On Jul 3, 2014, at 2:25 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> You don't think a bright 3rd grader can understand the concept of acceleration? That some accelerations are greater or lesser that others, and that the magnitude of acceleration itself can vary over time? And, that understanding most definitely does not have to be "formal".
Imo, that isn?t calculus, and I don?t think they can understand the notion of an acceleration or velocity varying over time. They can understand viscerally what acceleration is and that it is different than their visceral understanding of velocity. And then it goes blank.:) The varying by time is too formal. You asked with regard to understanding. Do I think there is benefit to calculus milieu? Probably, sure, but I don?t think you have to pretend to do it for that, just talk about it once in awhile.
Earlier, in the congas discussion, you remarked on and exhibited a level of appreciation much deeper than what I was using the term for, or had thought of. I equated it to an intensity of appreciation that can only occur at the performance level, not simply a study. I was trying to identify what that looks like in mathematics. Originally, I supposed that advanced music, like Beethoven, would be comparable to advanced mathematics, like calculus, or Newton if you wish.:) But that didn?t seem right because I also believe that talents progress similarly in time and age, regardless of whether its music, math, or whatever. This is a corollary to the ?10 years to become an expert? notion that was discussed awhile back. And these kids are playing Beethoven at least well enough that if this was math and Beethoven was calculus then these kids would pass an AP exam. But we know that is not reasonable, well maybe that 5 year old would pass AP calculus, but not the rest. Back to the draw! ing board.
While I was looking at the kids performing it dawned on me that I should be comparing level of effort in order to make the parallel. In that sense, the talented 6th grade music student has put in close to 1000 hours of training since starting to play, which would compare to a similar amount of time a talented 6th grade math student might spend. The only activity that would come close to consuming this much time in a student so young is arithmetic (and all its precursors) and thus I believe the parallel at this level is with arithmetic. In other words, whatever you want to call these performances, an equally talented elementary math student who practiced that much would be very good at arithmetic. Not algebra or calculus. So the talented musical students are either not playing Beethoven, or if they are playing Beethoven, then Beethoven isn?t a parallel to calculus, or if Beethoven is a parallel to calculus then it isn?t just playing Beethoven that is a parallel to calculus, i! t is understanding Beethoven that is a parallel to calculus, which makes just playing Beethoven a parallel to arithmetic. How would you rather understand Beethoven? Playing it, or listening to it?
I know it isn?t the answer some people keep looking for. And some people might look at these kids and say they don?t have enough theory, which floors me, considering how good they are (at what ever it is we agree they are doing). It doesn?t floor me because I think they have enough theory, they probably have very little. It floors me because as good as these kids are, what?s the point in analyzing how much theory they have at this point? I want to know how they got this damn good!