I don't think the study of fractals will make or break an elem or middle school mathematics curriculum but I believe this area offers a nice context to explore more important concepts: rational number arithmetic and similarity. It also broadens a student's view of mathematics (as a field that deals with more than just numerical computations).
The length of each generation of the Koch curve offers a good application of arithmetic with fractions or decimals. At a higher level, students can generate an expressioon for the length of the n-th generation Koch curve.
Constructing self-similar objects and recording area and volume of each generation strikes me as a worthwhile exploration involving similarity. Cubes can be used or tetrahedrons to construct curiosities such as the Menger Sponge or the Sierpinski Arrowhead. Tiles can be used to make a Sierpinski carpet (and if you spill your Minkowski sausage on your Sierpinski carpet you can clean it up with your Menger sponge -- see fractals are practical as well as fun).
The grades 5-8 Addenda Series book on Understanding Rational Numbers has suggestions for this type of lesson. Articles by Barton (vol. 83 pp 524) and Camp (vol. 84 pp. 265) in the Mathematics Teacher have ideas as well.
In message <199505041223.HAA13725@informns.k12.mn.us> Eileen Abrahamson writes: > I have been spending lots of extra time in meetings related to designing > math staff development for elementary instructors, and while we were > planning one of the sixth grade teachers brought up fractals. Basically > she was asking are fractals a topic that should be addressed in the > intermediate grades. >