Thanks for those fascinating thoughts on "Enhancing the Cognitive impact of Calculus Courses".
I believe that, by and large, you've done something most valuable indeed. It could, I think, be quite easily developed into a 'system'.
Some hasty remarks in overview:
1. I suspect that the 'cognitive challenge' has to be taken up long, LONG before the student arrives at Calculus. Because, long before the student ever gets to Calculus, he/she has already developed a 'fear and/or loathing of math'. (And once such an 'aversion for math' has captured the mind, it is extremely difficult to counter it/ escape its malign grip).
The above is the case for a great many students, except for those students in the conventional math educational system who've had the good fortune to have been taught by a sensitive and empathetic teacher (who had a genuine appreciation of math); and those students who've been through a 'real Montessori school and have thus been properly introduced to the power and beauty of math rigfht from the start. We observe that many of even those students who've been adequately interested in/ enthused by math for a start are still often 'turned off' by the 'traditional math education'.
I would strongly recommend that teachers should think in terms of creating a few systems models via the 'One Page Management System' (OPMS) for a 'Mission' like: "To ensure that my students do not develop a 'fear and/ or loathing' for math - that they come out of my class with a healthy interest/ liking for math as a most useful tool in real life". (Every teacher should be encouraged to to develop his/her own statement of a Mission that would be of adequate interest). This kind of 'Mission Statement' by teachers could help them develop an adequate 'system' to 'enthuse' students lifelong about math.
2. What about students who've NOT had the good fortune to be exposed to initial math with a sensitive, empathetic teacher who himself or herself loved math or via the Montessori system?
Well, I am NOT a math teacher, and therefore would hesitate to 'prescribe' to real math teachers who are 'in the trenches', so to speak. However, I did once encounter a freshman college student who had done very badly in math right through his school career - never having got above 45% in any of his exams, etc. When he came to me, he was seriously concerned that his poor understanding of math was going to ruin his college career, and he was willing to do the hard work to try to change things around by using the 'One Page Management System' (OPMS).
He articulated, with a little help from me, his 'Mission' as follows:
"To understand all topics of my math syllabus, and THEREBY to improve, very significantly, my results in my math exams, tests, quizzes".
I explained to him that I would give him NO MATH TUITION AT ALL - that I would only guide him ib develop his own OPMS for the above Mission. That condition caused him quite a few doubts. Below, i have a brief explanation of how he got over this.
At that time, I didn't have my prototype OPMS software available, and he therefore had to learn how to do the 'systems modeling' involved without computer assistance. That was quite tiresome indeed (for him!); now the prototype OPMS software makes it a whole lot less tiresomely difficult to create an OPMS for any Mission.
Anyway, that student worked on his Mission, with guidance from me for an hour or so each day, over a period of about 40 days, at which time I had to leave on a long-term assignment. I left him with plenty of 'homework', mainly instructions on 'how to interpret the models he would be creating', and 'what to do next' at various stages of development of the OPMS for the chosen Mission.
[For beginners, the 'interpretation' and continuing development of OPMS models can be somewhat tricky at times: it's not technically difficult at all, but we're in general entirely unaccustomed to 'systems thinking'].
One issue requires some clarification, I believe. How exactly did that student manage to get over the real difficulties that he would have confronted in his 'math learning' (without 'math tuition')? Well, his OPMS model did contain a couple of 'elements' as follows:
- -- "To get needed help from my classmates who are very good at math"; - -- "To get needed guidance from my math professors and math lecturers". (Those two elements were ample for him to get ALL the math help/tuition he needed - from exactly the right people who could best give it to him! It did take him some time to recognise that his mind was actually telling him to DO exactly those two things!!)
Those days there was no Internet and email readily available, so I lost touch with him for a while - till about 8 months later he wrote to me saying that he had done all the 'homework' I had set, and that he was now consistently scoring above 75% in all his math exams, etc. (That was the very first successful OPMS model made by someone other than myself).
More in due course if you/others are interested in looking at this approach in more detail.
I have looked at your 'complete post' at http://bit.ly/1B9dyvD, shall hope to be *studying* that in detail (i.e. doing a little 'systems modeling alongside reading it. I note that such modeling would be most effective if we could jointly work on them). Unfortunately, I currently do not have ready access to a university library, so I am unable to look at the many references you've provided.
To respond to the question in the title of your thread: Yes, indeed it can (and you are probably accomplishing this to a pretty good extent). To accomplish this 'maximally well', I'd suggest the development of an OPMS for the Mission: "To enhance the cognitive impact of calculus course".