Property:AnnotationComment
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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. +
In other words, if there are several kinds of disturbance and, consequently, several kinds of error signals, the system has to discover which of the activities in its behavioral repertoire is most likely to correct a particular error signal. On the simplest level this can be achieved only through inductive inference. +
It allows us to proceed much as a bricklayer, who can devote all his energy and attention to the creation of a wall or an arch, without ever stopping to ask where the bricks he is using came from or how they were made. And just as the characteristics of the bricks (e.g., shape and size) make it impossible for the bricklayer to build certain structures, so the ready-made conceptual building blocks impose constraints on any future construction. +
To refer once more to the feedback model, one might say that assimilation, insofar as it adjusts sensory signals, reduces the generation of error signals. Accommodation, on the other hand, occurs only when there is a discrepancy or disturbance for which the organism does not yet have an established remedy. +
There seems to be no way around the assumption that, as far as the organism is concerned, an “object” must be a construct, actively abstracted from a number of experiences by holding on to a somewhat flexible constellation of characteristics and allowing each of them to vary within a certain range. +
Hence this use of an invariant scheme is by no means a manifestation of the concept of object permanence, because its invariance arises from and consists in the repetition of an activity and does not yet involve the invariance of an independent object. +
Now, if the invariant can be used on the representational level, without an activity, it becomes like a program or a subroutine that is invariant in that it is stored somewhere in a memory from which it can be retrieved. It is this change of status that gives rise to the concepts of permanence and of identity, a further step in the construction of permanent objects. +
For many five-year-olds, for instance, the sun today and the sun yesterday are not yet one and the same individual (Piaget, 1971, p. 87). +
Suppose a very young child applies the word dog to every four-legged creature he sees. He may have abstracted a limited set of attributes and created a large category, but his abstraction will now show up in his vocabulary. Parents will not provide him with a conventional name for his category, e.g., quadruped, but instead will require him to narrow his use of dog to its proper range...
The child who spontaneously hits on the category four-legged animals will be required to give it up in favor of dogs, cats, horses, cows, and the like ... The schoolboy who learns the word quadruped has abstracted from differentiated and named subor- dinates. The child he was abstracted through a failure to differentiate. Abstraction after differentiation may be the mature process, and abstraction from a failure to differentiate the primitive. +
Here I shall confine myself to pointing out that the kind of knowledge our simple organism acquires by installing connections between error signals and activities is, indeed, a form of construction, and since it deals exclusively with the proximal data of the organism’s own subjective experience, one would be justified in calling it wholly subjective. +
In other words, we can come to know only what we consider to be in some sense separate from our knowing selves. By questioning something, by the very act of asking what it is, we have already set our self, the questioner, apart. +
