Chandy, I read through your document about how a child learns. I am supportive of what you say there. Questions are important, and one should not stifle them. Questions lead to the magnificent realm of knowledge which is as yet unexplored. That's how I treat them in myself, and how I learn so many interesting things, some of my own discovery. Students' questions ought to be treated similarly.
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. 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 side! s of the x coordinate of the static point.
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.
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.