>A bridging figure might be Knuth. He's really into >discrete math but isn't shy about the calculus. He >thinks so-called O-notation, prevalent in computer >science, used to measure the relative efficiencies of >algorithms in general terms (e.g. "polynomial time" >vs. "logarithmic time" kinds of curves) might rear its >Gamera-like head in K-16 more consistently.
Funny you should mention O-notation. Many calculus textbooks will describe the tangent as the best local linear approximation to a general curve, but they do not explain why this is so or exactly what this means. Yet, Richard Courant gives the fairly simple explanation using---you guessed it---the concept of order of magnitude. More interesting, Courant clearly expects the student to be familiar with the idea. I remember getting "a thrill up my leg" (http://youtu.be/no9fpKVXxCc) when I read Courant. I can only wonder, and regret, how this piece of the puzzle fell out of the calculus pedagogy.
>Looking at history, a lot of good deeds and bold ideas >get insufficient press at the time. People just don't >have the context for it.
Too true, generally, but not this time. Uhl was working on a project, Calculus and Mathematica, that was part of a major nation-wide movement to improve the calculus curriculum. The people involved were all in close communication with each other over a span of at least 15 yrs. They met, they spoke, they wrote learned papers, etc. Had Uhl achieved a breakthrough, it is hard to see how or why he would have been ignored.
>Basically: what was passing for "knowing calculus" in >the USA didn't cut it in Russia, and this was apparently >(by his analysis) because the requirement for oral exams >elicited a different quality of comprehension in >students.
I am sure that is right. I have long thought that American math professors are among the first and most exuberant practitioners of grade inflation. Students fall out of the calculus curriculum not so much because professors give them "F's" (although that does occur), but because the students themselves know they are not learning much, and they find the calculus experience so miserable and demoralizing they just do not want to continue, despite the fact that this means life-altering, and not necessarily welcome, changes to their courses of study.
>You may find it easier to get back to calculus via some >much longer detour into a host of other math topics, say >in number and group theory. > >It's not like we're dealing with a strictly linear >terrain.
Basically, I agree, although I would put it differently. The calculus works at a high intellectual level, and the student has to bring with him into the classroom as much mathematical background as possible. It is interesting to note that many of the architects of the calculus also made important contributions to many fields, from optics to number theory. This gives one a sense of the intellectual furniture that decorated their minds. At this point, it strikes me as preposterous to suppose that the very thin gruel of
high school algebra-->geometry-->trigonometry
can possibly be enough intellectual preparation for the calculus. There is so much superb pre-calculus mathematics students could do, if only the schools did not waste so much time. Number theory, a sans-calculus treatment of linear algebra, sans-calculus sequences and series, even statistics. I am sure there is more, and all of it could be done with as much rigor as you can stand. And, of course, some decent science courses would not go amiss.
>You mentioned Economics above, as if economics had no >calculus, but then of course it may have, depending on >whether a fluid mechanics metaphor is being used >(velocity of money etc.). Or am I mistaken about that?
I was not clear on this point, sorry. Many colleges used to have (maybe still do) two tracks in economics just as in physics: a calculus track and a non-calculus track. And, as with physics, you could have some fun in a non-calculus economics course, but you would certainly not become an economist that way.