One more childhood story.
My father is a mathematician. I was interested quite early in numbers
and mathematics. My father answered all my questions counting on the
fact that I would just stop listening when I get bored.
I would estimate that when I was about 11 or 12 years old I had the
opportunity to take part in a voluntary afternoon class about
mathematics in school, where we were exposed to actual mathematical
problems. These problems were of course quite simple but of a flavour
that it was not just applying some previously learned trick or
technique but rather one needed an actual mathematical idea.
This is where my story starts. I remember that I was in the beginning
quite desperate because it was totally unclear to me how one would
find such new ideas or tricks. What was the standard procedure to get
to them? I could easily follow such tricks once they were presented to
me, but how should I find them myself???
Of course, I asked my father. And to my big surprise he simply
answered something along the lines of: “You have to try more ore
less random ideas and just see which of them work!”
And indeed, my problems were gone and I knew what to do and I could
relatively quickly solve “proper” mathematical problems myself.
I could imagine that lots of students in school who have difficulties
in maths never had this kind of experience with “problem solving”.
In my experience most of them implicitly assume that mathematics is
all about learning some techniques and then simply applying them
verbatim, without any trial and error. And therefore they get stuck as
soon as a “non-standard” problem comes up.