Student and Teacher
Student and Teacher
One effect that the creation of NASA had on me was that it inspired me to enter briefly into experimental work, and thereby lose my fruitful relationship with Dr. Katzoff. It was years later that I looked back and realized that I had not expressed to him the gratitude that he was due because of what he had taught me. Perhaps “taught” is not the precise term because it was not so much direct instruction as “causing me to learn”. He told me which books and papers to study in order to develop an understanding of theoretical aerodynamics and fluid mechanics.
And it was, in a way, a remarkable learning process. I was pleasantly surprised to discover the amazing relationships between math and physical processes. I had worked simple textbook problems relating to problems like particle motions, and that eternal algebra bugaboo about two men digging a ditch. But now, in studying far more complex problems, I began to see how the mathematics that allows us to describe a process numerically also helps us to gain an understanding of the physics itself. Even more enlightening was the reverse concept: i.e., understanding a physical process can lead to new mathematical methods. For example the mathematical method of images for treating boundary effects is based entirely on physical considerations.
Those first few years at NACA/NASA were a time of intense study, and almost none of this study was wasted on material of purely academic interest. All of it had the potential for practical application. I learned in at least four ways. First, I studied the material that was relevant to my current research project, as I described above. I also studied textbooks on applied math, aerodynamics, and applied science to fill in the technical background that was missing from my formal education. Third, I took evening courses on some of these same subjects. These were UVA and VPI extension classes. Fourth, I taught classes in Advanced Calculus and Advanced Engineering Mathematics.
It may seem strange that I would undertake to teach material that had not been part of my own academic studies. The most advanced applied math classes that I had taken before coming to Langley were First Year Calculus and Ordinary Differential Equations. They represented merely introductory material to the math that I was to teach. But I wasn’t particularly concerned about that problem. I had already found that learning that kind of math was not difficult compared to trying to comprehend the abstract mathematics of graduate school. All I had to do was stay at least one chapter ahead of the students.
I actually developed a reputation as a pretty good teacher. And it wasn’t in spite of the fact that I was teaching material that was new to me. It was because of it. You see, a professor who has taught the same material for years becomes so familiar with it that he loses the ability to see it from the standpoint of a student. But I didn’t have that problem, because I had been “in the shoes of” a student myself just a few days earlier. I spent extra class time on concepts that had given me trouble and I tried to unravel textbook wording that I had found confusing. As I taught a concept, I would simultaneously work through an example on the blackboard, so that the students could see the concept in practice. And I encouraged the students to ask questions.
Also there was another factor that made that study easier for me. I was truly interested in the math in those textbooks. In college I was always faced with the question as to whether the mental effort was worth it. Would I ever have any use for this knowledge? But now, that doubt had vanished. As I encountered the techniques that were introduced to me in each chapter, my mind would race forward, imagining a variety of applications to which they might apply.
We tend to remember those things that truly interest us. For example, my mother had an extensive knowledge of flower names and cultivation procedures, not because she took classes in the subject, but because she was so enthusiastic about it she remembered almost everything that she read or heard about it.
My experience with the Dini series taught me that a golden nugget of a math technique might lie hidden in an unexpected place; so I began each new chapter with an eager anticipation that here I might discover kind of magic key that would unlock an engineering problem. Or if not, then it would at least provide a new weapon to add to my growing arsenal to attack theoretical problems.
I also took classes in theoretical gas dynamics and theoretical aerodynamics; and the physics courses that I had taken in high school and college now began to demonstrate that they were not a waste of time. But most of the physics that I learned was self-taught through study of background material for the specific problem that I was attempting to solve.
So, while I cannot truthfully say that that learning process was fun, because much of it involved extensive and often frustrating mental struggling, I at least had strong motivation. Not only was there the reward of increasing my mental arsenal, but each new insight into the understanding of a new mathematical or physical concept brought a kind of sensation - a thrill - of triumph, and increased confidence. In school, learning had been for basic preparation for work. Now it became part, a rewarding part, of that work.
I believe that in those two or three years I learned more applied math than the average applied math doctoral candidate learns in his doctoral program. In fact I served on the doctoral examining committees for several students.
But it was rarely easy for me. I don't have the kind of sharp mind that grasps difficult concepts quickly. I have known a few people with that kind of intelligence, and I often envied them that gift; but judging from that small sample that I knew personally, I perceive that the gift is not always an advantage. None of them had strong originality and an innovative mind, although they were quick to grasp the work that others had done. And none possessed a breadth of knowledge. I suspect that the problem is that learning is so easy for them that they are never motivated to develop the discipline and patience that is required to undertake and see through a major mental project. For me, it takes time for the material to “sink in”.
But, because of the rewards that learning brought, it became a habit, and extended beyond math and science to other subjects that interested me. Later, I will describe some of these subjects that in one way or another involved my mind - or, at least, my inner life. Study took over a large part of my life; but it was not always productive, because I was slow to realize that study does not always equate to learning. It was later in life that I undertook to clarify and solve that problem.
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