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Anyway, the more sophisticated view of the reflex enabled Piaget to take the tripartite pattern of perceived situation, action, and result as the basis for what he called ‘Action Scheme’. It provided a powerful model for a form of practical learning on the sensorimotor level that was the same, in principle, for animals and humans.
Studies of animal behavior had shown that even the most primitive organisms tend to move towards situations that in the past provided agreeable experiences rather than towards those that proved unpleasant or painful. +
To believe that the future affects the present is no doubt a superstition, but to declare that purpose and goal-directed action must be discarded because they are teleological notions is no better. It shows an abysmal ignorance of the difference between empirical and metaphysical teleology. +
If you consider that in the context of the Darwinian theory of evolution, “to be adapted” means to survive by avoiding constraints, it becomes clear that, for Piaget, “to know” does not involve acquiring a picture of the world around us. Instead, it concerns the discovery of paths of action and of thought that are open to us, paths that are viable in the face of experience. +
Hence, when we intend to stimulate and enhance a student’s learning, we cannot afford to forget that knowledge does not exist outside a person’s mind. +
Knowledge, then, could be treated, not as a more or less accurate representation of external things, situations, and events, but rather as a mapping of actions and conceptual operations that had proven viable in the knowing subject’s experience. +
At best one may observe that in a given number of situations their constructs seem to function in the same way, i.e. they seem compatible. +
If two people share a room, there is one room and both live in it. If they share a bowl of cherries, none of the cherries is eaten by both persons. +
Hence, no matter how one looks at it, an analysis of meanings always leads to individual experience and the social process of accommodating the links between words and chunks of that experience until the individual deems they are compatible with the usage and the linguistic and behavioral responses of others. +
Hence, when Piaget speaks of interaction, this does not imply an organism that interacts with objects as they “really” are, but rather a cognitive subject that is dealing with previously constructed perceptual and conceptual structures. +
That is to say, no matter how hard investigators try to adapt their analyses to the “foreign” ways of children, the model they build up will always be a model constructed out of concepts that are necessarily the investigators’. Because children’s ways of thinking are never directly accessible, the investigators’ model can never be compared to a child’s thought in order to determine whether there is or is not a perfect match. The most one can hope for is that the model fits whatever observations one has made and, more importantly, that it remains viable in the face of new observations. +
In other words, they must come to share some basic ideas on the process of education and the teaching of mathematics in particular. +
if one wants to generate understanding, the reasons why a student operates in a certain way are far more indicative of the student’s stage of conceptual development than whether or not these operations lead to a result that the teacher finds acceptable. Only when teachers have some notion of the conceptual structures with which students operate, can they try to intervene in ways that might lead students to change something in these conceptual structures. +
therefore, they need first of all a plausible model of the conceptual structures with which students are operating at the time. +
In order to formulate even the most tentative model of cognitive change, educational scientists must witness the growth of mathematical knowledge in particular children and clarify and substantiate their interpretations by means of deliberate interventions. Conceptual analysis alone is simply not sufficient as a source of insight in model building. It is only on the basis of models of particular children, that a more general model can eventually be abstracted – and the models of particular children are a natural bridge between educational scientists and the teachers. +
In adopting the new, cognitive paradigm, then, it becomes imperative that both teachers and researchers acquire some theoretical notions of how this “making sense” can be conceptualized. +
Working with children is in many ways like working with foreigners with whom one has only fragments of a language in common. +
A model, then, “simulates reality”; it is a conceptual construct that is treated as though it gave an accurate picture of the real world, but has the actual function of making experimental results and other experiential elements compatible with the general assumptions that are inherent in the research program’s core. +
Just as the interpretation of a piece of language is always guided by the individual interpreter’s experience and expectations, so the interpretation of what one observes is always governed by some theory one has in mind and a goal one has chosen. +
Teachers, therefore, need an at least partially generalized theory and a model of the learner that is general enough to serve as a basis for the establishment of more than one individual model. Ideally, then, the teachers’ models of individual students will be instantiations of the educational scientists’ more general model of mathematics learning; and conversely, the individual models the teachers construct for individual students will be a continuous testing ground for the theoretical assumptions the scientists have incorporated in the more general model. +
When I visually distinguish a hand from the writing pad and the table on which it lies, I carry out exactly the same kinds of operations as when I distinguish the coffee cup from the table on which it stands, or the picture from the wall on which it hangs, or the cardinal outside my window from the branch on which it happens to be perched and from the rest of the landscape. +
