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Originally Posted by Joe Durnavich
(1) The model consists of the steps he performs to calculate the next number in the sequence.
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What you would choose to call those "steps" would actually represent making a model of the process the student used to make his model. The actual model is more like the instructions for what steps to take, whereas what you mean by "steps" sounds too subjectively connected to that individual. Probably what you would end up doing is applying behavioral model-making, and the whole problem with behavioral model-making is that it's a very clunky way to learn about his model, if that's all you want to know. If all you want to know is his model--
just ask him what it is.
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(2) The model is an active agent inside him that works out the next number in the sequence for him.
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Only if you don't take that picture too literally would it be useful. For example, there's no reason to animate the model, nor to keep the model "inside him". He can easily write it down for all to see, and then it's "in" anyone who chooses to learn it.
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(3) You (Ken) can describe only your (Ken's) model of the student.
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My goodness, that would be the clunkiest of all! This is exactly what I mean about the uselessness of behavioral models if you want to understand how someone is generating the next prime number. It's all about maintaining objectivity and not bogging down unnecessarily in subjective issues, which is precisely what behavioral approaches fail to do. Hence, such approaches are only even remotely advisable when subjective issues are the focus of the investigation.
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However, whenever you see him correctly list sequences of prime numbers, at some point you will pipe up an say, “See, he has a model of the primes.”
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I may do that, if the student is my point of interest. If it's instead his model, I will probably not be interested in that moment.
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Are any of these correct?
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The first is the closest to the usual meaning of a model. But one can, as with the other two, give it an unnecessarily complicated behavioral interpretation of a question whose focus is just a lot simpler. I am not interested in how the model was generated or at what moment I would say that it has been, I merely know that it was, and my interest will be in what the model is. You see, it is perfectly obvious that any behavior by humans involving models is going to be an example of human behavior involving models, so it says nothing interesting to point that out. The interesting question is, what is the model, and what does it do for us. Science is very much a process of knowing what you don't have to pay attention to, so you can focus on what matters to a given issue.