On Mar 20, 2014, at 4:57 PM, kirby urner <email@example.com> wrote:
> The meaning of "gravity" was changed within the context of general relativity, i.e. "the idea of gravity" changed from what it had been, in Newtonian mechanics.
Yes, over time we understand more and more about gravity (the phenomenon). And likewise would any other civilization that makes it this far. We would both have gravity and both have f = hmm/r^2. And if they make it as far as relativity, so forth. Both civilizations are working the same puzzle. What you aren?t getting is how ?working? works. I think Lou is getting it, because I sense him heading towards a ?what is is? argument, but not you and Joe. Joe is citing scripture and you keep changing the topic. I am much more keen in recognizing this behavior in you two than I was a month ago.
Take for example your prior example about the usage of the word ?screwdriver?. It is pretty easy to talk to someone about screwdrivers if they have experience with screwdrivers. You can even say ?that thing you turn with your hand? and they will get it. It is a whole different story if they have no experience with screwdrivers. Even if you use the correct word, ?screwdriver?. Even if you define the word ?screwdriver?. The fact is, you are not going to have a deep discussion about screwdrivers, no way, no how, unless they can relate it to something they have experienced.
That is what is happening here. I am talking about mental processes that neither of you have experience with. I have compiled several pages of notes these last few days days that I still need to sort though, but the gist is that Piaget?s last cognitive stage, the formal stage, was not the actual last stage. There is a higher stage and that stage is where new theories and the elements of those theories are born. The formal stage that is documented is actually the ability to understand formal theories and apply that understanding to problems. That activity though does not engage an even higher level of thinking that is required to distill new theory. Essentially, Piaget?s theory is lacking an explanation for Piaget?s theory.
In Joe?s case for example, I was noting at least a year ago that there was something ?off? in his take on mathematics. But I couldn?t put my finger on it. It was a ?mental feeling?. Over time the clues kept coming in and I am the type that can?t let unsolved riddles go. I put out my theories regarding the illusion of pedagogical effectiveness in mathematical illustrations and visual examples. Joe?s response? ?Says you!?. Well, yeah, it?s my theory, so yes, I said it, but that isn?t the response I get from most people when I describe it. And this has continued and the vitriol has increased. But there is a silver lining to this cloud. I finally, a month or so ago, solved the riddle. I realized that Joe cannot understand art (another word for this higher cognitive stage). He can read books and apply some of that to problems, but he doesn?t understand how it all is made. And when I say art, it isn?t the art of problem solving. The art I am talking about draws on those facilities! but it goes much deeper than that. When you solve problems at this stage you change the course of the subject itself.
But one doesn?t have to change the course of the subject to gain the advantages of this higher stage of thinking. This higher stage of thinking affects your experience with mathematics whether you discover the next paradigm shift or not. And understanding this higher stage of thinking is crucial to understanding all of the lower stages of thinking. It gives you that necessary insight to understand the development of mathematical ability in a student.
I accept that in this case I can never convince Joe that this is not a trick nor that I am just making this up. And I aways realized that school math stops short of this higher stage, probably because it is rare, but only now realized the significance of that. I thought maybe they should add another course devoted to this art but we know how that would turn out. One of the reasons for such poor ideas in math education in the lower stages is that few people even make it to those stages. The rarity of this higher stage would only compound this further. I hope my work in AI, when published, will break through that somehow. As Lou has shown, people have a better go at it when the theory is complete and the ontology nice and neat. But no theory will ever solve the politics of education.