I might have misused the term—I thought up to 8th grade and perhaps even 12th grade was “grade school”? I got my sister to think algebraically with non-geometrical problems, and then apply that successfully to a novel geometrical problem with perimeters and volume when she was 10...but she wasn’t able to retain it after a week. Later on when she learned it in school at an older age, she did retain it. I suspect attentional control is the limiting factor, rather than abstract thought.
But you’re right, this should be tested. She’s technically not a teen yet so next time she has a long holiday of free time I’ll see if she can learn about basic logical properties and set theory. (It seems way easier than the graphs simultaneous linear equations she’s doing right now, so I am optimistic).
I see. I’m used to it being a synonym for primary school, but according to Wikipedia, that’s apparently ambiguous or incorrect.
I agree that trying to teach it and seeing what happens is the way to go. :) Although I guess there is probably a lot of individual variation, so a school curriculum based on what works for your sister might also not generalize.
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.)
I might have misused the term—I thought up to 8th grade and perhaps even 12th grade was “grade school”? I got my sister to think algebraically with non-geometrical problems, and then apply that successfully to a novel geometrical problem with perimeters and volume when she was 10...but she wasn’t able to retain it after a week. Later on when she learned it in school at an older age, she did retain it. I suspect attentional control is the limiting factor, rather than abstract thought.
But you’re right, this should be tested. She’s technically not a teen yet so next time she has a long holiday of free time I’ll see if she can learn about basic logical properties and set theory. (It seems way easier than the graphs simultaneous linear equations she’s doing right now, so I am optimistic).
I see. I’m used to it being a synonym for primary school, but according to Wikipedia, that’s apparently ambiguous or incorrect.
I agree that trying to teach it and seeing what happens is the way to go. :) Although I guess there is probably a lot of individual variation, so a school curriculum based on what works for your sister might also not generalize.
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.)