h a l f b a k e r yNot so much a thought experiment as a single neuron misfire.
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In the elementary schools, we are asked to solve equations with
assumption
that
operators are already known. For example: 4 + X = 5, or 6 - X = 9, etc.,
mostly
with the already defined operators, like PEMDAS. This is is damaging to
our curiosity and creativity, as our minds are forced to
live in someone's
single
computational universe.
However, what's more useful in real life, is the creativity and operation
search.
So, instead of providing "+", "-", "*", ... as given, teach kids to do operation
search
to satisfy conditions defined by the equality sign.
For example, 4 ? = 5, meaning, -- what can we do with 4 to get 5. Or 6 ? =
9 --
what can we do with 6 to get 9? Now, replace the numbers with real
situations in
kids' lives, and teaching operation search math can help them, fostering
their
creativity rather than life in one axiomatics of mathematics.
And, perhaps the question mark is not the best symbol here, as it is
directional. Something like an infintie-dimensional circular question mark
around one side of equation would be more appropriate...
[link]
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It's going to generate confusion and ambiguity, and may actually inhibit learning. |
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The problem is that the notation and symbology isn't confined to mathematics. Those symbols are used in computation, as arguments to search engines and the like. |
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You either add new symbols (demanding more learning) or assign additional meanings to existing symbols (confusing and context-sensetive). |
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// confusion and ambiguity |
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I don't think so. The practical solutions to their everyday lives may
actually make them get those solved, and then have time for other kind
of equations. Solving what matters, not what's on textbook, is top
priority, and most rewarding, thus, facilitating learning. Like, usefulness
of learning to ride a bike, encourages further learning... |
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// Those symbols are used in computation, as arguments to search
engines and the like. |
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// additional meanings to existing symbols (confusing and context-
sensetive) |
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Can you elaborate on these, as, for your pre-trained mind, the symbols
may already have some meaning, unlike for a kid without much
knowledge, say, in GOFAI and grounding propositions. |
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And, making kids useful to how OUR specific minds are working (boxing
them into our thinking), so they can work for us, is not a very noble
cause. I'd go for diversity, and letting kids discover their new ways. |
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And, if you read the bottom line of the idea more closely, you see, that what I proposed is not a reuse of
particular symbol, but "a symbol" for operation search requirement, where "?" is not the best option. |
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//This is is damaging to our curiosity and creativity// |
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No it isn't. In fact, quite the opposite. I remember first
coming across propositional logic (and then first-order
predicate logic) as an undergraduate, and thinking "Where
have you been all my life?" That's because, from quite
early in childhood, I had had trains of thought which could
be described in the language of formal logic - but, because
I had not been taught this language, I had no way of
sharing these thoughts with others, nor even of developing
them in my own head. |
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Now, if I had been taught the language of formal logic at
the same age at which I had been taught the language of
arithmetic, I would have been able to take those thoughts
and run with them, instead of just stewing in frustration. |
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If you don't give children tools of this kind, you're not
liberating them; you're crippling them. |
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I get that you're trying to stimulate higher orders of
reasoning (meta-reasoning, if you like) - but, to do that,
you need first to establish lower orders of reasoning to
build on. Have you read "Goedel, Escher, Bach"? |
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// No it isn't. In fact, quite the opposite. [...] if I had been taught the
language of formal logic at the same age at which I had been taught the
language of arithmetic, I would have been able to take those thoughts
and run with them. // |
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I could say the same about integral calculus or differential equations.
But, I could not say that about my friend <Q>. She would have found
the valence bond theory and molecular orbital theory empowering her
creativity. Yet learning of something else, like integral calculus,
premature and not as empowering, perhaps making her ask "why do I
learn that?". |
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Knowledge being instrumental, it is an imperative to teach the most
general-purpose tools that we have in our arsenal, and the standard set
of operations are not them. Search for operations is. We do that every
time we think of new idea here on Halfbakery -- an idea is essentially
some operation X to be applied to a situation F to obtain an outcome
Y, i.e., X, such that F X = Y, that is a solution to F ? = Y. |
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// I get that you're trying to stimulate higher orders of reasoning (meta-
reasoning, if you like) - but, to do
that, you need first to establish lower orders of reasoning to build on. // |
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It is my experience, that to learn modern mathematics, you need to
follow the prerequisites very
thoroughly, because results build one on top of another, and unless
you're a genius that can reinvent entire fields of
mathematics spot-on, or have read literature on the background topics,
you'll be unable to follow. This idea is not to refute that pyramid of
knowledge. It's to extract
certain results, that, after all, do not
require knowing the lower orders of reasoning, because operation
search is so fundamental to
intelligence and life, that it coincides with search for actions ("what to do
to..?"), and would be
natural for kids to comprehend. |
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