This is true. It would just be a test case for whether pre-teens can learn these concepts.
If I was designing a hypothetical curriculum, I wouldn’t use high-sounding words like “axiom”. It would just be—“Here is a rule. Is this allowed? Is this allowed? If this is a rule, then does it mean that that must be a rule? Why? Can this and that both be rules in the same game? Why not?” Framing it as a question of consistent tautology, inconsistent contradiction as opposed to “right” and “wrong” in the sense that science or history is right or wrong.
And “breaking” the rules, just to instill the idea that they are all arbitrary rules, nothing special. “What if “+” meant this instead of what it usually means?”
And maybe classical logic, taught in the same style that we teach algebra. (I really think [A=>B ⇔ ~B=>~A]? is comparable to [y+x=z ⇔ y=z-x]? in difficulty) with just a brief introduction to one or two examples of non-classical logic in later grades, to hammer in the point about the arbitrariness of it. I’d encourage people to treat it more like a set of games and puzzles rather than a set of facts to learn.
...and after that’s done, just continue teaching math as usual. I’m not proposing a radical re-haul of everything. It’s not about a question of complex abstract thought- it’s just a matter of casual awareness, that math is just a game we make, and sometimes we make our math games match the actual world. (Although, if I had my way entirely, it would probably be part of a general “philosophy” class which started maybe around 5th or 6th grade.)
(I’m not actually suggesting implementing this in schools yet, of course, since most teachers haven’t been trained to think this way despite it not being difficult, and I haven’t even tested it yet. Just sketching castles in the sky.)
This is true. It would just be a test case for whether pre-teens can learn these concepts.
If I was designing a hypothetical curriculum, I wouldn’t use high-sounding words like “axiom”. It would just be—“Here is a rule. Is this allowed? Is this allowed? If this is a rule, then does it mean that that must be a rule? Why? Can this and that both be rules in the same game? Why not?” Framing it as a question of consistent tautology, inconsistent contradiction as opposed to “right” and “wrong” in the sense that science or history is right or wrong.
And “breaking” the rules, just to instill the idea that they are all arbitrary rules, nothing special. “What if “+” meant this instead of what it usually means?”
And maybe classical logic, taught in the same style that we teach algebra. (I really think [A=>B ⇔ ~B=>~A]? is comparable to [y+x=z ⇔ y=z-x]? in difficulty) with just a brief introduction to one or two examples of non-classical logic in later grades, to hammer in the point about the arbitrariness of it. I’d encourage people to treat it more like a set of games and puzzles rather than a set of facts to learn.
...and after that’s done, just continue teaching math as usual. I’m not proposing a radical re-haul of everything. It’s not about a question of complex abstract thought- it’s just a matter of casual awareness, that math is just a game we make, and sometimes we make our math games match the actual world. (Although, if I had my way entirely, it would probably be part of a general “philosophy” class which started maybe around 5th or 6th grade.)
(I’m not actually suggesting implementing this in schools yet, of course, since most teachers haven’t been trained to think this way despite it not being difficult, and I haven’t even tested it yet. Just sketching castles in the sky.)