Math by Default

 

Math by Default

When I completed my high school education, I set my sights on college studies. That, in itself, was not a simple matter. My parents had to be convinced by a couple of my teachers that it was worth the cost, as it would be a considerable financial strain for them. And I was not entirely enthusiastic about it because those same teachers emphasized to me that college level courses would be much harder than those I had been taking. They seemed hopeful – not confident – that I could handle it. And they recommended some choices that I could make that would ease the burden the first year of the transition.

Four years later, when I planned to continue my education at a graduate level, I had a similar experience. My professors warned me that the advanced courses would be much more difficult and recommended some means of making the transition easier.

Now all of these advisers meant well, but I could not, or would not, take any of the advice. Some of it was simply not viable. For example, one teacher advised that, since I had three years of Spanish to my credit, I should select Spanish as my college foreign language elective. However, Spanish was not an option for a science major. The only other modern languages taught were French and German, so I chose German – which turned out to be the more difficult but more useful choice.

The dire warnings about the difficulty of the material turned out to be entirely correct, but they didn’t cause me too much concern. I reasoned that lots of other students had made it through these curricula, so it certainly wasn’t impossible. All I had to do was work hard.

I think that there is a principle involved with these decisions – perhaps a lesson to be learned. It is that we should listen to, even seek, advice from those with experience, but ultimately set our own standards and goals. However, in doing so, we should not choose the easy way but set our sights so high that they will always represent a challenge. If we don’t, we will always carry within us (perhaps deep within, at the subconscious level) a lack of self-respect that will manifest itself in unpleasant ways at the conscious level.

However, at the time my choice of a difficult curriculum was not based on any philosophical or psychological principle, but on more practical factors. The primary consideration was the absolute necessity that my education should lead directly to economic selfreliance, with no further dependence on my parents. This consideration meant that there would be no room for “crip” courses that would require little time or effort - that would provide a break from the work and stress of real courses - but that would ultimately provide me with no practical information.

However, as it turned out, that first year of college level course work really did surprise me with its difficulty – but there were special circumstances that conspired to compound that difficulty. The primary factor was that Roanoke College, along with many other colleges at that time, had a surfeit of applicants. That situation resulted in large part from flood of WWII veterans taking advantage of the GI Bill of Rights, but also because the vast advances in science and technology that took place during the war gave a clear signal that there would be great opportunities in those areas. Roanoke College was a liberal arts college, but it did offer good basic programs in chemistry, biology, physics and math.

With an oversupply of applicants Roanoke College could have accepted only the top applicants, but the college chose to handle the situation in a different way: accept more first year students than could be reasonably handled in the more advanced courses, and then skim off the cream of the crop by flunking out the rest. One strategy that the school employed was to administer a sadistic grading system: D: 75-79, C: 80-90, B: 91-95, A: 96-100. Another strategy was to employ such an extensive list of course requirements for graduation that there would be little room for easy courses that might ease the burden of study.

These strategies worked. At the time I thought the system was heartless, but in retrospect, I can see some justifications for it. For one, it provided an opportunity for many veterans that would not have qualified under the conventional system. But it also eliminated that class of students that get admitted by an impressive high school record, but are unable to adapt to the rigorous study discipline that would be required for a degree that would equate to a meaningful education.

Since my life profession turned out to be that of an applied mathematician, I should briefly describe how that circumstance came about, because it certainly was not my goal at the time that I enrolled in college. My vision was to be a chemist, or chemical engineer, or just a chemistry teacher. Unfortunately, I didn’t realize that Roanoke College chemistry department at that time had an outstanding reputation, and as a result the first year class was filled by the time I tried to sign up for it. Furthermore, there was no viable way for me to overcome that loss. Summer classes were not an option because I had to work in order to contribute to my tuition and other related expenses. I couldn’t take simultaneous courses because they had to be taken sequentially.

With great disappointment I dropped back to my second (distant second) choice, Biology. The first semester was on Zoology, the second on Botany. I thought the subject would be easy because I had taken a high school Biology class, but that background proved to be of little help, as there was little overlap between the two courses. The information on the carbon cycle was interesting and proved to be helpful later on in discussions of the science of solar energy transfer, which subject involves a myriad of concepts from biology, chemistry, and math (especially mathematical modeling and statistics).

But the memorization of seemingly endless lists of classifications of flora and fauna, and the detailed descriptions of the inner organs of a few organisms bored me to exhaustion, so try as I might, I could not develop the kind of enthusiasm for biology that I had desired.

The second year I got in my coveted course in General Chemistry. I can’t honestly say that it was fun because it required such hard work. The study of the periodic table required extensive memorization. Some of the concepts took time and concentration to grasp. (e.g., the relation between a gram molecule, Avogadro’s number, and stoichiometry.) The lab experiments required lots of time largely because of the detailed report that was required for each one. But we had a good lecturer, and I can’t say that one minute spent in that class was wasted, even though I never performed a single chemistry experiment during my later scientific career. But chemical processes are taking place around us all of the time – in nature and at home with cooking ingredients, cleaning products, etc. As a result of taking that one course I now understood how and why those processes worked, how to take full advantage of them, and how to avoid problems with them.

It was during this Chemistry course that I experienced, more than ever, a problem of the mind that I had often experienced before, and would often experience again. It often happened during this course, and in other courses that involved conceptual understanding (especially math classes). I could follow the lecture easily, and could do the homework with little difficulty, but somehow have trouble with tests and exams. I dealt with this problem for years before its cause finally became clear to me. I could follow the lectures and do the homework problems, because the examples that were worked out in class and in the textbook fit smoothly into the formulas or procedures that were being taught. However, when I was faced with a problem that was not sufficiently similar to those examples, and I couldn’t follow a standard procedure, I was frustrated and often baffled.

Eventually, it struck me that my problem was my failure to distinguish between knowing and understanding. I can know the material in the sense of knowing the formulas and procedures, and being able to apply them to most problems; but if some nonstandard situation arises, then I can attack it only if I truly understand the theory that gave rise to those procedures. For example, I can solve many chemical reaction problems with a periodic table, a bit of basic algebra, and a knowledge of fractions. But I can fully comprehend why that procedure works only if I understand the relation between a gram molecule, Avogadro’s number, and stoichiometry. Similarly, if I forget a needed mathematical formula, I can derive it if I truly understand the theory.

But Chemistry still remained off the table as a major subject. The third year, I signed up for Physics. I had no enthusiasm for the subject, but I felt that it would be required for almost any area of applied science in which I might later be engaged. However, in a repeat of my high school experience, I surprised both the professor and myself by cruising through the course at the top of the class. And that course was a thorough survey of General Physics, devoting a full detailed chapter to each topic: basic concepts (force, momentum, energy, etc.), static forces, dynamic forces, acoustics, optics, electricity and magnetism, wave theory, etc., in addition to a three hour lab each week and a lengthy afternoon test each week to assure that none of us would fall behind and be faced with the near-impossible task of trying to play catch-up.

The professor took me aside, and almost begged me to major in Physics, but that idea remained in the realm of the impossible, inasmuch as it would have required me to take summer classes after both my junior and senior years – which I could not do because I had to work during the summers. However, I did manage to fit in two more semesters of Physics in my senior year, which was sufficient for a minor.

At this point, my only option for a major subject was Math, since I had taken a math course every semester throughout my four college years. It was far from my initial goal when I began my college education, but ultimately it proved to be a good thing.

Future events, which I could never have anticipated at the time, led to successes that would not have been possible if I had pursued a different track. And on that basis, I am tempted to speculate that the events that forced me along this track were guided by Providence. On the other hand, I have no way of knowing either what I might have been able to accomplish as a chemist, or how much personal satisfaction that profession would have provided.

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