Ron Teitelbaum wrote:
My mental model for arithmetic is a clock face, which is very clumsy in base 10 (and worse in base 16!)
This is very interesting. Something I had not thought of, but it makes perfect sense that a picture of numbers would help.
My clock face model is clumsy and prone to errors (except when calculating times which is fine). However it seems more useful than memorising relationships between numerals, which I think some people do.
I didn't purposely invent my clock face model or consciously try to adopt it. Unfortunately It just emerged in my head in the first year or two of elementary school and dominates my thinking about number. I suspect a more tactile and flexible model, like an abacus or cuisenaire rods, would have made mental arithmetic much easier. Perhaps it could have made the beautiful structures in numbers more accessible to me, such as primes, the fibonacci sequence, irrationality, perfect numbers, fractals and so on. Anyway - sexagesimal arithmetic is trivial for me. And I love hearing about others' internal models.
Meanwhile I still remember reciting multiplication tables out loud, and those facts (reinforced by auditory and oral/verbal memory) are also invaluable tools for my mental arithmetic. Whether they help me to think mathematically is a different question which I don't think I am able to address. This experience suggests to me that while neither numerals nor the names of numbers seem to be very useful in grasping the basics of number, they become useful later for acquiring facts that lead to other skills and areas of understanding.
Etoys uses numerals a lot (in tiles and watchers) but also uses some other very compelling representations of number, such as polar coordinate vectors, sound and movement. It would be interesting to add more representations to Etoys, such as tally charts, rods, an abacus, tesselating shapes, liquids: to see if they help children become fluent in logarithms, polynomials, statistics, complex numbers and other things that bring back bad high school memories for my generation (or indeed fun and enlightening math things that are not traditional school math, like fluid mechanics, statistical mechanics, Shrodinger's cat, cellular automata, developmental biology ....). Forgive me if I mentioned someone's completed project that I am unaware of.
Best, David