## Sunday, January 15, 2012

### When Math Nerds Dream. . .

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?

1. Hey, you write pretty good for an engineer (and better'n me). I think, maybe with this post, you've out Suldogged Suldog.

On another note, I was one of those kids who never applied whatever aptitude for math I had until I could see a practical reason for it. Hence, I finally took plane geometry in college (after the Navy had put it to work for me for a few years.

2. Third in a statewide math competition? Wow, and you're not even Asian :)

I can't say as I've ever dreamed any sort of math!

3. I dreamed about having a wife... does that count?

Seriously, the only thing I ever dreamed was that homework would magically get done while I was asleep. Sometimes I would finish it after I woke up.

4. ok, first off, i think i deserve a great big pat on the back for reading the entirety of such a long post about MATH. *turns shoulder to present it for patting* thanks.

now that's out of the way, may i say i wish i had more mr. lewis' and fewer of the ilk of math teachers i was plagued by. perhaps i'd come away with less disdain for the subject and more skill. i was put in the group for advanced math instruction in jr hi. god knows why. basically it meant i got algebra a year earlier than the "not advanced" and could eventually get through calculus in high school if i so desired.

algebra was awful. i vastly prefer that my letters and numbers be kept separate. it just makes no sense when you go mixing them up. too abstract. then you start adding various things like parenthesis. it's unnatural and disturbing. algerba 1 teacher was an idiot. i used to manage to distract her from the lesson by whistling through my front teeth in such a way that it sounded like a bird chirping. she'd waddle over to the window to look for the sweet widdle birdie making the songs and so that was always good for getting out of a few minutes of aggravation. algerba 2 i had a smart teacher who did not abide such nonsense and who was not going to let me think i couldn't get through the material. she even had the gall to make me come in for tutoring during homeroom and study hall. the nerve! she'd augh when i sat sullenly glaring at her as she wrote on the board, "lime, i can feel those daggers you're throwing into my back, hehehehe!" (thank you mrs. eidel for not giving up on me)

then i got to geometry and suddenly the clouds parted, the sun shone, and angels descended as they strummed their harps of gold. this was math? couldn't be. it made too much sense. it wasn't abstract. it was elegant. i got As without even trying. the teacher wasn't even a particularly good one, the material just made sense. i was happy.

(part 2 to come)

5. then came trigonometry. aaaarrrrgggg.....the flashbacks. my PTSD is kicking in! a truly spine decalcifying experience. first day of class we were introduced thusly, "good mawning cwass. my name is wussell miward. i teach twigonometwy." yeah, you know what my response to that was. two boards of equations with arrows going back and forth between them and zero instruction. copy the equations and here's your homework. what.the.hell???? i was hopelesly lost from day one on. i had absolutely no idea which end was up, ever. shoot me now. it's the only class i ever gave up in completely. i asked for help and the guy looked at me like i had to be the biggest moron on the planet. i got nuthin. very end of the year was the first time all year we ever saw a picture of a cone with the various ways of slicing it and a corresponding graph of curves. until then it was all equations. i saw the cone and graphs at the end and the lightbulb went on. someone said it was the same stuff as we'd been doing in the hideous equations all year. i asked very angrily why the hell no one showed me this stuff at the beginning of the year because if it was all the same THIS actually made sense and might possibly have helped interpret that crap that never made any sense to me. again the looks conveying my utter idiocy.

here's me, fewer than 10 years ago. i meet a friend who is a total math nerd. dreams about math like you do. totally in love with math. we start talking high math and my eyes glaze over. the jaw goes slack, a string of drool snaps in the breeze. i tell him my experience with math and he doesn't look at me like i am a special brand of retarded. he just says, "well, there are 2 ways to learn math, algebraically and geometrically. you clearly have a geometric brain." he proceeds to demonstrate the difference between how you'd instruct the same concept the 2 different ways. he told me i'd just never been taught correctly for the way my brain processes things because most math nerds are very algebraic but of course then there's me...outside the norm.

i told him he needed to get his butt into a university and start teaching math teachers how to teach their subject because that's a pretty critical thing to know in order to effectively convey a concept.

but to answer your ultimate question. yes, i have dreamed solutions...never to math...but to other things.

and i seriously cannot believe the longest comment i have ever typed out was about MATH!

6. and i even spared you the stories of math in college...

but when mr. lime changed from teaching special ed to shop class one thing he had to teach was mechanical drawing. he was going over his new curriculum and making sure he could produce the various geometric shapes correctly in the first unit. the last one was a 5 pointed star inside a circle. he could not figure it out. he threw the book on the table in disgust and said, "see if you can figure this stupid thing out and then show me if you do." guess who figured it out. oh yes, that would be me.

on a recent standardized test i had to take i got some dumb word problem i know every math teacher i ever had would say to set up in an equation to solve. not this girl. i did it in a chart of my own invention that let me see the relationships of the numbers to each other.

7. Skip - Awww, thanks. But let's not get carried away. Suldog is the Man. . .

;)

Bijoux - Back when I was in school, Asians hadn't been discovered yet. . . And certainly not as far north as I lived. . .

IT - Your dream sounds even more fun than mine. . .

Lime (all 3 of you) - (*pat, pat*) Thank you; I am honored (honestly, I had no idea it would go so long; you are a loyal friend. . .). And even more honored that you invested such a, uh, thorough comment on my humble post. . .

Mr. Lewis was one of the treasures of my educational experience. I finally took the opportunity, a couple years ago, to write him and thank him for his influence on my life. Your Mrs. Eidel sounds pretty special, in her own right. . .

That 'algebraic/geometric' thing really resonates w/me. The Abstract Algebra class in college just did me in, and pretty completely flummoxed me (which was a pretty new experience for me, all by itself), and of course, Engineering math is pretty thoroughly 'geometric'. The funny thing is, I liked my Algebra classes WAY better than Geometry in high school - the Geometry class was really mostly a class in Logic, which took me a while to figure it out. . .

And, as my kids would say. . . YOU'RE a 5-pointed star. . .

;)

8. Nope. That is not to say that I haven't had some interesting dream experiences, I most certainly have, but none ever involved math problems. Ever ....

9. Xavier - I'm not so much wondering about other folks' math dreams (I've got a decent sense of just how strange that is), as I am about dreaming solutions to knotty problems that you couldn't figure out when you were awake. . .

10. First, let me say that I'm both impressed and flattered that you remember when my birthday is. I'm lucky if I can recall when my relatives have birthdays, let alone relative strangers.

I've never solved any math problems in my sleep, but I have resolved some life problems. The most notable instance I can recall is my former fear of dogs.

There were a number of nasty dogs in my neighborhood during my youth, so I had every reasonable reason to fear them. I had been attacked by them a few times; nothing that resulted in major injury, but still enough to make me fearful every time I saw a canine looming in my path. And those fears carried over to my dreams. I often had a dream wherein a dog attacked me, and I would wake up breathing hard and sweating each time.

What finally happened was that I confronted the mean dog in my dreams. I don't know why I chose to do so, but this time, rather than running away or freezing or whatever other reactions I had previously had, I stood my ground. And when the dog lunged at me, I reached into his mouth, took its upper jaw in one hand and its lower jaw in another, and I pulled in opposite directions until I snapped the dog's jaw (with a most notable and profound noise accompanying that action.)

The dog flapped its broken jaws, but I knew it was powerless to truly harm me now. And I have never had a nightmare about a dog since then, nor have I had anything other than normal reactions to them in real life (that is to say, realistic caution if the dog is menacing in some way, but never a feeling of sweaty fright or an urge to run away.)

By the way, reading this solved a minor problem for me. I was going to post something tomorrow, concerning the on-line Jeopardy test which I'll be taking, but I've decided to re-run something, instead, using it as the major part of the text and mentioning Jeopardy as an aside. This is now your fault, and you can either take the credit or the blame depending upon how much you enjoy or loathe it.

11. as a kid i did logic puzzles for fun so proofs and theorems in geometry were just plain fun. also with the algebra classes and solving equations the teachers seemed to be of the mind that there was only ever one solution and even if i got the right answer but in a slightly different way they'd get on my case. proofs and theorems, the teacher said straight out, as long as it follows logic it's right, 10 of you could conceivably have 10 different answers, all of them right. took pressure off.

12. Nope, never solved a problem in my dreams though I have had some interesting experiences. Up until my college years I would almost always wake up part way through my dreams and then could 'steer' the direction of the conclusion of my dream by saying out-loud what I wanted to happen and how I wanted it to end, then falling back asleep. It felt like I was creating my own mini-movie.

My weirdest dream was one that I did not wake in the middle of. The short of it: I ad some friends were running down a gully chased by a boulder. I managed to scramble out, then watched as each of my friends got hit (but only injured) by the boulder as it caught each one.

Within 5 years each of those friends experienced a similar serious injury in one fashion or another.

I try to not dream about friends when possible ..... :-)

13. Suldog - Well, being that your birthday is the day before mine, it's not that hard to remember. . .

And that's a pretty cool 'therapeutic' dream, there. . .

Lime - I did eventually get the hang of the whole 'logic/proof' thing. It was just so different from what I'd seen previously, that it took me a while to get the hang of it.

And the idea that an algebra problem can only be solved one way is just silly. I think there are something like 163 different proofs of the Pythagorean Theorem; I know I've seen at least half a dozen of 'em. . .

Xavier - Can you do me a favor, and banish me from your dreams?

;)

14. I think the only time I ever had a math related dream, it was one involving figuring out how not to get my engineer father frustrated with me when he was trying to help me figure out rather simple, in hindsight, math matters over my evening homework. I was math challenged very early on, and yet somehow have had a hand in creating two kids rather smart in the ways of an equation...which is good, because seriously, I can not be of any help with math homework!

15. faDKoG - I'm tempted to say that it explains a lot that you are the spawn of an engineer.

Sometimes critical traits like math, skip a generation. . .

;)

16. Having been a math major myself, I found your post fascinating, if not just a bit long. But while I was good in math, I majored in it for probably the wrong reasons. 1) I was good at all subjects; how to choose? 2) I hated writing term papers and my math classes didn't have term papers. 3) I loved computers but my college only offered a computer minor, not a major, back in the day.

I loved the applied math classes, but hated the theoretical ones. I struggled for a bit in my modern algebra class until I got the hang of doing proofs.

SO... after two years of work in a math-related field, I decided the only part of my job that I really liked was the part dealing with computers, and I switched to programming. Which my math background really seems to complement well.

And yes, I have had several times where I spent the day struggling with how to handle a difficult problem, and then dreamed the solution at night.

17. Me - Thanks for commenting!

You know, there are plenty of times when I feel a bit out-of-place, like a mathematician in a sea of engineers. I mean, 'mechanical aptitude' is not really one of my main strengths (my wife is usually more willing to tackle home repairs than I am *sigh*). I mean, I didn't grow up tearing down lawn-mower engines, or building go-carts, like many of my co-workers. But I do get to actually do an integral or two, from time to time. . . ;)

I, too, loved the applied classes and hated the Abstract Algebra classes; never did get the hang of all that 'ring/group' stuff. Did well in Advanced Calculus, which is as full of proofs as anything. Maybe my Algebra profs just weren't very good. . . (my math-major roommate left me with this, tho - "what's purple and commutes?" "An Abelian grape.") (sorry)

Funny, too - my math-major roomie has been a programmer for his whole professional career, even tho, when we were rooming together, he professed loathing for anything that wasn't 'pure' mathematics. A man's gotta eat, I guess (and women are more eager to marry guys who actually get paid).

And it's nice to know that I'm not the only weird math nerd dreamer out there. . . ;)

18. What a great gift to have a teacher early on who inspired you to find your inner drive and inspiration in that work... although giving you those problems which were "feeding my own sense of having fun with math." would sound to my then-young-high-school ears like some kind of special, torturous punishment. But then, me and math weren't on the best of terms in high school.

We did have some nice moments over Junior High algebra before Math got all weird and aggressive and started calling me names. :)

And the dreaming of solutions at night? Pure awesome. I have dreamt of story line and plots for novels, but only one of them really stood up to daytime scrutiny. Apparently my brain is not at it's best when I'm snoozing. lol

19. Hi, Flutter! Yeah, I've long-since resigned myself to the idea that 'fun with math' marks me as some kind of social misfit. . . ;)

Dreaming solutions IS awesome (or at least it was; it hasn't happened for many a year). But for my money, the best part was that, in the dream, I reminded myself to wake up and write it down. . .

20. Well, now that I've stumbled onto your blog, let me just observe, apropos of some of the comments above, that geometry was the most compelling thing about math for me for most of my academic career. And then I took the abstract algebra series my senior year of college, and it was great! Groups and rings were just fun!

21. Hi, Fed; welcome!

Groups and rings. . . sounds like a dance class, or something. . .

;)