Panel discussion: Can the American Mind be Opened?

> From: David Corking
>
> My mental model for arithmetic is a clock face, which is very clumsy
> in base 10 (and worse in base 16!)
>

Hi David,

This is very interesting.  Something I had not thought of, but it makes
perfect sense that a picture of numbers would help.  I found something very
similar playing Sudoku.  It's a lot of fun.  When I first started playing I
would find missing numbers by counting from 1 to 9 and looking for the
missing numbers.  It became obvious to me that I didn't need to do that and
that I could just look at all the numbers at once and for what ever reason
the picture itself allowed me to figure out what was missing.  The mind
works in very amazing ways!

Ron


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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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On Thursday 29 November 2007 7:04 pm, David Corking wrote:
> My mental model for arithmetic is a clock face, which is very clumsy
> in base 10 (and worse in base 16!)
A clock face has many interesting properties too. It incorporates the concept
of magnitude (countables) and angles (directions/turns), "feeling" time
(kairos) and periodic time (chronos).

1. The clock face has two hands of different magnitudes - small and big
2. The small hand moves in steps of one. The big hand moves in steps of five.
3. Reading a clock means reading the magnitude traversed by the small hand and
then the magnitude travelled by the big hand.
4. Even though the "magnitude" of big hands falls to zero, the time is larger
because the "magnitude" of small hand has increased (concept of place value).
5. The hands move but still stay in the same amount of space. This turning
around a pivot is an "angular movement". (Note: Alan's car demo shows how
accumulation of linear and angular movements leads to a circle).
6. The angular separation between small and big hand makes interesting shapes.
When they are farthest apart, it is like someone cut the clock face into two
same pieces. When they are like room corners, then you can have four such
pieces.
7. The clock chimes whenever the big hand points "straight up". The big
hand "triggers" the sound. (The concept that an event is triggered when a
specific combinations of events happen is the basis of kairos. A seed remains
dormant till rains arrive to germinate. Kairos has no "magnitude" between
event occurrences. A ten-minute wait in a long queue feels like an hour while
a hour-long video game session feels like a minute).
8. The thin red hand (second hand) moves very rapidly and makes a regular
ticking sound like water dripping from a faucet. Our heart races when
exercising or when scared, but red hand always makes sixty ticks to complete
one turn. (tick-tock time is chronos time. It is not subjective and involves
magnitude. Pulse beats or dripping droplets are approximations. Galileo used
his pulse to time chandelier swings in a church).

I will stop here and hope you got the drift. A first-grader amazed me one day
by reading out the clock correctly. With my curiosity provoked, I got her to
explain the process to me gradually over the next few days. The language may
come across as a bit strange because I tried to use her own words as much as
possible.

Subbu

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