© 2013 Stan Yack

In my initial passage through academe, from elementary school through the first of my incomplete Masters degrees, I was a math and science nerd. Short, pudgy, bespectacled, unfashionably dressed, my appearance and social ineptness blessed me with freedom from many typical, time-consuming teenage distractions, like sports, partying, and girlfriends. With time to spend on learning, I came to enjoy playing the role of nerd.

My studies focussed on math and science, especially physics. Graduating with a mid-80s average from my high school's grade 13, I enrolled in an "honours" math, physics and chemistry course at the University of Toronto. I did well enough in first year, maintaining a first class honours average, though for the first time my marks put me in bottom half of the class. But things fell apart in second year, when I was distracted by a female student (who was both better looking, and much brighter than I was). My marks weren't high enough for the U of T to let me continue in the third year an honours course, but the University of Waterloo did. I spent the next two years in Waterloo, and after earning a B. Math, I was accepted into a graduate program studying the Mathematical Theory of Relativity. Before the end of one semester I realized that I had reached my level of academic incompetence, and a couple of months after dropping out I landed a full-time job in information technology. (See my story The Good Old Days of Computing.)

Many years later I broadened my education to include the "softer sciences" of philosophy, psychology, language, and human cognition. In that later-life return to learning, I occasional paid to register for courses (which by the late 1980s at the University of Toronto cost more per one-term course than had my entire first two-term year in 1966). But I soon discovered that I could attend classes without registering and paying fees (or taking tests). So each new term I browse the current course calendar and a class schedule, and walk into classrooms where I've decided that I might hear something or someone interesting. In a large class I make sure to sit on the aisle close to the door for easy escape. In what turns out to be an ill-chosen class is a small one I'll sit quietly and try not to squirm while waiting for the first lesson to end.

If an initial lecture whets my appetite, I'll present myself to the instructor and ask for permission to attend. No instructors have ever rejected my request, though one whose class I'd already decided to skip, stopped as he himslef was leaving (probably when he spotted my out-of-place gray hair), and cautioned me that his uninspiring, derivative material "was copyrighted" (by implication warning me not to try to "steal his stuff").

Over five decades of learning, I remember many thrills of discovery facilitated by star performers in the teachers guild. I have attended lectures and seminars by sparkling University of Toronto residents like Mark Kingwell, Keith Oatley, David Olson, and John Vervaeke, and talks by visiting researchers and educators, some of them world famous. I've learned about memory, insight, wisdom and foolishness, intelligence (both the human and artificial kind); and about research in the newly legitimized field of the development and origins of language (which until the late 20th century had been proscribed as an academic pursuit).

I recall that as a young student I always did well on tests, including those that were said to measure not my knowledge, but my aptitude for learning. But I had learned a trick in a highschool "enriched math" class given by the great educator W. W. Sawyer, which let me solve a typical class of IQ test question as mindlessly as does a computer. You probably remember those question yourself, which had the form "What is the next number in this sequence?" Some sequences were easy, like 1, 3, 5, 7, 9, ... (the next number is of course 11) but others were harder, like 6, 9, 18, 27, ...

The sequence 6, 9, 18, 27, is generated by the polynomial (n

^{2}+ 2n + 3), and the next number in the sequence, for n = 5, is 25 + 10 + 3 = 38.Those problems were composed to trigger organized thinking to discover the solution. Sawyer taught us highschool kids a much faster way to find the next number in the sequence, using what I would later learn was an "algorithm". No deep thinking was involved; the algorithm was so simple that by 1975 it could be programmed into a pocket calculator. That's because almost all of the "What is the next number?" IQ test questions were based on polynomial expansions, so they could be answered using that algorithm. (Is that still the case in IQ tests today?)

The algorithm we learned is called the method of finite differences (which Sawyer explained to us more clearly than does the reference in the link I've provided in this note).

Sure, I had to have some aptitude to learn the algorithm, and my correct IQ test answers did reflect that aptitude, but for years my knowledge of that algorithm "unfairly" boosted measures of my IQ, and advanced my academic and later my professional career. My learning of that algorithm gave me a preview of the formal language world that I would encounter on my later career as a computer softsmith.

Learning about the finite differences algorithm helped me gain an understanding of the grounding metaphors of mathematics. To see what I think about the importance of metaphor in language, read my essay "Literally Metaphorical".