On Mar 3, 2014, at 5:14 PM, Joe Niederberger <email@example.com> wrote:
> R Hansen: >> Remember that conversation about imaginary numbers and AC circuit analysis. I said something like ?Some bright person saw the mathematics of complex numbers and connected it to the mathematics of AC circuits and poof, reduced the mathematics of AC circuits to arithmetic!? > > I won't accept your memory here; post a reference and I'll look at it again. I have a different interpretation of our fundamental differences in point of view: you are an either-or guy and I'm a both-and guy.
The thread(s) I am speaking of in this example are here?
And I tried many different words in that exchange (and many after), to convey what I meant, to no avail. And when you weren?t saying ?Says you!? you were saying things like this (from that last reference above)...
"You keep talking about "the significance of X" -- last time it was variables. So, in a nutshell, what is the "meaning and significance" of continuity? How does it differ from an understanding of continuity??
I even got close to seeing the issue behind this communication gap in this post...
It only fully came to me in this discussion about programming. This issue between us is art. Whenever the underlying theme of what I am talking to is the art of what I am talking about, your radio goes silent. And it seems to not matter the subject. Programming, mathematics, or even music. In those discussions above, when I point out a pedagogical failure that involves the art not coming across, you hear ?blah blah blah blah blah?. Artistic autism. You are incapable of recognizing the art in all of this.
My ultimate goal, and maybe it is too ultimate a goal for everyone, is that the student experience mathematics, not simply understand its conclusions, or have a semblance of understanding them. And I am not talking about ?thinking like a mathematician? because that notion has been fully polluted by non-artists. To try to appreciate the meaning of ?experience? versus ?understand?, take every sentence where you might use the word ?understand? and replace it with ?personally experience?. Rather than understand a derivation, personally experience it. Rather than understand what an axiom is, personally experience it. Have the same experience that the original artists had. You don?t have to relive their entire history, you just have to have a particular grasp of the art that is going on in all of this.
When you have done that, you don?t have arguments about what ?is? is. You don?t have those arguments because you are sharing an experience, not debating some semblance of an understanding based on formal definitions. And there is no difference between understanding and just a semblance of understanding. The former is just a better semblance than the latter. Without actually experiencing the subject and connecting to its art, you are just an observer. You must place yourself within its actual creation. This is the source of my insistence on the importance of *development* in pedagogy. Because you cannot possibly develop the art unless you do it in the proper sequence. Many people don?t get this because they are not connected to the art. They see only the surface, the semblance of understanding. Other than empirical evidence for a progression, they don't understand or feel the actual reasons for the it. Thus, they try all sorts of strange reforms, that make no sense to artists.
Take for example the discussion about *choosing* something other than the unit square as the unit of area. I thought about it deeply and replied to the effect ?If a thousand alien civilizations developed mathematics they would all arrive at using the unit square for the unit of area.? Your reply, as usual, ?Says you!, How do you know that?? I know it because I connect at the art level, not the semblance of understanding level. I ran through the scenarios and the art of mathematics would never develop away from the unit square as we know it. The human experience of mathematics is quite different than the science of mathematics. The former is not a static result like the latter, to be gotten to by any *logical* direction. The former is art and art progresses through layering each successive generation of thinking on top of the previous generation of thinking. Whether it is in a whole civilization or a single individual. Unlike in the science of mathematics, which is static and! things can be logically equivalent, in the art of mathematics which involves time, they are not.
At least now I know what I am dealing with in these discussion with you.