Addressed to the participants of the "Conference in Honor of Stanley Tennenbaum”, held at the Graduate Center of the City University of New York on April 7, 2006.
By Jonathan Tennenbaum
I was unfortunately not physically able to attend your conference, but I should like -- and I think it appropriate -- to contribute some remarks, not on my late father's mathematical work, but concerning the subject to which he devoted the greater part of his life's energies: education.
My father had to do with an enormous number of people; and each of them, in their own way, has a piece of the story. I limit myself to the period in which I was more or less intensively involved in his educational projects: the period prior to my leaving the United States in 1973. Those were different times, and my father was also in many ways a different person, from the one people encountered, who got to know him in the 1980s or 1990s. I should ask you to forgive possible inaccuracies concerning some events and places, as I have not yet had the opportunity to cross- check my memory with documents.
Over the years many people have spoken and written about the disaster of school education in the United States. But I am sure nobody studied and understood the nature of the problem more deeply than my father, including the ominous implications of certain changes in education, introduced beginning in the late 1950s and early 1960s, for the political future of our country. Moreover, while many stood by and allowed the disaster to happen, he undertook serious efforts to do something about it.
Probably the quickest way to get to the heart of what my father was concerned about, is to contrast, as he often did, the 1960s so-called "New Math", and the classroom environment in which it was taught, with what went on in the old- fashioned schools of his own childhood experience.
I should preface the comparison by emphasizing, that for my father what really matters in education is not the curriculum per se, not what children learn or do not learn in school, but rather what happens to their minds. He was certainly not against learning; but, in his view, it were better children would learn nothing, than that their ability to think things through by and for themselves -- the sovereignty of their minds -- be destroyed. (I don't recall my father using that word, "sovereignty", but its seems at the moment the best term to express his idea as I remember it now.) So, he would often challenge me and others, to be rigorously honest with ourselves, whether we said or believed things just because we had heard or read them somewhere, or because the teacher said them, or whether we really knew them on the basis of working them out for ourselves. Something you have understood or truly discovered in that way, becomes yours in the fullest sense: it belongs to you forever, and nobody can take it away from you.
My father held that all healthy young children are essentially geniuses with regard to talent, at the point they first enter school. He could demonstrate that with actual children, and did so many times, for anyone who showed an interest. In nearly all cases, however, a child's talents decrease monotonically as a function of the number of years spent in the educational system. Only a very few arrive at the end of their "education", with something of their original brilliance still intact. He often liked to sum it up this way: "If we would teach people in school how to walk, we would be a nation of cripples."
My father emphasized that these were no casual observations, but conclusions grounded in many years of careful investigation. In these matters he was a passionate experimentalist. He spent an enormous amount of time, over the years, talking to children, visiting all kinds of schools around the country, teaching classrooms there, working with both "problem cases", "normal" kids and so-called "gifted children" of all ages and backgrounds.
In this process, my father developed his own educational method. Its roots go back at least to his early student days at the University of Chicago, when, after a deeply demoralizing encounter with Andre Weil, my father turned from his original interest in geometry, to philosophy. He spent the better part of the following ten years studying first of all Plato, whose dialogs he came to know "inside and out", and secondarily St. Augustine and a large number of other authors. Another very important influence was my father's experience working as a voluntary assistant in Bruno Bettelheim's school for autistic and schizophrenic children in Chicago. By understanding the extreme case of the damage done to those children, and how they could be helped, and comparing with many other experiences, he gained an extraordinary insight into the emotional side of education generally.
It was on this basis, I think, that he saw through the malevolent intent of the so- called "liberal reforms" in education in the 1960s most clearly, and identified the essential problem not in a lowering of the intellectual standards of education per se, but in a specific form of emotional "brain damage" which was inflicted on practically the entire generation that attended U.S. primary schools in the 1960s.
On the other hand, he demonstrated again and again, that children could progress with astounding rapidity in learning whatever they took an interest in, if only their sovereignty and intellectual self-confidence were restored, and a suitable environment provided.
I was also among of my father's many "experiments", one might say, albeit not in the way some people thought, who imagined, merely because I finished my formal education at a very young age (by present standards), that I had received extensive instruction from him. In reality he taught me only a very few, but extremely valuable things.
My father was convinced, for example, that an alert and healthy child could easily understand even the most "advanced" mathematical ideas, as long as they were real ideas, and they were presented in a suitable way. Sometimes he would try such ideas out on me, often things he was studying himself at the time. Once, for example -- I could not have been more than 7 years old or so -- he showed me Georg Cantor’s argument for the uncountability of the real numbers; or, as he put it at the time, why is impossible to make a list of all the points on a line. He explained Cantor's idea with a few scratches in a dirt patch during a walk outside our apartment in Hyde Park. I got it immediately, and it has stayed with me all my life.
Several years earlier, he had tried out a method for presenting Pythagoras' Theorem to very young children. It evidently made a big impression on me, as I remember the experience quite vividly to this day. He had prepared some very nice big squares and triangles out of solid wood: eight identical right triangles of one bright color, the square fitting on the hypothenus in a second color, and the two squares fitting to the triangle's shorter sides, in a third color. He put the squares before me and asked me, playfully, to imagine they were made of delicious candy, and I could choose between getting either the big square, or the two smaller squares together. That captured my interest. But which should I choose? I remember my hesitation: it was really not clear how to compare the amounts by sight or in my mind. My father then showed me, to my great surprise and delight, that, by adding four right triangles at a time, first to the big square, and then another four to the two smaller squares, in two different ways, we got larger squares of exactly the same size and area. In fact, we could superimpose them, one on top of the other. And then, by taking away the same four triangles again, from each of the larger squares, we could verify for ourselves that the remaining areas -- the hypothenus-sized square on the one side, and the two smaller squares on the other -- had to be equivalent.
For me, this surprising truth -- more physical than mathematical in the way he presented it -- was really sweeter than candy!
Among the few other things in mathematics, my father showed me as a kid, I should mention: the incommensurability of the diagonal and side of a square; Erathosthenes' sieve for the prime numbers; and the concept of a "Turing machine". He also used to talk to me about the idea of a Riemann surfaces, which he was fascinated with, although without going into specifics; and a bit also about the work of Cantor and Gödel, and the way the power of the human mind transcends even the most universal type of mechanical proceedure.
My father's educational method, however, is not fully expressed in such examples. The heart of it lay in his unequaled skill in a special sort of freely-improvised conversation, through which, starting from some question or problem posed at the outset as a theme, he evoked actual, sovereign thinking in people. Often, particularly with the older children and adults, thinking would only emerge after a more or less extended period of "battle", in which my father -- ruthlessly, but with a great deal of humor -- uncovered and beat back each and every attempt by them to get away with "bullshitting" (his technical term for sophism!). In this respect my father's method resembled the classical one of Socrates, but with at least one important difference, which he often emphasized. My father's approach was much more subjective, directed not so much at the content of the sophisms per se, but rather to exposing and overcoming the pathological state of mind, that causes people to become "bullshitters" in the first place. In that way, his educational method was much more a kind of therapy than the teaching of a specific subject- matter. Yet, it had nothing of the manipulative and often condescending manner associated with modern psychology, but was always playful, always focussed on ideas, ironical and humorous.
My father’s conversational method corresponded also to his ideal of a university, as basically a discussion process. Indeed, one could say, that the greater part of scientific education at the University of Chicago in the 1950s, did not go on in classrooms, but rather in places like Steinway’s coffeehouse on 57th Street, where my father used to hang out a lot of the time, often with me tagging along; and where young people could learn most of their physics, mathematics and chemistry by just sitting at the table with scientists, many of them already famous ones, arguing and fighting over ideas. My father attributed a particularly “American” character to this atmosphere: open, direct and free of the European tendency to bow down to authority.
After this preface I now come to the promised comparison between "Old" and "New" Math, and the ominous side of what was happening with school education in the United States, and which my father became more and more clearly aware of, going from the 1950s into the 1960s. It was these developments which caused him finally to launch the university projects, I shall describe later.