There is no track!!! Of course there is, because what we call mathematics education is broken beyond repair, but that's a crime.
Shutting down the student's curiosity about perhaps the only thing of mathematical interest that happened all day is not the best you can do.
Do you think you know what the basics are? Are you sure you know which example is confusing and which simple? Are you sure that ignoring confusing anomalies is the best habit you want the next generation of engineers or statespeople to learn as a reflex from an early age?
there's not much evidence that most humans would ever discover the concept of fractional arithmetic by themselves.
this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves.
there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning experience in general.
now, if you are homeschooling a child or somehow in a 1 on 1 (or at least, working with a very small teacher:student ratio), then perhaps a more freeform exploration of fractional arithmetic might be a wonderful thing.
doing so with a general class of kids? i strongly suspect you're wrong. they won't even understand that there is an anomaly, because they don't even understand the basic concepts that make it an anomaly. why shouldn't adding two fractions (never seen those before!) result in different answers? in fact, why is 2/3 different from 2/6 anyway? and so on.
this is not about shutting down curiosity IMO. it's about nurturing the basic concepts so that curiosity can grow amidst them in the (near) future.
I have a hard time accepting this. My daughter is in 3rd grade and seems to have a reasonable grasp of fractions. They teach her about them at (public) school and we've discussed them at home. Sure, "1/3 + 2/7" is outside her reach, but simpler stuff, things she can visualize, are well within her grasp.
> why is 2/3 different from 2/6 anyway
Because 2 pieces of a pie that you cut into 3 pieces is more than 2 pieces of a pie that you cut into 6 pieces. She understands that 4/8 == 2/4 == 1/2. Sometimes she needs to think about it a bit, but she does "get it".
No disrespect, but you read HN and think that your daughter's take on math and the home context you provide for it is a sensibly representative starting point?
My daughter's now 25 and has been quite the nerd herself through the years (eventually landing in linguistics and speech pathology), but I'd never assume that the fascinations she had instrinsically and that I helped foster as a parent were really typical. I wish they were - and hey, here I am reading HN too :)
I think the homework her school gives out is a sensibly representative starting point. Why wouldn't it be?
That homework expects a reasonable understanding of fractions. Enough that the child doing them can understand the difference between the number of slices in a pie being representative of the bottom number of a fraction.
Sure, I do math exercises with my daughter that aren't representative of what they teach in school (square roots, the fact that parallel lines _do_ meet at the vanishing point in the real world, etc). But those things aren't what I'm basing my assumptions are; the expectations the school has of her are.
There's incredibly strong evidence that the way to teach mathematics and the only way to learn it is to foster the environment in which you can rediscover the key insight for yourself. Obviously (to the younger students, to me, Lockhart, Dewey, and others, but not to most teachers, administrators, voters) there is no track and no curriculum for this.
They are not stumbling into it by themselves! They have a guide! The guide's job is to point out the works in the museum, and to make sure the kids don't get lost. It's not to stand in front of the art so the kids can't even see it and lecture about it!
There's nothing complex or sophisticated about recognizing that "+" has a meaning that we chose, and we could have chosen others, and in some cases other choices would be more natural.
You don't think students learning fractions know that there is one right answer, or that 2/3 and 1/3 can't both be right? Their short little lives have already been filled with enough test-taking to teach them that, at least.
And if they don't understand that 1/3 and 2/3 represent something different about the real world, and that 2/6 and 1/3 are different in a different way from 1/3 and 2/3, then why are we going on to teach them even more complex sophisticated concepts before they have grasped these most basic of basics?
There's never been a pedagogical program to shut down the curiosity of primary students. We do it by accident, by "nurturing the basic concepts" (the ones you learned at that age) so that curiosity can grow amidst them "in the future," which means, maybe, after they finish calculus, if they are lucky.
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.
Shutting down the student's curiosity about perhaps the only thing of mathematical interest that happened all day is not the best you can do.
Do you think you know what the basics are? Are you sure you know which example is confusing and which simple? Are you sure that ignoring confusing anomalies is the best habit you want the next generation of engineers or statespeople to learn as a reflex from an early age?