Peter Duveen posted Aug 15, 2012 7:21 PM: > I have a student who doesn't do well on tests. He has > completed geometry, and will be studying intermediate > algebra. I have decided to explore calculus with him. > > I hope that, by learning a little of something that > his other fellow students know little or nothing > about, his self-esteem and interest in the subject > can be kindled. > > My introduction consists of attempting to define and > determine the slope, or inclination if you will, of a > curve at a particular point. We use y = x^2. We take > two points on the curve and determine the slope of > the line that joins them. Then we move one point > close to the other. > > We say that the slope of the curve at a particular > point is the unique quantity approached as one point > moves closer to the other, taken as static. > > This gentleman may never even take a calculus course. > But I feel the slope of a curve at a particular point > is not difficult for him to grasp, in the context of > math he has already taken. He may also be able to use > his insights gained from this introduction to graph > equations. > > At any rate, this raises the subject of enrichment vs > teaching to the test. Is it possible that enrichment > can raise the level of cognition, and spark greater > interest in the subject, thus actually enhancing > performance on tests? One of course must have a > varied tool kit to approach the needs of different > students. > I believe Robert Hansen has correctly pointed out a couple of difficulties that may be 'associated' with your "enrichment" project.
However, I am all for enriching that student's "adventures in mathematics".
a) (for him): "To enhance my knowledge of all topics of my math syllabus and THEREBY to improve, very significantly, my results in my math exams"; and
b) (for yourself): "To enrich, very significantly, --- (name's) knowledge of math".
I recall that, shortly after I had developed the OPMS concept (but long before my prototype OPMS software had been developed) a student came to me after a 'Rotary Club evening talk' I had conducted, inquiring whether OPMS could help him improve his hitherto truly woeful results in math: right through his school career he had never gotten above 45% in a math exam, and now, as a freshman in college, he was doing far worse!
We took up the challenge together, he with his Mission more or less as articulated above, and me with mine (a little differently worded from the one noted above).
I worked with him for about 1 hour or so each day for a little over a month. As noted, in those days the OPMS s/w did not exist, so Peter had to learn how to do the needed modeling without access to a computer.
In about a month or so after I started with him, I got a long-term assignment with a company at Bombay (Mumbai, now). I left him with plenty of 'homework' for him to do independently, mainly related with:
a) how to 'interpret' models constructed; and b) "what to do next" at each stage of development of his OPMS for his Mission.
Those days there was no Internet available; and I more or less lost touch with Peter. However, about eight months later, I was delighted to receive a letter from Peter to tell me that he was now regularly getting well above 75% in all his math exams, tests and quizzes. That was the first real success I'd had with the OPMS, with someone else's Mission (other than my own, of which I had tried out plenty).
[I should emphasize that I deliberately gave him NO MATH TUITION whatsoever - he got whatever he needed of math knowledge from his college professors, his peers, and from his textbooks, etc. (There were such elements specifically stating that in his models)].