I have some memories from first grade in a California public school (in Santa Cruz) of "finding a number we can put in place of x so that x + 2 = 4" and how "at

I have some memories from first grade in a California public school (in Santa Cruz) of "finding a number we can put in place of x so that x + 2 = 4" and how "at some point you will learn how to deal with x + 4 = 2". In 3rd grade we explored prime factorizations, and I also remember an emphasis on clearing out greatest common divisors for fractions -- so, while it didn't encourage thinking of fractions as equivalence classes, it at least encouraged comparing things by canonical forms. I even remember around then an introduction to base four representations, probably to illustrate the "carry the ten" when adding. My experience, in contrast to yours, was well-meaning teachers drilling the algorithms while hinting at some underlying structure. I suppose it was enough inspiration to figure out what math was about, and I eventually found myself in a math PhD program. (Despite the fact I had a hard time remembering the so-called "math facts." During competitions, I would try to calculate, slowly, something like 7 x 9 using something like (8 - 1) x (8 + 1) = 8 x 8 + 8 - 8 - 1, since the only "facts" I ever remembered were multiples of five, 6 x 8 and 8 x 8. Those competitions did make me think I was a bit bad at math!) Though, let me share my story about fractions. In fifth grade, I got a day planner because the school pushed for everyone to get one, and in the front leaves there were various references, like a periodic table, lists of equations, etc., and one which caught my eye was "a/b + c/d = (ad+bc)/bd". For some reason I thought "oh, that is what fractions are" and I tried to explain to a teacher how this defines fraction addition, how finding a common denominator is just a way to calculate this, but instead I was told I was incorrect, and you just find common denominators. I can't say I didn't feel somewhat vindicated when I learned, much later, about the Grothendieck construction. I also remember trying to find a formula for triangular numbers in sixth grade, and I will never forget the look of horror on my "home-room" teacher's face as she backed away while I explained what I was trying to do.