Benoît Mandelbrot’s thoughts about mathematics education
There is a supply of unsolved, elementary problems that give students the opportunity to learn how mathematics can be done by enabling them to do new (if not necessarily earth-shaking) mathematics; there is a continuing flow of new results in unexpected directions.
From Chapter One of Fractals, Graphics, and Mathematics Education by M.L. Frame and B.B. Mandelbrot
In this book, Benoît Mandelbrot and Michael Frame remind readers of why students came away from fractal geometry courses ‘feeling they had understood some little bit of how the world works’ – they were given opportunities to learn to do certain things themselves, such as grow fractal trees and synthesise their own fractal mountains. Mathematics had been revealed to them as ‘an enterprise as full of guesses, mistakes, and luck as any other creative activity’, with amazing surprises waiting around almost every corner.
When students experience mathematics as a closed, finished subject they are put off. Mandelbrot saw that students were engaged by the ‘unfinished’ appeal of fractal geometry – they understood problems in it that are still unsolved today. He believed, as do other mathematics educators, that seeing mathematics as a ‘lively, growing field’ to which it is possible that they can contribute, motivates students. Few things bring home to students the accessibility of mathematics so much as seeing and understanding something new in mathematics that has been done by another student. And it’s very exciting to be able to show other students what you have discovered. Benoît Mandelbrot understood how vital these observations about ‘ownership’ are to any thinking about the teaching of mathematics.
He believed that students eagerly explore fractal geometry because it keeps alive, or re-awakens, natural, youthful curiosity. He knew that to teach mathematics so that students are curious and eager to contribute demands faith in students’ capabilities. While acknowledging that students cannot learn all mathematics by ‘reconstructing it from the ground up’, Mandelbrot and Frame observed:
“Generally, giving a student an open-ended project and the responsibility for formulating at least some of the questions, and being interested in what the student has to say about these questions, is a wonderful way to extract hard work.”
Benoît Mandelbrot also felt that the fascinating visual appeal of fractal structures motivated students – living in a world in which much communication is visual (television, video and the internet), the more explicitly visual are the ways in which we present and communicate mathematical ideas the more readily students may engage with them. In an interview in 1984, Mandelbrot said:
“Before, people would run a mile from my papers, but they could not run from my pictures. In the beginning, I used the graphics purely for this reason - to illustrate my ideas and to force people to accept them. But I soon realized that this method enabled me to go further and integrate into a single theory a collection of things that otherwise would have seemed unrelated. Now, very complex geometric shapes could be compared with one another and with reality. The equations behind the shapes were abstract, but the shapes themselves looked alive.”
In Chapter Three of Fractals, Graphics, and Mathematics Education, Mandelbrot observed ‘that fractals – together with chaos, easy graphics, and the computer – enchant many young people and make them excited about learning mathematics and physics. In part, this is because an element of instant gratification happens to be strongly present in this piece of mathematics called fractal geometry. The belief is that this excitement can help make these subjects easier to teach to teenagers. This is true even of those students who do not feel they will need mathematics and physics in their professions.”
Benoît Mandelbrot goes on to express his strongly-felt belief (and hope) that the unusual interest shown by the general public in fractal geometry might result in more people than do presently, regarding a good understanding of some mathematics as an essential outcome of everybody’s education. He comments that “vital decisions about science and technology policy are all too often taken either by people so closely concerned that they have strong vested interests, or by people who went through the schools with no math or science.”
He explains why he does not believe that the need for everyone to be better educated mathematically ought to be justified by purely utilitarian considerations – because we live in an increasingly technological society. “The lesson for the educator is obvious. Motivate the students by that which is fascinating, and hope that the resulting enthusiasm will create sufficient momentum to move them through material that must be studied but is less widely viewed as fun.”
Mathematics educators need to know, Benoît Mandelbrot fervently believed, how research mathematicians view their craft, and how that view has continually changed throughout history, particularly during the last century. In Chapter Four, Mathematics and Society in the 20th Century, Mandelbrot compares the ‘conservative’ view of mathematics – as up on a high hill, understood only by the few professional mathematicians, ‘looking down on’ ordinary people who do not aspire to try to comprehend it – with his own view in which mathematics is attractive to people who are not professional mathematicians, including school mathematics teachers and students.
The use of computers had a profound effect on the life and mind of Benoît Mandelbrot. In another 1984 interview he mentioned implications of computers for mathematics teaching:
“We have entered a period of intense change in the mood of mathematics. Increasingly many research mathematicians use computer graphics to enhance their geometric intuition, others cease to hide (from outsiders, or even from themselves) the fact that they had been practicing geometry. This return of geometry to the frontiers of mathematics and of physics should have an effect on the teaching of geometry in colleges, high schools, and even in elementary schools, because so much geometry which had been quite impractical can now be easily done with the help of computers.”
Perhaps students can be inspired by Benoît Mandelbrot’s reflections on the beginning of his own lifelong learning journey:
“During the last term at Polytechnique, I looked for ways to apply my gift for shape, and a growing knowledge of various fields, to real, concrete, and complex problems. I wanted to keep far from organized physics and mathematics and instead find a degree of order in some area – significant or not – where everyone else saw a lawless mess.”