On Aug 17, 2012, at 10:00 AM, Peter Duveen <email@example.com> wrote:
> Bob, I did not go into detail as to what my approach would be regarding the introduction of "some calculus." Basically, the way you interpreted it that phrase is similar to what I thought of it. I have not said much about the specifics of how I would introduce it, and continue with its development. Let me just say that there are more ways than one to skin a cat, and I think an attitude that we can learn from one another will foster productive dialogue. I appreciate your approach, which you spelled out in some detail, but which, I think you will admit, would be the likely approach of many of us here. (I see no reason to start with a circle, but I might examine that idea.
I am trying to learn. My attitude is fine. My focus is intense. I am thinking critically. Tell me how you propose to do the development I spoke of with a student that has yet to take algebra or has taken it and doesn't get it? How are you going to deal with the difference quotient?
> Mainly, you want to show that the line you construct as the tangent to a curve intersects the curve at one point only, and that the limit that the slope of the line passing through two chosen points on the curve, as one point approaches the other, is unique, and is the same approached from values on both sides of the x coordinate of the static point.
No, I do not agree that this is teaching calculus. In fact, I knew this was coming because I have been in this conversation enough times. That is why I included in my post the clause that "explaining the tangent to a curve is not teaching calculus". All you are doing then is teaching the student what calculus is about.
> I do not think that a sixth or seventh grader would merely behave as a monkey repeating tricks by wrote, but rather could clearly comprehend the thoughts put forward. Having mastered the idea of the equation of a line, its slope, and the equation of a parabola, seems to be a good foundation to begin the calculus. I shall do a little experimenting where I can to verify that sixth or seventh graders can actually comprehend such material.
I comprehend that Tennis is about hitting the ball to your opponent, in a legal manner, and such that they are unable to do the same. There! I know some tennis. Wait, I don't know any tennis. I know something about tennis. For you to say that you have taught them some calculus you had better show students that can DO some calculus. In order for that to occur those students must be able to DO algebra. If your 6th and 7th grade students are already well enough into algebra then I agree that you can teach them some calculus (my definition of some calculus). But if all they know is something about algebra then I do not think you can teach them some calculus. You can only teach them something about calculus. There is a huge difference. It wasn't until so many students, unable to cope with the subject, were pushed into these classes that we started to see pedagogies that teach "about" subjects instead of teaching "the" subjects. You go back 40 years or so, you don't find text book! s that teach "about" subjects like algebra and calculus.
If we think that there needs to be an "about" thread in public school, for students that are unable to cope with the actual subjects, then by all means, let's make one. Then we can avoid the confusion and strife.
> > When I was a sophomore in college, one of the students in my advanced mechanics physics course was a 13 year old. He was much better versed in the subject than the professor, who always tried to catch him on a problem, but never succeeded. One time the professor was quite sure he had caught the kid at making a mistake at the black board. He shouted, "You're wrong," with a loud, haughty voice. "No I'm not," responded the fellow on the verge of adolescence. Well, as it turned out, the 13 year old was not wrong. > > Now was this 13 year old a monkey repeating tricks? He went on to become a NASA scientist. I tell this story to some of my classes when I substitute teach, and it becomes a great buzz throughout their grade level (7th/8th grade). I'm eating lunch, and they will approach me about it. Then, later classes ask me to repeat the story. I want them to know that they can do more. I want them to think that the story of this boy is not atypical, but perhaps something they themselves are capable of.
Let's keep the details straight. No one was talking about 13 year old prodigies. We were talking about a student that can't get algebra. And we were talking about 6th and 7th graders that (I assumed) hadn't yet taken algebra.