adaptively with regards to this kind of arithmetic exercises. Second, we are well aware that our classification of factors that influence adaptive strategy choices is not without problems, and, more specifically, that the frontiers between the different categories of variables are blurred. Take, for instance, the semantic structure of the word problem within which a simple subtraction like 16-7=? is embedded and which has been shown to have a serious impact on children's strategy choices, e.g., on their choice of a direct subtraction strategy ('16 minus six minus one is...') or an indirect addition strategy ('how much do I have to add to seven to get at 16?') (De Corte & Verschaffel, 1987). Or, consider time pressure, which has proven to affect the distribution of the choices between addition and subtraction strategies in a numerosity judgment task (Luwel & Verschaffel, 2003). Depending on how the task is defined, these features can either be considered as task or context variables. Likewise, taking into account that the task and context variables do not (only) have a direct influence on an individual's strategy performance, but (also) indirectly affect that performance through the individuals' personal beliefs, perceptions, expectations, values, etc. (Ellis, 1997; Star, 2005), the distinction between the subject variables and the two other categories of variables is also less straightforward than one might initially think. A third and last important point of concern with respect to the above definition of strategy adaptivity is what do we really mean when we say that 'a student is using strategy x' or that 'strategy x is being taught'. According to authors such as Threlfall (2002) and Gravemeijer (2004), it is inappropriate to think of strategies as ready-made methods or techniques that are available in the repertoire of the children, waiting to be selected and applied in a particular situation - even if one does not envision choice as an explicit, mindñil, top-down process. Especially with young children, a more plausible way of thinking might be that they base their computations on their familiarity with certain number relations. For instance, Threlfall (2002, p. 41) argues that in a subtraction problem like 63-26=?, some or all ofthe following relations might be noticed about the two separate numbers involved: "63 is... 60 and 3, 2 less than 65, double 30 and 3, 50 and 13, 20 and 20 and 20 and 3, 7 less than 70; 26 is... 20 and 6, 1 more than 25, 4 less than 30, double 13, 10 and 10 and 6, just over half of 50". Moreover, as part of this thinking, there may also be "connections between what is being noticed about the numbers separately, like 65 is 40 more than 25, the 3 (of 63) is nearly the 4 (that the 26 is less than 30), there is a 20 in both, there is a 13 in both, 6 is double 3, etc.". And he continues: In this way, flexible mental calculation can be seen as an individual and personal reaction with knowledge, manifested in the subjective sense of what is noticed about the specific problem. As a result of this interaction between noticing and knowledge, each solution 'method' is in a sense unique to that case, and is invented in the context of a particular calculation - although clearly influenced by experience. It is not leariied as a general approach and then applied to particular cases. The solution path taken may be interpreted later as being the result of a decision or choice, and be called 'a strategy', but the labels are misleading. The 'strategy' (in the holistic sense ofthe entire solution path) is not decided, it emerges (Threlfall, 2002, p. 42).
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So, depending on the age and the instructional background of the children and/or the realm of the mathematics curriculum being addressed, the characterization of flexibility/ adaptivity as a choice between a set of identifiable alternative strategies available in the repertoire of the solver might be less straightforward than it seems. While we do not believe that the above observations and comments warrant the complete abandonment of the term strategy in (the psychology of) elementary mathematics education - whether the use of the term is appropriate is partly a definitional and partly an empirical question - the above critical reflections at least deserve serious attention from researchers and practitioners.
Educational considerations Underlying many current curriculum reform documents, such as the Curriculum and
evaluation standards for school mathematics of the National Council of Teachers of Mathematics in the US (1989, 2000), the Numeracy Strategy in the UK (Straker, 1999), the Proeve van een Nationaal Programma voor het Reken/wiskundeonderwijs in The Netherlands (Treffers, De Moor, & Feijs, 1990), the Handbuch Produktiver Rechenübungen in Germany (Wittmann & Müller, 1990-1992), and the Ontwikkelingsdoelen en Eindtermen for the elementary education in Flanders (1998), as well as many innovative curricula, textbooks, software, and other instructional materials based on these reform documents worldwide, there is a basic belief in the feasibility and value of striving for strategy flexibility/adaptivity. However, this basic belief, as well as some accompanying presuppositions about when and/or whom and how to strive for it, have not yet been subjected to much systematic and scrutinized theoretical reflection or empirical research. Engendering adaptive expertise: From when on andfor whom? First, there is the issue of the optimal moment for starting to strive for adaptive expertise. Departing from the generally accepted idea that without long-term memory of previously leamed facts, procedures, models, and representational tools, there can be no flexible/adaptive thinking (Warner, 2005; Warner et al., 2002) and also from the finding that many people who have had years of problem-solving experience in a given domain still fail to go beyond routine expertise (Adelson, 1984; Frensch & Stemberg, 1989; Hatano & Oura, 2003), several authors argue that it is better to teach, first and above all, for routine expertise, and only afterwards, change one's aims and pedagogy in the direction of adaptive expertise (see e.g., Geary, 2003; Milo & Ruijssenaars, 2002). This viewpoint is opposed by many advocates of the reform-based approach, who conjecture that the development of adaptive expertise is not something that simply happens after people develop routine expertise but that education for adaptive expertise should be already present from the very beginning of the teaching/learning process (see e.g., Baroody, 2003; Gravemeijer, 1994; Selter, 1998; Warner et al., 2002; Wittmann & Müller, 1990-1992), an idea that is nicely expressed in the following quote from Bransford (2001, p. 3): You don't develop it in a 'capstone course' at the end of students' senior year. Instead the path toward adaptive expertise is probably different from the path toward routine expertise. Adaptive expertise involves habits of mind, attitudes, and ways of thinking and organizing one's knowledge that are different from routine expertise and that take time to develop. I don't mean to imply that 'you can't teach an old routine expert new tricks'. But it's probably harder to do this than to start people down an 'adaptive expertise' path to begin with - at least for most people. Some authors go even further and warn against the rigidifying effects of years of diligent practice in routine expertise. According to Feltovich et al. (1997, p. 126), "there are effects on cognition that come with such an extended practice that could lead to reduction in cognitive
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flexibility - to conditions of relative rigidity in thinking and acting (while, we have noted, affecting other, more desirable goals, such as in efficiency and speed)". Some of the potentially rigidifying effects discussed by these authors are: viewing the world as too orderly and repeatable due to schematization/routinization (Spiro, 1980), the phenomenon of 'functional flxedness' as observed and described by the Gestalt psychologist Karl Duncker (1945), and the so-called 'reductive bias', i.e., a tendency among people to treat and interpret complex circumstances and topics as simpler than they really are, leading to misconception as well as to error and to limitation in knowledge use due to inertness (Coulson, Feltovich, & Spiro, 1989). Given the decrease in flexibility that might accompany increased experience in a certain domain, it seems very dsky to design teaching/learning environments wherein one strives (only) for routine expertise first and postpones engendering flexibility/adaptivity until the moment routine expertise has been established. Taking into account the existence of such opposing views on when to start striving for flexibility/adaptivity, there is a great need of frank, open-minded, comparative research involving different instructional approaches with respect to the moment at which one starts engendering flexibility/adaptivity and systematic comparisons of the leaming, retention, and transfer effects on a broad scale of cognitive as well as non-cognitive learner variables. Here again, to be theoretically and practically valuable, the instructional interventions involved in this comparative research need to be long-term. Closely related to the issue when to start striving for strategy flexibility/adaptivity is the question whether promoting flexible and adaptive strategy use is feasible and valuable across different levels of mathematical achievement, including the average and especially the weaker ones. Threlfall (2002, p. 40) refers to the frequently heard claim that "(because) only the more 'mathematically minded' children will be capable of leaming how to make good choices, (...) flexibility should be abandoned as an objective for the 'average' and 'below average'". Whereas there are studies that support the claim that lower-achieving children do not benefit from reform-based instruction that strongly aims for the development of adaptive expertise (Baroody, 1996; Baxter, Woodward, & Olson, 2001; Boaler, 1998; Fuson, Carroll, & Drueck, 2000; Geary, 2003; Milo & Ruijssenaars, 2002; Sowder, Philipp, Armstrong, & Schappelle, 1998; Woodward & Baxter, 1997; Woodward, Monroe, & Baxter, 2001), other investigations suggest that such reform-based instruction does enhance the mathematical performances of low-achieving children more than traditionally-oriented or direct instruction in one particular strategy does (Baroody, 2003; Bottge, 1999; Bottge, Heinrichs, Chan, & Seriin, 2001; Bottge, Heinrichs, Mehta, & Hung, 2002; Cichon & Ellis, 2003; Klein, Beishuizen, & Treffers, 1998; Moser Opitz, 2001; Van den Heuvel-Panhuizen, 2001). So, while there are some indications that warrant optimism (see e.g., Baroody, 2003; Moser Opitz, 2001; Van den Heuvel-Panhuizen, 2001), more research is needed to determine whether it is indeed feasible and valuable to design and implement instructional approaches aiming at (strategy) flexibility/adaptivity that are successful for all children, including the mathematically average and weaker ones. Like the needed research with respect to the 'when' question, this research should be open to alternative hypotheses, e.g., that especially mathematically weaker children might profit more from instruction that first aims at developing efficiency (i.e., accuracy and speed) in one arithmetic strategy and that only starts to teach for strategy flexibility and adaptivity once (a certain degree of) efficiency in the initially taught strategy has been reached. Moreover, it is important that this research is clear with respect to its focus. If the major aim is to obtain, in the short term, better gains in computational facility (operationalized as being able to solve familiar sums quickly and correctly, or, stated differently, as developing routine expertise), it might be more efficient to teach one single mental 'strategy' for each arithmetic operation, to be used with all such problems, without any concem for task, subject, or context variables (or even any concem for conceptual understanding at all). If, however, the instructional goals are broader and more long-term and comprise also strategy flexibility, good understanding of mathematical concepts and principles, pattern recognition skills, and appropriate beliefs, attitudes, and emotions towards mathematics (learning and teaching), then other, less routine-based instructional
ADAPTIVE EXPERTISE IN MATHEMATICS EDUCATION
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approaches might be more appropriate, also for the younger and mathematically weaker children. The question is not merely an empirical one, but is, to a significant degree, dependent on the value system underlying one's view of what mathematics education is for. Further, it may be regarded as an ethical question whether it is equitable to expose children of average or low ability (as assessed by traditional criteria) to a mathematics education that is intellectually less stimulating (Greer, personal communication). Engendering what kind of adaptivity and how to do it? This brings us to the last question: how to design and implement effective instruction aimed at adaptive expertise in (elementary) mathematics education? In the traditional 'mechanistic approach' (Treffers, 1987) or 'skills and/or conceptual approach' (Baroody, 2003) to elementary mathematics education, one strategy for solving all problems of a given type is imposed on children (with little or no room for alternative solution methods). For instance, the decomposition-to-ten strategy (e,g,, 7+8=?: 7+3=10 and 10+5=15; 16-9=?: 16-6=10 and 10-3=7) to solve all.additions and subtractions 'with a bridge over ten'^ or the standard-jump approach-(b',g,'^-34+29=?: 34+20=54 atid"54+9^63;61-57=?::61-50=11 and 11-7=4) to solve all additions and subtractions in the number domain 20-100, ''•'' •-by means of five different sti-ategiés) arid'by'stimuläting thërri'toîunderstand these-different'methods,-bütJthen to'seléct theií own preferential'Strategy or-strategies, and-to talk abotit'and reflect upon-theip selection's linder supervision-of'the'teacher; Never do'the au'thörs of' Däs'Zahlenbuch try'to establish fixed links';between -particular^ strategies'and particular-problem types; br to' impose aikind of 'nieta-strategy' 'for selectingi/ieimost-apprbpriate'solution- strategy fbr'every problem' type, as was'done in the'Flerfiish textbook series,''Ih'Oth'er^words,'although flexibility/adaptivitjy is seeh aS' very important, there' is'iho explicit teachihg towards 'the establishment' of-flxed 'associations between part'iCular st'rategies'andp'árticularproblenís^'-'í ) •i: •-' 'l^' /-jii • i'¡.ur. ' ¡ M , ! . v : ; ; ; ! r r ' i o ' - / .
•' ' So, if'we'want to make progress in our the'oretical ünderstari'ding''änid! prácti&ál enháncehieht of strategy flexibility/adaptivity'in'èlemehtàry- arithmetic,'there'is' á 'great need for continued research efforts'-'arid especially of'design "experiments'^"thatémbody'thé following' átt:ribütes.';First,' these studies should be ;dóne*in 'ecologically valid' settings, which rnearis that they are' do'né 'in regular classes 'with regular teachers and 'pupils; and involve analysés of textbooks and teacher behavior'ovei- longer periods of tirne'!'Sécohd, serious attention shoulü; be pài'd at tKèpptiniaîrnoniérit'to start' With^'^this' strive foi- strategy 'flexibility an'd'àdaptiyity as'well as/at the po'terit'ialij^ 'diffei^eiitiäj impact,on children of differeri't (matheniatical) abilities. Third, agoqd.bpáñcejs needed between^ the social and the'indiyidua'j perspective - or stated differently: between,the,socio-mathematical norms about,what it means to be, adaptive, on the,,one'jhandi..,ánd,the development,of,indiyidual,children's, strategy adaptiyity, on the other hahd. Fourth, given that strategy a4apti>;ity,in, elementary,arithmetic is, after all, frorn a,(reform-based) mathematics' education perspective,,,not a goal in itself,but essentially a,,vehicle for; reaching,more general and far-reaching goals, (such as,dealing with conceptual understanding,; nuniber sense, pattern recognition, problem solving, and affective goals,,see e.g., Baroody,;2003; Baroody & Rosu, 2006;.Kilpatrick,,SwaffoTdi,&,,!F;indell, 20,0;U Verschaffel et a l , 2007; Wittmann & Müller,. 1990-92), the assessment of children's,progress in¡these design experiments, can not.be restricted.,to their, procedural .competence in (mental) arithmetic,' but ¡should also icomprise their understanding of 'big ideas'^underlying, the strategies they use,' their, problem-solving and communication fskills in arithmetic, and their affects towards flexibility/adajjtivity and mathematics in'generali. • r , ! . • • . : . ..,!.'.,; (Only) when studies with these attributes have convincingly and repeatedly shown'that the reform-based approach that strives for adaptive expertise effectively leads to the intended outcomes, without resulting in significant loss in computational accuracy and fluency, will researchers will be in a good position to convince policy-makers, teachers, and'parents Ofithè feasibility and value of striving for adaptive expertise, from a young age on and for all children, rather than!reserving it as a'pinnacle' for those who have fii-st developed routine c o m p u t a t i o n a l expertise.'
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Notes '
Although we are aware that sotne authors define the terms 'procedure' and 'strategy' differently, in this artiele they will be used as synonyms.
2
Some researchers, such as Koriat (2000), have proposed an alternative view wherein it is suggested that metaeognition does not necessarily depend on conscious awareness and deliberate control (Cary & Reder, 2002). Of course, whether metaeognition implies conscious awareness or not is largely a matter of defmition.
•5 According to Siegler and Jenkins (1989), it is exactly this feature that distinguishes a strategy from a plan, which is always the result of a rational choice process that involves consciousness. "•
The 1010 procedure involves splitting off the tens and the units in both integers and handling them separately (e.g., 47+15=.; 40+10=50; 7+5=12; 50+12=62). The applieation of the GIO procedure, on the other hand, requires the child to add or subtract the tens and the units of the second integer to/from the first unsplit integer (e.g., 47+15=.; 47+10=57; 57+5=62).
5
Using NIOC for 62-29 means that the student does "62-30=32, then 32+1=33", while a AlO strategy for 62-24 involves the following steps: "62-2=60, 60-20=40, 40-2=38".
^
This analysis consists of a purely rational task analysis (wherein the procedural and conceptual complexities of the different strategies are compared), of an empirical analysis of the accuracy data from large groups of students for the different (types of) problems, or of a combination of both kinds of analysis. , . r, r, , ;,. . ;,
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Actually, the,selection.mechanisni iS|Somewhat more sophisticated, as the systetn also awards 'novely points', to newly disccjvered strategies, which inc'rease the chance that a new strategy will be attempted (regardless of its efficiency in terms o f s p e e d a n d accuracy).
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' " Here we have, 6f course, to be aware that people's verbalizations and explanations'are by no rneans ahelear and direct 'window' to the actual impact of certain sitüatioiial emphases oil their strategy choices.'- ^ •'••• ' • •
References ,,-,;.
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Adelson, B. (1984). When novices surpass expertsi'The'difficulty'of'a task iiíáy increase with expertisé:'youráa/ of Experimental Psychology: Learning, Memory and Cognition, 10, A%'i.^A95. , , > ,, ,-,. ' ' " • • I ' i. ) Anghileri, J. (1999). Issues in teaching niujtiplication and division. In I. Thornpson iEd.),Jssues in leaching numeracy inprimarvic/ioofe (pp! 184-194). Buckingham,'U.K.: Operi University Press.^, _. , ; . Baroody, A.J. (1996). An investigative approach to^ the mathematics instruçtioti of chiljiren elas^sified as le^arning disabled. In D.K. Reid, W.P. Hresko, et al. (Eds.), Cognitive approaches toJearning.disabilities (3rd ed., pp. 545615). Austin, TX: PRO-ED. .
,
..,.
.
,•
,
.
1
,:
,
;
,.
.
. 1 , . ;
.,;.
I
.:
-
,
'
.
,
^'
.'
; '
.
1
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Baroody, A.J. (2003). The development of adaptive expertise and flexibility: The integration of concepttial'and proeedurai knowledge, iti A.J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts ^and skills (pp. 1-34). Mahwah, NJ: Lawrence Erlbaum Associates. Baroody, A.J., & Dowker, A. (Eds.). (2003). The development of arithmetic concepts and skills. Constructing adaptive • .'• &rperi«e..Mahwah,'NJ:Lawrenee Erlbaum Associates. ' ' •• ' 'i if ••) i ^ i !.'..,''• Baroody, A.J., & Ginsburg, H.P. (1986). The relationship between initial nieaningful learning and niechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics '
(pp.'75-ll'2). Hillsdale.NJ: Lawrence Eribaum Associates:
- '
•
•'
•• •:• •"••
•'• " " ' • ' .
'.
'...'.'
Baroody, A.J., & Rosu, L. (2006). Adaptive expertise with basic addition and subtraction combinations - The number '' ' ' .ooi(ro/eí/ucaí/ona//).s)'c/io/ogv (pp. 15-46). New York, NY: Macmillan. ' • ' Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems. Learning and Instruction. 7, 293-307.
• '
••
• • '
'
'
•
Groen, G.J., & Parkman, J.M. (1972). A Chronometrie analysis'of simple iàdition! Psychological Review. 79, 329-343. ' Guilford, J.P. (1967). The nature of human intelligence. New York: McGraw-Hill. Hatano, G. (1982). Cognitive consequences of practice in culture specific procedural, skills. The,Quartely Newsletter of the Laboratory of Comparative Human Cognition, 4, 15-Í&. Hatano, G. (2003). Foreword. In A.J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. xi-xiii). Mahwah, NJ: Lawrence Erlbaum Associates. Hatano, G.,.& Oura, Y..(2003). Reconceptualizing school leaming using insight.from expertise researeh. ídMcaííona/ Researcher. 32ß),26-29. , i. • : . ... , . : Heavey,'L. (2003). Arithmetical savants.In A.J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts (pp. 409-434). MahwahjNJ: Lawrence Erlbaum Associates. •, • • • . .
354
L. VERSCHAFFEL, K. LUWEL, J. TORBEYNS, & W. VAN DOOREN
Hecht, S.A. (2002). Counting on working memory in simple arithmetic when counting is used for problem solving. Memory and Cognition, 30, 447-455. Heirdsfield, A.M., & Cooper, T.J. (2002). Flexibility and inflexibility in accurate mental addition and subtraction: Two case studies. The Journal of Mathematical Behavior, 21, 51-1 A. Hiebert, J. (Ed.). (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates. Ishida, J. (2002). Students' evaluation of their strategies when they fmd several solutions. Journal of Mathematical Behavior, 21,49-56. Jausovec, N. (1994). Flexible thinking: An explanation for individual differences in ability. Cresskill, NJ: Hampton Press. Kahneman, D., Slovie, P., & Tversky, A. (1982). Judgment under uncertainty: Heuristics and biases. New York: Cambridge University Press. Kaput, J. (1992). Technology and mathematics education. In D.A. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 515-556). New York: Macmillan. Kilpatrick,.J.y'Swafford,iJ.,'&findell,.B. (200\yMding,it •1'!"
' N a t i o n a l i A c a d e m y
Press.'11'íí,',iM...
'
(
;•''.
-u;!!
up. Helping children.learn >mathematics.iWashington, D.C: ,,,i'
,¡,y,
-f,
•;•,-)'.;•'•..'I
. L :.'. ..'¡'I
i 'i
ni
-ji.c, n , • ,
Klayman, J. (1985). Children's decision strategies and their adaptation "to task charèctèristics'.''Örganizaiiona/ßäavio/'.'••
. a n d M u m a n
Décision Processes,35',
\19720\'.i
I::':':.:'¡i.
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Kle'in;'Â.'s'., Beishuizèn'.'M.',' & Treffers, A':''(Í998)'.'The'empty' nutnb¿r"liñe' in'Dutc'li secônd'gradesV Realistic'verj«j g r a á u a i p r o g r s i m d e ú g n . J o u r n a l f o r R e s é a r c h ' i n M a t h e m a t i c s E d u c a t i o n , 2S,,443r,464....;, ¡ i v - i ) . 1 : .Í,;,;,,,H}I, . I Koriat, A. (2000). Control processes; in, remenibering. In E.,,Tulving,&,F-,I.M.n,Craik'((Eds,), The.Oxford memory (pp. 333-346). New York, NY: Oxford University Press. ' ; ' j . ' i , < i . • • i j : i : l ; ' - ; , w • . . < ! < ; ' i i , y i r . ! t ó , ¡ ; i..-!," l ! ! j , T , i - . l o v j -
l o i i : i l : . - , •• i n - . r i i - , • , . ! ; i /
( ' V Á W } .1 . i - u K l i o
handbook.of
,. V W . ' - , : )
. i
•[;•,/.:•'
Krutetskii, V.A. ^(1976): The psychology of.mathematical qbilities.,in\scho.ol cAíW/;e«.;Chicago:. .University-ofiChicago Press. .L ^
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Lave, J^ &\^enger,^B:..(\99\).'Situated,learmng:^Legitimate-,^peripherahparticipatiom Ça^^ University Press.
.^^¡.j,,,,:,-/ •• „.r-l. ; .,jriv,,,, j . 1 / d,. 7'l,:i/ .. I--,).....^Î ,,^,
Lemaire,,;P;.v,&.Lecacheur,¡ M., (20,0;l).:.Older^,and younger-,adults'i;strategyi;use, and:exeieution :in currency conversion - c \ ' tasks: .Insights.frpin French franc-,to Éurq\and;Eurg to,French, franc,,cpnyersipns..yoM™o/,o/,&pmmenio/ Psychology: Applied 7, 195-206. / - . n , , . ' i ; b •,i ; , •••'„'.•...•.•] ; !;..:.¡:,O 1 „ T H . , - ' ,••' •,-! ,si u^K ire, IÍ.,¡&^LsCaeheur,jM.i(2pO2).; Children's strategies.in computational estittiation. Journal,of Experiniental.ÇhiM Psychology, 82, 2»\-304. ' ' ,,ï,',',,•,;,',/,',„ I L.emaire, P., & Reder, L.|(I999,). What,effects strategy^seleçtipn^n.arithnieJic'i'Theiexatnpl^^^ product verification. Memory a«í/CogmVíon, 27, 3 6 4 - 3 8 2 . , - j . ; , ; i , , ,,- u,. \ (,u , i, i»»,, • . , , . , •
,, i
i U-'',&. H a z z a n , ¡ p . j ( 1 9 9 7 ) . ( T h e world according,to Jphny:i!A;coping perspective in matheijiatics education: Educational Studies in Mathematics, 32,'265r292.. , \,- , i; > : ,,i.i^...:\.-\ , '.,.. , • ,, ,'i \ , •.•.).'. „,,\\ Lssh,,R.,,&,Doerr,_H.M. (Eds.),,i(2pO3), Beyçrid, constnictivisrn., Aforfefa-a«rf; morfe/mg j:;e/-ípec;íi^^^^ ommathematical problem solving, learning and teaching. Mahwah,NJ. Lawrence Eribaum Associates. ' , , ¡ , : ^,^ Luchins, A.S., &,Luchins, ,E.H. .{1959). ^Rigidity.of behavior,- A,yariqtw.nal approachjtp^the ¡effect of einstellung. Eugene, OR: University of Oregon Books. i!iii
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' I I u i o i l i i w
Luwel, K., & Verschaffel, L. (2003). Adapting strategy choices tp situational factors: The effect of time pressure on " ''children's numerosity judgement stt^tègiès.fi>'c/!o7ogicaBè/g/co,Ù5,269-295.-""' • • n . . ' : ' . ( . ' ; ' n f ' i i) nri,!.!! Luwel, K., Verschaffel, L., & Lemaire, P. (2005). Children's strategies in numerosity judgment. Cognitive . . - . " • i y . , ' , ' / , ,"'i",T:!-f! • „ ! .,••,;, . 1 / i:,,vi¡.;f/
(•¡•/.•/
Development, " ( | ,
McClain, K., Cobb, P., & Bowers, J. (1998). A contextual investigation of three-digit addition and subtraction. In L.J. •"' ''Morrow!&M.'J.-KénneyJ(BdS;),"7'AeteacÁmg-aníí learning of>dlgófithmsHns'chobl''ihathemati¿s (pp.'141-150)! Reston: National Council of Teachers of Mathematics. .>''•'. !•
