Situated Cognition and Cognitive Apprenticeship ...

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Situated Cognition and Cognitive Apprenticeship: Implications for Classroom Learning Sara Hennessy School of Education, Open University, Walton Hall, Milton Keynes MK7 6AA Studies in Science Education, 1993, vol. 21, 1-41

INTRODUCTION This paper reviews recent research in the area of situated cognition, focusing on the domains of mathematics, science and technology. The key issue addressed here is the disjunction between classroom learning and cognition in practice. The notion of apprenticeship inside and outside the classroom, the potential scaffolding function of computer-based learning environments, the role of prior knowledge in learning, and the notion of a general thinking and problem-solving capability are also explored, and implications for pedagogy are considered. The first important implication for our understanding of knowledge construction and problemsolving processes comes from the previously established literature which characterises cognitive differences between novices and experts. This work indicates that predominantly through an interactive process of cognitive apprenticeship, experts spend years acquiring intuitive specialist knowledge and sophisticated mental models of their domain (e.g. Collins and Gentner, 1987). These models are influenced by the social and cultural context in which solving a problem takes place, including the physical structure, the purpose of the activity, the existence of collaborating partners and the social mileu in which the problem is embedded (Furnham, 1992). Rather than being an individual, purely experiential process, then, cultural transmission plays a major role in the construction of domain expertise. However, practical applications of expert/novice comparisons are somewhat limited. Since experts and novices do not normally share similar goals or constraints nor wish to carry out similar tasks, teaching novices what experts are assumed to know is not necessarily an optimal strategy. Direct comparison between expert problem solving and that of young novices in the form of schoolchildren is therefore unlikely to be illuminating. The previous tendencies of cognitive psychologists to characterise various forms of everyday knowledge as deficient by comparison with expert knowledge lack credibility in light of recent research concerning the construction of informal knowledge. The latter is successfully used in a range of domains - by children and adults - to meet the demands of everyday problem-solving situations, without recourse to school-taught strategies. Expert problem solving is similar to this ‘knowledge in action’ in that practitioners rarely find it useful to draw upon knowledge or skills attained during schooling. This is because schooling does not really prepare pupils for later life or for problem solving in the workplace; it can be viewed as a unique culture, a specialised practice with its own conventions, organisation and concerns, which are in fact of little value to society outside. The consequent gap between schooling, and everyday practice by children, adult expert practitioners and 'just plain folks' (Lave, 1988), is now well-documented and is explored below. SITUATED COGNITION Learning and problem solving in and out of school The relatively new theoretical framework which characterises 'everyday' or 'situated' cognition considerably widens our view of cognitive models of problem solving (a) to recognise the critical role of the social and physical circumstances in which actions are situated, when interpreting those actions (Suchman, 1987), and (b) to encompass thinking as a part of culturally organised activity which is carried out within a community of practitioners. In this view, learning is a process of enculturation or

2 individual participation in socially organised practices, through which specialised local knowledge, rituals, practices, and vocabulary are developed. The foundation of actions in local interactions with the environment is no longer an extraneous problem but the essential resource that makes knowledge possible and actions meaningful (Suchman, 1987). The context and content of thought are hence inseparable from the reasoning process. The implication that problem solving is not a process internal to an individual mind, but one grounded in social practice (e.g. Crook, 1991) has, in fact, increasingly been recognised in a wide range of domains, for example, by researchers working on gender issues in science and technology (Harding and Grant, 1984; Murphy, 1991). The situated cognition framework has important implications for our understanding of classroom learning. For example, the Piagetian view that children develop an ability to handle abstraction is now being revised in terms of their developing awareness of a set of cultural conventions for interpreting a task and communicating the answer (Mercer, 1992). The child’s task has become one of learning an implicit set of educational ground-rules - representing social conventions for presenting knowledge in school, and cognitive problem-solving procedures (Edwards and Mercer, 1987). These rules operate only in the traditional classroom environment, which is considered an alien culture to the practices and thinking which take place outside it. The main reasons for this disjunction are as follows: a According to the situated perspective on learning, knowledge moves from being private to being shared through engagement in socially shared activity and discourse. Wenger's (1991) study of insurance claim processing demonstrated that even the most routine jobs generate very complex social communities where learning is characterised by the sharing of informal knowledge and the construction and negotiation of meaning. One of the main purposes of formal schooling is likewise to develop a ‘common knowledge’ (Edwards and Mercer, 1987). Yet an individualist perspective is dominant in our education system, where a premium is placed upon what individuals can achieve by themselves and without external support (Resnick, 1987a), and where even when groupwork is apparently being encouraged, pupils work as individuals (Galton, Simon and Croll, 1980). b The incentives outside school lead to learning that is self-motivated or commercially driven and the problems encountered are hence authentic and relevant to the learner rather than artificially constructed, as, for instance, computation exercises are. Hence, everyday learning is goal-directed and often incidental or effortless, whereas much academic learning is deliberate, effortful and decontextualised, being carried out for its own sake (Reeve, Palincsar and Brown, 1987). An apparent lack of purpose and of explicit criteria for success divorces classroom learning from readily understandable goals such as the play goals of childhood or the work activities of adulthood (Bruner, 1972). c Informal learning fuses intellectual and emotional factors, which are separated in formal learning (Harris and Evans, 1991), where, for instance, a concern with technical solutions to technical problems can overshadow the values component of technology and its relevance to people and social issues. As in science, this alienates girls in particular (Harding and Grant, 1984.) d In school, almost all problems are pre-formulated and accompanied by the requisite data, whereas outside school, problems are seldom clearly defined initially and the information necessary for solving them must be actively sought from a variety of sources (Maier, 1980). Moreover, accuracy is defined by the situation and correctness is negotiable outside school (Hoyles,1991). e Although the same subject topics arise in both contexts, the methods used are quite different; school mathematics relies heavily on paper and pencil, for instance, and the greatest premium is placed on pure thought activities - what individuals can do without the external support of books, notes, calculators or other complex instruments. In contrast, most mental activities and actions in everyday life are intimately engaged with the physical world - with objects, events and with some form of tools. The resultant cognitive activity is shaped by and dependent upon the kinds of tools available (Resnick, 1987a). The mechanisms of informal learning include observation, imitation,

3 identification and cooperation, whereas formal knowledge is transmitted primarily through teacher talk (Scribner and Cole, 1973).

Everyday mathematics The major pioneering work in the area of situated cognition has focused on the relationship between mathematics learning in school and mathematics ‘in context’, namely in the workplace and in domestic life. (A global and up-to-date overview of this work is collectively provided by a collection of papers edited by Harris, 1991; Lave’s book, Cognition in Practice, 1988; Nunes’ analysis of mathematical development, 1992; and by the May 1988 special issue of Educational Studies in Mathematics.) A series of key research studies characterising problem-solving strategies in everyday situations provides further elaboration. This work spans several cultures: it documents the craft apprenticeship of tailors in Liberia (Lave, 1977; Reed and Lave, 1981), the everyday practice of arithmetic and manipulation of quantity relationships in grocery shopping, cooking, dieting and money management, (Lave, 1988; Lave, Murtaugh and de la Rocha, 1984), street vending in Brazil (Carraher, Carraher and Schliemann, 1985; 1987; Saxe, 1988), dairy workers' calculation strategies (Scribner, 1984), construction work (Carraher, 1986), and pottery making (Price-Williams, Gordon and Ramirez, 1969). This research describes forms of mathematics - sometimes known as ‘ethnomathematics’ (d’Ambrosio, 1985) - that vary as a consequence of being embedded in cultural activities whose purpose is not ‘doing mathematics’, and that differ radically from those used in school. Researchers in this tradition have consistently found that quantitative relations are dealt with inventively and effectively in everyday situations, and that arithmetic activity is structured into and by ongoing activity (Lave, Smith and Butler, 1988). Lave (1988) showed that personal methods are commonly invented and used successfully by adults in a practical situation - calculating the ‘best buy’ in a supermarket - with a very high degree of accuracy (98%), whereas the same people solved only 59% of similar calculations correctly in a written test. This finding and those of Reed and Lave (1981) and Saxe (1988) indicated that two distinct systems of arithmetic procedures and practices (symbol- or rule-based versus meaning-based) function independently within the same culture with different procedures and rates of success. This conclusion is corroborated by studies contrasting the calculation strategies employed by the same children in (a) selling produce in Brazilian street markets, (b) solving word problems and (c) computation exercises (Carraher et al., 1985; 1987). Similarly, construction workers have been found to demonstrate greater skill in applying proportional reasoning when interpreting blueprints than students who learned the principles in formal mathematics; the presence of physical objects in the builders’ environment rendered the task more meaningful to them (Carraher, 1986). Carraher’s work has also demonstrated the sophisticated understanding of proportional relations developed by fishermen in their everyday practice; they were able to invert their computational strategies and transfer them to new domains (Carraher et al., 1988). A further example is Scribner’s (1984) observation that dairy employees used a wide range of strategies in processing their milk orders, including the invention of units of calculation based on the structuring resources available in the materials of the situation. The apparent inconsistency in their strategies reflected rapid adaptation derived from the high priority assigned to economy of effort. By way of contrast, students asked to solve similar problems used a single, much less efficient algorithm. Scribner concluded that 'Skilled practical thinking is goal-directed and varies adaptively with the changing properties of problems and the changing conditions in the task environment' (1984, p.39). Saxe’s (1982) work with the Oksapmin of Papua New Guinea provides a classic example of such flexible adaptation; the introduction of a money economy influenced the indigenous numeration system which is based on body part counting. Conversely, the manipulation of abstract symbols (rather than concrete quantities) carries the burden of computation in a classroom situation and divorces mathematical operations from reality (Carraher et al., 1985; Reed and Lave, 1981).

4 To conclude, it is now widely recognised that most adults and children spontaneously invent their own, reliable mathematical procedures, and they rarely use the standard written methods outside school (Fitzgerald, 1985; Shuard, 1986b). These findings are not altogether surprising. Two large-scale survey reports on adult numeracy have concluded that formal teaching lacks relevance to mathematics as commonly practiced in daily life; many adults have forgotten those methods or they lack the confidence to use them (ALBSU, 1983; Sewell, 1982). Knowledge formed in practice, on the other hand, is often used to address problems in different contexts, and it can even be adapted for school-based problem solving. Saxe (1988) has investigated the mathematical understandings of schooled and unschooled child candy sellers and non-sellers in Brazil. The sellers were observed to construct and operate on problems that were influenced by cultural artifacts, social conventions and interactions. They developed increasingly complex problem-solving strategies that drew on the structure of their currency system and conventional pricing ratios. Some sellers who attended school worked towards adjusting their invented regrouping strategies to solve mathematics problems in school, although the opposite influence of schooling on out-of-school mathematics was minor. Saxe’s studies highlighted the interrelationship between social and developmental processes in mathematical learning and corroborated other findings from work on counting and measuring such as those of Carraher et al. (1985) and Saxe and Moylan (1982). Collectively, this work provides clear evidence for out-of-school development of abilities like transitive inference-making about units, and decomposition or grouping of numbers in ways appropriate to the activity, independently of a base structure in the particular numeration system. Note that not all out-of-school mathematical abilities are as sophisticated as those reported above, as Hatano’s (1988) comparison between Brazilian street mathematics and the routine expertise of Japanese abacus operation shows. Both kinds of competence are acquired without formal teaching in school and are used for commercial transactions, but the process of accelerating speed in abacus manipulation has the same consequence as the demand for high precision in executing mathematical procedures taught in school: the flexibility, comprehension and the clarity of meaning underlying each calculation step which characterise sellers’ strategies, are sacrificed. Everyday science The 'everyday' and 'street' mathematics literature is supported by characterisations of situated reasoning in other domains, including intuitive psychology, biology and physics (Hatano, 1990; Inagaki, 1990). Most prominently, there is a growing interest in 'street science' (George and Glasgow, 1988) and a wealth of research over the last two decades has focused on ‘alternative frameworks’ in children’s understanding of science (e.g. Driver, 1989; McDermott, 1984) and on everyday scientific thinking in general (Kuhn, 1989; see Furnham’s review in the previous volume of this journal for an informed overview of the area). Briefly, there is conclusive evidence that children construct intuitive beliefs about natural phenomena (such as heat or forces) which conflict with the scientific viewpoint and that these beliefs are highly resistant to counter-evidence and instruction. These frameworks, and similarly lay adults' mental models of science and technology (Wynne, Payne and Wakeford, 1990), often appear to be partial, incoherent or internally inconsistent (Champagne, Gunstone and Klopfer, 1985); typically there is no coordination of theory with evidence (Kuhn, 1989). This is because, as with everyday mathematics, pieces of knowledge or models are being drawn upon flexibly and according to their appropriateness and usefulness in a specific practical context. They provide a sensible framework for understanding and describing phenomena which fit with the learner’s experience; new information is integrated with existing beliefs and ideas about how the world works. Likewise, external scientific knowledge is defined by lay adults against familiar knowledge (Wynne et al., 1990). Alternative conceptions are continually reinforced through the mass media and daily conversation; the persistence of such socialised knowledge is unsurprising. Collectively, researchers engaged in studying situated cognition have built upon recent developments in the areas of mathematics and science education in challenging the previous assumptions of some educators that school is the central source of everyday practice in these domains

5 and that schooling needs to replace the inferior, intuitive knowledge acquired in the outside world (Lave et al., 1988). In fact, the following discussion implies that the reverse is more accurate!

The role of prior knowledge A key factor in problem-solving ability according to Schoenfeld (1985) is the learner’s resources the informal or intuitive knowledge an individual is capable of bringing to bear and his or her relevant competencies. Schema theory and related research indicate that as in the case of expert practitioners, children's previous knowledge and experience are a major determining factor in how new tasks are interpreted, what is understood and what they can go on to learn. The organised, abstract bodies of information which learners bring to learning determine whether new material will make sense (Shuell, 1986). The situated perspective elaborates upon this essential assumption of constructivism, portraying learning as a process of guided participation in sociocultural activity, which involves building bridges between what children know and new information to be learned, and supporting children’s cognitive development (Rogoff, 1990). A large body of research now shows that children - including preschoolers - are capable of participating productively in activities involving mathematical and scientific thinking; they have significant implicit understanding of many concepts and principles before encountering instruction that enables these to be made explicit (e.g. Gelman and Gallistel, 1978; Greeno, 1992; Hughes, 1986). However, while quite a lot of encouraging progress is being made as curriculum developers revise their views of learning and re-evaluate their aims, conventional teaching in complex domains such as mathematics and science nevertheless continues to impede the development of understanding through (a) over-emphasising formal problem-solving procedures whilst neglecting the appropriate conditions for applying them (NSF, 1983) and (b) ignoring children's existing knowledge and experience. Many children are consequently unable to cope with novel problem-solving situations and to bring their own resources to bear. Ultimately they cannot bridge the gap between school-taught procedures and everyday practice and thinking. An unfortunate disjunction clearly persists between theory and current practice. Constructivist theories of learning, which hold that the forms and content of knowledge are constructed through active interaction with the environment, have long been established in the research literature and are now part of conventional educational wisdom, yet few of today’s classroom situations encourage pupils to perceive what they are doing as the construction of knowledge. For example, the views of many science teachers conflict with the constructivist stance taken in the non-statutory guidelines for National Curriculum Science in England and Wales (NCC, 1989c) and in teaching schemes based on investigation (such as CLIS, 1987). A study by Aguirre et al. (1990) investigating student teachers’ conceptions concerning the nature of science, teaching and learning produced depressing results; almost half of those questioned believed in a transmission model, namely the passive accumulation of a body of knowledge which has independent reality. Evidence for the mismatch between theory and practice is widespread across domains. For instance, many educators now believe that a combination of calculator work and a varied repertoire of reliable mental methods of computation is sufficient for most practical purposes and that there is little future for the traditional written methods (e.g. Shuard, 1986b). The use of out-of-school mathematics in school is generally no longer viewed as the intrusion of inappropriate and primitive strategies that need to be replaced by formal mathematics. Significant progress has accordingly been made in developing mathematical curricula which build on children’s existing knowledge and foster a flexible problemsolving approach (e.g. Hobbs, 1989; PrIME, 1990; Resnick et al., 1987; Shuard, 1986a; Stacey, 1991), and the National Curriculum recognises the limitations of the highly complex conventional written algorithms (NCC, 1989b). Nevertheless, the learning and practice of those algorithms remains a dominant activity in primary classrooms (in 1986 it took up 80% of the total time spent on number: Shuard, 1986a). The reports by Sewell (1982) and ALBSU (1983) indicated that one consequence of this outdated approach is an exorbitant rate of adult innumeracy in the U.K., leading to unexpected

6 difficulties affecting home life and employment prospects. The surveys both concluded that one in four British adults could not successfully solve problems such as calculating the change from £5 for one item. One reason why some children and adults cannot perform even the most straightforward calculations mentally (Cockroft, 1982; Sewell, 1982) may be because they are using mental versions of formal column arithmetic procedures (Hope and Sherrill, 1987). Those procedures often appear 'magical' and meaningless to pupils (Hart, 1981), and according to Lampert (1986, p. 340), many school leavers believe that ‘mathematics is an esoteric body of memorized knowledge enjoyed by a small and strange minority’! It should be stressed that pupils’ apparent difficulties in understanding mathematics and operating with numbers are probably not conceptual in origin; some may derive from confusions concerning notational conventions, as indicated by Nunes’ work on the development of understanding of signed numbers in everyday life (Nunes, in press). Similarly, formal teaching of science neither builds upon nor dispels children's robust, informal beliefs and gives little opportunity for the development of necessary qualitative reasoning. Its predominantly quantitative approach in the physics domain can instead hinder children in acquiring an understanding of underlying scientific principles (cf. White and Horwitz, 1987). Indeed, the same informal theories are found in undergraduate physicists (Viennot, 1979). The dominant conceptual change approach to science learning involves facilitating the restructuring of children’s conceptions through deliberate instruction. (While some of the most primitive ideas may need to be replaced, others can probably be fruitfully built upon.) However, the ‘situated cognition’ view suggests that co-existing alternative models may be appropriate in different contexts. Learning then becomes a process of distinguishing when particular conceptions are appropriate rather than one of exchanging faulty prior conceptions for scientific ones (Solomon, 1983). Since everyday notions of science are necessary for communication, the demand is for pupils to think and operate in two different domains of knowledge and to be capable of distinguishing between them, yet crossing over from one domain of meaning to another is exceedingly hard. Solomon’s work shows that solving physics problems deliberately set in familiar contexts can cause pupils to relinquish their grasp on learnt symbolic knowledge, reverting to familiar everyday systems of explanation. This phenomenon was researched by Murphy (1989) who found a significant performance difference when pupils were set the same problem in a familiar context and in an overtly scientific one. Her finding has been replicated by recent research concerning investigative work in science (Foulds, Gott and Feasey, 1992). (Note that the biological domain is less problematic, since rather than hindering the development of school-based scientific expertise, everyday biological knowledge appears to provide a positive foundation for it: Inagaki, 1990.) A further consequence of ignoring children’s prior knowledge is that there are frequent mismatches between actual learning experiences and outcomes, and teacher-intended ones (Simpson, 1988). An extensive investigation of language and mathematical learning at primary school level by Bennett et al.. (1984) found that an astonishingly high proportion - over half - of the tasks presented by teachers were mismatched to children's level of attainment, and teachers remained unaware of the cases where they had underestimated. In a similar vein, the mathematics education literature is replete with examples of the ways in which children ignore newly taught formal procedures, secretly adhering to their own intuitive methods whilst presenting a 'veneer of accomplishment' (Lave, Smith and Butler, 1988). Their behaviour is understandable and sensible; informal arithmetic methods - typically based on mental regrouping of terms - are usually more accurate and easier to execute than traditional written ones (Hennessy et al., 1992). Another tremendous advantage is their great flexibility; invented techniques can be adapted according to the numbers involved, and to the minds and purposes of individuals. Although an enormous variety of reliable methods is in general use and they share the same underlying logical principles as those taught in school (Carraher et al., 1988), these informal methods are in fact often devalued in the classroom context (Hart, 1981), and outside by both children and adults (Lave, 1988). My previous research indicates that the failure to build upon children's informal knowledge impedes acquisition of the formal calculation methods and precludes children from bringing their implicit conceptual understanding to bear: for example, children's invented versions of written procedures frequently violate their knowledge of basic arithmetic principles (Hennessy, 1986). The

7 teacher’s job accordingly becomes one of trying to reconnect principled conceptual knowledge with procedural knowledge, i.e. connecting procedures which have become mindless to contexts that give them mathematical reasonability (Lampert, 1986). Children are also known to construct their own intentions and conclusions regarding school science activities (Wittrock, 1977). These often differ from the teacher's agenda and may influence what is learned from an activity in unintended ways. Children's prior conceptual knowledge significantly affects their predictions, explanations and perceptions of novel phenomena and problem situations. Its interference with development of their understanding of some unfamiliar scientific concepts may give rise to conceptual confusion. 'Confirmatory bias' (seeing what one expects to see) is very common and prior beliefs are thus often reinforced through teaching which merely tries to overlay new ideas. Another consequence of this teaching method is that much of the knowledge acquired in many domains is inert. Remedial teaching is generally no more effective than initial teaching since it usually repeats earlier instruction whilst ignoring pupils' well-formed procedures and models, which may be incorrect or inappropriate in certain contexts. The critical importance of considering pupils’ different experiences outside school and how these affect their perceptions of tasks and learning situations in school emerges from Murphy’s (1990) discussion of the lessons learned from the Assessment of Performance Unit science project. Numerous examples (spanning the full range of tests and age groups) are offered which show that like other activities, assessment activities are situated: their content and context are highly significant variables affecting outcome. Indeed, the ways in which pupils access their knowledge depend on many things, including the perceived purpose of a task. Confusingly for them, pupils are often expected to apply their knowledge selectively in different circumstances. Murphy concludes that to construct meaning in pupils’ responses, assessors must understand that children’s models of the world depend on whether and how they have linked the different experiences gained in and out of school. Unfortunately, such understanding is often lacking and formal assessment in the classroom is rarely diagnostically revealing; it ignores everyday knowledge and tends merely to confirm what teachers already know (Simpson, 1988). Many teachers are consequently under the illusion that children know what they have previously been taught, but this is often not measured effectively. For example, Brown (1981) has observed that secondary school children have a much more limited mathematical knowledge base than teachers believe, and building upon it in fact necessitates re-teaching much of primary mathematics. It is obvious that merely presenting children with new information and experiences in the classroom is insufficient to promote learning. To avoid confirmatory bias and move pupils' primitive thinking forwards, new experiences need to be articulated and reasoned about, related to their informal conceptions, and generalised to other similar situations. An increasingly large body of research shows that social interaction contributes to children’s cognitive development, and that collaborative problem solving is a key means of significantly increasing the chances of these outcomes. Current work on computer-supported learning shows that tasks which encourage joint decisions not only can promote conceptual change (Howe et al., 1991), but they can even improve learning outcome on subsequent individual tasks (e.g. Blaye et al., 1990). There are many advantages of working in collaboration. For instance, it highlights the significance of individual elements in the learning process within the context of a meaningful whole, rather than promoting the practise of small pieces of skill in isolation. It also motivates learners, providing: encouragement to try new approaches, support for their partially successful efforts, and opportunities for appropriating shared thinking for their own uses. The notion of learning through participation in collaborative thinking processes is at the root of apprenticeship, as discussed below.

Apprenticeship

8 Cognitive psychology research into the processes involved in situated cognition and learning through collaboration with others is heavily influenced by the work of Vygotsky (1962; 1978) and other activity theorists in his tradition. Cognitive development is portrayed as the internalisation of cognitive activity originally experienced in social contexts, whereby the learner’s existing knowledge and skills are extended through appropriation of shared cognitive processes. This work shows that learning can be facilitated through a series of processes such as modelling, coaching, scaffolding, fading, articulation and encouraging learners to reflect on their own problem-solving strategies (Collins, Brown and Newman, 1989). These processes are the components of apprenticeship, which essentially involves providing help in developing an appropriate notation and conceptual framework for a new or complex domain and allowing the learner to explore that domain extensively, then gradually withdrawing support. This process makes normally hidden mental processes overt. Exposing learners to alternative viewpoints and providing counter-examples encourages articulation -and argumentation concerning the utility - of particular solutions or models (Vygotsky, 1978). The ultimate aim is to give learners control over their own learning processes and the confidence to engage in critical analysis. The apprenticeship process usually begins with a competent other person - the tutor - making explicit their tacit knowledge or modelling effective strategies through demonstrating desirable ways of problem solving in authentic activity. It then continues through the social sharing of tasks, supporting the learner's attempts to execute the task, and allowing knowledge to build up bit by bit. Fading then involves a gradual withdrawal of help and learner participation increases - according to the needs and learning pace of the individual - as independent thinking and practical skills are developed. Scaffolding refers to the help which thereby enables learners to engage more successfully in activity at the expanding limits of their competence, and which they would not have been quite able to manage alone, i.e. within the ‘zone of proximal development’. The latter extends to 'the level of potential development as determined through problem solving under adult guidance or in collaboration with more competent peers' (Vygotsky, 1978, p.88). Subsequent interpretations and applications of the notion of apprenticeship have without exception focused on the tutor’s implicit theory of the learner as being a crucial element of the scaffolding process. It is now considered critical that the tutor possesses some understanding of - and displays sensitivity to - the learner’s current needs, knowledge structure and performance characteristics. This understanding interacts with the tutor’s theory of the task or problem, which should incorporate an understanding of the skills and knowledge needed to handle the situation independently. It is necessary for generating feedback and devising situations appropriately tailored to the learner at any given point in task mastery. Developing the fundamental ability to generate hypotheses about a learner’s hypotheses and interpretation is difficult because problem-solving activity often has a deep structure that may not be apparent until a long sequence in process is near completion (Wood, Bruner and Ross, 1976). (For instance, the tutor cannot always be sure whether a child is ignoring a suggestion or systematically misunderstanding it.) Modification of the tutoring approach according to the learner’s responses entails a subtle evaluation and re-evaluation of the learner’s readiness and the level of participation of which s/he is capable; the learner always handles the problem before the tutor intervenes. Classroom teachers may carry out this kind of on-line diagnosis using information about the timing and length of a student’s responses or nonverbal cues, as in the case of undergraduate tutorial interactions as analysed by Fox (1988a; 1988b). The essential features of scaffolding in a classroom setting have not yet been conclusively identified, although very recent work is making strides in this direction. Teacher’s scaffolding strategies in mathematics, science and technology instruction are currently under investigation by Bliss and Askew (1992), and analytical frameworks for examining classroom interaction and the practical implications of the scaffolding concept have been put forward by Maybin, Mercer and Stierer (1992) and by Fleer (1992). Another important addendum to Vygotsky’s theory is provided by Bruner’s (1985) interpretation of the proposed development of consciousness and control. According to his analysis, the tutor serves the learner as a vicarious form of consciousness until the learner can master an action through achieving conscious control over a new function or conceptual system; it can then be used as a tool. Before that

9 point, the tutor effectively performs the critical function of scaffolding the learning task to make it possible for the learner to internalise external knowledge and convert it into a tool for conscious control. Wood, Bruner and Ross (1976) have provided a useful elaboration of the scaffolding functions of a tutor during joint problem solving. These include first recruiting the learner as a tutoring partner, then controlling those elements of the task that are initially beyond the learner’s capability, thus allowing her to concentrate on only those elements within her range of competence. The tutor continues to direct the problem solving, marks critical features of the task, and controls the learner’s frustration. (Fewer interventions per action are needed with older and more experienced learners.) The tutor also demonstrates solutions once the learner can recognise them, interpreting any discrepancies. Finally, the tutor stands in a confirmatory role until the learner is ready to act independently. A prerequisite in this analysis is that the learner must recognise the properties of a solution to a particular class of problems before moving towards producing it unaided (linguistically, comprehension must precede production). Tutorial interactions are a fundamental feature of learning in infancy and childhood. While Vygotsky’s work focused on the acquisition of language, children’s efforts in developing socially, physically and intellectually are in fact also assisted by more skillful others. Rogoff’s (1990) exhaustive treatise of work in this area discusses the processes of guided participation in which caregivers and children collaborate in arrangements and interactions that both tacitly and explicitly support children in learning the social and cognitive skills and values important in their culture. Socialisation through guided participation in relevant problem-solving activities is a universally observed process (although there are cultural variations in arrangements for and comunication with children). The caregiver subdivides tasks into manageable goals and gradually increases the child’s participation and responsibility for activities, thereby providing a natural bridging which extends the child’s familiar knowledge and skills to a higher level of competence. Rogoff’s thesis is that interactional cues are central to the achievement of a challenging and supportive structure for learning that adjusts to the learner’s changes in understanding. Children themselves are normally eager to seek and share meaning; they take a significant, creative role in structuring instruction (except in explicit teaching situations) and in influencing the nature and direction of scaffolding. This role increases with age. Rogoff stresses the importance of achieving a shared focus of attention, with children’s participation and social guidance building on the child’s perspective. She offers examples of mothers intentionally teaching their children and consciously adjusting their input and demands (according to the degree of skill or hesitation exhibited), as the children simultaneously adjust their level of participation and request assistance, greater responsibility and greater involvement. Support in such situations provides both challenge and sensitive assistance to the child, whereas insufficient familiarity with the child’s background precludes sensitive support and leads to difficulties for the child. In sum, interactions are finely tuned, and both adults and children actively manage their contributions to guided participation through collaboratively ensuring that the child works at a safe but challenging level, namely within his or her zone of proximal development. Note that children's learning may also be supported without deliberate attempts at doing so; guided participation includes tacit structuring of communication. Simply observing expert partners plays an important role too and has been shown to promote learning (Azmitia, 1988). Thus, learning is mediated by a learner’s own initiative in imitating and making suggestions as well as by expert guidance. In the case of craft apprenticeship, learners usually establish - through observation - criteria by which they can judge their own progress and they often correct their own errors (Lave, 1977). Rogoff’s analysis covers collaboration involving both symmetry and asymmetry of status, levels of skill and understanding of partners and their responsibility for adjustment to each other. Asymmetrical situations have the advantage that skilled partners can use their clearer ideas of the eventual goal and sophisticated means of reaching it to provide direction in problem solving and may assist the novice in appropriating relevant new information. The key prerequisites of communication between any partners include both common ground and differences in perspectives and ideas; for example, the interaction of

10 a young child with a peer - particularly an older sibling - may challenge both partners to stretch their understanding and take account of each other’s perspectives since they are relatively unskilled in supporting others’ communication. Another advantage of asymmetry, then, is that skilled partners themselves often gain understanding of the process they attempt to facilitate, and of the needs and skills of the children with whom they interact (for example, experience and troubleshooting strategies acquired by new parents in testing hypotheses regarding how to handle their first child are useful in handling their second child). Both partners individually appropriate the jointly produced products, so that information and skills are not transmitted but transferred in the creative process of appropriation. Rogoff (1990) has clarified this notion of the appropriation of shared activity as reflecting an individual’s understanding of and involvement in the activity rather than the taking of something directly from an external model. She adopts Leont’ev’s (1981) view that 'internalisation is not the transferal of an external activity to a preexisting, internal plane of consciousness: it is the process in which this internal plane is formed' (p.57). In these accounts, interpersonal aspects of an individual’s functioning are internalised integrally with the individual aspects. In conclusion, development is not spontaneous but is chanelled through sociocultural activity, in which children and their partners are interdependent. Social exchanges are continuous and essential bases for advances in individuals’ ways of thinking and acting. Communication and shared problem solving inherently bridge the gaps between old and new knowledge, and between partners’ differing understanding of the values and tools of the culture, which itself is revised and recreated as they seek a common ground of shared understanding.

CLASSROOM APPLICATIONS: COGNITIVE APPRENTICESHIP Introduction Our increasing understanding of everyday cognition has significant practical implications for improving learning in the classroom situation. To summarise, work in this field shows that learning is most successful when embedded in authentic and meaningful activity, making deliberate use of the physical and social context. Expertise is acquired through both the spontaneous invention of personal, highly efficient procedures in response to the needs of a situation, and through apprenticeship. The latter is actually the normal means of informally indoctrinating novices in the workplace in ours and many other societies. Previously, this mode of teaching was formalised in the form of the guild system for vocational training. The master taught the novice in a realistic environment, so that professional skills were acquired through interaction with the same tools used by experts; the master’s control was faded as the learner developed expertise (Pieters and deBruijn, 1992). Unfortunately, formal technical skill training and professional education have moved towards a largely unproductive blend of schooltype instruction (theoretical explanation), unstructured observation and practice, with inadequate engagement with the tools and materials of work (Resnick, 1987a). Apprenticeship has almost disappeared in the Western educational system, although it is still current practice, for example, within science research groups where young scientists successfully learn informally from more experienced ones. However, our relatively recent system of formal schooling is incompatible with this model and has led to an unfortunate decontextualisation and abstraction of many kinds of skills and knowledge from their uses in the world. Insofar as they address processes, conventional pedagogic practices still tend to emphasise routines for solving 'textbook' problems, and domain (conceptual and factual) knowledge and procedures. Heuristic, control and learning strategies - the crucial elements of problem solving (Schoenfeld, 1985) - are usually ignored. Consequently, the metacognitive awareness that is critical for mathematical and scientific thinking (Kuhn, 1989) does not develop. At present, problem-solving activity by experts - mathematicians and scientists engaged in everyday practice - much more closely resembles that of ordinary people in everyday situations than what takes place in school. Cultural transmission through schooling is ineffective and distorted, and

11 success within this culture often has little bearing elsewhere. The problems which engage and motivate learners in school classrooms are dilemmas about their performance rather than problem-solving dilemmas. For example, the rhetoric of the relatively new curriculum area of design and technology education assumes that pupil motivation is provided by the posing of 'real' problems (e.g. designing a new cycle lock), whereas previous experience indicates that such aspects of the everyday culture cannot be so readily transposed to the artificial environment of the classroom. Attempts to make classroom activity more meaningful and contextual, for example by introducing mathematics word problems, have tended to fail dismally. Such ‘problems’ have little in common with life outside school; they are merely 'coated with a thin veneer of ‘real-world’ associations' (Maier, 1980, p.21). Faced with such problems, learners continue to rely heavily on their knowledge of standard textbook patterns of problem presentation rather than their knowledge of problem-solving strategies (Schoenfeld, 1985). The former knowledge provides cues for students in many domains where narrowly-focused problems require correct solutions, including study at undergraduate level (Miller and Parlett, 1974). A more promising current development is the ‘Enterprising Mathematics Project’ which aims to develop concepts and techniques from investigations and problem solving in real contexts, and to relate mathematics to other school subjects (Hobbs, 1989). It remains to be seen whether such approaches can succeed in overcoming students’ cue-consciousness. The critical factor is provision of authentic dilemmas and these may be imaginary or real (Lave, 1992). Because totally different ways of learning are usually imposed in the classroom (Brown, Collins and Duguid, 1989), misconceptions of what practitioners do are common. Rennie’s (1987) survey showed that children hold misconceptions of what technology is and of its pervasiveness in everyday life. A pilot study carried out as part of the ongoing PSTE (Problem Solving in Technology Education) project at the Open University investigated perceptions of problem-solving activities undertaken by children outside school and by expert practitioners, and their relationship with school-based activity. The preliminary results signify that the link is in fact minimal in both teachers' and pupils' views; our subjects saw a link only where there was a one-to-one correspondence between activities (e.g. kitemaking) or use of skills. Of course, pupils' experiences are almost inevitably remote from those of practitioners. Nevertheless, teaching which does not even reflect the real world of technological activity is unlikely to be successful in developing children's awareness and appropriate use of technological thinking, action and vocabulary. In order to realise the pedagogic intention that learning design and technology prepares children for living and working in a changing technological society, then, teaching needs to be informed by knowledge of the nature of technology as it is actually practised (Medway, 1989). For example, Petroski (1985) portrays the process of developing a successful idea as a steady improvement upon unsuccessful ideas and learning from past failures - one's own and others' - by avoiding weaknesses observed in existing artefacts/systems. Yet, children often start from scratch without evaluating the latter. Prototype modification and refinement is indeed a more realistic and less frustrating mode of working than the present demand for the artificial generation of several design ideas. Research shows that existing assessment procedures can lead pupils to omit unsuccessful designs from their final project folders (Anning, 1992) and to describe a logical, systematic procedure rather than the actual development of design ideas (Jeffery, 1990). The Hayes report (1983) indicates that professional designers too have been criticised by their employers for not offering alternative solutions to a design problem, and for starting from scratch or persisting with an unsuccessful idea. These imposed constraints have the same consequences outside and inside school: professional designers can be observed to doctor their portfolios in the same way that children do.

Cognitive apprenticeship programmes There have, however, been a few significant exceptions, whereby craft apprenticeship methods have been adapted for the guided learning of thinking and problem-solving skills in a classroom context: this is the notion of cognitive apprenticeship, which has been successfully applied to a variety of school subjects, notably mathematics. Lave, Smith and Butler (1988) suggest that it is crucial to

12 engage with children in argumentation about real dilemmas in the area of focus. These should provide opportunities for discovery and invention of problems as well as solutions, and understanding in patterns of ongoing activity, rather than specific problem procedures and a compendium of experts' truths (cf. Papert, 1980). This approach is encapsulated in the mathematics curriculum offered to primary school children at the University of Irvine's 'Farm School'. Similar examples include teachers and pupils negotiating and making sense of mathematical notations and procedures such as long division (as described by Newman, Griffin and Cole, 1989), and the collaborative thinking about mathematical concepts and principles arising during Lampert’s (1986) classroom instruction in multidigit multiplication. The latter entails whole-class discussion in which children invent and justify solutions. The teacher models interpretive problem solving, making an explicit connection between the concrete/intuitive and the computational, and requiring children to use their own ways of deciding whether procedures are mathematically reasonable. Resnick and her colleagues have developed an initial mathematics programme built entirely on children’s invented procedures and informal knowledge about quantities and their relationships (Resnick, Lesgold and Bill, 1990). This makes a link with the formal language of mathematics by using standard notation to record public discussions and conclusions, and draws upon everyday problem finding; children generate and bring in problems from home, and their homework concerns events and objects in their home lives. It has successfully been shown to foster both number sense and computational competence, and can therefore replace rather than merely supplement the standard curriculum. The basic notions of cognitive apprenticeship have been successfully embodied in two further instructional programmes which mimic the conditions of natural learning and engage students in constructing and interpreting meaning (a concise overview and critique of these programmes is provided by Collins, Brown and Newman, 1989). In Palincsar and Brown's (1984) 'reciprocal teaching' of reading comprehension, a progressive transfer of responsibility takes place as the teacher’s demands for student involvement increase and students begin to perform parts of the task until they finally produce strategic behaviour which resembles that initially modelled by the teacher. The students and teacher take turns playing the role of teacher and the students gradually serve as experts to each other; they pose questions about texts, summarise them, offer clarification and make predictions. The programme has been found to promote significant and lasting improvement in reading performance after 8 weeks. (Resnick and Nelson-Le Gall, 1990, ] are currently attempting to adapt the method of reciprocal teaching for mathematics learning). In Schoenfeld's mathematics courses, problem-solving heuristics are deliberately taught, in a contextualised way that makes contact with the students’ knowledge base. These heuristics themselves are critically assessed as the students use self-regulatory strategies to assess their own solution paths; they ask themselves questions such as ‘What am I doing now?’, ‘Why?’ or ‘Am I making progress?’ (Schoenfeld, 1985). Schoenfeld (1991) outlines three further programmes of mathematical instruction which share an underlying goal with his own programme and with Lampert’s: that students develop a deep, rather than a superficial understanding of mathematical processses. These programmes have specifically overcome the problems inherent in importing the concept of scaffolding from its context of investigation - the linguistic and cognitive development of very young children - to a practical classroom context (cf. Maybin et al., 1992). Indeed, they have successfully tailored the characteristics of everyday practice and of craft apprenticeship to the needs of pupils in school (including socially disadvantaged children and slow learners). For example, by beginning with a task embedded in a familar activity, Schoenfeld's approach shows his students the legitimacy of their implicit knowledge and its availability as scaffolding in apparently unfamiliar tasks. All of the programmes outlined strive to promote reflective thinking about the meaning of what children are doing and its relation to solving real problems, and they are based on the notion that development of meaningfulness comes from a social process of interaction and negotiation. Brown and Campione (1984) have analysed the mechanisms of transfer which underlie such successful instructional programmes. They stress the importance of appropriate transfer across domains, of adaptability and flexibility (rather than routine expertise), and of deliberately preparing for transfer by seeking

13 analogies, performing thought experiments and self-questioning. Good learners are aware of what they know and do not know. They facilitate and reflect upon their own learning using other metacognitive strategies such as planning, predicting outcomes, and managing time and cognitive resources efficiently (Brown, 1978). Another important feature of successful programmes is that they build a sense of ‘empowerment’ and confidence in the learner’s own knowledge and mastery of the domain. Children need to realise that there are inevitably multiple ways to solve any problem, and to think of themselves as reasoners who are able - indeed expected - to discover some of those ways (Resnick, Lesgold and Bill, 1990). The frequent tendency to present a preconceived view of a ‘correct’ way of mastering a mathematical procedure, for example, may be singularly unsuccessful with some children; mathematics must be matched to each individual and built upon his or her existing knowledge base (Hart, 1981). A fundamental difference which is evident between conventional teaching programmes and those based on apprenticeship is the degree of control taken by teacher and pupils; in one case, they together engage in deciphering the demands of a textbook, and in the other, they collaborate in building a culture of sense-making in the classroom. This means assessing the adequacy of their shared understanding of the purpose of their mutual pursuit - which must obviously be clear to pupils - and sharing responsibility for ascertaining the legitimacy of procedures, actions or solutions (e.g. by reference to known mathematical principles). There may be a lesson to be learned here from Japanese methods of evaluating the products and processes of students’ problem-solving efforts, which commonly involve children in presenting inadequate solutions to the class for discussion of ways of correcting them; this is a far cry from the predominant Western obsession with praising students who perform well (Stigler and Perry, 1988). This highlights again the crucial importance of providing a target for reflective comparison through the discussion about problem-solving processes and solutions which underlies many effective apprenticeship programmes. Comparison between one’s own and one’s peers’ ideas helps develop awareness and means of modification, and the teacher’s guidance is also critical. In sum, learning experiences and activities can be rendered useful and meaningful by the sense made of them by classroom talk (which is generally under-valued as a learning tool: Edwards and Mercer, 1987). The apprenticeship model involves successive approximation of mature or expert practice. However, its success depends upon its emphasis on teaching not only the skills and domain knowledge which experts possess but more importantly, the actual processes they use to handle complex tasks. In the case of cognitive apprenticeship, this tacit, strategic knowledge includes both cognitive and metacognitive processes - ideally incorporating (a) problem-solving heuristics, (b) control strategies with monitoring, diagnostic and remedial components for managing problem solving, and (c) knowledge about how to learn, including general strategies for exploring a new domain and local ones for reconfiguring knowledge (Collins, Brown and Newman, 1989). (These are rarely observed in the average classroom.) Cognitive apprenticeship programmes promote situated learning by giving students the critical opportunity to observe, engage in and invent or discover expert strategies in context. As with other forms of apprenticeship, the emphasis is on developing the learner’s own resources rather than teaching recipes or algorithms. Ultimately, support is withdrawn and students are gently eased into a mode of independent problem solving; teaching exploration strategies is critical if they are to explore a domain productively and pose problems that are interesting and that they can solve. It is also important that teachers relinquish control and refrain from dominating the agenda and discussion so that pupils can function independently of the precise context of the classroom activity (Edwards and Mercer, 1987). In contrast with traditional apprenticeship, cognitive apprenticeship programmes aim to extend situated learning to different settings, generalising acquired knowledge across a range of applicable contexts. This serves to help learners to cope with novel problems and to progress from embedded activity to general principles of the culture. Mathematics, science and other areas of problem solving are considered to be practices or forms of authentic activity, rather than bodies of content which must

14 be assimilated. Students are thus exposed to a culture's ways of thinking and solving problems, and to its conceptual viewpoint. According to Brown, Collins and Duguid (1989), this is essential if the knowledge and conceptual tools they acquire are to be robust and useable rather than inert, like most knowledge acquired through conventional schooling. The outcomes of learning are said to be recognising opportunities or problem finding, knowing when and how to apply skills that have been learned in other contexts, and exploiting properties of the presenting situation (Collins, Brown and Newman, 1989). Note that an interaction between domain-specific and strategic knowledge is critical, and that there is no disjunction between conceptual knowledge and problem-solving activity, i.e. between knowing and doing, in this analysis. The assumption that they can be separated ignores the ways in which situations structure cognition. Activities are not carried out with the explicit purpose of determining principles separate from acting in the world. Computer-supported learning environments In classrooms where cognitive apprenticeship programmes operate, pupils usually derive help from an adult (teacher) or more skilled peers. Other children may fill different, important roles to those taken by adults and they are more available. They offer unique possibilities for discussion, exploration and collaboration when they carefully consider and challenge each other’s perspectives (Rogoff, 1990). However, the advent of sophisticated computer technology in today's classrooms now offers an additional resource of a related kind (Hoyles and Noss, 1987) and this enables us to realise ‘intelligent’ apprenticeship learning environments that were previously impossible or impractical (Collins, 1991). The recent trend towards development of computer-based learning environments based on 'guided discovery' and experimentation (Elsom-Cook, 1990; Hennessy et al., in press[a]) illustrates how we have begun to exploit this resource, further extending the notion of scaffolding through guided participation in social activity. Some of the tools now available have been designed to provide procedural facilitation; i.e. they enable learners to carry out more complex and sophisticated operations than they could otherwise carry out (without providing the substantive knowledge required); research involving children using a LOGO environment exemplifies this (Hoyles and Noss, 1987; Noss, 1991). One line of research is exploring the potential of hyper- and multi-media for helping learners master thinking skills which allow them to access and use information effectively (Dede, 1992). A key example is a hypermedia tool called HyperAuthor developed by Carver, Lehrer and their colleagues for scaffolding students’ acquisition of an extensive set of design skills (Carver et al, 1992). HyperAuthor contains both hypermedia construction tools and reflection tools that help students focus on the organisation and structure of their ideas. Its underlying model of design skills is used as a basis for a wide range of assessments of students’ understanding and use of their skills. Other ongoing developments include computer-simulated environments for technical/ professional education (e.g. Johnson, 1990; Lesgold et al., 1986) and Pieters and deBruijn’s (1992) exploration of heuristic coaching (offering the learner global advice rather than a solution). The aim here is for students to schematise their strategies and use them in another appropriate learning phase or in a similar context. One of the most promising and ambitious developments in this area is the Sherlock system which combines instruction and dynamic, diagnostic assessment in the context of troubleshooting. Lajoie and Lesgold (1992) assert that the system provides a platform for assessing: structural changes in learners’ knowledge organisation, transitions in their mental models, the efficiency of troubleshooting procedures, whether learners know the information required and whether they interpret given information correctly. Finally, Sherlock is said to be able to foster metacognitive skills (by explicitly modelling which plans a learner has entered up to the point of an impasse) and to assess the use of those skills. Design principles for effective computer-supported learning environments have been derived from the key conditions of apprenticeship by Scardamalia et al. (1989). They include: making knowledge construction activities overt through objectifying knowledge and encouraging comparison, criticism and cross-fertilisation of new ideas and information; maintaining attention to cognitive goals;

15 examining and maximising existing knowledge; sustaining perceptible progress through knowledge operations which have relevant consequences; supporting individual learning styles; giving pupils responsibility for contributing to each other’s learning. These principles have been implemented in Scardamalia and Bereiter’s cross-curricular CSILE system, which is an impressively flexible, networked hypermedia system with a student-created communal database. The latest version of CSILE is based on the idea of a knowledge-building community and is aimed at restructuring classrooms as places for collaborative inquiry (Scardamalia and Bereiter, 1992). It constitutes a prototype general-purpose environment which enhances the value of public and private knowledge. Activities in this environment engage children in the conscious, cooperative development of shared knowledge, in the same way that research communities work. Students contribute to the advancement of the group’s knowledge (in contrast with the usual judgment of schoolwork using internal criteria) and their activities are not separated from either the curriculum or the social life of the classroom. Discreet environments have now been created for different kinds of operations such as hypothesis-testing, exploring analogies, identifying causal relationships and underlying mechanisms, and research using on-and off-line reference material. Scardamalia and Bereiter acknowledge the dangers of computer intervention in complex and delicate cognitive processes and they have consequently moved toward designing whole environments to be supportive of certain kinds of mental activity, rather than depending on specific procedural facilitations (hints or structural devices triggered by user actions). While many computer-based tools incorporate a model of the learner, they actually encounter similar difficulties to those of human tutors in recognising hypotheses underlying long sequences, particularly at the most critical point - the ‘disordered’ mid-phase of learning. There are indeed too many complexities for either human or computer programs to take into account at this point, where formal programmes of individualised teaching are consequently most difficult to realise. Fox (1998b) argues that the sophisticated, tacit processes of diagnosis and repair used by human tutors in sensitively adjusting their support are altogether beyond the capability of a computer program, which lacks the necessary cognitive flexibility and multiple interpretations of ongoing interaction in context. Suchman (1987) similarly highlights the disparity of the richness of the resources available in situated interaction, maintaining that machines lack access to most of their users’ actions and circumstances. Merrill et al. (in press) are more optimistic in their comparison of the guidance and support offered by human tutors and intelligent tutoring systems. They conclude that human tutors offer more flexible and more subtle support, but that the differences in terms of monitoring problem solving and guiding students in error recovery are smaller than previously argued. Future development work will probably lead to more interactive and effective computer tutors, but it must be recognised that the scope of interaction between machines and their users presently remains substantially limited in some ways. SITUATED VERSUS DECONTEXTUALISED KNOWLEDGE AND THE NOTION OF GENERAL PROBLEM-SOLVING CAPABILITY Expectations and evidence of transfer A fundamental implication of the research on learning through apprenticeship in everyday situations is that learning of knowledge or skills takes place in the context of their intended use. Nevertheless, many teachers and educators work on the unvalidated assumption that universal cognitive skills of thinking and problem solving can be taught (often independently of the acquisition of prior subject matter) and immediately and flexibly applied to a variety of contexts (e.g. Argles, 1988; Bonington, 1988; Wharry, 1988). Such skills have the appealing potential to transfer across subject boundaries as well as to activity outside school and in adult life, whereas pure content-focused study is for the most part accepted as having extremely limited application in everyday life. Domain knowledge tends to remain inert when it is acquired in isolation from realistic problem contexts (Collins, Brown

16 and Newman, 1989). The assumption of domain-independent cognitive skill underlies the development of programmes of instruction in higher order skills, such as de Bono’s CoRT (Cognitive Research Trust) Thinking Program (this and other such programmes are outlined and reviewed in Segal et al., 1985, where the effectiveness of directly teaching thinking skills is also questioned). The CoRT programme uses generally familiar knowledge and is intended to foster problem-solving, interpersonal and lateral thinking skills, including metacognitive skills (such as planning) and a wide variety of evaluation techniques and idea- and solution-generation techniques. Although the programme has been widely used in USA and elsewhere for over 15 years, very little systematic evaluation of it has taken place and there is minimal support for its lofty claims (although anecdotal evidence is often positive and short-term gains have been reported by Edwards, 1991). There is some other evidence to support the notion of general cognitive skill development (Adey and Shayer, in press); however, the limitations of transfer theory and the context-dependence of the application of knowledge and skills have increasingly been recognised and demonstrated empirically (Glaser, 1984). This evidence derives from several sources. To begin with, the first substantial evaluation of de Bono's classroom materials by the Schools Council (Hunter-Grundin, 1985) found little evidence for transfer of learning, despite the programme’s ambition for students to extend their new thinking skills to ‘a variety of real-life situations’ (de Bono, 1985). Secondly, a considerable body of work, beginning with Gay and Cole’s (1967) cross-cultural study of estimation tasks, and including that of Donaldson (Donaldson, 1978; McGarrigle and Donaldson, 1974), has shown that cognitive performance is affected by task familiarity and implicit contextual cues. It can be improved through making more explicit the nature of a task - changing its focus or symbolic features. (For example, Roazzi and Bryant, 1992, have shown that children’s approach to, and their performance on, a quantitative task is determined by the extent to which they realise it is a quantitative task.) This work presents a substantial challenge to Piagetian theory. In the same vein, the situated cognition literature discussed above clearly provides some further evidence against the notion of a general problem-solving capability. Specifically, for instance, the study of young street vendors in Brazil by Carraher et al. (1985) found that problems embedded in a familiar context were solved most easily. However, careful manipulation of task conditions has indicated that transfer and generalisation from an initially limited range of contexts and contents may be mediated by tasks set in familiar contexts (Schliemann and Magalhaes, 1990). Lave (1988) has additionally reviewed research in cognitive psychology that, despite manipulation of experimental conditions and problems, cannot demonstrate a strong transfer. The problems posed have typically required correct, experimenter-determined answers, whereas authentic and everyday problem solving is characterised by the resolution of dilemmas. In fact, the situated cognition framework diminishes the importance of the notion from more conventional learning theory that knowledge exists independently of individuals and is to be transferred to novel situations and activities. The latter are exceptionally rare in everyday life and when they do occur, individuals are unlikely to encounter them alone. According to Lave (1988), attempts to represent everyday cognitive activity as a sequence of ‘recognising a problem, representing it, implementing a resolution and evaluating the results’ ignore the multitude of ways of tackling a problem and the fact that some activities take place simultaneously or structure each other differently on different occasions. We can conclude that a simplistic, linear all-purpose problem-solving process is inherently unproductive and most unlikely to be successfully appropriated and generalised. Yet, such a process forms the backbone of the design and technology component of the present National Curriculum for Technology (in England and Wales), which explicitly goes beyond the specific qualities which technology education in particular might be expected to foster. It intends to develop general practical capability and to prepare students to handle complex problems in their future personal and working lives (NCC, 1989a, 1.47). Technological capability is defined in terms of the ability to employ the processes of: identifying and clarifying tasks or problems; investigating; generating and developing solutions; evaluating (NCC, 1989a). Design educators maintain that these broad processes are universal, and proponents of this approach make ambitious claims about the variety of potential

17 outcomes; these comprise personal, social, cognitive, metacognitive, creative and practical skills. Examples include discovery, critical assessment, decision making, problem solving, planning, evaluating, reflecting, and collaboration (deLuca, 1992). Eggleston (1992, p. 24) predicts that such skills will have a 'wide general applicability in the adult life likely to be experienced by the students', and deLuca argues that technological problem-solving activities simultaneously teach students how to apply knowledge learned from experiences in and out of school. Sellwood (1990) asserts further that the value of the process approach lies in its structuring and organising of thinking skills, and that it will ideally become second nature to pupils (and teachers) to organise a means of successfully achieving objectives which can then operate at all levels and in all situations. As yet, there is no evidence to support these claims (see Hennessy, McCormick and Murphy, 1993, for a fuller discussion of the ‘general problem-solving capability’ notion in technology education). As in other domains, it seems likely that direct teaching for transfer will be necessary to realise the desired aims. This could incorporate some of the features of the successful cognitive training programmes described earlier, as identified by Brown and Campione (1984): namely, task-specific skills training, self-regulation training, and awareness training. Instructional programmes which instead leave the problem of transfer up to the learner are inevitably less effective. Direct teaching for transfer can also be successful in facilitating children’s conceptual development. This is exemplified by my own recent work addressing children's alternative conceptions of science using interactive computer simulations (Hennessy et al., in press[b]). The body of research in this field indicates that, like the prominence of adults’ scientific and technological knowledge in practical contexts (Wynne, Payne and Wakeford, 1990), children's conceptions are perpetually context-bound (Driver et al., 1985; Solomon, 1983). Our intervention showed that an approach which both explicitly addresses specific difficulties and highlights qualitative relationships between variables, can, however, help children to develop a coherent set of rules which transfer across physical contexts, including both real ones and those not directly experienced (such as space). Nevertheless, transfer of newly acquired concepts and procedures across the cultural barrier between the classroom and everyday problem solving can be problematic. One obstacle is that formal knowledge, such as that taught in science and mathematics, is not in a form that can simply be ‘applied’. Layton (1991b) reminds us that formal scientific knowledge needs to be reconstructed, integrated and contextualised for practical action in everyday life. (This means re-introducing real-life complications, emphasising lost relationships between components of scientific knowledge, and reducing the level of abstraction.) Resnick makes a similar case for mathematics: 'the packages of knowledge and skill that schools provide seem unlikely to map directly onto the clusters of knowledge people will use in their work or personal lives' (Resnick, 1987a, p.15). This is undoubtedly true in both directions. Carraher (1989) proposes that representations of certain aspects of mathematics encountered in everyday life have become incorporated in natural language and differ from those used in school for the same concepts; arithmetic problem-solving behaviour and understanding is re-organised under the influence of schooling and the learning of formal representations. Our demands on pupils to transfer knowledge and skills across domains may also be unreasonable within the context of school. An example is the belief that pupils will bring knowledge from other subject areas to bear in technological problem solving. The design process embodied in the National Curriculum presumes that pupils will apply conceptual knowledge from across the curriculum especially from science and mathematics - in conjunction with procedural knowledge specifically from technology in tackling realistic problems, designing and making artefacts and systems (Layton, 1991a). However, the use of scientific knowledge in technological activities is not at all straightforward, as McCormick (1992b) has outlined; at the very least, some degree of restructuring or translation is required (Layton, 1991b). Informal observation accordingly indicates that children have difficulty in bringing in knowledge and skills from other domains, even when they have recently been demonstrated. Conversely, Layton (1991b) implies that design parameters are often specific to particular situations and have little validity outside these fields.

18 Narrowing the focus further to the domain of design and technology itself, we find a clear expectation placed upon pupils to appropriate an all-purpose design process and apply it across five very diverse subject areas which previously had little connection in the curriculum. The notion of transferable general problem-solving skill appears unrealistic once again in this context. Research has shown that failure to transfer often derives from a lack of the conditions needed for transfer rather than from domain-specificity (Perkins and Saloman, 1989). However, there is actually no evidence that technology teachers are effectively or even explicitly assisting pupils in acquiring general skills and in making a convincing link between the five subject contexts. Teachers are not usually trained to teach across contexts and we know that their previous subject traditions strongly influence the new ways of working required by design and technology (McCormick, 1991). The specific activities within different subjects may actually be projecting different views of the ‘design process’. This is exacerbated by the use of loose overarching themes within which the subject areas work, and may preclude pupils from developing a coherent view of the process. Experience indicates that children engaged in ‘food technology’ activities, for instance, perceive that they are ‘doing cooking’ rather than ‘investigating’ or ‘evaluating’; this is hardly surprising. Links across contexts are probably tenuous and pupils are consequently unlikely to be able to recognise, reflect on and use the prescribed subprocesses. Subject matter knowledge and problem-solving skills: a balance While most forms of everyday problem solving require context-specific forms of competence, and the traditional, strong generalist position now lacks credibility, intuition nevertheless leads us to believe that there are some commonalities in children's thinking. Global problem-solving strategies which may have a role to play in knowledge acquisition include general heuristic, control and learning strategies (Collins, Brown and Newman, 1989). These are especially helpful when experts face novel problems in a domain (Perkins and Soloman, 1989). Situation-specific learning by itself can be very limiting, precluding transfer when familiar aspects of a task are changed (as shown by Schliemann and Acioly, 1988, in their comparison of schooled and unschooled individuals' ability to invent new procedures for calculating bets). The power inherent in decontextualised knowledge was recognised early on by Bruner (1972) who argued that the process of reorganising knowedge into formal notational systems results in far greater flexibility. When the solving of particular problems becomes a mere instance of much simpler general problems, the range of applicability of knowledge is increased. In mathematics, furthermore, the power of expressions derives from their divorce from the situations to which they refer. In this sense, the meaning of algebra is encountered within the formal system, although algebraic expressions and transformations can always be interpreted in terms of the quantities and relationships to which they refer (Resnick, 1986). Much work has focused on schooling’s counterproductive requirement that students use symbolic representations and procedures before they fully understand them. By studying how people come to make sense of mathematical ideas outside school, we can now perhaps learn how to promote that understanding, hence increasing the utility and power of mathematical knowledge. Although general performance components in complex tasks are difficult to specify (Gardner, 1985), the situated perspective on knowledge construction in everyday contexts is sometimes unconcerned with these. Its focus on embedded learning thus puts it in conflict not only with prevalent adaptations of cognitive science views of problem solving to educational research, but also with Piagetian stage theory. The latter emphasises the role of cognitive invariants in the structuring of local contexts; the situated approach conversely stresses the importance of context in the construction of cognitive invariants and questions the power of formal thinking. Ackermann (1990) helpfully smooths over the differences by offering an integrated perspective which maintains that distancing oneself from and then re-engaging with a situation is sometimes necessary for gaining deeper understanding. Like Bruner, she asserts that decontextualised knowledge helps us master complex situations, giving them form. Replication of the Piagetian water-level experiment led her to the compelling conclusion that

19 cognitive development depends on both coordination of local knowledge and differentiation from general rules. Perkins and Saloman (1989) have also attempted to reconcile the differences between the conflicting context-free and context-bound positions, by focusing on general heuristic strategies which function in contextualised ways to access and deploy domain-specific knowledge. A major conclusion of an extensive literature review by Alexander and Judy (1988) was similarly that problem-solving strategies cannot be utilised efficiently in the absence of a foundation of domain-specific knowledge. Collectively, the above findings concerning lack of transfer are in keeping with this conclusion, and a recent collection of papers on enhancing thinking and learning skills in science and mathematics (Halpern, 1992) indicates that the present trend is indeed towards embedding instruction within specific academic disciplines of the curriculum, rather than isolating thinking skills as separate topics. (Isolated decontextualised skills are problematic in many ways, not the least being that they do not usually engage pupils’ interest.) Unfortunately, cognitive skills tend to be driven out altogether by a demand for teaching ever larger bodies of knowledge, with the idea that their application to reasoning and problem solving can be delayed (Resnick, 1987b). Educational practice swings periodically between knowledge-oriented and general process- or skill-oriented teaching, with an accompanying de-emphasis of subject matter knowledge. This is precisely the situation in which technology education finds itself at present. The value of an all-pervasive design process and the accompanying diminished role of craftwork skills in the present technology curriculum is highly controversial. For example, the recent report on technology education sponsored by the Engineering Council deplores the fact that practical skills have become a secondary concern in design and technology, which now constitutes 'generalised problem solving without a specialised knowledge base' (Smithers and Robinson, 1992, p.6). This controversy has already led to an increase in weighting of the 'planning and making' component and revision of the Statutory Order for Technology is underway. The new formulation will probably raise the prominence of specific skills, knowledge and understanding (NCC, 1992). This oscillation is likely to prove unproductive since research on problem solving in knowledge-rich domains shows that subject matter knowledge and reasoning processes are intimately connected. Instruction needs to accomodate this connection so that thinking and problem solving arise in the context of acquiring structures of knowledge and skill that comprise the subject matter of schooling (Glaser, 1984). Encouraging steps have indeed been taken in this direction in the form of the recently publicised work on ‘cognitive acceleration’ through science learning (Adey and Shayer, in press). Focusing on cognitive conflict, metacognition and bridging, this intervention has led to increased achievement in mathematics and English language as well as science. In sum, an important implication of the research literature is that instruction needs to achieve a balance between subject matter knowledge, problem-solving strategies and strategies for effective learning. CONCLUSION The work reviewed here indicates that cognitive activity is socially defined, interpreted and supported, and that prior knowledge and experience are crucial factors in the learning process. The situated cognition literature shows that the thinking of ‘experts’ and lay people alike is intricately interwoven with the specific problem-solving context and sensibly adjusted to meet the situation's demands. Problem-solving strategies are shaped by the structuring resources available in the situation (Lave, 1988), and all acquired knowledge and understanding is ‘situated’ in the sense that it is partly a product of the activity, context and culture in which it is developed and used (Brown, Collins and Duguid, 1989). A major conclusion from this literature and related work concerning scientific ‘knowledge in action’ is that there is a fundamental distinction between problem-solving processes in everyday and classroom situations. Specifically, in real life problem solving, a situation becomes a problem in the course of activity in a particular setting (Lave et al., 1984). Uncertainty, flexibility,

20 improvisation, contingency and adaptation to uncontrolled factors are normal, and values, goals and beliefs are taken into account (Wynne, 1991). This provides a contrast with (a) the culture of school mathematics, whose emphasis on formal procedures precludes application of sophisticated informal knowledge, and similarly, (b) the culture of science with its assumptions of coherence and manipulability, as perceived - and often rejected - by both schoolchildren and lay adults (Wynne, Payne and Wakeford, 1990). The curriculum emphasis on acquiring algorithms, formulae and other quantitative techniques seems to override 'doing science' (Jenkins, 1992), 'doing maths' or indeed, 'doing technology’; we can now add (c) the culture of school technology, where an over-emphasised and narrowly perceived problem-solving process may again result in a mechanical approach to solving artificial 'problems'. The research discussed demonstrates the effectiveness of situated learning in supporting the emergence of productive thinking. It has moved us radically forward from the wisdom of 20 years ago when school-based knowledge was perceived as being powerfully abstract and context-free rather than ‘encapsulated’ as it is presently characterised (Inagaki, 1990). It is now recognised that knowledge acquired outside school may attain a far more sophisticated level and display generativity that was previously unknown (Cole, 1990). Although school is an artificial setting in many ways, it is nevertheless the way in which our society chooses to organise its education of young people. It is a ‘real world’ of its own for children (and teachers), absorbing most of their waking hours for many years; school activity is situated practice. Within this framework, there are in fact opportunites for training pupils to develop practical expertise and for preparing them for the world of work and everyday problem solving. While it is impossible (as well as probably undesirable and impractical) to provide direct job training for school students since job requirements now change so quickly, the potential is there instead for work which steps back to consider and evaluate the everyday world through intellectual processes of reasoning and reflection (Resnick, 1987a). Resnick chides us to take up these opportunities and to aim to prepare people to be good adaptive learners, able to perform effectively in unpredictable and changing situations. 'One important function of schooling is to develop the knowledge and mental skills students will need to construct appropriate mental models of systems with which they will eventually work... (These mental models) permit flexibility in responding to unexpected situations' (p.18). Adaptability is crucial in science and technology education since society is continually developing with respect both to science and technology itself and to our understanding of them. In sum, the focus of schooling needs be to redirected to encompass more of the features of successful everyday functioning and apprenticeship programmes, as outlined above, in order to help children become strong out-of-school learners, able to participate and function effectively in social, personal and political life. Schooling must therefore move away from individualised forms of competence, tool-free performance, decontextualised skills, encouraging efficiency in solving routine problems, and the urgent need for external reinforcement. Specific suggestions for how teachers can overcome some of these crippling features of conventional educational practice are outlined in the following section, with an emphasis on extending the pioneering work in mathematics education to science and technology.

21 IMPLICATIONS FOR TEACHING Cognitive apprenticeship Implementing an apprenticeship programme in the classroom involves pupils in working with and observing an ‘expert’ solving problems. This represents a departure from the norm whereby teachers demonstrate particular skills rather than participating in activity and modelling problem-solving strategies. Teaching through cognitive apprenticeship can help make explicit the largely tacit knowledge most expert practitioners possess about their own problem-solving processes. Pilot attempts to adapt it to a classroom context show that it can help children acquire a culture's tools and vocabulary, and the means to discuss and evaluate conventional procedures, collaboratively. The work in this tradition (reviewed above) focuses predominantly upon the curriculum area of mathematics and to some extent, reading and writing. These domains are thought to be particularly well-suited to cognitive apprenticeship methods because they involve cognitive and metacognitive processes that are basic to learning and thinking more generally (Collins, Brown and Newman, 1989). Recent changes in our vision of science and - especially - technology education mean that potentially its central notions are especially applicable to these domains. Johnson and Thomas (1992, p.10) have pointed out that instructional approaches like Schoenfeld's could feasibly be adapted to the technology classroom as follows: ‘Teachers need to act like technologists. They need to solve unfamiliar problems for students and not be afraid to make errors or have difficulties finding solutions. By serving as a role model, technology teachers can show students how to collect and use information to solve technological problems and help them realise that not all problems have straightforward solutions.’ A potential problem here is that teachers are not, on the whole, actually technologists; this is partly because unlike mathematics, technology is not an identifiable domain outside school, but a conglomeration of numerous technological practices. (A similar situation holds for the practice of science.) Yet this may not be an impassable obstacle: Ursula Franklin (1991) urges teachers simply to 'become learners along with their students and be willing to share their own personal expertise and life experiences'. Another potential problem is that the economic bias of the workplace means that useful and meaningful skills will be acquired rapidly in traditional appprenticeship settings, whereas the scope of problems which can be transposed to a classroom context, and the range of solutions which can be formulated, are necessarily limited (in this case by practical constraints rather than an economic bias). However, a tentative step towards overcoming these problems has been taken in the form of the Neighbourhood Engineers scheme, whose most productive aspect for pupils turned out to be the opportunity to work alongside professional engineers who played an advisory role (Bridges et al., 1991). The purpose of activity of this nature is to teach directly cognitive skills such as problem solving, decision making, planning, evaluating and reflecting. Johnson and Thomas (1992) remind us that instructional goals which highlight conceptual learning and thinking processes are essential in technology education, and these are the kinds of goals which are best reached through cognitiveoriented techniques. Apprenticeship can also take place in the context of peer interaction and collaboration. A situation where pupils are working together is both beneficial to learning and reflects most activity outside of school. Discussions and negotiations in groupwork situations will provoke a more meaningful engagement with the problem-solving processes that teachers want to encourage. For example, if technology teachers believe that a general design and problem-solving process is useful and can be transferred across many different contexts, then they must endeavour to make it explicit as pupils will not otherwise appropriate it; collaborative problem solving can play a crucial role here in getting children to reflect upon the process they have used. (To overcome the problems of individual assessment as part of a group, specific roles and division of labour may be necessary.)

22 Bridging from prior knowledge It is clear from the discussion in section 1.2 that teaching must build on children’s existing knowledge and experience so that these are no longer downgraded in the classroom while expertise is glorified. A successful teacher therefore requires a model of the learner’s knowledge structure. S/he must be aware of the kinds of other subject matter knowledge the pupils have encountered and developed, and of the knowledge requirements of tasks being presented. Rather than ‘providing’ knowledge, the teacher must make a deliberate effort to help children access and use their prior knowledge appropriately in solving problems in the new domain under mastery. It is also critical that deliberate attempts are made to bridge from intuitive to formal knowledge. This is illustrated successfully by the mathematics programme developed by Resnick et al.. (1987), which draws upon everyday problem finding; this approach has great potential for science and technology too. Apprenticeship programmes like Resnick’s radically challenge our view of schooling to become one of providing a context for knowing, acting and sense-making, in which children participate daily and actively in a socially valued process. This means devising problem situations which are ‘authentic’.

Authenticity A critical insight derived from the situated cognition research is that problems emerge out of dilemmas and learning arises when means are sought to resolve those dilemmas. The implications are that formal educational settings need to encourage active intellectual engagement in mathematical, scientific and technological thinking, and that tasks should relate to those encountered in daily life. We should encourage schoolchildren to formulate, attempt to solve and communicate their discoveries about questions arising in their classrooms, playgrounds and homes, i.e. in their own environments, in the same way that proficient lay people do. Lave (1992) points out that ‘mathematizing’ everyday experience is more productive than attempting to couch classroom problems in everyday terms. We must recognise that 'emergent problems' will arise while students are carrying out complex tasks in a rich problem-solving context. In technological problem solving, for example, the process of solving a problem can often be observed to change its nature quickly. According to Collins, Brown & Newman (1989), knowledge about the instructional designer's goals and simple debugging techniques are insufficient when solving emergent problems; students must learn to refine these problems iteratively, using the constraints of the embedding context to help solve them. In sum, invention, discovery and refinement of problems are the hallmarks of the most successful instructional programmes, which thereby strive to promote pupils’ ‘ownership’ of problems. Problems should be ones pupils want to solve, which are real and relevant to them, which engage their interest, and for which they can take responsibility. ‘Problem solving’ now comes to denote the resolution of meaningful problems and dilemmas in the context of guided social interaction and negotiation with teachers and peers. Metacognition Finally, the ability to reflect on one’s own knowledge is a critical factor in learning and there is thus a need to develop learners’ metacognitive strategies. One effective technique is that adopted by Schoenfeld (1985) in teaching mathematical problem solving: he encouraged students to ask themselves questions continually about what they were doing, what they were trying to achieve and what they would do next. The questions were eventually internalised and the students’ problem-solving performance improved. Requiring children to assess and monitor their own progress and performance in this way can help make pupils aware of what they are doing and why.

23 ISSUES FOR RESEARCH The main implication of the situated cognition perspective for research is that we should seek to ground theories of action in empirical evidence, generalising from records of particular, naturally occurring activities. This will entail explication of the relationship between structures of action and the resources and constraints afforded by the physical and social context (Suchman, 1987). When this context is a conventional classroom, education may fruitfully be construed as a form of situated activity and discourse (Edwards and Mercer, 1987). Close examination of this process whereby teachers and children interact to some supposed common purpose permits identification of the ways in which that purpose is achieved or lost. The research reviewed here provides a clear framework within which to examine the structuring of activity in the classroom and to assess whether teaching is in fact achieving a productive level of specification - in terms of projects and dilemmas rather than mechanical procedures - with clear aims. The PSTE project is currently pursuing this research goal; our focus is on characterising the degree of mismatch between teachers’ and childrens’ perceptions, agendas and intentions concerning a holistic design and problem-solving process, and investigating the possibility of a ‘veneer of accomplishment’ in technology learning. This type of research allows us to observe how much and what kinds of help children need to complete a task successfully. It will hopefully yield a constructive outcome in terms of information about what kinds of problems are motivating and what kinds of task structure and presentation are most productive. New styles of problems are emerging in mathematics as researchers pursue ways in which teachers may more easily employ co-operative learning strategies so that strong goals for learners are linked with an improvisational approach for reaching them (Lave, 1992). Research must not ignore the real dilemmas of teachers who are attempting, within the system of schooling, to sustain such an approach to the learning of mathematics, science and technology as practices.

NOTE The references cited here form a subset of related references included in a subject-indexed bibliography compiled by the author and available for distribution. Thanks are extended to my colleagues Patricia Murphy and Bob McCormick for their helpful criticisms of an earlier draft of this paper.

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