New Smithee manuscript:
Fundamental Program of the Calculus
[To download PDF FILE click here].
Alan Smithee, Fundamental Program of the Calculus (2007) Prepublication BETA version available |
 | Note from A. Smithee (February 5, 2007): A hyperlinked PDF file is now available. This is not another DRIP manuscript, but more a full treatment of the natural integration theory on the real line in a development that we would tend to consider at a senior undergraduate level. This is a very preliminary version and readers are asked to write back with criticisms and suggestions for correcting errors and omissions. Send comments to calculusprogram@gmail.com. Note from Web Site Authors (February 6, 2007): The Smithee manuscript was written [discovered?] by the author who has asked for a web site on which to offer his preview version. Without endorsing or encouraging writing of this kind we have offered this space here for this purpose. |
PREFACE to FUNDAMENTAL PROGRAM OF THE CALCULUS.
On a holiday (to Vienna and the Czech Republic) I encountered a time-travelling student from the 22nd century who was taking his spring break in our era before returning to resume his calculus studies. This fact might not have arisen were it not for a situation, embarrassing to me, in which I offered to help him with his calculus homework and found that I was entirely unfamiliar with the material, this in spite of many years of teaching the subject. He was equally embarrassed since it was considered, naturally, an ethical breach to be travelling with copies of time-sensitive material. Nonetheless, he graciously gave me a copy of his classroom notes and it is these notes that I present here as being of some possible interest to the mathematical community in general and the designers of calculus curricula in particular.
Not fully confident in my ability to assess the notes correctly, I approached several of my colleagues. The first, a young woman in whom I had the greatest confidence, took them from me and returned them later in the afternoon. ``Curious,'' she said. I could elicit no further opinions from her and she wandered off deep in thought, about something else no doubt. I could not help but agree with this one word commentary and my confidence in the notes was somewhat shaken. I considered then that I needed to find a more outspoken, but equally distinguished colleague. I approached one, timidly with the notes in hand, and asked for his assessment. He glanced impatiently at his watch but agreed to take a look at them.
As he read some passages from the notes, he began sputtering and grew incandescent with rage. ``What ...what? Riemann's unfortunate integral! The Lebesgue integral? Not the correct integral? Correct integral indeed!''
He then threw the notes at me and yelled ``The writer is an ignoramus. The Lebesgue integral is the correct version of the integral, as anyone who knows anything at all about mathematics would surely realize. Does he know nothing of modern probability theory, functional analysis, harmonic analysis, ....'' He stormed off continuing to list applications of the Lebesgue integral as he disappeared from view. I could not help but agree with him and my confidence in the notes was quite shaken.
I decided then that, since these were in fact calculus notes, I should approach my good friend who was recognized as the best teacher in the department and who had spoken at numerous conferences on mathematical education and calculus reform. He was pleased to be consulted and agreed to read the notes overnight. The next morning he returned them to me with an indulgent smile.
``Rather interesting, but seriously flawed. This is no way to teach the calculus! The most important idea to get across is the mean-value theorem of course. Look here ... no fanfare, and the theorem is never used. Never used! Imagine. Granted we never give a full proof, but to treat it this way ... well!
``Then these covering relations. I confess to being absolutely baffled by these things, full covers, fine covers and so on. But really ... partitions are covering relations? These covering relation things really obscure the true nature of a partition. You see really what the partition is, is the set of points where the subdivisions are made. In studies students have been videotaped in small group settings and the consensus has emerged that partitions are clearly understood if presented in this way, especially with appropriate teaching aids.''
He then rushed to a blackboard where he described to me with various pictures how partitions were sets of points, and then there were associated points too, and the fineness of the partition was given by certain distances, not at all the strange notion of ``fineness'' in the notes, and so on. I could not follow the argument and got lost in the maze of pictures, but I could not help but agree that his was by far the better way to teach the subject. He ambled off continuing to list further pedagogical defects. By now my confidence in the notes was entirely shattered.
Not knowing what to do next I considered destroying all of the material. But I reasoned that, just as historians are always searching for documents from the past to analyse, so too there might be a dual version of history in which documents from the future play a role. Thus I offer the notes here, with only some minor editing to make the language sound rather more contemporary.
I thought for a time that I could perform an historically useful function and track the source of the ideas and refer to the authors of these ideas. I have been assured by an expert that, while the ideas are well-known, the authorship is murky and many are certain to take offense. Most likely, he said, all of the integration and variational ideas can be found at least in the papers (published and unpublished) of Ralph Henstock from the 1950s and 1960s. The task, however, of reading the many papers and finding which ideas are not so attributable would take more dedication to the project than I can imagine applying.
A.S.
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