It seems that you did a lot of problems, and gained no insights. Perhaps you were never intended or destined to be a mathematician. This is no insult - I genuinely believe that different people think in different ways, and the balance is important
Indeed. It was Knuth who said that mathematicians and computer scientists think in very different ways. With that I don't disagree.
How can we make kids experience more discoveries if they won't actually play with the underlying basics?
I'd argue that we should let them play with calculators and other tools. My discoveries came as I did programming. In 4th grade I wrote an arbitrary precision number package. I learned more about arithmetic doing that than all of the rote drills combined.
It's not the arithmetic you're trying to get insight into - it's the structures.
I have a feeling I'm never going to be able to convey what I'm trying to because you're not a mathematician, and I'm not a good enough writer. It's not just about becoming better at the arithmetic, it's about finding and creating structure, order, relationships and mappings.
But I'm going to stop now. It's clear that I'm just not expressing myself well enough to make the point, and I've spent far too long on it. I regret not being a good enough writer - and perhaps not a good enough mathematician - to explain in a way so as to make it clearer.
Indeed. It was Knuth who said that mathematicians and computer scientists think in very different ways. With that I don't disagree.
How can we make kids experience more discoveries if they won't actually play with the underlying basics?
I'd argue that we should let them play with calculators and other tools. My discoveries came as I did programming. In 4th grade I wrote an arbitrary precision number package. I learned more about arithmetic doing that than all of the rote drills combined.