I went to Montgomery Blair High School in Silver Spring, Maryland. Silver Spring is just outside Washington, D.C., in Montgomery County, which later on would acquire the reputation of being the richest county in the U.S. The really wealthy part of the county was at the other end, though, where students went to Bethesda-Chevy Chase High.
Nevertheless Blair was a pretty good high school and I certainly had some good classes and good teachers (along with a few losers). In particular, my senior year English teacher (Miss Robinson, as I recall) was very influential in getting me interested in writing fiction.
The intellectual climate in my circle of friends in high school was very different than what I have found in the rest of the country since that time. I'm not sure to what extent this is because of geography and to what extent it's just that times have changed, but my friends and I took our education very seriously. We listened to classical music, we read Hemingway and Faulkner, we worked on the school literary magazine and won prizes in the annual high school writing content sponsored by the Washington Evening Star. (Of course some of my friends were more interested in mathematics and science than literature. I tended to have a foot in both camps.) When we were assigned term papers to write, we went down to the Library of Congress, which admitted high school students at that time.
We all took it for granted that we would attend an Ivy League school, or something comparable -- such as Swarthmore, or St. Johns College in Annapolis, where they had the Great Books program. I myself was accepted by five Ivy League schools and had originally intended to go to Brown. But then I got involved with some very interesting literary people in the Washington area and decided that I'd rather not go too far away, so I applied to Johns Hopkins in Baltimore, which was also financially advantageous, but which turned out to be an enormous mistake. (In retrospect, I realize that the school I really should have gone to was Columbia. But my father, in his usual unanswerable fashion, had completely ruled that out. ``You wouldn't like that at all. It's right in the middle of New York City.'' In retrospect, I realize that that's precisely what I would have liked.)
``Were you intending to become a medical student when you started?''
Although Johns Hopkins is rightly known for its fine medical school and hospital, it has a lot of other programs as well. At the time I went there, it had an undergraduate enrollment of 2,500 and a graduate enrollment of 2,500 as well. The overwhelming majority of the undergraduates were engineering majors. At that time, there was a widespread belief that there was a drastic shortage of engineers, and the State of Maryland had given Hopkins quite a bit of money to devote to an engineering program.
Academically, going to Hopkins for my freshman year was very useful. The students were all pretty good and the courses were fast paced. General Chemistry took only a semester rather than a year, so I got to take Qualitative Analysis the second semester. (The main thing I learned from Chemistry, though, was never to take another chemistry course.) I escaped ROTC almost completely, being forced to take only one semester later at Arizona, until I became a junior. And, most important, I acquired six units credit in English Composition with a grade of A by taking a test one Satuday afternoon, and those were accepted as transfer credits later by Arizona, so that I never had to take freshman composition.
One of the attractive things about the Hopkins program was that they had a major called Engineering Physics. This would have meant, if I'd stayed there, that I could have learned lots of physics and still wound up with an engineering degree. (My becoming an engineer was mostly my father's idea, since his primary concern was that I study something I could make a living at.)
One Professor Morrell had written the text for calculus and seemed to be generally in charge of the calculus sequence. Although as a matter of principle one could get credit for any course by examination, Morrell made it clear that he would give me a very rough time if I tried to pass out of Calculus I. So it seemed simpler to just register for the course but only come to class for the tests. Since Calculus I was only given in the spring and Calculus II only in the fall, this meant taking Calculus II before Calculus I. Nobody seemed to notice my doing this.
German was a perfectly reasonable course for an engineering student at Hopkins to take. And since I was already signed up for all the prescribed freshman engineering courses, nobody cared if I also signed up for classical Greek. (By the second semester we were already reading Plato. My two years of high school Latin enabled me to master the grammar without too much difficulty.)
During my second semester at Hopkins, I was extremely depressed. I calculated that I was actually registered for 26 actual hours of class per week, of which I regularly attended 14.
Socially, Hopkins was a hell hole. The biggest problem was that it was, on the undergraduate level, a men's school. (And obviously a freshman was not likely to attract any interest from the few female graduate students.) The dormitories reeked of sexual frustration. Once or twice a month we had a horrible event called a mixer with Goucher College, a nearby girls school, or Townsend State Teachers College, which at that time had an overwhelmingly female student body. At these mixers, masses of boys and girls who didn't know each other would stand around a ballroom, while the boys tried to get up the courage to ask girls to dance to Pat Boone and other insipid Fifties music. (No rock and roll was allowed, of course, not even Elvis.)
I actually did have a girl friend for a short while, met at one of those mixers. I actually took her to a football game. It was probably the third time in my life I'd been to a football game, and definitely the last time. But without a car, and being underage, there were very few choices for what to do on a date.
In any case, I lost that girl friend when I made the collosal blunder of showing her a short story I'd written in which the boy and girl go to bed together.
In retrospect, it would have been an extremely smart move for me to have joined a fraternity, if I could have found one that would have me. The fraternities gave parties at which there was lots of liquor and to which lots of women came. But at the time, I was much too much of an intellectual snob to even consider that option. (I was also much too puritanical to drink very much.)
General Chemistry Analytic Geometry Calculus 2 First Year German First Year Greek Phys. Ed.
Qualitative Analysis (Chemistry) Calculus 1 Calculus 3 First Year German First Year Greek Phys Ed. Six Hours English Composition (Credit by Examination)
In a lot of ways, the theatre courses were the most important part of my college curriculum. I spent an immense amount of time working on the university productions, and people in the theatre department sometimes assumed that I was a drama major. (On the other hand, some of my history professors assumed that I was a history major.)
I could have majored in a foreign language or in physics, but these would all have involved my taking a number of courses that I didn't want. (I especially didn't want all the physics labs.) The math major, on the other hand, had very minimal requirements (four math courses beyond calculus) that left me with lots of electives for the other courses I wanted.
I decided to satisfy my core requirements in humanities by taking upper division history courses. I had to talk my way into Chinese history, since I was still a sophomore when I enrolled for it. (I had to work my ass off in that course, too, as I did the next year in Greek and Roman history.)
During my junior year, the professor in my Greek class (Dr. Best) showed me that by taking one more year of Greek, which I pretty much wanted in any case, I could get a second degree with a major in Classics.
During my senior year, I loosened up a little. I decided to take epistemology, since I was becoming obsessed with the question of how people know the things they do. I had to talk my way into that course too, since I'd never had any other philosophy courses.
My final semester, I really loosened my ``no soft courses'' policy by signing up for existentialism. To my disappointment, the professor had a very Christian orientation and spent a whole lot of time on Kierkegaard (without ever teaching us how to pronouce the name) and almost none on the French existentialists.
I also loosened up my ``only applied math'' policy by taking topology. I'd heard a lot of people talking about it as something exciting and important, but I couldn't make any sense at all out of the few books on it in the library. In retrospect, it turned out to be probably the most valuable mathematics course I took as an undergraduate.
Second Year German Second Year French Real Analysis 1 (Advanced Calculus) Chinese History Western Civ 1 Stagecraft ROTC
Second Year German Linear Algebra Partial Differential Equations (audit) Vector Analysis and LaPlace Transforms Physics 1 (Mechanics) Stage Lighting
Conversational German Second Year Greek Second Year Latin Beginning Literary Chinese Greek History Electricity and Magnetism
Conversational German Second Year Greek Second Year Latin Beginning Literary Chinese Roman History Western Civ 2
Optics and Acoustics Intro to Relativity and Quantum Theory
Third Year Greek Numerical Analysis Differential Geometry Russian History Epistemology Scene Design
Third Year Greek Numerical Analysis Topology Soviet History (audit) Existentialism Scene Design Stage Make-up
On the other hand, I'd done a hell of a lot of reading. It's just that it was a bit eccentric. I'd read Zariski-Samuels (Commutative Ring Theory) fairly carefully, and I'd gone through more than half of Hilton & Wylie (Algebraic Topology), a book which had fascinated me when I first saw it in the bookstore at Arizona. I'd read Knopp's little series on complex variable theory and Pontryagin's book on topological groups.
I was damned if I was going to sign up for any stupid undergraduate courses, so I started reading furiously some of the books recommended for the prelim: Kelly, Apostol, and Hoffman & Kunze, and Chapters 1, 2, and 9 of the Bourbaki Topology.
Since I started in the spring semester, I asked Professor Goldhaber if I could take Algebra 2 without having had the first semester. I had expected to have to argue with him a lot, but after hearing I'd read Zariski-Samuels he said, ``Why not just skip Algebra 2 as well?'' I wasn't sure that was a good idea. It did turn out that his graduate algebra course was mostly a review, but it was a very useful one.
Academically, Maryland was a good place for me to start my graduate work. There were a large number of courses available, and they didn't worry much about which ones students took. They didn't give me any static, for instance, about signing up for Algebraic Topology without having had General Topology first, which at Maryland was a separate two-semester course.
While I was at Maryland, I was completely passionate about learning mathematics. One reason was that I wanted to get a degree as fast as possible, but the demonic need to learn mathematics went beyond that. I especially wanted to learn algebraic geometry, which I'd been attracted to ever since reading Zariski-Samuels. And I guess I wanted to learn every other form of geometry as well -- differential geometry, for sure, and differential topology (which I never did get to take a course in), and geometric topology. But I was always fairly sure that I'd eventually wind up specializing in algebra.
I worked extremely hard on all my courses at Maryland, and also spent an incredible amount of time in the library reading books on other aspects on mathematics. I learned an incredible amount of off-beat and important mathematics reading library books. Essentially it was the only thing I had to do on campus, aside from going to classes and studying for them. (Despite being a graduate assistant, I didn't get an office, only a library carrel. I had the use of a shared office three hours a week to meet with my students.)
My selection of books to read was rather bizarre. I read a large part of Bourbaki, in French (and eventually would read almost all the rest, except for the set theory). I was reading Godement's book on sheaf theory, primarily because it was written in French. And I read a lot of Corps Locaux, which as I recall is only partly in French. And Jacobson's AMS notes on ring theory.
For the most part, my system was to look through the mathematics section of the library until I found a book that seemed exotic and fascinating to me and then see if I could read it.
Algebra (second semester) [Goldhaber] Real Analysis [Nieto] Differential Geometry [Chu]
Beginning Russian
Algebraic Topology [Guido Lehner] Algebraic Number Theory [Kuroda] Complex Analysis [ ? ] Algebraic Geometry (out of Lang's book) [Greenburg] (audit) Second Year Russian (audit)
Algebraic Topology [Holzager] Algebraic Number Theory [Kuroda] Real Analysis (second semester) [Horvath] Local Rings (out of Serre) [Greenberg] (audit) Second Year Russian (audit)
I decided to transfer to UC San Diego (in La Jolla) purely because one of my fellow students had been there and said that it was an incredibly friendly place: all the graduate students were friends, and many of the faculty were very friendly with the students. I looked at the UCSD catalog, saw that they listed some really exciting courses (most of which I subsequently discovered were never actually offered), and sent them an application, not applying anywhere else. (I didn't even apply to Berkeley, although going to Berkeley had been my dream for several years, because I foolishly took it for granted that they wouldn't accept me.)
The students at UCSD were bright, and mostly pretty likeable. (One of them, incidentally, was Vernor Vinge, who had already published his first science fiction novel.) A woman named Connie who had been a fellow computer programmer at Sylvania was now a student there, so I arrived with an instant friend. (By the time I left UCSD, we were no longer really friends. That was almost completely my fault, because of the classic problem I have in being friends with women. Rent the video of When Harry Met Sally for further details.)
Whereas most of the students at Maryland struggled to get passing grades in the basic courses, the students at UCSD were not satisfied just to pass their courses. They wanted to really understand everything. And consequently, although there was not the diversity in course offerings that there had been at Maryland where the program was much larger, the level of the courses was a step up. At Maryland, category theory -- which was then the big new fashionable thing in mathematics -- had been rather hesitantly mentioned in a couple of my courses. Mostly I had learned about it from books in the library. But at UCSD, even some of the most basic courses were completely organized around category theory. (Partly this was because of a brand new textbook on abstract algebra by a guy at Columbia named Lang that was to become, as it were, the industry standard.) I felt that I was really learning things on the cutting edge, and this was very exciting for me.
While still at Maryland, I had received the UCSD list recommended books for comprehensive exams. I had worked through Halmos (Measure Theory) very thoroughly, although when I actually got to La Jolla it turned out that all the graduate students had been taught Real Analysis out of Rudin, just like me. (Except that at Maryland Horvath had taught us the second semester of Real Analysis out of Bourbaki.)
I realized that I should have taken two semesters of Complex Analysis, but my new friend David Minda had a notebook containing several past years of Complex Analysis comprehensive exams, with very carefully written out solutions for all the problems. This notebook was a revelation to me, because for the first time I realized that things like Rouché's Theorem could actually be useful for doing calculations. My Complex Analysis course at Maryland (taught out of Ahlfors) had never taught us how to use any of the theorems for anything practical.
Connie lent me her very conscientiously taken notes from the graduate Algebra course, which included lots that hadn't been taught at Maryland, although I already knew a lot of it from reading Bourbaki and Godement (Sheaf Theory) and Serre and Jacobson (the AMS ring theory notes).
I suppose she also lent me her topology notes. The topology graduate students took at UCSD was mostly algebraic topology, so it wasn't that important that I'd never had a really intensive course in general topology.
We had to take four written comprehensive exams. I took the ones in algebra, topology, real analysis and complex analysis. Oddly enough, the only one I didn't do really well on was algebra, but I passed them all.
UCSD was on the quarter system. I can't remember most of the courses I was taking very well. I guess the courses at UCSD weren't very important to me.
Although I was learning a lot at UCSD, I eventually realized that there was really nobody to work with as far as writing a dissertation in my chosen area (abstract algebra). So after two years I took my Masters degree and left for a temporary teaching job at Humboldt State University.
Functional Analysis [Holbrook]
Functional Analysis [Holbrook]
Sheaf Theory [Rohrl] Geometric Topology [Morton Brown] (audit)Winter, 1969
Algebraic Geometry [Fillmore]
At Humboldt State I was treated as absolutely equal to the other faculty. There was only one respect in which my status was inferior -- without a Ph.D., I could never be given a permanent position. Furthermore, since I didn't have a permanent position, I couldn't even take an unpaid leave of absence to go back and get my Ph.D.
So I had to leave Humboldt State. I looked through a lot of catalogs and applied to four universities which seemed to have some reasonable faculty in algebra and where it would be at least technically feasible for me to get my degree in one year. New Mexico State University was my first choice. For one thing, I'd never lived in New Mexico. (I'd gone to Arizona as an undergraduate, though, and liked it.) And one of the older professors at Humboldt State had family in Las Cruces and told me that the faculty at NMSU were reasonable human beings, even if a little on the crazy side. (I rated the latter as a plus.)
NMSU turned out to be a cow college, but the science departments -- especially mathematics -- were excellent.
The graduate students at New Mexico State were not exceptionally good -- maybe comparable to those at Maryland. They didn't take mathematics seriously the way students at UCSD had, but they were reasonably okay people. They were certainly not socially challenged.
The Math Department was good enough to waive comprehensive exams and the language exam for me, since I'd already taken them at UCSD, but they made me take a good stiff comprehensive oral.
On a more amusing note, the graduate dean was not willing to waive the GRE exams, which I had never taken anywhere. I was told that I was accepted for graduate study in any case, but that after I arrived in New Mexico I would have to take the GREs, as a pure formality. I strongly resented that requirement, mainly because the exam started at eight in the morning, and sleep has always been a really major issue for me. Besides that, it was totally stupid that with a straight A record at places like Maryland and UCSD I should be required to take this exam designed for beginning graduate students.
I took the GREs, of course, but I decided that under the circumstances it would be more interesting not to do my best. So I answered about one out of every four questions wrong, which I thought would make my results about average. As it turned out, though, when the results came back I had got about the equivalent of the C minus.
A year later, when it came time for me to take my final orals, the graduate dean was on my committee as the outside member. And he took the opportunity to let me know that he was extremely pissed off about what I had done on the GREs. Somebody later explained to me that he had in his office an enormous chart showing the GREs of all the graduate students in the university along with indications of their subsequent academic performance. He had been looking forward to having my score as an illustrious data point to ornament his chart.
Aside from telling me how pissed off he was, though, there was no way he could penalize me. The Math Department would have raised hell if he'd tried to delay my getting my Ph.D. on such a stupid pretext.
When school started, I fairly quickly chose Fred Richman as a dissertation advisor, since I found Elbert Walker's brash Texas manner just a bit intimidating. This turned out to be a very wise choice, and I quickly realized that leaving UCSD had been an even wiser choice.
After being around Richman and the other NMSU algebraists for a little while, I realized that no one at UCSD had really had an understanding of what algebra is about. They had known the concepts and the theorems, but they didn't have any real understanding of what it is that one looks for when one does research in algebra -- what questions one asks. They had not been real algebraists in the way that the faculty at New Mexico State were. (The year after I left UCSD, though, they hired a very good non-commutative ring theorist named Lance Small, so maybe the algebra situation there improved after that.)
Fred Richman suggested a couple of problems I could work on, although he didn't think they were very promising. I started off, though, working on a question that occurred to me on my own, and he agreed that it was worth investigating.
Fred never set up appointments for me to come see him but left it to me to stop by when I had time and see if he was free. I quickly learned not to do this unless I was able to spend the rest of the afternoon with him, and I realized that I'd better restrict myself to one afternoon a week if I was to have any time left to do anything else.
I was always a little embarrassed about coming to see him, because it seemed like I hadn't accomplished anything during the preceding week. But I'd tell him some of the things I'd thought about, which didn't seem to get me anywhere, and he'd say, ``Oh, that sounds rather interesting. Why happens if you [do whatever]?'' And then I tried to give an answer to his question without revealing that it had never occurred to me to wonder that. And he'd lean back in his chair and sink into deep thought, and pretty soon we'd have spent the whole afternoon trying to figure out the implications of the stuff that I'd originally thought scarcely worth mentioning.
After three or four weeks, I was in the library once more looking at one of the papers by Nunke which very strongly related to the question Fred and I were thinking about, and I suddenly realized that the answer to this question was right there in the introduction to the paper.
I was totally mortified by having wasted so much of Richman's time by having failed to notice something that I should have seen immediately, before we even started work on the question. But there was no possible way to avoid telling Fred.
When I did tell him, he just shrugged, and said, ``Oh, well, that sort of thing happens. Have you thought about something else you might want to work on?''
As if that hadn't been bad enough, essentially the same thing happened with the next problem I chose. Fred told me about a theorem that he had tried to prove unsuccessfully, and wasn't very encouraging about my prospects with it. But as it so happened, I saw a relationship between it and a homework problem Walker had assigned in the abelian groups class, and I was actually able to come in and give Fred a counterexample showing that the theorem was false.
He didn't think that the counterexample closed the investigation, though, and suggested some questions I might ask about it. We spent a few weeks trying to see if there was some reasonable theory that could be developed about examples like these, when one day I suddenly realized that my counterexample was wrong. Once again, I had to go tell him that I'd been wasting our time on something that was incorrect, and once again he shrugged his shoulders, saying that certainly it had convinced him, and that that was the nature of doing research.
He then suggested that since I'd come to understand the situation reasonably well, maybe I should try proving the theorem. Which I was eventually successful at.
And Richman and Walker seemed to consider my proof of this theorem a major accomplishment. I suppose that was because they'd tried fairly hard to prove it themselves. Their enthusiasm was certainly very useful to me, because it resulted in letters of recommendation that got me job offers. But to this day, to me that damned theorem seems like a minor result. I'm certainly glad that I was later able to prove some things that I'm much more proud of.
During my second semester at NMSU, Dave Arnold offered a course in torsion free groups, a topic that had been minimally covered in Walker's abelian groups course. In the process of taking Arnold's course, I began to realize that the realm of torsion free groups was the land of opportunity for a young algebraist. I got interested in some of the examples of direct sum pathologies Dave presented in class, and actually managed to prove a very minor theorem which earned me a footnote in the new edition of Fuchs's book. Within a month or so, I had improved this result and started blending it with some of Dave's own new ideas, and he suggested that we write a joint paper. Although it took about two years from the time we finished this paper until it was published, it eventually turned out to be one of my most popular pieces of research.
By the time I left NMSU, I knew that my future would be devoted to torsion free groups.
Fred Richman had taught me how to do algebra. Dave Arnold showed me where to find some promising subject matter for my research.
The course offerings at NMSU were pretty limited, but there was a course in abelian groups, my dissertation specialty (which, in anticipation, I'd pretty thoroughly studied on my own anyway). It was taken for granted that everyone in the course would write their dissertation in that specialty, and although the course started with the basics, it was very much directed toward current research. Other than that, I didn't really need much in the way of courses at that point.
I think I may have only taken one course during my first semester at NMSU. I know that I wanted to take differential equations, but it didn't have enough enrollment to make. So maybe I just signed up for six credits of dissertation work.
Abelian Groups [Walker]
Abelian Groups [Walker] Torsion Free Groups [Arnold] Some Course in Applied Math [Zund]
It would be impossible to include here all of the one-evening workshops I have taken, often in someone's living room. Also, I won't try to list the various mathematical workshops I've been through from time to time during summers.
I'm simply going to try to include those courses and seminars, mostly outside the traditional academic framework, which have had a significant impact on my life. (The Indiana Summer Russian Intensive and the Clarion Science Fiction Workshop were formally given as summer school courses at standard universities. Academic credits were given and grades were awarded for these courses, however these credits and grades were irrelevant for most of us participating in them. Milton Diamond's seminar on sexuality is given in the Medical School here, but is not much like a standard university course and rarely has students actually enrolled for credit. The Institute for Advanced Studies of Human Sexuality and More University are licensed by the State of California as non-traditional graduate schools, but are not comparable to ordinary universities.)
When I was six or seven I took lessons from a neighbor who was a music teacher, and at that point I was able to accept the idea that learning EGBDF and FACE was a necessary preliminary, and within a few weeks I was able to play something sort of like a tune with one hand.
My mother would call me in from play every day to practice my piano lesson for half an hour. This was a real drag, so after a year or two (long enough so that I had learned to play with both hands) I decided that I'd lost interest in the piano.
Then in seventh or eighth grade, I started playing the piano again, but teaching myself this time, from some of my mother's sheet music -- Beethoven and Schubert and Schumann and whatever. I'm sure I spent more than half an hour per day on the piano then, but it was more fun because I was figuring things out for myself, and also I was practicing whenever I felt like it, not when my mother told me to.
Eventually I was playing some fairly complicated Beethoven sonatas and Chopin waltzes. It sounded pretty good to me, although I didn't have any idea of how these pieces were supposed to sound.
LP records were new then, and the only phonograph my family had only played 78s, and we didn't have any records of classical music. The few times I did get a chance to hear recordings of Chopin by Horowitz or whoever, I didn't like them because I thought that the painist played much too fast, so that one didn't have time to really listen to all the individual notes. In fact, I still think that about a lot of concert pianists today.
My parents knew that my playing was very bad, so they finally convinced me to go back to a piano teacher, but this time one who was very good. His name was Gilmore McDonald and he had been a student of Gieseking (a name that meant nothing to me) and he lived about six blocks from me.
These piano lessons are to me a good example of the difference between the way teaching and learning ought to be, and the way they exist in schools and universities. I went to Gilmore McDonald because there was something I wanted to learn from him. We thus had an objective in common, namely for me to learn. He told me things to do that he believed would help me learn what I wanted. I didn't always completely agree with his approach, but I accepted the idea that it was more appropriate for him to make the decisions on how I should learn than for me to make them. He never gave me any grades or credit. I did the assignments he gave me because I believed that that would help me achieve our mutual goal of my learning the piano.
It was quite clear that for him, the appropriate ultimate goal would be for me to become a concert pianist. I could never quite believe that this was what I would wind up wanting to do with my life, but that wasn't really terribly important.
By the time I went to college, I had basically achieved my objective, which was to learn to play the piano. Once I left home, there was no piano available for me to play, but that didn't seem so important. Learning to play was what had been important. Later on, when I had pianos available from time to time, I didn't really have much interest in going back and re-learning how to play Bach and Chopin. I was more interested in learning what Gilmore McDonald had never taught me anything about -- how to just sit and the piano and make up my own music. The fact that it wasn't really especially good wasn't that important.
Numerous years later, I decided that I wanted to learn to play the guitar and bought a guitar and several books. The first books I bought were intended for students learning to play classical guitar. I learned to play an kind of okay guitar style, mostly folk guitar, with a touch of blues. Oddly enough, the one thing I never learned to do was the one thing that almost everybody who plays a little guitar does, namely thumb strumming. I was much more oriented toward the type of finger-picking that is mostly used in classical guitar.
The guitar books I learned from were later useful when I had a piano available again, because the guitar books explained harmony from a practical point of view, and that was useful to me in playing improvisational piano.
Many years later, I took a weekend course here in Hawaii from a guy who was visiting from the Mainland, on how to play blues and boogie-woogie piano. The course was advertised for people who did not necessarily have any previous experience with piano. In any case, taking the very simple principles he taught, and listening to the cassette tape he sold, I was able to learn to play an elementary type of blue and boogie woogie on an electronic keyboard.
Now, though, the thing that I regret is that I never learned to play jazz.