All children (and most adults) crave attention, and the giving and taking of it is perhaps the most powerful tool that a parent and/or teacher has in the arsenal.
I've always thought a lot of parents do it totally wrong...they give the misbehaving kid lots of (albeit negative) attention while the quiet, well-behaved one they ignore.
My parenting style was the exact opposite...I always stroked good behaviours and tried my best to ignore bad ones.
The same goes for schools, of course. The misbehavers earn the attention of both teachers and students alike, and if they do it enough, even that of their parents and the administrators. The promising students, on the other hand, are often expected to do their work in near-monastic solitude: usually, the only reason for special attention is if a student is performing weakly.
This is necessarily the case given that a teacher having to divide their attention between twenty students and the current priority structure of our schools. Regardless, that's a clear area for improvement.
But is it? Don't we all remember that "one teacher" who seemed to give us that little extra attention or push when we needed it the most?
Good teachers (at least my good ones) all seem to understand the attention-tool intuitively, but it just became glaringly obvious to me raising my three children and seeing young children behave badly to obviously get their parents attention, and many parents obliging them without thinking.
The rather odd part, for me, was, when mentioning this to parents in social groups, I was often looked at like I was nuts...like the idea of bothering a well-behaving kids with attention was beyond the realm of reason.
This is not uncommon now and is kind of the standard bearer for how to parent.
Grrr...mixing metaphors and it's not even New Year's Eve yet.
The comment “What an idiot I was,” I thought. “That was just an axiom, it is called commutativity. One doesn’t prove axioms.” is interesting. What's chosen as an axiom, and why, is an advanced question. Unless you get into foundations of mathematics, that question is seldom addressed. It's way beyond most pre-college math teachers. It's the sort of question that occurs to smart kids, but there's no easy answer you can give them. The usual answers are theological, and boil down to "shut up, kid". Here's a discussion on Stack Exchange of that subject: http://math.stackexchange.com/questions/127158/in-what-sense...
(If you get into automatic theorem proving, you have to address such issues head-on. Adding an inconsistent axiom can create a contradiction and break the system. This leads to constructive mathematics, Russell and Whitehead, Boyer and Moore, and an incredible amount of grinding just to get the basics of arithmetic and number theory locked down solid. In constructive mathematics, commutativity of integer addition is a provable theorem, not an axiom.
I once spent time developing and machine-proving a constructive theory of arrays, without the "axioms" of set theory. The "axioms" of arrays are in fact provable as theorems using constructive methods. It took a lot of automated case analysis, but I was able to come up with a set of theorems which the Boyer-Moore prover could prove in sequence to get to the usual rules for arrays. Some mathematicians who looked at that result didn't like seeing so much grinding needed to prove things that seemed fundamental. This was in the 1980s; today's mathematicians would not be bothered by a need for mechanized case analysis.)
This is the probably the most insightful part of the AMS narrative:
>OK, if it is so hard to teach kids the notion of a number, what am I trying to do? What is the point of my lessons? I said it many times and I am going to say it again: the meaning of the lessons is the lessons themselves. Because they are fun. Because it’s fun to ask questions and look for the answers. It’s a way of life.
If math pedagogy is your main interest any of the MSRI Math Circle Library books are worthwhile. This includes "Circle in a Box" which is a Math Circle starter kit freely available here http://www.mathcircles.org/GettingStartedForNewOrganizers_Wh...
You could have a whole set of sessions with children exploring different arrangements of coins and noting that no matter how many you add within the hull, you don't get any 'more coin' (altering the plurality may help adults understand this problem.) If you have some button[s] and much more coin[s], can you add just one coin so that you have more coin than button? How far away do you need to add it?
don't be fooled by the title of the book. The content is beyond 7 year old kids in the US.
I really loved his note about Dima who asked everyone to swap chairs around. The group had one more person than there were chairs, yet with each shuffling he was bewildered to find they were still short one chair. The child wasn't dumb, and in fact is occasionally clever, but this isn't the sort of problem to be curious at - it's baffling. (Though that's not to say Dima didn't reflect on it at a later date - I'm only halfway through.) The fact Dima didn't quite understand the problem is the part of the interesting point being made; something seems off, but Dima didn't understand to dig into it and figure out right then why this situation felt wrong, at least not yet. Indeed, shortly after on page 28 he notes that this sort of misunderstanding could happen to an old person, too, despite their longer lives and observations.
I think part of the lesson here is that the lessons don't even really stimulate curiosity. Rather, it gives the children more information, more opportunities to have that special serendipitous 'aha!' moment. To your point, it's more akin to bringing the child to a curious place and letting them do their natural thing, rather than foisting stimulation onto them.
Kids are curious out of the box, but if you gently guide them, they are capable of asking amazing questions and understanding things that will blow you away. My kindergarten nephew and my two year old inspire me this way.