You're stretching what I said. I should note that I'm a lifelong fan of radical education (particularly John Holt), and I don't think that the way we teach children most things has a lot to recommend it.
However, your conception of how this could work starts from the supposition that the kids are actually interested enough to wonder. I don't doubt that there's something that will get every child wondering (and likely, more than one thing). But if we're going to actually require the teaching of fractional arithmetic (implying that we're requiring the learning of it, really) then we need to accept that we'll be teaching it a great many children who are not interested in the conundrum, and who even in the presence of a great teacher will remain not interested in it.
Self-led discovery learning is without doubt the best kind, but its not compatible with the current goals of education in a (post)industrial highly structured society, because it will naturally lead to people who for their own reasons chose never to learn things that we consider vital. I might be entirely willing to agree that they are not vital, and even that a (post)industrial highly structured society may also be a bit of an issue, but pretending that every child will just be naturally curious about 1/3+1/3=2/6 vs 1/3+1/3=2/3 is, IMO, not ground in reality.
In the original article, the guide isn't "standing front of the art". She's just taken them over to a corner that has a piece called "no mammals lay eggs", and then noticed that right next door to it, there's a platypus. She's wondering what to say next, or whether to say anything.
Yes. Every child I've ever met is curious about something, and that's what they should be learning. The idea that in year X every student should learn Y during hour Z every day is one of the reasons I said the system was irreparably broken. That is how schooling gets in the way of education.
Does every student learn to understand fractions under the current regime? Not in my experience.
Edit: And yes, you should say something about the platypus, because understanding that models are simplifications and incomplete can be enough of an escape hatch for the smart kids (the ones that usually hate math class) to notice that even the teacher knows that there's always more, and that can be enough to keep them from throwing it all away in disgust as a useless mishmash of arbitrary, conflicting, and incorrect rules.
However, your conception of how this could work starts from the supposition that the kids are actually interested enough to wonder. I don't doubt that there's something that will get every child wondering (and likely, more than one thing). But if we're going to actually require the teaching of fractional arithmetic (implying that we're requiring the learning of it, really) then we need to accept that we'll be teaching it a great many children who are not interested in the conundrum, and who even in the presence of a great teacher will remain not interested in it.
Self-led discovery learning is without doubt the best kind, but its not compatible with the current goals of education in a (post)industrial highly structured society, because it will naturally lead to people who for their own reasons chose never to learn things that we consider vital. I might be entirely willing to agree that they are not vital, and even that a (post)industrial highly structured society may also be a bit of an issue, but pretending that every child will just be naturally curious about 1/3+1/3=2/6 vs 1/3+1/3=2/3 is, IMO, not ground in reality.
In the original article, the guide isn't "standing front of the art". She's just taken them over to a corner that has a piece called "no mammals lay eggs", and then noticed that right next door to it, there's a platypus. She's wondering what to say next, or whether to say anything.