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
they should.]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.
Mathematics under the Microscope 
Dear Sasha,
I’ve just discovered your blog and downloaded your book.
I’m in no way an expert of education (mostly, I teach calculus to engineering students), but I had a recent experience teaching in the local elementary school some games one can do with pencil-and-paper (unfortunately, I only know their names in Italian). No big deal: it was exactly the kind of games we used to play as children, mostly in the rainy days. An enthusiastic teacher told me afterwards that her students had now a better attitude towards mathematics. “How comes? I didn’t really do mathematics, and they already knew most of the games.” She explained me that there is no learning without an emotional investment, and playing games gave them an emotional boost.
(As research mathematician, I must confirm that without strong emotional involvment one does not go anywhere: we all know that, but I never stopped to think about it).
Greetings,
Nicola
By: nicola arcozzi on February 5, 2010
at 11:43 am
I think for many mathematicians, perhaps even a majority, the appeal of the subject lies in being able to solve particular, special problems, such as logic puzzles, and being able to use (and know) special tricks for doing so. Those are the sort of mathematicians who excel at Olympiads and enjoy mathematical games.
For others of us, however, all that emphasis on puzzles and tricks was – and is – a complete waste of time. I always thought those puzzles were about as interesting as crossword puzzles or card games, activities which I always found “mind numbing and soul destroying”, to use the words of Yevtushenko. How is the world a better place after solving a cross-word puzzle? What have you, the solver, gained from solving a logic puzzle? Absolutely nothing, as far as I could tell.
My mathematical interest was always in the large-scale formal, abstract systems and structures of the subject, not in particular puzzles or tricks. I remember being thrilled at about age 14 to learn that “y = f(x)”, that two different concepts which my teachers had tossed around loosely for some time, the vertical axis “y” in cartesion co-ordinates and a function of some variable “x” were in fact the same thing. At last these two disparate subjects were connected, and lots of different things made sense as part of the same over-arching picture. The same experience happened a year or so later when we were first taught the differential calculus, and I could finally see a general theory behind what we had been doing with all those discussions of tangents to particular functions.
Some mathematicians certainly do like examples and particular problems and clever tricks. Others of us, however, think top-down and not bottom-up, and so desire general frameworks and abstract structures. Any effective education system should recognize the existence of at least these two styles of thinking, and not assume that all students are bottom-up thinkers.
By: peter on February 6, 2010
at 10:09 am
I agree with Peter: I found logic puzzles soul-destroying, as well as problems about identification of a false coin using a balance, etc. — essentially the entire mathematics recreation field was for me an intellectual wasteland. And I became a mathematician.
By: Alexandre Borovik on February 6, 2010
at 3:24 pm
Yes; “hear, hear” to Peter.
I still don’t see the deeper theory behind cranking out Ramsey numbers. Why should I care?
By: John Armstrong on February 6, 2010
at 5:37 pm
Dear Sasha, I feel flattered that you have included a reference to my article and some ideas from it into chapter 9 of your book. Still, on page 88 you present the idea that differentiation can be treated as factoring as “yet another didactic trick.” I totally disagree with your assessment.
Differentiation can be viewed as factoring f(x)-f(a)=(x-a)p(x,a), the derivative being f'(x)=p(x,x). Of course we have to specify what kind of factors we want to deal with. To differentiate polynomials, we want p to me a polynomial. Classical differentiation is recaptured if we want p(x,a), as a function of x, to be continuous at x=a. Continuous (uniform) differentiability means that p is uniformly continuous in both variables. Uniform Lipschitz differentiability means that p is Lipschitz in both variables, etc. It allows to understand all sorts of differentiation within the same algebraic scheme and makes differentiation rules obvious.
On page 89 you say: “Later, this ‘factoring’ approach can be rigorously justified,” as if there were something non-rigorous or illegitimate about it, while in fact it clarifies the subject and makes it easier to learn and understand.
I also have a minor complaint about the typography. The small type and very wide margin makes it difficult to read.
I have not read the whole book yet, but I find it highly entertaining and interesting. As for the title, my interpretation of it is probably different from what you intended. I don’t consider the latest fashions in mathematical formalism as “the truth,” and the ideas that are encountered in elementary mathematics as “the shadow.” To me it is the other way around.
By: misha on February 10, 2010
at 10:36 am
The uniform estimates approach allows us to treat differentiation independently of real numbers, continuity, compactness and other “ghosts of departed quantities” from classical analysis. It is mostly due to Hermann Karcher. It has already found its way into serious teaching and mathematical literature in the form of his German lecture notes with an English summary and an innovative text Real Analysis: a Constructive Approach by Mark Bridger.
By: misha on February 11, 2010
at 8:06 am