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 _______________________________________________ IAEP -- It's An Education Project (not a laptop project!) IAEP@lists.sugarlabs.org http://lists.sugarlabs.org/listinfo/iaep