On Nov 10, 2012, at 2:19 PM, kirby urner <email@example.com> wrote:
>> The biggest problem, and Clyde's problem as well, is that the vast majority >> of teachers do not have longitudinal experience. They don't know how this >> process works because they have never seen the whole process work. They do >> not teach mathematics to a student or a class of students for six straight >> years, they are only responsible for one of those six years. This is >> something I would address first. If teachers had more longitudinal >> experience they would understand the progression. Can you imagine the effect >> this would have on a teacher's experience? >> >> Bob Hansen > > This sounds too linear, as if all math topics were on this timeline in > a sequence and the job is to hit them in the right order, of > increasing sophistication.
I don't care what it sounds like. The preponderance of evidence says it is the truth. I am not saying that each and every topic is fixed in concrete, but the order is fairly well defined. And this isn't really about math. It is about the natural path to formal reasoning. Teachers are so anxious to put the cart before the horse and I posit that this is because they don't have much experience with the whole race and how it is won. Even I have this anxiousness, but I also have sense and the advantage of seeing the whole race.
> > In actual fact, it's more like a network of multiple highways and > byways.
No, actually it is not. If there were an ounce of truth to this statement then it would have shown up in the 2 million exams I studied. I mean, surely some percentage of students would have found this hidden network of highways and wormholes, even if just by accident. I would have seen kids heading straight for the fraction store before they could count to 10. It would show up all around us. Yet, it doesn't show up at all. Isn't what you're suggesting based on a philosophical imagination rather than reality? If all of these alternate paths exist, then why don't we see that happening?
> Fragments of group theory fit easily in Algebra 1 and make > both better, but because Group Theory as a whole is quite > sophisticated it gets thrown out until after Calculus.
I am good with the extra for experts inserts, but you have to get the algebra first before you can teach the student the why behind it, otherwise it is just pretend. Besides, Clyde will tell you that since they are doing algebra, and since algebra is controlled by group theory, they understand group theory. So, no reason to teach group theory twice.:)
> Junior is saddled with the same > linear sequence is grandparents had. Is that a good thing? By > definition?
Well, Junior is just human, like his grandparents. Naturally, the progression would be the same, right?