On Sun, May 4, 2008 at 12:55 AM, Hilaire Fernandes hilaire@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 age.
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 arithmetic.
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 describe.
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
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