<div dir="ltr"><div class="gmail_quote">On Mon, Feb 22, 2010 at 22:29, K. K. Subramaniam <span dir="ltr"><<a href="mailto:subbukk@gmail.com">subbukk@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<div class="im">On Monday 22 February 2010 09:54:47 pm Edward Cherlin wrote:<br>
> Your version requires the preliminary step of proving (a+b)^2 = a^2 +<br>
> 2ab +b^2 geometrically,<br>
</div>This "equation" is a symbolic way of showing "growth". Spatial concepts are<br>
introduced before the concept of relation between two spaces. Pythagoras'<br>
observation took a different route (integer triples) and is of historical<br>
interest today. </blockquote><div><br>Or the Egyptian priests who used 3:4:5 right triangles to survey the banks of the Nile. Yes, the questions what did they know and when and how did they know it is of perennial interest.<br>
</div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">Euclid observed that the relation holds good for any similar<br>
shape, not just a square and the law of cosines is extended it to any triangle<br>
not just a right angled one. Curiously, this loops us back to the starting<br>
equation.<br>
<br>
BTW, proof is a strong word to be used in this context. The exposition is<br>
elucidating but not elementary. The observation is true only for Euclidean<br>
surfaces (e.g. paper but not orange peel or flower petals).<br><div class="im"></div></blockquote><div><br>Any proof requires a context of axioms and rules of inference. There are other such relationships for non-Euclidean surfaces of constant curvature, and beyond that in differential geometry.<br>
</div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"><div class="im">
> or at least pointing out that your diagram<br>
> includes that proof. Caleb Gattegno has demonstrated that all of the<br>
> essential ideas of algebra can be taught in first grade, or even<br>
> kindergarten, using Cuisenaire rods, so this is not an obstacle.<br>
</div>By essential, do you mean precursors to symbolic arithmetic or symbolic<br>
arithmetic itself? </blockquote><div><br>Precursors. For example, a+b = b+a by placing pairs of rods together in reverse order. a×b = b×a by building rectangles from two lengths of rod along different axes.<br><br>In my childhood, variables and the rest of symbolic algebra were considered 8th or 9th year subjects. However, we know that 3rd year elementary-school students can use variables in programming. We therefore need to investigate bringing symbolic algebra into 3rd year learning, possibly even earlier.<br>
</div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">This appears ambitious to me. The cognitive base of first<br>
graders (in general) is insufficient to deal with symbolic arithmetic (as<br>
algebra is known in Indian subcontinent). I don't rule out the possibility but<br>
such cases are exceptional rather than the norm. First graders are just<br>
building a cognitive understanding of quantity and its conservation. </blockquote><div><br>That may be common experience, but it does not take into account what will happen when we start to design software for preschool learning. We know that two-year-old children take well to the camera on the XO laptop. We do not know at what age they can learn to read and write with a laptop helping them, although the Edison Talking Typewriter gives us a hint (two years old), or at what age math concepts will make sense in some appropriate form. We do know that, as in astronomy, every new instrument will produce astonishing observations of previously unknown phenomena. We also know that simply asking the question and making careful observations also gives astonishing results, as, for example, in the careers of Maria Montessori and Jean Piaget. Also<br>
<h2><span class="mw-headline">Jerome Bruner</span></h2>
<ul><li> We begin with the hypothesis that any subject can be taught
effectively in some intellectually honest form to any child at any
stage of development.
</li></ul>
<dl><dd>The Process of Education (1960)
</dd></dl> </div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">Concepts<br>
like product (a*b), square, square root, symbols to represent quantity and<br>
manipulating them will take some more time. The constructional technique<br>
adopted by Julia Nakajima is so beautiful because it uses growth instead of<br>
symbols.<br></blockquote><div> </div><div>Can you give me a URL for that?<br> </div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
I apologize in advance if I have misunderstood your statement.<br></blockquote><div><br>Not at all. I appreciate questions that show me how better to explain my ideas.<br> </div><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
Subbu<br>
</blockquote></div><br><br clear="all"><br>-- <br>Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin<br>Silent Thunder is my name, and Children are my nation. <br>The Cosmos is my dwelling place, the Truth my destination.<br>
<a href="http://www.earthtreasury.org/">http://www.earthtreasury.org/</a><br>
</div>