Changing Conceptions of Human Intelligence and Reasoning: Implications for the Classroom

ion, embodiment and displaced knowledge We have been slowly building a rough outline of an alternate account of the cognitive abilities that support the competence we hope our students will gain. We discussed very basic emergent properties of connectionist networks in relation to learning and began an account of what representation, abstraction, and category formation might be taken to mean under an interpretation based on network theorising and experimentation. We discovered that connectionist representations could not, alone, be expected to support the full weight of this enterprise, since they did not indicate a way to decide which of the many regularities in the world would be selected as the basis for category formation. Out of concern for 262 Australian Journal ofEducation educational relevance, we directed our attention to the displaced knowledge that schools usually emphasise, passing lightly over the physical regularities that some have made central to their theorising (Inhelder & Piaget, 1964). Here we have pursued the thesis that socially ordered practices constitute the regularities that we seek. In doing so, we have reached deeper into the properties ofconnectionist representation, emphasising particularly the instability that is the darker side of connectionism's valued flexibility. The particular forms of instability exhibited by connectionism's representations have led us to consider the possibility that much of what we call symbol processing is actually manipulation of an environment we have created for such purposes and brought into our social and individual worlds as practices. We have perhaps pushed connectionist insights as far as they can take us at this point by revealing the need for a socially organised theory of practices. Connectionist theorising continues to constrain our approach to social cognition in what follows, but it cannot offer the same positive guidance that it did when we were discussing the relationship between representation and learning. The path we have followed to this point has led to seeing practices (which are learned as patterns and integrated into both activity and perception by connectionist principles of representation) as the key to understanding the sorts of displaced knowledge that the community values and the schools are established to teach. Knowledge built on such a basis could be expected to exhibit certain peculiarities. It would, for instance, be context dependent in particular ways. Thus we would expect that simply learning the practice of multiplying three-column figures would not transfer to a situation in which the problem is laid out horizontally. Such a presentation would be a different practice for the uninitiated student. Staying at the level of practices, we can see how a new practice made up of specific patterns of multiplication and addition could be mapped onto the new, horizontally organised problem. One could especially easily imagine a practice that transforms the horizontal problem into a vertical one. But none of this activity produces the kind of understanding the school generally strives to achieve. It will be too easy to say that the abstract 'principles of mathematics' knowledge that schools value does not and never did exist in the forms that the tradition gives us. Although both connectionists and researchers in a Vygotskian tradition might each for their different reasons endorse such a suggestion, this conclusion will not be sufficient to support an educational theory that deals adequately with the issue. That conclusion, although it may be well supported, fails to deal with the existence and utility of what has been called abstract knowledge. Connectionism and practice theories do not deny the existence of a more broadly powerful way of knowing. They deny that it takes the logical form that the tradition describes, and that it is learnt in the manner that would be appropriate to that form. The community's evident desire that students learn a powerful, generalisable form of knowledge is not somehow obviated by new research. More powerful understandings of this sort do exist and seem as inordinately valuable as the community apparently understands them to be. The task before educators is not to dismiss such Human intelligence and reasoning 263 abstract, displaced knowledge but to explain it on a different, more educationally adequate basis (see Whitson, 1997, pp. 97-149). This will be clearer if we return to the concrete example of three-digit multiplication examined earlier. Educators generally agree that the mechanical, rote production of the answer, even the correct answer, produced by the practice described does not constitute the sort oflearning that is sought. Coming to underdisplaced knowledge but to explain it on a different, more educationally adequate basis (see Whitson, 1997, pp. 97-149). This will be clearer if we return to the concrete example of three-digit multiplication examined earlier. Educators generally agree that the mechanical, rote production of the answer, even the correct answer, produced by the practice described does not constitute the sort oflearning that is sought. Coming to understand the role ofpractices in the production of the student's competent response is an advance. But it is not all that we wish to understand. As mathematics teachers will testify, simply labelling the various positions that the numerals hold as 'place value' does not result in the broader understanding that is sought. The student may be able to label the 'ten's place' correctly without being able to use this concept in any other situation. Given the story that connectionism tells, this is not surprising. The student has learnt to label a part of a practice--and does not possess what we would call an abstract concept. The label signifies only within this practice. It is just at this point that our usual apparatus for explaining learning and knowledge most obviously fails us. A student who can repeat the definition and apply it appropriately is believed to understand. He or she is said to possess the concept but apparently fails to 'transfer' or 'apply' it to new situations. The prominence of application problems within the traditional research literature of education implicitly supports this claim. This problem is generally understood as a failure to access the conceptual object stored in memory. ' Within the standard framework, a framework which posits objects (defined in terms of necessary and sufficient features) as the atoms in a logical calculus of thought, there is seemingly little that can be done. In this story, the apparent context dependence of our retrieval mechanism is simply a source of failure. The most likely route to more accessible recall is to somehow strengthen the memory-a task to which repetition and practice are thought to be well suited. In the view we are developing here, the generally powerful abstraction that is sought is not inaccessible; it simply does not yet exist, regardless of the ability to repeat a definition or label a portion of a practice. There is therefore no failure to transfer or apply the concept, and context dependence is not the source of this non-existent failure. But granting that powerful generalisations do exist--even if the traditional definition is inadequate-how are these generalisations achieved? It is at this point that the developing position we take here has something radically different to say that is relevant to the practice of teaching. It would suggest that this generalisation is but another species ofcategory formation and is learnt in the same way as other categories are learnt, through the socially organised practices which led us to recognise difference as the same. Recall the child and her mother. The mother presents a different set of sensory impressions at every new perception of her and yet the child learns to recognise each as her mother and to act appropriately. The pedagogical issue suggested by this is one of abstraction. The child has the label ofmother (admittedly prelinguistically), which is itself an abstraction from particular encounters, but does not have the next level of abstraction, which we 264 Australian Journal ofEducation may suggest is that of parent. The further, potentially educationally valuable, abstraction 'parent' is built on the basis of a previous understanding of 'mother' and 'father'. Analogously, in a more familiar educational context, the label 'ten's place' located in the context of a single practice is not the abstraction which educators seek but it is properly understood as based on earlier 'concrete abstractions' . At this point, we return to Rumelhart and Smolensky's intuition that their work is consistent with Vygotsky's understanding ofthe relationship between concept formation and language as Vygotsky (1986) argued it in Thought and language. In the course of discussing their understanding of the way the same network that produces perception and activity could also produce the anticipation ofactivity and thereby a functional model of the world (which they call mental simulations), Rumelhart and Smolensky (1986) remark: There are a number ofways ofcontrolling that sequence. One way involves the running of 'mental simulations'. Another way involves recycling linguistic inputs. Suppose that the interpretation that led to the production of the internal speech was much richer than the linguistic forms could possibly suggest. Thus, the linguistic forms pick out aspects of the entire interpretation to emphasize. Once this emphasis has taken place and the new input has been processed, the next state will be strongly affected by the new words we choose to express our first idea. (pp.43-44) Although this formulation captures Vygotsky's (1986) suggestion of a pivotal role for language and particularly his presentation of the role of the formal definition of the word in scientific concept formation, it misses a central point in Vygot