Ever since I was a pretty young kid, I've been a lover of mathematics. Numbers, and the many and varied things that can be done with them, caught my fancy at a young age, and never left me. By the time I was in college, I came to perceive something stately, elegant, and intrinsically TRUE about mathematics, as though it was built into the Universe on a foundational level. As someone once said, Mathematics is the language with which God called the Universe into being. . .
I remember learning to add and subtract pretty quickly, when I was in kindergarten or first grade, then going to my dad and asking him what came after adding and subtracting. So he taught me how to multiply, and how multiplication was just repeated addition, except you could do it all at once, without having to go and do all those additions, which I thought was pretty cool, and really powerful. Then he showed me how to divide, and how it was 'reverse multiplication', similar to how subtraction was 'reverse addition'. When I asked what happens when you try to subtract a larger number from a smaller one, he showed me negative numbers (I didn't know the 'minus sign' notation, so I would just write an 'n' next to the number, to show it was negative).
Once I'd mastered multiplication and division (at least enough to convince myself that I understood how they worked; I wasn't doing five-digit division problems just yet), I was pretty happy with myself, and figured I must have learned just about all the math there was to know. So I went to my dad (I was in the second grade at the time) and asked him, heh-heh, if there was anything else besides adding, subtracting, multiplying and dividing, half-expecting him to tell me that, no, that was about all there was. So imagine my surprise when he showed me how to raise numbers to powers, and how it was repeated multiplication, just like multiplication was repeated addition. At that point, I was starting to suspect that there might just be more math available for me to learn than I was going to master in the next few days, at any rate. Once I'd gotten the hang of doing powers, I showed my teacher how to do it; 'cuz, you know, all she ever did was add and subtract, although she did seem to have some inkling of how to multiply and divide. She looked at me (2nd-grader that I was) a little funny, and expressed a degree of amazement at what I was showing her. For my part, I was just happy to help her out. . .
(As a side note, I eventually got around to the idea of extracting roots as the 'opposite' of taking powers, much like division is the 'opposite' of multiplication. I learned how to extract square roots by hand, which actually looks pretty similar to doing long division. I'm told there's a similar method for extracting cube roots, but I've never seen it. In the fullness of time, I learned about imaginary numbers, which arise from the problem of extracting roots of negative numbers in like manner to how negative numbers themselves arise from subtraction. Man, there's no end to this stuff. . .)
Bookish kid that I was, I read a lot, and my parents always kept me supplied with new and interesting books to read. But my favorites (aside from anything by Dr. Seuss; who, by the way, shared his birthday with my friend Suldog) were always the math books - the ones showing how the ancient Greeks and Egyptians used math to prove the earth was round (and to make a pretty good estimate as to its size), or to build the pyramids, and all that good stuff.
From that point on, I was just a voracious math nerd. When I was in 6th grade, our school district (backwoods northern hicks that we were) started a pilot program to identify kids with math talent and run them through an 'accelerated' math program. So, when I showed up for my first day of 7th grade (which was,coincidentally, also the first day that I had separate 'classes' taught by different teachers, where we students had to move from classroom to classroom in the course of the day), I went to my math class, and the teacher handed us 8th-grade math books. At first it seemed a little scary, like we were going to be in over our heads all year (and bright kids really hate feeling like they're in over their heads; they don't like it at all), but the teacher reassured us that we were gonna be just fine, and in the fullness of time, we were.
When I was in 8th grade, we had algebra, and I had one of the best math teachers I ever had (Mr. Lewis, if you're reading this, thank you). When we were doing a unit on graphing, and I was just loving it (the visual aspects of math have always held a special fascination for me). Once I'd gotten the hang of graphing lines and parabolas, and absolute-values, and all that stuff, he took me aside and gave me an equation that we hadn't seen in class, and asked me if I could graph it. I took it home and worked on it, and played around with different approaches to the problem (as much as my 12-year-old brain was equipped to do), and eventually figured out that it was a circle. My teacher was pleased that I'd been able to figure it out, and gave me a few more circles to practice on. Then he gave me another equation, a little different from the circles I knew how to do, and had me play around with that. I spent probably a week or so, but I couldn't figure it out, so I went back to my teacher, and he showed me that it was an ellipse, and then he gave me a few more ellipses to play with. The entire school year was like that - every couple months, Mr. Lewis would give me something to stretch what he was teaching us in class, and just let me play with it, feeding my own sense of having fun with math.
As the years went on, the number of kids in the Accelerated Math program got smaller; some kids just weren't all that interested in math, regardless of whatever 'aptitude' they might've had, and some of the 'marginally gifted' kids (if I can say it that way) just didn't want to run with such fast company. By the time we got to high school, there weren't enough of us left to fill a whole class anymore, so instead of having separate sections for the 'accelerated' kids (the kids from the accelerated program; we didn't move any faster than anyone else; if anything, we were probably more sluggish, as far as that goes; but, I digress), they just put us in classes with the older kids. Which was a little weird, at first; especially when I walked into my first day of Advanced Algebra, and the teacher, who was also the basketball coach, addressed the class. "I see," he began, with a slightly menacing tone, "that we have (here he paused for dramatic effect) sophomores in the class this year." (he said 'sophomores' with an air of utter disdain) "Well, let me tell you my philosophy of what to do with sophomores - Fail Them. Fail Them ALLLLL. . ." Of course, it was all a joke. Heh-heh-heh. Funny guy. But once again, there was the slightest sense of wondering if I was getting in over my head, again. But again, once we got used to the new surroundings, we were fine. My senior year, I finished third in a statewide math competition, which won me a small scholarship for my university studies.
The Accelerated program basically put us a year ahead of the 'regular' math program, which meant that, when we finished our junior year, we'd taken all the math that there was to be taken at our high school, and what to do with us as seniors was an as-yet-unresolved question. The resolution was to dual-enroll us in the junior college, so we could take Calculus at the college our senior year. Which was all sorts of cool. First, we got to leave the high school campus in the middle of the day, to drive across town to the JC. And we were taking a real, live, bona-fide college class, taught by a real, live, bona-fide college instructor. And again, immediately, on the first day of class, I was confronted with the fact that this was something new and different than what I'd seen before. I was young, even among my own class - 16 at the beginning of my senior year. And sitting across the aisle from me was a 29-year-old Viet Nam vet. At least, at the high school, everyone was within a year or two of my age (and social maturity level, altho that might have covered a slightly wider range). But this was like my first step out of the 'protected' world of school-as-I'd-known-it, and into something more like Real Life. Which, once I'd gotten used to it, was really pretty exciting. I also found that college classes move along at a significantly quicker pace than I'd been used to in high school - when our family moved, two months before graduation, I was placed into the high-school-level Calculus class at the new school, but I was already considerably farther along than they would be by the end of the school year, so I basically ran an independent study with the teacher on the side, and acted as a tutor in the class.
When I finally got to the University, I started as a Math major. It was what I liked, and I was good at it, so it seemed obvious. By virtue of my year at the JC, I was already through all the freshman math classes, so by the end of my own freshman year, I was starting to take classes for my major. Without trying to bore you all to tears, I'll just say that I encountered my first Abstract Algebra class, and I realized that, if I was going to be seeing significantly more of this stuff (and I assuredly was), then Math wasn't really what I wanted to study after all. Looking around for another plausible field of study, I settled on Mechanical Engineering. I considered studying Physics, but when I thought about it, that could take me into realms just as abstract as Mathematics. So Engineering, I reasoned, would involve me in lots of good math problems, of a suitably concrete nature, that I might even get paid to solve, someday. I kept taking math classes 'on the side' (with my 'elective' credits), and by the time I finished my Bachelor's Degree, I had more math credits on my transcript than my roommate, who was a Math major (I've occasionally thought about going back to see if I could finagle a 'second major' in Applied Math, or somesuch, out of the classes I'd already taken).
(Another side note. . . when I took my GREs, in preparation for applying to graduate school, I was initially undecided as to whether I wanted to study Engineering or Applied Math, so I took the exams for both, and had the results sent to the corresponding departments. By the time the results came back, I'd decided in favor of getting another Engineering degree. But the Math department still got my test results; and I did well enough that I got a letter from the Math chairman, saying that they'd gotten my test results, and they were really good, and they wanted to admit me, but there was a small problem - I hadn't applied yet. So. . . would I please apply? Which was nice for my ego, but I'd already made my choice. I probably should have written back, explaining my decision - or heck, just walked across the street and told him myself - but I was still a little too green for that. *sigh* )
All of which makes for a nice story, and a nice insight into my life and psyche (if you're remotely interested in such things; God only knows why you would be), but it's really only background for the story I set out to tell you (I hope you don't feel deceived) (and Lord knows, this post is already long enough). . .
When I was in high school, I rather enjoyed my reputation (such as it was) as the school's 'Math Whiz'. But, as noted above, I mainly did it out of my own enjoyment and love of math. I did all the 'Extra Credit' problems, and sometimes, just for my own interest and challenge, I'd do the 'hard' problems at the bottom of the page, even if the teacher hadn't assigned them. Of course, those 'challenging' problems were often, um. . . challenging. Even to me (hard to believe, I know, but it happens. . .). And sometimes, I'd become the least bit, uh, obsessive about 'conquering' them. Sometimes, I'd spend an hour or more, trying as many different approaches to a problem as I could think of, in order to crack the problem, and make it give up its answer to me. And sometimes, I'd go past my bedtime banging away on a problem, without success, and go to bed frustrated that I hadn't been able to beat the problem into submission.
I don't remember the first time it happened, but I clearly recall several of them, all when I was in high school. There might have been a few in college, but I clearly remember the times it happened in high school. I went to bed, and fell asleep, still agitated that I hadn't been able to solve the problem. Then I began to dream. And I dreamed the solution to the problem I'd been working on. I remember those dreams, even today. I'd be right back where I'd been, at my desk, grinding away at the problem, staring at the page in front of me. Then I'd have a crucial flash of insight, and work the problem through to solution. I'd check and double-check my work, until I was satisfied that what I had was really right. Then, at the end, still in my dream, I'd remind myself to wake up and write it down, before I forgot it. Then I'd wake up, excited, still remembering the 'key insight' that had come to me in my dream, and write down the solution, which was invariably correct. Shades of Kekule?
I don't know if that's indicative of how deeply I was obsessing over the problem, or if, once I was relaxed enough to sleep and dream, my brain (mind?) could work more efficiently, or what. But I still get a chuckle from the very idea of dreaming the answers to math problems. . .
Has anything like that ever happened to any of you?