## John L Kelley, Once over lightly

The American Mathematical Society asked John L Kelley to write an autobiographical account and he readily agreed. His article, 'Once over lightly', was published in

*P L Duren and U C Merzbach (eds.), A Century of Mathematics in America***3**(American Mathematical Society, 1988), 471-493. After his death in November 1999 it was agreed by the American Mathematical Society that it could be republished and it appeared as Once over lightly,*TopCom***5**(5 March 2001). We present below a version of this first part of the article.**John L Kelley: Once over lightly**

I am a member of a threatened species. For the first thirteen years of my life my family was not urban, nor suburban, but just country. We lived in small towns, the largest with fewer than 2500 inhabitants: the roads were unpaved, we had no radio and television hadn't been invented. I was born in my family's house (there was no hospital in town) and about the only hint of modernity at my birth was that I was an accident, the result of a contraceptive failure. But I was a genuine, twenty-four-carat country boy, a vanishing breed in these United States.

My schooling began in Meno, Oklahoma, which was then a village of a few hundred people, two churches, one general store, a blacksmith's and a one-room school. There were no electricity and the town centre was marked by a couple of hundred feet of boardwalk on one side of the road. I went to school at a very early age because my mother was the school teacher and there weren't any babysitters. I remember my first day at school; I got spanked.

There were about thirty students in the school, spread over the first eight grades. Most of the time was devoted to oral recitation, reading aloud. spelling, and arithmetic drill, with various groups performing in turn. We were supposed to study or do written work while other groups were reciting, but listening wasn't forbidden and we often learned from other recitations (simian curiosity is not a bad teacher). The first couple of years of arithmetic were almost entirely oral, quite independent of reading. We recited the "ands" and the "takeaways", as in "seven and five is" and "eleven take away three is", and we counted on our fingers. Eventually, we got so we could do elementary computations without moving our lips, but it was a strain.

The arithmetic I was taught by my mother during the two years in Meno, and thereafter by a half dozen different teachers in four of five other small towns, was mostly calculation. Compared with today's programs: there was more oral work then, and less written; the textbooks then were unabashedly problem lists with a minimum of explanatory prose and they weren't in colour, but then and now not very many students read what prose there was; the textbooks then were much shorter. Then and now, most teachers assumed that boys were better at arithmetic, especially after the third or fourth grade; and the end result, then and now, of the first half dozen or so years of arithmetic classes was the ability to duplicate some of the simpler answers from a five-dollar hand calculator. Of course we didn't have hand calculators so this seemed much more important than it does now.

Perhaps it's worth recounting that the mathematics program I was taught in the first six or eight years differed from that taught my father. Somewhere about the seventh or eighth grade there used to be a course called "mental arithmetic", which was problem solving without pencil and paper or, in my father's time father's time, without slate and crayon. He also studied practical arithmetic where they learned about liquid measure and bulk measure, liquid ounces and ounces and ounces avoirdupois, bushels and pecks, furlongs and fortnights, gallons and pints and gills, interest and discount, and other esoteric matters. Some of these subjects still appear in the late elementary math curriculum, but even though the French Revolution did not overrun England, its system of measurements is conquering the world.

But the mental arithmetic course has apparently vanished from our schools. I regret its demise. A modest competence in mental arithmetic and a five-dollar calculator would, I think, ensure arithmetic competence as measured by the usual standard tests, as well as saving an enormous amount of student and teacher time.

But to return to my own schooling. After arithmetic and a rather muddled study of measurement, I entered high school and an algebra class. The former experience was frightening: the latter devastating I didn't understand why letters at the beginning of the alphabet were called constants and those at the end were variables, it seemed odd to me that a variable could take on a value, or several different values if it wanted to; I didn't know what a function was, and why a string of symbols should be called an identity some of the time and an equation, or a conditional equation, at other times, and disastrously, I decided that our teacher, who was inexperienced, did not understand these things either. This was quite unfair although it was comforting and the real difficulty was probably my own pattern of being literal-minded (or perhaps simple-minded) in times of insecurity. But the mathematics was abominably organized, and the quantifiers "for every" and "there exists", weren't mentioned, so no one without prior information or divine inspiration could tell an equation from an identity. At any rate, my teacher indicated by her grading that she agreed with my assessment of my understanding of the course.

The following year I took my last high school mathematics course, geometry. It was a traditional course, very near to Euclid; it talked about axioms and postulates, defined lines and points in utterly confusing ways. The woman who taught us had a chancy disposition and she had been known to throw erasers at inattentive students. It was the loveliest course, the most beautiful stuff that I've ever seen. I thought so then; I think so now.

One would suppose that I having fallen in love with geometry, would immediately have pursued mathematics passionately, and one would be wrong. The mathematics course that, then and now, follows Euclidean geometry is algebra again. In my junior year in high school I decided to be an artist (we had a sensational art teacher that year) and in my senior year I decided to be a physicist (I had a sensational physics teacher).

It is time to pause a bit, with me proudly graduating from high school, to explain what was going on with the rest of the world. We had moved to California in 1930 along with the rest of the "okies" and so my last high school year was in a downtown high school in Los Angles. Times were hard. One-third of men in LA County were out of work and no one counted how many women needed work. But women weren't neglected. There was considerable rumblings about women taking jobs away from men that needed them and, for example, the state legislature in Colorado passed an act denying teaching jobs to married women (this was one of the reasons we emigrated from Colorado); but women had not yet advanced to the dignity of unemployment statistics.

We were poor and it was not a good time to be poor. One summer a couple of years later I worked with my father trucking oranges from the LA basin up to the central valley and peddling them, buying potatoes and fruit in the central valley and peddling it in LA. I remember the Los Angles basin with stacks of oranges a hundred and fifty feet long with purple dye poured over then so people couldn't steal them to eat or sell; and I remember the camp outside Shafter where hundreds and hundreds of "okie" families lived and everyone, including children of four, picked up potatoes and sacked them following the potato digging machine. There was food rotting, and people hungry, and my view of the glories of unrestrained capitalism became and remains a trifle jaundiced.

But I digress.

One of California's truly great educational innovations was tuition-free junior colleges. I entered Los Angeles Junior College in 1931, at the bottom of the depression, faced only with a three-dollar student activity fee and a block-long line to see a dean for permission to pay the fee with four bits down and four bits a month. But the fee included admission to LAJC's little theatre productions, football games and many other goodies and my sister worked in the bookstore and got books for me, so I really had it made.

Besides four semesters of physics (I was still going to be the great physicist). I took of necessity Intermediate Algebra, College Algebra, Trigonometry, Analytic Geometry (even the words have archaic significance) and finally a year and a half of calculus. Calculus was almost as nice as geometry (analytic geometry wasn't really geometry, since Descartes muddled over what Apollonius discovered). And experimental equipment displayed a distinct antipathy towards me. So I entered UCLA, well-trained by very good teachers at LAJC, wanting to be a mathematics major and wondering just how a mathematics major made a living.

As far as I could find out, there was very little market for mathematically trained people. Teaching, actuarial work, and a very few jobs at places like Bell Labs, seemed to be the size of it. I had no money so graduate school seemed unlikely, and high school teaching looked like the best bet. Consequently I took three courses in education in my first three semesters at UCLA in order to prepare for a secondary credential. The courses were pretty bad and besides, the grading was unfair, e.g. I wrote a term paper for Philosophy of Education and got a B on it; my friend Wes Hicks, whose handwriting was better than mine, copied the paper the next term and got a B+, and our friend Dick Gorman typed the paper the following term and got an A.

Of course teaching is a low prestige field in this country. The prestige of a field of study is apparently a direct function of the technical complexity of the surrounding society. Engineering, and especially civil engineering, seems to be the prestige field at a relatively early developmental stage (e.g., pre- World War I U.S., pre-World War II India), to be overtaken by chemistry and chemical engineering as technology develops (World War I was a chemist's war), followed in turn by electrical engineering and physics (World War II, radar and nuclear weapons). It has been said that the last war will be a mathematician's war, so mathematics is now deadly and hence reasonably prestigious.

Fortunately for history, the precise time that mathematics acquired prestige among students at Berkeley is recorded. My student Eva Kallin explained to me that during her first couple of years at Berkeley she suppressed the fact she was a math major when talking to an interesting new man; later it was OK to be a math major, and a little later it was a very definite plus.

Back to UCLA. Los Angeles itself was then a gaggle of small towns held together by a water company, and UCLA was on a new campus, plopped down on the west side of town in the middle of an expensive real estate development. Too expensive for most of us students, so we drove, hitchhiked car pooled or bussed from our homes to the school. There were about 4500 students and the math department was on the top floor of one wing of the chemistry building. It was definitely not a prestigious location. But mathematicians were usually viewed with an uneasy mixture of awe and contempt like, say, minor prophets. Our prophetic powers were used: math courses were prerequisites for courses in other fields, and math grades were often used to section physics classes into fast and slow groups. But mathematics was scorned as being irrelevant to the "real" world.

E R Hedrick, of Goursat-Hedrick,

*Cours d'analyse*, chaired the department - he later became chancellor. I enrolled in the last term of calculus, won the departmental prize for a calculus exam ($10), then blew the final on my calculus course and got a B (Wes Hicks said they should have offered a fifty-cent prize). I got shifted from my part-time job in a school parking lot to a part-time job in the math department office keeping time sheets for readers, recording grades, and whatever. I took all of the courses in geometry, mostly from P H Daus, admired Hedrick's flamboyant lecturing style, conceived quite a fancy for my own mathematical ability, and quit taking education courses, thus abandoning a career as a high school teacher. (I could always go back and get a teaching credential if I had to.)

In midyear 1935-36 I graduated and was given a teaching assistantship in the department at $55 a month, which was enough to live on, and so became one of the multitude feeding at the public trough at the taxpayers' expense. Of course I could never have gone to college except at a public school - I could barely manage to cope with UCLA's $27 per semester fee - so I like public schools, and the public trough is just fine. The fall of 1935 was notable for another event: I received my first college scholarship. It paid $30.

During my last year at ULCA I began to learn how to teach (I was terrified) and I was first exposed to the R L Moore method of instruction, which was fascinating. W M Whyburn, who took his degree at Texas, introduced me to the Moore methodology in a real variable course, told me I had to leave to get a Ph.D. (I didn't even realise that ULCA had no mathematics doctoral program), and arranged a teaching assistantship to the University of Virginia for me. In 1937 I was granted an M.A. and headed for Virginia.