I appreciate your idea to summarize the fractal discussion, Eileen.
I'm not in agreement with the statement concerning "in-depth discussion" because I feel the basic concept behind a fractal (starting with a pattern and re-iterating it) _can_ be discussed in-depth with middle grades (and younger) students. Perhaps what you meant was formal mathematical discussions are not possible.
The issue of the definition of a fractal seems to be arising. I like Mandelbrot's statement (p. 361 of his _Fractal Geometry of Nature_):
"Although the term fractal is defined in Chapter 3, I continue to believe that one would do better without a definition (my 1975 essay included none). The immediate reason is that the present definition will be seen to exclude certain sets one would prefer to see included."
Mandelbrot explains in the first chapter of this book his thinking when he coined the term "fractal." I think this informal definition related to irregular fragments can lead to in-depth discusssions. One of my favorite points is that fractal's Latin derivation is etymologically opposite to the Arabic derivation of "algebra" from "jabara" which means to bind together.
My interpretation of the quote above is that a great deal can be gained by not locking ourselves into a formalism (especially in an emerging field of study). I think we gain very little by limiting our interpretation of fractals to "sets for which the Hausdorff Besicovitch dimension stricly exceeds the topological dimension." I think this is true in the educational domain (exploring fractals with middle grades students) and, I believe Mandelbrot argues in the chapter from which I quoted, that it's also true within a mathematical research perspective.
In message <199505050302.WAA11360@informns.k12.mn.us> Eileen Abrahamson writes: > Thanks for the stimulating discussion - I am going to summarize what I > think you told me about fractals so that I can tell if I got the idea. > Please correct me if I have it wrong :-) > > Fractals are self-similar objects/patterns such as a line of telephone > poles in perspective, rivers branching, or an artery structure, except that > they branch infinitely. At the intermediate level any kind of in-depth > discussion can not take place because there is some significant math > involved in the generation of fractals, however, if nothing else, the kids > could be exposed to what they are and could do some simple activities > involving them. Sometimes Self-similar patterns do not lead to fractals. > The word `fractals' mean that they have non-integer fractal dimensions. >