My father taught me binary in the early seventies when I was still in elementary school, with black marbles and a grey egg carton. I got it right away. Numbers were one thing, representations of numbers was another thing, and these could be whatever you found convenient, so long as you obeyed certain rules (I wasn't so accelerated that I immediately started banging out Euclid's Elements on the piano).
Then I thought really hard one Saturday afternoon about fractions (on the unit interval, which I thought of as positive integers with the numerator greater than the denominator), and discovered that even though there are a lot of them, it is possible to enumerate them exhaustively, though not by the traditional "counting up" procedure, which got me hooked into the problem of the common divisor thing.
The next project I recall was to exhaustive write out the Tic Tac Toe game tree. Since I was a lazy bastard (always have been) this involving thinking very hard about something somewhat like symmetry groups.
Over the annual summer visit to my grandparents—small town prairie Badlands without the cool geography, though often we managed a trip to see the hoodoos—I played a lot of solitaire on the golden-green shag carpet which Puss Puss—the duodecarian house cat who lived in the shadows under my grandparent's bed (the short duration of our visits was probably for her sake)—sometimes preferred in her dotage over asking out into the Canadian winter. Quite undeterred by the sticky and/or stinky patches, I managed to clearly formulate the concept of a "decision procedure" and that such a thing could be unambiguously specified; furthermore, I worked out (at first empirically) that the greedy algorithm was provably not optimal for Klondike (for me at that time, all Solitaire was just "Solitaire", though I knew several).
At age ten, the boundary between empiricism and proof is still a fuzzy one.
In grade five, I spent a lot of time (by myself) trying to puzzle out the rate-limiting step in long-hand square root. I had by then also discovered E=IR and P=IE. Pretty soon I had determined that this generates 4 choose 1 times 4 choose 2 simple algebraic forms. But for an entire painful week, some kind of thick cloud entered my brain and I couldn't reliably write all the forms down without a lot of mucking around; this I knew to be completely bogus, and a permanent blot on my record. By the time the cloud passed, I was pretty good at substitution and gathering. Later, when I first encountered a matrix (don't recall), I immediately went to myself "oh, that's just algebra, better organized". At least something stuck.
Now, during this entire period of my life, I was in a constant state of deeply repressed rage about this thing called "school", with all the inherent stimulation of Puss Puss waiting out the daily bedtime / ultimate final departure of the grandchildren (geriatric cat yay!) from the furthest dark remove under the master bed.
Grade six came as a shock. For the first time I experienced a math teacher who believed in letting kids learn at their own natural rate. He quickly put four of us a private work program. We could go as fast as we wanted, but the rule was we had to do all of the tedious exercises at the end of every chapter. Many of these exercises were heavy on the pencil work, so I only made it through grades six, seven, eight, and nine. My fingers put in about 90% of the work (this is not actually a bad thing), and my brain put in the other 10% (this being 100 times more than 0.1%). Awesome!
So I was armed, locked, and loaded for bear when I showed up at the beginning of grade seven. I figured I could knock off ten, eleven, twelve by Easter, and still have a month left over for real math at long last.
Problem: my grade seven teacher thought my purpose in life was to sit enthralled by his boring lectures. Shields up! I don't recall a single thing he wrote on the board in math class the entire year, and I just sat there doing stupid pet tricks with numbers—no useful development whatsoever.
So eventually that year we have this weird event day outdoors, and one of the girls has been asked to demonstrate her figure skating. She was jumping! And spinning! And throwing one of her legs around without falling down! (On skates, I was still working my way from three legs to two.) Wow! Some adult somewhere actually gives a shit about her natural abilities, and gives her not only the opportunity, but also coaching, and even a pat on the back. How is this possible?
That was the day I realized I was a tent-camp refugee in the world of math phobia.
By this point, whatever natural ability I had was on a fast track to nowhere. My the miracle of moving from one province (relatively good school system) to another (not so good school system), it turns out that my grade nine school year is spent repeating my grade eight school year. Back in grade six, the grade-nine math book had only challenged my pencil, and this was now my third tour of duty.
My grade nine math teacher surely recognized that I was paying him 1% of my full attention, out of 1% of one corner of one eye. Sometime mid-year, I hear from a classmate that there's this thing called a "math competition". "Oh," I said, waking up from a long coma. "That might almost be interesting." Later that day I go up to my math teacher (this being our longest point of contact for the entire year) and say "I heard there's this math competition thing." He says, "there's no point bothering, you wouldn't be good at it." He wouldn't even tell me the room where it was held. Revenge? Or just a cockroach sucker?
Funny he should think that. Two years after my parents finally wake up and send me to a private school, I was ranked nationally. This after a four year hiatus with my parking brake engaged. So, while this is a story about opportunity wasted, it's not a story about being ruined—you can only be ruined if you let it happen.
But what did happen is that my ability, under my random self-tutelage, folded back in on itself. Lacking a curated challenge, I posed my own quirky challenges, and I spent a lot of time thinking about myself thinking about myself. I became very good at thinking about myself, and I finally matured into an adroit, adept, meta-cognitive gadfly. Substance about substance, not anchored to substance.
No worries. I figure this will all pay off at some point in my seventies, when the world is adrift with cognitive agents. "Somebody ... please! ... is there meta-cognitive specialist in the house? Our pets are running wild!" Well, had my early education gone a little differently (you know, with any structure at all), I could now be the guy building the metacognitive agents, instead of cooling my jets sitting around waiting to fix them.
Whatever. It all works out in the end.