The full schema model

This chapter describes the full model of schema implementation in arithmetic problem solving. In current terminology, the model is a hybrid model of cognition, utilizing both production systems and connectionist networks to represent schema knowledge. 

Identification knowledge has to do with recognizing patterns. The question of interest is whether the stimulus problem contains a pattern of elements sufficient to activate an existing schema. The pattern recognition is accomplished by a connectionist component of the model. 

Elaboration knowledge, on the other hand, has to do with deciding whether the necessary elements are provided in the problem, after the pattern has been recognized as characteristic of a schema, so that the full schema can be accessed and used. This is a question best answered by a production system, but there is also a connectionist part to elaboration knowledge. In a single problem, several potential patterns may match the recognized schema, and the most reasonable or most likely one for solution needs to be selected. This is a special case for competitive performance by all candidate patterns to determine which pattern most strongly reflects the identified schema. 

Finally, execution knowledge involves the step-by-step execution of already-learned algorithms, which again calls for a production system. Execution knowledge comes into play only when the plans call for it. 

As described in part II, both SPS and PSE were designed around schema theory. In particular, they were developed so that each of the four components of knowledge could be isolated and evaluated as students acquired their schema knowledge. The results of the experiments using these systems were given in part III. The importance of the experiments for the present chapter is that they remain the best source of empirical evidence against which the computer implementation of the hybrid model can be evaluated. 

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Chapters 13, 14, and 15 describe particular models developed from the schema theory and problem-solving studies discussed in previous chapters. Chapter 13 contains details about a full computer simulation of the first experiment of Chapter 7. Chapter 14 describes a back propagation connectionist model for the same conditions. And, finally, chapter 15 contains the full hybrid model of schema implementation. 

An important criticism of Marx s development of the reproduction schema in Capital, volume 2, is that he fails to consider the importance of competition. For Howard and King (1985 191), Marx s reproduction models do not relate to a full competitive capitalist system, in which capital mobility and credit flows operate to equalise profit rates and bring about prices of production which deviate from labour values. As a result, Marx may have drawn unwarranted conclusions as to the difficulties involved in achieving fully co-ordinated production in capitalism (ibid. 191). This key limitation of the reproduction schema is addressed here. 

How are schemas created We have made a substantial beginning in understanding how schemas develop, primarily through the studies described in subsequent chapters. The schema development observed in experimental subjects progresses from a few details of examples to full abstract characterizations and strong mental models. 

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