Graduate School

 

Graduate School

Now I had to make a crucial decision. I needed to qualify for some profession, and the only one that seemed within reach at this point was the teaching profession. I could easily, with a few evening classes, qualify for a teaching certificate. The major drawback was that teachers’ salaries at that time were pitiably inadequate, especially if one needed to support a family. I learned that a master’s degree would command a premium salary as a teacher, so I decided to make that effort. I wanted to understand advanced mathematics, and I had learned that the University of Virginia math department had a good reputation. But when I discussed that possibility with one of my undergraduate math professors, he warned me that the UVa graduate math curriculum was extremely difficult, as it focused on “pure, abstract” math, as opposed to applied math. His recommendation was that I apply instead to the Graduate School of Education with a concentration on math.

Consequently, I made an appointment with a member of the Education School faculty to discuss a possible course schedule to lead to a master’s degree. That appointment led to another change in plans. If I pursued that path, I would learn a minimum of real mathematics, because the course list was filled with titles like, “The History of Education in Virginia Public Schools”. I tried to think of the best way to approach my future and finally, with considerable trepidation, applied to the graduate math program at UVa.

In graduate school a course schedule of three subjects was considered a full load, as compared with the undergraduate requirement of five courses, including labs. Even so, after sitting through the first lecture of each of my three courses, I was convinced that I was in over my head. I fervently wished that there was some way that I could alter the decision to undertake this plan. But there wasn’t – for many reasons. I didn’t qualify for admission to any other graduate department, except for Education, and I had already ruled that out. I couldn’t drop out of school altogether because I was now married, and I needed the income that a graduate degree would command. I had committed myself to an apartment rent contract, and a fellowship that would provide me with a stipend for grading undergraduate test papers.

I reminded myself of my philosophy that had led me here: if others can do it, so can I. And I soon found that the few other students that were starting into this graduate department also felt overwhelmed by the difficulty of trying to comprehend mathematics on an abstract level. So my days became a routine of sitting in a classroom struggling furiously to copy the notes that a lecturer was writing out in chalk while trying to follow some of the lecture; then spending long hours poring over and over the notes to try to make sense of them. To me, the study of math had always meant learning the techniques and formulas for obtaining the numerical solutions of real-world problems. But now we no longer saw numbers, and rarely even saw the word “number”. We were more likely to be dealing with sets of points, but even the term “point” was not specific. It might be an actual point – a zero-dimensional locus, or a number, a variable, a function of a variable, an operator, or any of a number of other possibilities.

Algebra became an exercise in proving theorems about “groups” and “fields” and other entities with simple names but complex definitions.

But Geometry was one of the few bright spots in my two-year work for a master’s degree. I want to dwell on this course just a bit, because it had a special significance for me, even though it occupied just one semester in my second year of graduate study. The course began with a set of axioms for a two-dimensional non-Euclidean space. Actually, only one of the axioms differed from those of Euclidean geometry, but it was a crucial one - one that threw the reasoning for this geometry out of whack with our intuition. During this entire brief course, the professor never lectured. After he laid out the initial set of axioms, he gave us a theorem to be proved from those axioms and dismissed the class until the next geometry class meeting. At the next class meeting he would point to one of the students to go to the chalk board and write out his proof. If the proof were correct, he would assign another theorem to be proved and dismiss the class. If the proof contained a flaw, he would halt the presentation, and choose another student to present his proof. This procedure continued until a valid proof was obtained, at which point he would assign a new theorem to be proved; but now the theorem that had been proved was available in addition to the original axioms. In this way a body of “truths” about this geometry was accumulated.

I did have some background for this course, not from my undergraduate studies, but in terms of my high school geometry class, which had been taught as a class in logic, beginning with Euclid’s original set of axioms. Of course, the high school teacher did plenty of lecturing, but I finished that course with a pretty good conception of logic and of what constitutes a valid proof. Consequently, I knew when I had a valid proof, so I never submitted a proof that the professor could criticize.

But the real turning point occurred when our sequence of proofs finally hit a brick wall. This theorem stumped everybody for a full week of classes, and it began to appear that we would complete the semester without a proof. But I was fascinated with the problem, and I kept trying different approaches until I finally found one that worked! After I finished writing out my proof on the chalkboard at the next class session, the professor didn’t hesitate, but immediately said, “That’s right! It’s not the way I did it, but it’s right”. At last, I began to have that modicum of confidence that I so severely needed. For the balance of the semester, the professor never called on me for a proof, and wouldn’t even allow me to give a proof when the rest of the class was stumped. And when the time finally came for the faculty to decide on my qualifications for a degree, he became a strong advocate for me.

There was one more thing that I learned by taking that course. It was that I had made the right decision in not taking math courses through the Department of Education. I found that out because of a little quirk in the requirements for a graduate level degree. Each student was expected to take one course (generally just one semester) in a subject that was different from, but related to, his major. For me that subject was a semester entitled, “Applied Mathematics”, which title was misleading at best, because the course was taught at such a theoretical level that it barely resembled any of the applied math that I had encountered in my high school and undergraduate classes. Now, the Education Department graduate level students who were minoring in math were expected to take one class in the Mathematics Department. Essentially all of them chose that Geometry class, for the reason that it did not require a single preparatory course credit. Consequently, the class of about twenty or so included no more than four or five math students with the remainder visitors from the School of Education. However, at least half of those students dropped out before the deadline for quitting without taking an “F”. And more opted out before the end of the semester. They simply were not prepared for the kind of logical and abstract thinking that the course required. It may have been unfair, and possibly undeserved, but I seemed to detect a little ego boost among the math students every time one of the education school students resigned.

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