Joe N says: > I don't have empirical evidence to suggest one ratio versus another, but my opinion is this is far too severe.
R Hansen says: >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
You are right. Perhaps I've muddied the waters by taking Kirby's division into "recog" and "recall" and calling them "appreciation" and "performance". I consider all the time spent understanding the subject "appreciation". Performance is just that - time spent mastering quickly and easily performing operation that you now know adequately well.
If a problem halts you in your tracks and you have to search for some new understanding to proceed, I would consider as being toggled back into appreciation (recog) mode. When you've gotten your "mini-ah-ha!" and can proceed again, you are back into performance track.
Performance is doing arithmetic fluently, taking tests, working at the board etc. Working the exercises when they don't make you stop and think too much. Drills to increase speed and accuracy. The dreaded "rote" tasks. Near effortless understanding of situations described in arithmetical terms is "recall" as well, or in my translation, performance. You've already "got it", you are just using by now established capabilities.
R Hansen: >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?
I'm talking special tracks for the obviously superior performers, which is a tiny percentage, probably around 10%. For the rest, as I say, what is the goal of grade school math education? Is it really (as in another thread) just a delivery vehicle for teaching logic, or generic thinking skills"? I don't buy that. It has to be to help them understand the world they live in and the place of numbers and math in creating that world. Some of those in the "average" group will persist and their interest in attaining higher performance may emerge later, so I'm not saying this is do nothing or play time track. But making these kids who are not so performance oriented (at such-and-such point in time at least) do tons of performance exercises hoping somehow to stem the slide of the USA into math mediocrity, is just ludicrous, a fools game based on fear.
R Hansen: >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.
I would simply have a normal track, (not a non-performing track!) and special help for those at the very top or bottom. The normal track, as I say, is geared to educating people in the meaning of math for their real lives, and as a litmus test I would say means being able to understand the math content of say, the New York Times. As I've tried to clarify above, that's actually "performance".
The 80% in the middle could be further subdivided, but not as distinct classes. I'm also talking specifically about recognizing this idea of performance/appreciation. There may be kids in the normal group who could be performing higher, but have other priorities at this moment in their life. I think its fine for individuals to set their own priorities.
R Hansen says: >All of these students couldn't contend with mathematics and thus began the era of pretend mathematics.
There is no time for pretend, everything done should be for a valid educational purpose, not as a babysitting service or to satisfy someone's political brain fart.