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Mathematics is alive and well


Fractals in education
Benoît Mandelbrot Mathematician
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And last, but by no means least, is the extraordinary usefulness of fractals in education. This first came as a complete surprise. That is, friends reading my books reported having spoken to children, small children, and asking them the various questions which my books ask. How long is the coast of Britain? The children were much less prejudiced about that than the adults. They soon realised that the question was one that didn't have an answer. And other questions were asked of children, and the children actually enjoyed these questions. The movement spread, more and more high school teachers gave fractal problems to advanced students, had great fun, won various prizes, and then the movement continued and moved down to younger and younger children. At this point it seems that actually very young children, before primary school even, or in primary school, are those for which fractals are the most, in a certain sense, interesting. A very interesting question is raised - not that anybody wanted to raise it, but I would say it raised itself. The way in which mathematics was taught to me, it was taught starting by the oldest organised mathematics and moving forward. Since I studied mathematics for a long time- very irregularly for a long time, I went to Newton, well beyond Newton, to even Laplace and then to the middle of the nineteenth century and even further. But it was understood that the sequence in which humans have mastered mathematics was the proper sequence for teaching it. Then there was briefly a period of modern mathematics in which they tried to teach it starting with the latest works. But in each case one tried to begin teaching mathematics starting by simple phenomena. The idea is that somehow simplicity is more attractive than complexity, that simple rules of geometry have been put at the root, at the basis of geometry and that students would learn in the same order in which axiomatics has been provided. This proved to be very widely unsuccessful. To the extent that fractals are used now in various countries, in various schools, sometimes in school systems, sometimes in individual classrooms, they consist in changing the sequence of events- not trying to induce an interest and awareness of mathematics by forcing the students to remember what Euclid and his predecessors found simple but to begin with shapes, which in a certain sense the students found friendly, at the same time mysterious, and in which they saw all kinds of symmetries. The symmetries were not obvious like in a circle or in an interval but hidden behind all kinds of complicated non-symmetric, semi-non-symmetric backgrounds. Children are much more interested in studying the symmetries of shapes in which symmetry does not scream at them like in a circle or as in an interval, than they are the most simple symmetries. After all, one doesn't begin life by looking at simple shapes. One begins life by looking at shapes that are around us, and those shapes, mountains, flowers, trees, are very complicated. For them, many of the simple rules of analysis don't apply. In a certain sense, I say many times, Euclid is very seldom present in the shapes which nature offers us. What is present is a great deal of complication which, however, in some cases, namely for fractals, can be tamed, can be reduced to simple rules of construction, not complete chaos in the sense of a complete mess, but a chaos having very precise rules. And children seem to be interested in it.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Daniel Zajdenweber Bernard Sapoval

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Duration: 4 minutes, 20 seconds

Date story recorded: May 1998

Date story went live: 29 September 2010