There are many people who ask me, "how do you make math interesting?" The key to the answer is that math is magic; and magic is inherently interesting. The real trick is, making math UN-interesting, by taking the magic away from it.

Here is the "Magic Phone", which I received as a birthday present for my 4th birthday from Dr. Y. Chr. Grigoryan (a friend of our family, and my very first teacher of math and physics, and a great influence, to whom I am indebted with getting the taste for exact sciences and the corresponding mode of thinking):

The toy comprises a disk cut out of cardboard, and attached to a rectangular cardboard back.
You start by resetting the arrow to point to the right — opposing the arrow "start". When you dial a word, the arrow will point to the corresponding picture; e.g. dialing "CAT" will turn the arrow to point to the cat).

(Of course, this is an adaptation for my English readers. The actual toy had Cyrillic letters combining into Russian words. I can not guarantee that the objects were the very same, but it is close enough to represent the Zeitgeist. And of course, I did make sure that the mod 28 arithmetic does work with the English words in the diagram).

The first thing I did — I tried out all the relevant words. They all worked.
"Wow, nice machine! I am curious what the mechanics of it are...— wait, what if I dial my name?"
The disk did turn, but ended up pointing in a direction bearing no relation to my name.
"Ok, whatever. Let's see how this thing operates."
I turned the toy over, preparing for a long joy of working on disassembling the genius mechanism and sorting out how it works. I wasn't even sure what to expect, what kind of super-clever machinery would produce such intelligent results. To my shock, there were no cogs, no motor, no battery, NOTHING. The disk was holding to the cardboard by a pin (obvious from that front side — inspecting the back added no new insight). That was it. No mechanism.
My first reaction was bitter disappointment. What, no hours of joy of trying to figure out how the cogs work together?!! But wait a minute. How DOES the thing work? —
Pause. A long, looong pause. A stretch of emptiness, of void, of not letting any noise enter the consciousness — just listening to any answer that might enter the mind. As a mathematician, now I know this was the first experience of concentrated thinking, when facing a mathematical mystery. And the sheer feeling of that void, was the most wonderful thing on earth.

Oh, of course I know how it works. It's a schitalochka (counting-out game), "eniki-beniki..."! Modular arithmetic: the arrow advances by as many objects (or kids) according to each piece of input; and when it makes more than a full revolution, it starts over.

Now this, is the mathematical insight. And interestingly, in this particular example, we have immediate anticipation of the digital toys such as those pictured at the top of this page.

When once you have tasted flight, you will forever walk the earth with your eyes turned skyward, for there you have been and there you will always long to return.

Leonardo
Codice sul velo degli uccelli