[Etoys] Some thought about Operational thinking and Squeak

Edward Cherlin echerlin at gmail.com
Sun May 4 06:16:35 EDT 2008

On Sun, May 4, 2008 at 12:55 AM, Hilaire Fernandes <hilaire at ofset.org> wrote:
> http://blog.ofset.org/hilaire/index.php?post/2008/05/01/Operational-thinking

It would be much easier to evaluate this contribution if it included
specific examples.

I have been working on some examples in DrGeo, and I disagree with the
author on its unsuitability. Certainly you can't expect children to
discover much with DrGeo if left entirely to their own devices. The
question is what guidance the teacher gives to the child in discovery.

I can build geometric models to illustrate a wide variety of concepts,
and then let children vary the diagram in many ways to see which
relationships remain the same through all variations. For example,
take any triangle and connect the midpoints to divide it into four
smaller triangles. What are the necessary relationships among them? If
you move any vertex of the original triangle, you change its shape and
size. What of the four smaller triangles? Which relationships change,
and which remain constant?

Similarly for many other constructions, and for symmetries,
tesselations, and other forms that lead to fundamental concepts of
math and science. We will not teach primary schoolers the details of
Emmy Noether's theorem that every symmetry in physics is equivalent to
a conservation law, but we can and should lay the groundwork for a
deeper understanding of this essential discovery at an appropriate

I have the outline of a practical Kindergarten Calculus program, in
which we would teach concepts visually without the algebraic and
numerical apparatus that is essential for calculus calculations. It
can all be done in DrGeo, as well as with physical objects.

The deepest understanding in math and physics, and in many other
areas, comes when we can see and use two or more representations of
the same ideas, and also see why they are equivalent, and how to turn
any of them into the others. The whole recent proof of Fermat's Last
Theorem came down to an instance of this called the Taniyama-Shimura
conjecture, now proven as the Modularity Theorem, that all elliptic
curves over the rational numbers are modular. This gives us mappings
between three realms: elliptic curves, modular functions, and
L-series, that were once seen as quite distinct. We can't even explain
what the theorem is about to young children, or even to most adults,
but we can show them other such mappings within geometry and

It turns out that in physics, it is necessary to connect the two quite
different realms of mathematical models and experimental results in a
fairly specific way in order to have an effective theory. One of the
greatest and at the same time most familiar and most misunderstood
examples is how the shift from Galilean to Einsteinian relativity,
based on the single painstakingly tested experimental result that the
speed of light is the same for all observers, requires the equivalence
of mass and energy.

If any of this fails to make sense to you, I recommend that you look
on that fact as a sign of some of the greatest failings in
conventional education. For anybody who would like an explanation of
any of this, I can answer some questions and refer to to excellent
published expositions for many more. I will not attempt to walk your
through the proofs, but I can demonstrate the relationships I

What we mostly don't have is a path by which children can be guided to
discover much of this themselves. But we have bits and pieces of that
path in work by Alan Kay, Seymour Papert and many others. I have
thought of a few other bits that I hope will add to the enterprise
when I get a chance to work them out in more detail and try them out
on children.

I think that the hard question is how to get teachers to discover
enough of this to be able to use is effectively. Nobel laureate
Richard Feynman said that we don't really understand a subject unless
we can create freshman lecture on it. Mathematicians suggest trying to
explain ideas to your grandmother. I propose that we find out how much
of what we think we know we can explain to children and to teachers.

>  Hilaire
>  --
>  http://blog.ofset.org/hilaire
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Edward Cherlin
End Poverty at a Profit by teaching children business
"The best way to predict the future is to invent it."--Alan Kay

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