> On Aug 17, 2012, at 10:00 AM, Peter Duveen >> > 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?
In the current presentation I am working on, quotients drop out before the limits are taken. So there is no need to explain the limit of the quotient of two quantities, both of which are approaching zero at the same time.
Now I will admit, that a conventional approach to the calculus is fraught with difficulties, and much of that is due to the notation and the fact that one is teaching a subject that had two primary founders with two rather different approaches and two different sets of notation. The notaion of the calculus, Leibniz's notation, is extremely challenging and confusing for the beginner. I would not use such notation at the beginning. Leibniz believed in infinitesimals, whereas Newton had the more modern approach of limits. Leibniz speaks a different language, one that is useful but must be approached with caution, and in an historical context. I would steer clear of all of that for beginners in the early grades. I would focus on the problems: tangent to a line, and maybe area under a curve. > > > 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 would endeavor to teach students about the slope of a curve, and how to calculate it, and how one can use this knowledge to graph functions. That would be the beginning. It is tangible knowledge. Call it whatever, calculus, or something else, that's what I would be teaching them. But I do like to think that it is an introduction to calculus, not just teaching about calculus. And, of course, if students can plot a line and a parabola, they can plot many other functions/equations. > > > 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.
Well, should we assume that this boy was a prodigy? Perhaps his parents merely encouraged his interest in the field, so he became proficient at an early age. Therefore, such proficiency may not be entirely beyond the grasp of a larger portion of the student population that we generally imagine.