Afterwards I asked him how it was possible for me, the kid who had failed all her math courses, who still did not know her times tables and who still had to count on her fingers, who could never remember which fraction you invert when, how this same person could have enjoyed and aced his course, which many other mathophobes had dropped out of.

How could I have ever understood and enjoyed a course in modern math?

His reply: "In grade school and high school they were teaching you arithmetic, which machines can do. I was teaching you mathematics, which are concepts only the human mind can comprehend!"

This is why the educational system in the USA is failing, because arithmetic is taught by teachers fresh out of school with only a general BA and no special aptitude in mathematics. None of them could ever reach my mind.

Years later I was telling this story to someone who had graduated from the same university where my modern math professor taught and my listener told me: "Oh yes! He was one of our best teachers!"

Only specialists should teach children.

Cheers! digits

Fundamentally, playing with mathematical structures needs to invoke at least one of these emotions to be properly interesting. Everybody is different, so they will find these emotions in different ways. Thus when thinking of educating a child, I believe that we need to begin solely with the first: Fun. Numbers and patterns need to be fun so that when a child sees them, he or she instinctively and intuitively treats them like a collection of toys to be played with, in any way shape or form possible. Later one needs to develop Joy: that is, a deep enjoyment of the structures and patterns that everybody can see in numbers. Personally I believe that only the times tables should be learned by rote and even then we have yet to find a universal method of making times table learning fun and enjoyable (just as you can do with scales on a musical instrument.) Finally there is beauty, and this for me comes from simple clear proofs such as Euclid's proof of Pythagoras' theorem or from equations such as e^{i\pi}+1=0, which tie what appear to be fundamentally separate concepts together in a stunningly clear way.

I am still thinking through my understanding of how I learned maths and how I think others can or should learn it, so I'll stop here.

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