Posted by dcorking on Nov 30, 2007; 1:31am
URL: https://forum.world.st/Panel-discussion-Can-the-American-Mind-be-Opened-tp114756p114769.html

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

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