R Hansen says: >As far as the mix of appreciation and performance, I am sticking with my 75/25 rule.
I don't have empirical evidence to suggest one ratio versus another, but my opinion is this is far too severe. I've used the word "milieu" in past discussions. In the current context I would suggest milieu makes up for a great *lack* on the appreciative, or in Kirby's terms "recog", elements of school learning. But of course it only does so for those students who are in that milieu. Perhaps a lot of what one easily sees as inherent or natural talent or interest is not really so inherent.
For those students who really *take off* and fly with the performance aspect, as in music performance, they absolutely need a separate track so they are not bogged down by those less smitten with the performance aspect. I would be happy if in general 80-90% of the population could read the New York Times and readily understand most of the mathematical content found therein (fractions, ratios, basic statistics, etc.) That doesn't require a high degree of performance. It would mean a lot more "recog" in the non-performance track.
Just as an aside, and continuing the music analogies, I took up a keen interest in really understanding Afro-Cuban and West African rhythms in the last few years. There is a great deal to learn and appreciate, beyond just listening. As I began to read a lot more I began to listen a lot better. And while I can beat a half-way convincing tumbao on my conga, I'll certainly never be a quinto soloist. However, the most surprising thing I've learned on this little journey is how *impossible* it is for some of my friends (who have BA degrees in music performance) to even *get* the most basic aspects of these rhythms. Listen to Afro Blue by Tito for example: http://www.youtube.com/watch?v=TjyiG2oaRO8 Can you hear and count the underlying 4 beat? Surprisingly my friends cannot, because of the strong triplet cross beats on the bass, and so insist that its not their.
Lack of certain essential "recog" or appreciative elements is an absolute show stopper in may types of learning. I think Kirby is really onto something here in relation to math.