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Originally Posted by Ken G
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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If I asked him what his model is, he would answer with the steps he takes to determine the prime numbers. That's why I said "steps." In that context, I don't see a difference between steps and instructions. As for being connected to the individual, well, you had me look at the student's success in identifying 19 as a prime as evidence of him having a model.
A troubling circularity is lurking here. If the student's model is what he recites to me when I ask him what his model is, then his model doesn't explain unification or understanding of primes. His model is then a bunch of words, which by your account, requires another set of models for intellectual backing, and then we go on an infinite regress. A model could never explain the student's ability to list or identify primes.
There is another way of noting this circularity. You said to me of my drawing a circle:
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This is quite easy to prove. Have you ever drawn a circle, using that so-called muscle memory you are describing, and then looked at it and erased part and redrawn it better? Did you do that because of some muscle memory in the process of drawing it? No. Did you do that because you referred to something you were told was a circle and noticed a problem? No. I know perfectly well what you did-- you compared your result to exactly that inner concept of a circle that you claim you don't have. How else would you be led to redraw it?
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If I did compare my drawn circle with an inner model, I would need to know which of the many models I possessed was a circle. But I could only "bring up" the circle model if I already knew what a circle was independent of the model. Your notion of model doesn't explain what you want it to explain; it assumes what you want it to explain.
A related troubling aspect is if a model provides understanding of circles or primes, how can you build the model in the first place? You must know when to stop building; you must know that you have accomplished the tasks required to finish a circle or prime number model, and this knowledge must be independent of the model under construction.
We very well may model, but understanding (and unification) is found only in the wider context involving both the person, the environment, and achievement in that environment.
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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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The only ones we would say who have learned it, however, are those who can demonstrate proficiency or who we can reasonably expect to demonstrate such proficiency based on past demonstrations. There are no special inner criteria such that one can immediately, directly, and instantly "see" that one understands it. But note that one could privately run through some primes and be confident in one's understanding. We do that before tests. It's the proficiency based on public criteria that's relevant. There is no gazing upon an inner model and knowing right away, "I understand it!" This should be no surprise. Understanding is not just about private contentment while sitting back with a fine cigar and brandy, but, more broadly, about mastery of one's life in the world.
I said:
(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.
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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I wrote that because you said to me things like,
"What you are missing is that Newton is not "discussing" anything, he could do that until he was blue in the face, with long treatises, perhaps waxing poetical. No, what Newton did was to create a model, pure and simple." And,
"We never understand toys, we only understand our models of toys." I took your position to be that we never can get beyond the models in our heads, which made me wonder how you knew the student built a model, or what gave you a basis to declare to me how things were in the world and how my writings did not correctly describe them.
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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.
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Feel free to focus then on the model. Your talk of a person telling me what his model is, a student identifying 19 as the next prime, or of me correcting a drawn circle takes us right back to behavior.