# [IAEP] [etoys-dev] TED - Alan Kay - Example(8:44)

K. K. Subramaniam subbukk at gmail.com
Mon Feb 22 22:29:36 EST 2010

```On Monday 22 February 2010 09:54:47 pm Edward Cherlin wrote:
> Your version requires the preliminary step of proving (a+b)^2 = a^2 +
> 2ab +b^2 geometrically,
This "equation" is a symbolic way of showing "growth". Spatial concepts are
introduced before the concept of relation between two spaces. Pythagoras'
observation took a different route (integer triples) and is of historical
interest today. Euclid observed that the relation holds good for any similar
shape, not just a square and the law of cosines is extended it to any triangle
not just a right angled one. Curiously, this loops us back to the starting
equation.

BTW, proof is a strong word to be used in this context. The exposition is
elucidating but not elementary. The observation is true only for Euclidean
surfaces (e.g. paper but not orange peel or flower petals).

> or at least pointing out that your diagram
> includes that proof. Caleb Gattegno has demonstrated that all of the
> essential ideas of algebra can be taught in first grade, or even
> kindergarten, using Cuisenaire rods, so this is not an obstacle.
By essential, do you mean precursors to symbolic arithmetic or symbolic
arithmetic itself? This appears ambitious to me. The cognitive base of first
graders (in general) is insufficient to deal with symbolic arithmetic (as
algebra is known in Indian subcontinent). I don't rule out the possibility but
such cases are exceptional rather than the norm. First graders are just
building a cognitive understanding of quantity and its conservation. Concepts
like product (a*b), square, square root, symbols to represent quantity and
manipulating them will take some more time. The constructional technique
adopted by Julia Nakajima is so beautiful because it uses growth instead of
symbols.

I apologize in advance if I have misunderstood your statement.

Subbu
```