First critical words about my draft book Shadows of the Truth (where I analyse childhood stories sent to me by readers of my blog). They came from good colleagues of mine, real experts in mathematical education.
Your material may be like using Olympiads to analyse mathematical development: those who perform at this level in their teenage years certainly exist, and one feels should be nurtured by the mathematical community to keep them fresh, so that their mathematical ability grows naturally into whatever they may do in adult life, leaving them free (if they choose) to graduate from adolescent problem solving to more serious mathematics.
[At present too many of those who feature in olympiads, move away
from mathematics for bad reasons! And too many of those who move
from Olympiads to more serious mathematics never really mature as
But 95+% of mathematicians do not feature in Olympiads during adolescence – reflecting the greater importance of other “affects” (stubbornness, persistence, delight in global features – rather than local problems, …).
So we suggest that the “healthy norm” for mathematical development may be to reach 16 or 17 or 18 without noticing anything in particular, other than a sufficient affinity for the mathematics one has met to want to continue.
It seems bizarre to suggest that it is “normal” for kids to philosophise consciously about what goes on in the classroom. Healthy schools are simply not like this: the classroom is the least important part of what goes on in school (the communal, the physical, the food, contests – good and bad – in the the playground or on the sports pitch, the sheer rhythm of the timetable, etc. are all so much more important at the time). Most kids, including those who might subsequently be seen as culturally / mathematically important, simply need a cocoon which allows them to emerge as stable young adults, who are also literate, moderately knowledgeable, and mathematically competent.
In particular, why should any kid be surprised that some things do not make sense at the time? For most kids that is normal. So your precocious examples
(i) may be unusual in coming from backgrounds that encourage (either actively or incidentally) this kind of precocious reflection;
(ii) may be extreme (See Chapter 17!!!) in experiencing this kind of puzzlement *so rarely* that it leaves almost a scar!
If so, then you are missing out
(a) those who *never* experience this in a domain that matters to them,
(b) those who accept this feature and learn to handle it quietly while getting on with the things that seem to matter, without ever becoming fully conscious of what they are doing.
In short, those who are precocious in their inner reflections strike us as being a rather small minority – though [like those who stand out in olympiads] you are quite right to explore what their memories and experiences have to tell us, as long as we do not make the mistake of thinking that such self-conscious reflection is to be viewed as “good”, or “normal”.
Dear Reader, if you have your opinion about my book, I will be happy to hear it.