Figure 1413

The number of pieces correctly recalled by masters and beginner chess players, from Chase and Simon (1973b), when they were presented with "normal" as opposed to randomised board positions.

second chess board with the first board still in view. They recorded the number and type of pieces subjects placed on the second board after a glance at the first board, where successive trials (glances) were made to build up the full board of pieces. The chunking hypothesis predicts that players should only place a few pieces after each glance and that they should form some coherent whole. In each of the three players studied, Chase and Simon found that the average number of pieces taken in at a glance was small in number (i.e., about three) and similar in content. However, better players used significantly shorter glances to encode a chunk. Chase and Simon also discovered systematic differences in the number of pieces encoded in a chunk as a function of expertise. The strongest player encoded about 2.5 pieces per chunk, whereas the weakest player encoded only 1.9 pieces per chunk. Recent research suggests that this may have been an underestimate of the chunk-size of masters (Gobet & Simon, 1998b). So, Chase and Simon's results showed that expert players can recognise chunks in a board position more quickly and can encode more information in these chunks than novice players.

Chase and Simon also found similar effects to those of DeGroot on the encoding of random board positions; using a presentation time of 5 seconds there was no difference between their three subjects. However, recently, this finding has been questioned in a review of similar studies. Gobet and Simon (1996a) showed that across a dozen or so studies there is a correlation between skill level and recall performance of random positions, although it is rarely statistically significant. They maintained that strong players do have a superiority in recalling random positions but that it has been overlooked by the lack of statistical power in most experiments because the sample size is so small (e.g., three subjects). This proposal does not defeat the chunking hypothesis, it merely suggests that portions of even random positions may be encoded by the chunks possessed by experts.

These results have acted as an important target for theories of expert memory and the computational modelling of thinking. Simon and Gilmartin (1973) produced one of the earliest instantiations of the chunking theory, called the Memory-Aided Pattern Perceiver (MAPP). The model contained a large number of different board patterns and encoded a "presented" board configuration into its short-term memory, by recognising various chunks of the total configuration. Simon and Gilmartin produced one version of the

Theories and models of chess expertise program with more patterns than another version (1114 patterns vs. 894 patterns) and ran both versions on board-reconstruction tasks. They found that the version with fewer patterns performed the poorest. Thus, the model provided concrete support for the proposal that board-position knowledge was the key to understanding novice-expert differences in these tasks. Extrapolating from the model, Simon and Gilmartin estimated that master-level performance required a long-term memory of between 10,000 and 100,000 patterns.

Beyond chunking theory there are currently three other theories of chess expertise (see Ericsson & Lehmann, 1996; Gobet, 1998a, for reviews): the SEEK theory (Holding, 1985, 1992), the long-term working memory theory (Ericsson & Staszewski, 1989; Ericsson & Kintsch, 1995) and the template theory (Gobet & Simon, 1996b, 1998a, 1998b, in press). The SEEK theory proposes that three elements are central to chess expertise: search, evaluation, and knowledge. Masters search more and better than weaker players, produce better evaluations of positions, and have much more knowledge. Although many aspects of the theory are ill specified (it has not been modelled computationally) there is evidence to support the importance of evaluative knowledge, as opposed to just board-position knowledge (see Charness, 1981b, 1991; Holding, 1985, 1989). Holding and Reynolds (1982) presented players, rated as being of high- or low-ability, with random board positions for 8 seconds and then again for 3 minutes to evaluate the strength of the position and decide on the next best move to make. They found that high-ability players produced better quality moves. So, even though the subjects had no specific schemata for these random board positions, they had other knowledge that allowed them to generate and evaluate potential moves from that position.

Ericsson and Kintsch's (1995) long-term working memory theory, which was reviewed earlier as a general account of many memory effects (see Chapter 6), critically makes use of similar assumptions to template theory (Gobet & Simon, 1996b). Both theories propose the idea that board-position knowledge is mediated by schematic retrieval structures; these are structures that are more general than actual board positions. In template theory these schemas, called templates, are an abstraction of a whole set of similar positions; the template encodes some invariant pieces and slots for other pieces that can change for this type of position (see Chapter 9). Importantly, the schema would also have associated knowledge on possible moves and general plans that follow from this type of position. These two theories are quite successful in dealing with a wide range of evidence in chess expertise; specifically, their proposed retrieval structures help to account for chess players' ability to give high-level accounts of the nature of a board position.

The mix of knowledge proposed in these theories (i.e., board positions, abstract schema, evaluation knowledge) shows that expertise is not just about memory for routine problem solving (see Green & Gilhooly, 1992). Hatano and Inagaki (1986) have made a crucial distinction between different types of expertise; routine and adaptive expertise (see also Holyoak, 1991; Lamberts & Pfeifer, 1992). Routine expertise manifests itself in the ability to solve familiar, standard problems in an efficient manner; and probably relies on schemata that encode the routine, like standard board-position knowledge. Adaptive expertise works best on non-standard, unfamiliar problems and allows experts to develop ad hoc procedures and strategies for solving such problems. Evaluation knowledge seems to underlie adaptive expertise. It comes into play when a problem situation deviates from a known situation. In general, board-position knowledge allows the player to find out what parts of the board are relevant, whereas evaluation knowledge helps to develop moves from these positions and evaluate the consequences of these moves.

Physics expertise

Anyone who has studied physics will recall (possibly with dread) problems like the following one:

A block of mass M is dropped from a height x onto a spring of force constant K. Neglecting friction, what is the maximum distance the spring will be compressed?

People solve physics problems by selecting appropriate principles from the physics domain and deriving a solution through the application of these principles. A problem solver must analyse the problem, build some cognitive representation of it that cues relevant principles, and then strategically apply these principles to solve it. Clearly, if someone represents the problem incorrectly they are less likely to solve it. If we follow the hypotheses from chess research, we should expect experts to have a larger repertoire of problem-solving knowledge than novices. In physics, this knowledge takes the form of schemata that link problem situations to principles (see Chapter 9). Without this knowledge both groups should fall back on more heuristic knowledge similar to that used in puzzle problems (e.g., means-ends analysis; see earlier).

Evidence of novice-expert differences in physics

It has been proposed that expert physicists build better representations of the problem than novices based on their schematic knowledge (Heller & Reif, 1984; Larkin, 1983, 1985). Chi, Feltovich, and Glaser (1981) asked novices and experts to sort problems into related groups and found that the two groups classified problems differently. Novices tended to group problems together that had the same surface features; they grouped two problems together if they used pulleys or ramps. Novices were led by the keywords and the objects in the problem. However, experts classified problems in terms of their deep structure. That is, they grouped problems together that could be solved by the same principles, even though these problems had different surface features (Chi, Glaser, & Rees, 1983; see Figure 14.14).

Chi et al. (1981) also discovered that even though experts solved the problems four times faster than novices, they spent more time than novices analysing and understanding the problems. Unlike the novices who waded into the problem immediately applying equations, the experts elaborated the representation of the problem by selecting the appropriate principles that applied to it. Experts carried out a complex categorisation of the problem situation using their available knowledge.

Strategic differences have also been found between experts and novices. Experts tend to work forwards to a solution whereas novices tend to work backwards (Larkin, McDermott, Simon, & Simon, 1980). When they have analysed the problem, experts apply the principles they have selected to the given quantities in the problem. These principles generate the unknown quantities needed to solve the problem. This planned working-forward strategy is both efficient and powerful. Novices, in contrast, have an impoverished repertoire of available principles. Typically, they take the goal (e.g., what is the maximum distance the spring will be compressed?), and find a principle that contains the desired quantity and usually no more than one other unknown quantity. They then try to find this new unknown quantity and hence work backwards to the givens of the problem statement. Symbolic and connectionist models of physics skills

Several computational models of physics problem solving have been produced to model physics experts and the shift in expertise from novices to experts (Elio & Sharf, 1990; Lamberts, 1990; Lamberts & Pfeifer, 1992; Larkin, 1979; Priest, 1986). Most of these models are conventional symbolic models, like production systems (see Chapter 1). However, recently Lamberts (1990) has produced an interesting hybrid model that mixes connectionist and production-system ideas. Lamberts noted that physics expertise seems to be a mix of knowledge of previous problems and strategic reasoning (e.g., forward reasoning). In his model, a connectionist memory encodes previous problem-solving experience and a production system handles the strategic reasoning.

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