For people who want to learn or teach mathematics on the basis of reason …
About the author
This page can be safely omitted by those able to read a novel without first having to read a biography of its author, a habit that I find perverse, if not pernicious. On the other hand, it is sometimes helpful to know the background of an author in order to form an opinion on her/his credibility. Certainly, were I to announce a proof of the Riemann hypothesis, it would be idiotic for anyone to read any further. In any case, here it goes.
I came from France to the University of Pennsylvania in the Fall of 1965, on a Fulbright,
with an ABD in Mechanics, a discipline that, in Europe, traditionally sits between Mathematics and Physics. It was both i. to learn from Peter Freyd who had just published Abelian Categories, an Introduction to the Theory of Functors, category theory being then a new area of mathematics of which I thought it might provide a better basis than set theory from which to learn mathematics and ii. to work as a Research Fellow at the University of Pennsylvania School of Education on issues that had been suggested to me by Z. P. Dienes. I never got around doing much of either but, while W. Lawvere eventually did the former surpassingly well with the introduction of topoi, unfortunately, nobody ever had a chance to do the latter after Z. P. Dienes fell out of favor, a totally innocent victim of the New Math disaster.
That I would switch from research in mechanics to research in learning mathematics had of course been rather unlikely but one thing had led to another. First there had been the realization that, while, strictly speaking, I was prepared for the use of asymptotic expansions in my dissertation on hypersonic shock-layers, in reality, my background in analysis and geometry was quite insufficient for my doing so as comfortably as I would have liked. Second, when, instead of getting my degree, I thought of using my aerodynamic competences in industry, I had had an obscenely high offer from one of the very few French aero-spatial companies. Presumably, this was because they thought, on the basis of my All But Dissertation, that I could easily and rapidly be trained to help design the geometry of the warheads that De Gaulle wanted in order to give France international standing by way of a "force de frappe". Since participating in the design of warheads did not appeal to me, I got a job teaching mathematics at one of the Napoleonic Lycees to tide me over financially. But, third, I then got so intrigued by the issue of why mathematics was so "hard to learn" for "most people" that I sought to learn some psychology to understand where the difficulties were. However, if, theoretically, I could enroll in the graduate Psychology department, practically, I found that this was somewhat frowned upon in the case of people who already had a graduate degree. So, instead, I eventually joined an effort led by Pierre Greco, an associate of Jean Piaget, to develop programmed instruction in the theoretical context of Genetic Epistemology rather than in the original one of Skinner's Behaviorism. Fourth, F. Schremmer Mattei, my wife and a real mathematician, was given a one-year grant to spend anywhere in the world she wanted. She thought the U. S. might be a good idea and it was right after I had mentioned to Greco that I would be going there the next year that I got the offer from the Fulbright people in Paris. A coincidence?
At my request, the School of Education at the University of Pennsylvania set me up working an hour every morning with lowest track third-graders at an associated Elementary School with Dienes' Attribute and Multi-Base Arithmetic blocks. However, fascinated by the children's reaction to my very clumsy efforts at letting them go through the Dienes four-cycle, I immediately forgot what I had meant to investigate and let myself be carried by the experience. The results were of course diverse but two incidents have stayed in my mind. Once, I had left the classroom for a few minutes to find at the door, on my return, a very agitated administrator who explained to me that what I had done was completely irresponsible and quite dangerous as such students as I had could, left to their own devices, be expected to do just about anything, including setting the school on fire. Looking through the window, though, it seemed as if the students had hardly noticed my absence and were just continuing to do with their blocks whatever they had been doing when I left. That did not mean much for the agitated administrator but may have perhaps accounted for my not being thrown in jail. The other incident was my visit to the principal, towards the end of the academic year, because Russell, a quiet loner, had written down, completely on his own and not even at my suggestion, base-independent rules for the four operations! As imperfect as these rules of course were, I thought it most remarkable that he would have even thought of it, let alone that he had made a respectable job of it. In consideration, I suppose, of my being at the University, the principal very courteously looked up Russell's file. According to all the scores that were in it, though, Russell had been placed in the lowest track very appropriately. Nothing I could say about what I saw Russell doing every day could change anything. I was not familiar with the system and, not knowing what to do, left it at that. But over forty years later, I still wonder about Russell with sadness and guilt. As the Negro College Fund billboard ads used to say at the time, a mind is a terrible thing to waste.
The next year, after my wife and I decided to stay a while more, I got a job teaching at Community College of Philadelphia which had half started just the year before. The "open door" concept was a fascinatingly new one to me and completely in accord with my political views. In France, if education was free, including beyond secondary education, and theoretically reserved to talent, it was in fact almost exclusively for the bourgeoisie. To be sure, in France, the elite did not coincide entirely with the privileged classes as, historically, the latter has always been very keen on recruiting a managerial class from the lower classes. Lebesgue for instance was the son of a blacksmith whose elementary school teachers got him a cost of living scholarship for him to go to a secondary boarding school in the provincial capital. His teachers there got him a cost of living scholarship for him to go to the university in Paris. But, at least in the 1960s, students of blue-collar origin still accounted for only about 1% of all post-secondary students.
It didn't take me long, though, to realize that what we were teaching at Community College of Philadelphia was what appeared to me to be, at best, a very watered-down version of the first two years of 4-year colleges. Yet, dealing on a day-to-day basis with these students rapidly convinced me that there was no need for the inanity of the Mathematics for Liberal Studies that we were teaching them and I proceeded to write a text, Elements of Abstract Mathematics, to be used in a three-semester sequence with an initial group of about one hundred students in three classes. The paper I sent to the Monthly was promptly rejected on the basis of two reviews, one concluding that it was exceedingly controversial and the other that it was absolutely trivial. Being new to the game, it left me very sore. Eventually, though, I moved away from the Definition-Theorem format and from left-invertible pseudo-groups etc.
In the meantime, I had learned some model theory from Peter Freyd in a course loosely based on Bell & Slomson's Models and Ultraproducts, and about natural deductive systems from John Corcoran in a course more or less along the lines of Stoll's Set Theory and Logic. So, with the zeal of neophytes, I wrote a Model Theoretic Introduction to Mathematics with a colleague, Alfred Brown, whose main role was to prevent me from getting carried away as was my tendency. The idea was to start from "small situations", for instance situations involving a few people together with a few of their characteristics, from which to abstract "very small structures" to be represented by "very small languages". First order predicate logic took precedence over sentential logic with quantifiers introduced before connectors. Then, given a language, the second part introduced entailment and tautology relative to small sets of interpretations of that language. The third part introduced, in the name of Gödel Completeness Theorem, a natural deductive system as a syntactic counterpart of entailment and tautology. The course was taught, rather successfully, by a number of instructors to a couple of thousand students until, after a few years, the Provost put an end to it after having read Morris Kline's Why Johnny Can't Add; the Failure of the New Math. Instead, we were to start teaching "Remedial Arithmetic" and "Remedial Algebra". Still, the idea to keep separate what we do in the real world and what we write on paper to represent it was to remain with me to this day and will be readily apparent in the FMTo texts.
From almost the first day, my obsession had been to make the "open door" concept a real one as opposed to the revolving door that Community Colleges turned out to be for the most part. So I went back to the idea of starting basic arithmetic with Dienes' Multi-Base Arithmetic blocks. For a couple of years, A. Brown and I would generally start with base THREE, then work with other bases more or less in random order and end, in the last few weeks, with base TEN. By and large, we never "showed" the students how to do anything and we would only suggest issues of interest and discuss with them what they were doing. Predictably, the students did rather well. But, for instance, if, when faced with the legendary 1/2 + 1/3, they would take their time to figure out an answer which was usually correct as long as the base was less than TEN, when the base turned out eventually to be TEN, they would almost instantly write down 2/5. We had little opportunity really to investigate the matter and none to follow up the effects of what the students had done, if any, in further courses: The Provost saw to that.
I was also convinced, as I still am, that the lack of continuity inherent in semester courses was part of the problem. So I wrote a text for a sequence, Geometric Differential Calculus, that was based on some of the same premises as Osserman's Two Dimensional Calculus. But, while students did not have any particular difficulty with it, the faculty in the Physics Department, who were responsible for the Engineering Program, did and had no problem convincing the Provost to let them teach the calculus to their own students. So, the sequence died without having had any chance of being seriously worked on.
Then I thought that a systematic use of polynomial approximation, aka asymptotic expansions and first developed in Lagrange's Theorie des Fonctions Analytiques, might provide a more conservative approach to the Precalculus-Differential Calculus issue, one that should be more palatable to the Physics Department. However, again, if polynomial approximations worked well with the students, they did not pass muster with the Physics Department. For instance, I once showed to one of my colleagues I. Bivins' article in the College Mathematics Journal, What a Tangent Line is when it isn't a Limit, for which Bivins had received two prizes. The committee's citation
for the Polya prize read in part:
By defining the tangent line as the best linear approximation to the graph of a function near a point, [Bivins] has narrowed the gap, always treacherous to students, between an intuitive idea and a rigorous definition. The subject of this article is fundamental to the first two years of college mathematics and should simplify things for students.....
I also pointed out the difference in the usual definitions of the derivative in dimension 1 and in dimensions 2 and 3. Nothing would do and my colleague remained steadfastly unconvinced. You might say that he had a vested interest, though.
After running repeatedly into this kind of, let us say, lack of support, I thought I would keep to myself and just take care of my students. Then, in 1987 or so came a new Division Director who talked me into applying for the first calculus grant from the NSF which, to my surprise, we got. The project, Lagrange Differential Calculus, was for an integrated sequence of two four-hour semesters, to parallel Precalculus I, II and Calculus I. Again, the initial materials were not all that they should have been but the students' increasing mathematical ambitions as they got into the second semester were strikingly noticeable and we had some stunning
successes. As, together with people from a few other institutions, we were working on improving the materials, the NSF then decided to fund only big, prestigious consortia and, in spite of several proposals, we were never able to get another grant. Much later, I was struck by how much my experience had resembled that of D. Hestenes:
After the first couple of tries I was convinced that it would never get funded [by the NSF], but I continued to submit 12 times in all as an experiment on how the system works. I found that there was always a split opinion on my proposal that typically fell into three groups. About one third dismissed me outright as a crank. About one third was intrigued and sometimes gave my proposal an Excellent rating. The other third was noncommittal, mainly because they were not sure they understood what I was talking about.
And then, as before, the sequence came under heavy fire from the Physics/Engineering Department and, somehow, it just so happened that academic advisors started discouraging students from taking the integrated sequence on the grounds that taking Precalculus I, II and Calculus I in 8 semester hours obviously had to be much harder than doing so in a 10 semester hours. And, finally, the Division Director was fired for having supported such a preposterous idea. All this in spite of the school's Office of Institutional Research having reported that
Of those attempting the first course in each sequence, 12.5% finished the [conventional three semester 10 hour] sequence while 48.3% finished the [integrated two semester 8-hour] sequence, revealing a definite association between the [integrated two semester 8 hour] sequence and completion (chi2(1) = 82.14, p < .001).
So, I though that, if I were to keep on working on unconventional things, this would henceforth have be completely on my own and within the framework of the conventional courses that I was now teaching, Arithmetic, Basic Algebra and Precalculus I. After all, that is what tenure was for.
Eventually, though, a report on a longitudinal study came out from my school's Office of Institutional Research showing that less than one quarter of one percent (0.24%) of 1732 students starting in Arithmetic had passed Calculus 1 which I thought I couldn't possibly ignore and which is what made me start on From Arithmetic to Differential Calculus (A2DC).