Date: Mar 8, 2010 8:21 PM
Author: Allan Turton
Subject: Re: Inclusive and exclusive definitions... again!

From Walter Whiteley:

Allan
Your comment on the materials reminds me of an action the math educators in Australia took. They approached the publishers of picture books and asked them to give a variety of orientations, and shapes, to pictures labeled 'triangle'. The publishers did. The point is to expose children early to a range of examples. This flexible development of a space of examples, and non-examples is important over in the development of conceptual understanding (see the work of John Mason). The research of Doug Clements with young children reminds us that it is easier for children to learn a flexible version of the concept early (age 5 or younger) than to switch from a fixed image, where even orientation matters, later on (around grade 3). So we need to flexible range of materials. BTW, it has also been observed that children have a better chance of ignoring the orientation of a shape if the shape is in a circular frame, rather than a rectangular frame! If the frame orients you and your perception, and you don't have a flexible approach to ignoring the frame, or turning it (physically or in your mind) then the frame captures some of your mental space.

Froebel (the man who invented the word and the program for Kindergarten) started with 3-D objects for his first 7 rounds of materials and activities (he also invented wooden building blocks - but a bigger variety than we have now). I don't think it was a accident that when he got to 2-D, he had all the shapes the same color. That alone would help the teacher who wishes to play with the ideas of a variety of shapes which deserve the same name.
I have observed children in a grade 3 class comparing shapes who could gesture with their hands about squeezing one or another dimension to indicate a rectangular prism shared key properties (number of edges, vertices, faces) with a cube. Unfortunately, the teacher in this lesson study did not pick up that this was perhaps the most effective (and flexible) response in the entire class!

Walter