Most math beyond arithmetic in our school was absolutely unrequited to understand most other subjects - even in physics, it was a taxi from A to B, with the implied idea that the end state was determined completely by the initial conditions. Math didn't animate, it just let us obtain answers in a gradeable way. The idea of a pattern inalienable from sense, of efficiency and precision belonged to the domain of poetry: in math, precision did not matter, you were either wrong or right. At some point, we began studying materiel that didn't serve any purpose even in imagining situations in physics, it simply was, and for people like me - hungry teenage girls with disillusioned math teachers, affable language teachers and brilliant biology teachers - that was when it died and was dutifully buried. I still have a tiny feeling of 'setting affairs aside to execute social duties' when I have to solve something.
But on occasion, very rarely, I come across a description that makes those long-forgotten mental muscles twitch. I collect them, to offer to my own son when he grows enough, and would appreciate if you shared your examples, if you have any.
1. The sharp change in the steepness of steps leading down to our cellar - that one time when my husband had to re-apply Pythagorean theorem.
2. This place in Jerome K. Jerome's Three Men in a Boat:
I knew a young man once, he was a most conscientious fellow, and, when he took to fly-fishing, he determined never to exaggerate his hauls by more than twenty-five per cent.
“When I have caught forty fish,” said he, “then I will tell people that I have caught fifty, and so on. But I will not lie any more than that, because it is sinful to lie.”
But the twenty-five per cent. plan did not work well at all. He never was able to use it. The greatest number of fish he ever caught in one day was three, and you can’t add twenty-five per cent. to three—at least, not in fish.
So he increased his percentage to thirty-three-and-a-third; but that, again, was awkward, when he had only caught one or two; so, to simplify matters, he made up his mind to just double the quantity.
He stuck to this arrangement for a couple of months, and then he grew dissatisfied with it. Nobody believed him when he told them that he only doubled, and he, therefore, gained no credit that way whatever, while his moderation put him at a disadvantage among the other anglers. When he had really caught three small fish, and said he had caught six, it used to make him quite jealous to hear a man, whom he knew for a fact had only caught one, going about telling people he had landed two dozen.
So, eventually, he made one final arrangement with himself, which he has religiously held to ever since, and that was to count each fish that he caught as ten, and to assume ten to begin with. For example, if he did not catch any fish at all, then he said he had caught ten fish—you could never catch less than ten fish by his system; that was the foundation of it. Then, if by any chance he really did catch one fish, he called it twenty, while two fish would count thirty, three forty, and so on.
Linear progression in all its glory.
3. This passage in J. Lockwood Kipling's Beast and Man in India:
A live crow is spread-eagled on his back, with forked pegs holding down his pinions. He flutters and cries, and other crows come to investigate his case and presently attack him. With claws and beak he seizes an assailant and holds him fast. The gypsy steps from hiding, and secures and pinions the second crow. These two catch two more, the four catch four more, and so on, until there are enough for dinner, or to take into a town, where the crow-catcher stands before some respectable Hindu's shop and threatens to kill the bird in his hand. The Hindu pays a ransom of a pice or two and the crow is released.
Notice that at any iteration, the number of secured crows can be adjusted to serve the exact goals of the catcher, and there is always the fastest way to do it.
Two things I like about these snippets is that, without posing any specific question, they allow me to tweak assumptions, and to provide the underlying reasons for those assumptions, stretching back in time, and that 'in-world precision' matters.
What mistake did the cellar-digger most likely do when he did his calculations initially? Is there something to gain by making the steps different (like having place for flowerpots deeper under the surface, where the overwintering plants are less affected by the cold air bursting in every time I open the door)? At which point will the fishers' boasting become useless as a tool for signalling? (For example, a man says he only caught one fish; would that not make others believe him, because he could not have arrived at this number by common transformations?) How would a change in circumstances make the gypsy combine the exponent with the linear function (like, for example, stopping on his way to the disgruntled Hindu's shop to catch some more birds?)
I don't mean such trifles bring any deep insight; but they make it easier to turn the search for the answer into a game.