On Jun 29, 2014, at 12:11 PM, Joe Niederberger <email@example.com> wrote:
> I don't have empirical evidence to suggest one ratio versus another, but my opinion is this is far too severe.
Well, we haven?t really defined these parts very well. The 75% represents all of the time spent developing and understanding the mathematics. It includes homework, study, lectures, questions, discussions, and problem solving in class, and exams, both reviewing and taking. The 25% represents the rest. I would consider most of what Kirby posted in his latest post of activities to be part of the 25%. For example, going through the buttons on a scientific calculator might have inspirational value, but you can?t expect the students to solve trigonometric problems just because they know where the sin button is on a calculator. Obviously a lot of development and expertise in trigonometry must take place first. And before that, in algebra and geometry. And before that, arithmetic and especially, fractions.
Do you still think my 75% is too much?
> 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.
Maybe it is because of globalization but I have seen this natural talent in a very wide variety of cultures lately. One team member had to fly back to Zimbabwe two weeks ago. An arduous journey of three days. And back. In any event, the talent looks the same, even if the culture is different. And, when I did that deep dive into the TIMSS exam results, I found no discernible difference in the progress on the exams, across dozens of countries and every imaginable social factor.
I know that some are in better circumstances than others to recognize and take advantage of their talent. Some are in circumstances so bad that they might as well not even have the talent. That would have been my life before the algebra class. But when you examine a group of students, all with sufficient circumstances and exposure, then (I claim) it is inherent interest/talent that is the deciding factor. As far as *sufficient circumstances*, the common denominator I have found across all those cultures is that their talent was supported at home and at school. Mine wasn?t supported at home but was supported at school, once I got into the right track. And it wasn?t easy in my case. For example, we often didn?t have electricity. So even though I made it without support at home, that was exceptional. Generally speaking I would say that support at home and at school is necessary and sufficient.
> 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.
Can you clarify somewhat, because you implied more than one track. Let?s assume (for the sake of discussion) that all of the students master arithmetic and fractions by 6th or 7th grade and then when we start algebra only a percentage of the students show sufficient interest, talent, performance or whatever to continue. The students that do not continue, but accomplished arithmetic, are you saying that they are still on some sort of math track?
The reason I ask is that this is how pretend math, like Fibonacci coloring lessons, came to be. Arithmetic was always one of the three R?s, and as Lou said, arithmetic (when taught as an R) is not mathematics. In the not so distant past (40 years ago or so), algebra was not a required subject, and mathematics was not required all through high school. And even when required, more arithmetic like math for daily living or business math was sufficient. But then there was a push to make mathematics (not just arithmetic) mandatory for all students for all 12 years. Namely algebra and algebra 2, but also a push for calculus and AP as well, though not mandatory. All of these students couldn?t contend with mathematics and thus began the era of pretend mathematics. Also, this was the driver to stop teaching arithmetic as an R and start teaching it as mathematics, which was a not well thought idea, because now those same students can?t contend with arithmetic.
In any event, I am curious, what this math track for non performing students is supposed to do, if indeed you are suggesting a second 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.
I have had discussions with my musical friend like this a lot. I often have a lot of trouble relating music to its form (even with the sheet). I also often test him by finding something exotic and placing sheet music it in front of him, and damn if he doesn?t come very close on the first pass. His ability comes from (besides his musicality) a lot of study of different forms and structures and lot of self training, like you are doing. I don?t think all musicians are quite that deep into that but I don?t have any idea of how many are. If I was a musician I think I would be that type of one, but I am certainly not a musician and never will be (the talent is just not there). In fact, I often use my conversations with him in some of my thoughts on math pedagogy. A case of me trying to understand something much deeper than I have feelings for. It is strange though that a musician with a BA degree hasn?t had sufficient training to pick out that rhythm, but that is probably all it i! s, they haven?t had the training.